Download MEASURING PRODUCTION AND INCOME, Chapter 2

A COURSE IN MACROECONOMICS
Topics:
1. INTRODUCTION. Plots over Swedish Macro series.
2. NATIONAL INCOME ACCOUNTING I
Main lesson: The production of a country is roughly the income of its citizens
FLEXIBLE PRICE MACROECONOMICS:
3. THE PRODUCTION FUNCTION AND FACTOR MARKETS
Main lesson: A higher capital-to-labor ratio (K/L) increases the real wage and decreases
the return on capital
4. THE LABOR MARKET
5. NATIONAL INCOME ACCOUNTING II:
National Saving, Domestic Investment, Trade Balance and the Current Account
6. ANALYSIS OF THE ECONOMY IN THE LONG RUN:
The capital stock (K) is flexible: Flexible Prices leads to full employment of physical
capital (K) and labor (L).
A. Growth accounting
B. The Solow growth model
Output is determined by the amount of inputs.
Main lessons: Increased saving increases the capital-to-labor ratio (K/L) and thereby
production per capita. Decreased population growth increases the capital-to-labor ratio
(K/L) and thereby production per capita.
Better technology increases production per capita.
7. MONEY AND INFLATION IN THE LONG RUN.
Main lesson: A high money growth leads to a high inflation rate, to a high nominal
interest rate, and to a depreciating currency.
8. THE CLASSICAL MODEL = The Keynesian model in the long run. Assumption: The
levels of capital (K) and labor (L) are fixed. Prices are flexible which leads to full
employment of K and labor input (L): Actual output = potential output.
A. The closed economy: The interest rate is flexible. Main lesson: Increased government
spending or lower net taxes (which implies a higher government budget deficit) increases
the real interest rate and thereby lowers private investment to the same extent so that
aggregate demand and output are unchanged.
B. The small open economy with its own currency: The real interest rate is fixed. Main
lesson: Increased government spending or lower net taxes increases the real exchange rate
and thereby lowers net exports to the same extent so that aggregate demand and output are
unchanged.
9. Introduction to the Keynesian model in the short and the long run. Assumption: Capital
is fixed (more or less): The amount of physical capital is assumed to be fixed but in a
recession some of it may not be used.
Lessons: 1. A higher aggregate demand (AD) stimulates output in the short run but not in
the long run. 2. Higher nominal wages or temporary higher oil prices increases the price
level and decreases output in the short run but have no effect in the long run.
9.1. Keynesian model in the short run= Prices are fixed. The SRAS-curve is horizontal.
Aggregate demand determines production.
9.1.1. The simple Keynesian model for the closed economy without its own currency: The
interest rate is fixed. The multiplier effect: Increased government spending by 1 dollar
Increases output by more than a dollar.
9.1.2A. The closed economy with its own currency: The interest rate is flexible. Main
lesson: Fiscal policy: Increased government spending or lower net taxes increases
aggregate demand and output but increases the real interest rate and thereby lowers private
investment. Monetary policy: A higher money supply decreases the interest rate and
thereby private investment, aggregate demand and output.
9.1.2B. The small open economy with its own currency. The interest rate is fixed but the
exchange rate is flexible. Main lesson: Fiscal policy is less effective in a small open
economy. Increased government spending or lower net taxes increases real exchange rate
and thereby lowers net exports. Monetary policy: A higher money supply decreases the
real exchange rate and increases net exports, aggregate demand and output.
9.2.Keynesian model in the short and long run when the SRAS-curve has a positive slope.
Aggregate demand and supply jointly determines output in the short run. The AD-curve is
consistent with equilibria in the goods and money market.Main lesson: Fiscal and
monetary policy increases aggregate demand, which leads to a higher price level and to a
higher level of production in the short run because A higher price level decreases the real
wage which leads To increased employment. In the long run the real wage is restored so
that the employment level reverts to the full employment level.
9.3. The Phillips-curve is the SRAS-curve in percentage terms. Main lesson: A given level
of expected inflation, fiscal and monetary policy may lower unemployment in the short
run at the expense of a higher rate of inflation. Higher rate of inflation decreases real
wages and thereby Increases employment and production.
10. Stabilization policies. Main lesson: It is difficult for the government to pursue
stabilization policies in order to dampen business cycle fluctuations.
Exercise: Debate of pros and cons of active stab policies
FLEXIBLE MACROECONOMICS:
11. CONSUMPTION.
Main lesson: Expected future income may impact current Consumption.
12. GOVERNMENT DEBT
Main lesson: A budget deficit implies higher taxes in the
future which may lower private consumption today.
13 Investment, cost of capital. Not included
14 An alternative graphical representation of the Keynesian model in the short run for the
closed economy with its own currency:The IS-LM-model. Not included in A-level.
15 Microeconomics behind labor supply.
NOT included in A- or B-course
16 The life-cycle model: A mathematical example. Not included in A-level
17 The life-cycle model with endogenous labor supply. Not included in B-level course.
18 A thourough mathematical description of the Solow model with technological
progress.Not included in B-level course but in C-level course
19 Endogenous Growth Models, required for C-level but not for B-level
20 Fertility choice; not included in B-level macroeconomics
MACROECONOMICS: STUDIES THE ECONOMY AS A WHOLE
Questions studied:
Why are some countries growing faster than others?
Why is there unemployment? Why inflation? Why is the interest rate high?
Why are there recessions and booms?
Recessions are periods in which aggregate production falls mildly-- and depressions are
periods when aggregate production falls more severely.
Booms are periods when aggregate production increases rapidly.
INTRODUCTION: EXOGENOUS AND ENDOGENOUS VARIABLES
Economists use models to understand what goes on in the economy.
Here are two important points about models: endogenous variables
and exogenous variables. Endogenous variables are those which the
model tries to explain. Exogenous variables are those variables that a
model takes as given. In short, endogenous are variables within a
model, and exogenous are the variables outside the model.
Price
Supply
P*
Demand
Q * Quantity
This is the most famous
economic model. It describes
the ubiquitous relationship
between buyers and sellers in
the market. The point of
intersection is called an
equilibrium.
The basic demand and supply diagram for pizza.
Demand: Q  10  2 * income  3* P
Supply: Q  2  2 * wage  3* P
Which variables are exogenous variables and which variables are endogenous variables.
Endogenous variables are the variables whose values are determined within the model.
Holding constant the exogenous variables, what will the price of a pizza be in equilibrium?
Assume: Income=2, and wage=1.5.
What happens when income increases to 4?
Show in diagram, and calculate the new equilibrium price and quantity.
Can the actual price be above or below the equilibrium price? Are prices sticky? In other
words, does price adjustment take some time?
NATIONAL INCOME ACCOUNTING I (NATIONALRÄKENSKAPER)
MEASURING PRODUCTION AND INCOME
Gross domestic product (GDP) is the market value of all final goods and services produced
within an economy during a year.
In Swedish: Bruttonationalprodukt (BNP) till marknadspris
GDP can be measured in 3 ways:
Method 1. through expenditures on final goods and services.
Method 2. through income (wages and capital income).
Method 3. through production. GDP is calculated by adding up value added in all sectors of
production. Value added = total revenue – expenditures on inputs other than capital and labor.
These inputs are intermediate goods (insatsvaror). Value added = förädlingsvärde
In Swedish:BNP till marknadspris kan mätas från:
1. Utgiftssidan. 2. Inkomstsidan. 3. Produktionsidan.
In an economy without a government sector and without international trade
(Exports=imports=0).
Income, Expenditure
And the Circular Flow
There are 2 ways
of viewing GDP
Total income of everyone in the economy
Total expenditure on the economy’s
output of goods and services
Income $
Labor
HOUSEHOLDS
supply the factors of production (capital and labor)
to FIRMS which use
Households
Firms
these factors of production to produce final goods and services that are sold to the
HOUSEHOLDS.
Goods
Expenditure $
For the economy as a whole, income must equal expenditure.
GDP measures the flow of dollars in this economy.
Method 1 is to measure GDP from expenditures on goods, method 2 is to measure GDP from
income.
Households supply the factors of production (capital and labor) to the firms. In other words,
the households own the firms. Firms use factors of production to produce final goods and
services that are sold to the households.
The households’ expenditures on final goods and services equals factor incomes (wages,
interest, profit) from the firms in this economy.
Example of method 3; that is, measuring GDP by adding up value added in
sectors of production. Assume one final good in the economy, bread:
Sector:
Total revenue=
Cost of inputs
Value
P*Y
Other than K
Added.
And L
FörädlingsFörsäljningsvärde
Kostnad
för värde
insatsvaror
Peasant: Wheat
100
0
100
Miller: flour
150
100
50
Baker: bread
150
50
200
Sum:
200
Method:
1
3
the different
Capital and
Labor
income
100
50
50
200
2
Comparing GDP over time
Nominal and real GDP: GDP in current and constant prices
In Swedish: BNP till marknadspris i löpande och fasta priser.
The value of final goods and services measured at current prices is called nominal GDP. It can
change over time either because more goods and services are produced or because there is a
change in the prices of these goods and services. We calculate real GDP to see whether the
country is producing more or less goods and services over time.
Compute GDP in current prices (BNP till marknadspris i löpande priser) according to method
1. Assume two goods in the economy: Q1 , Q 2
1
1
2
2
Nominal GDP in 2000: GDP2000  P2000
 Q2000
 P2000
 Q2000
Real GDP 2000 (in 2000 year prices) = nominal GDP 2000.
1
1
2
2
Real GDP 2001 (in 2000 year prices) = P2000
 Q2001
 P2000
 Q2001
If real GDP 2001 > real GDP 2000, then the economy produced more goods and services in
2001.
Often in news papers and statistical reports you get data on GDP in current prices and data on
a price index. How do we calculate GDP in constant prices?
Real GDP in 2005 (in 2000 year prices) =
nominal GDP in 2005/ price index in 2005
Year
Nominal GDP=
Price index =
Real GDP
GDP in current
GDP-deflator: P
in 2000 year
Prices: P*Y
prices: (P*Y)/P=Y
2000
2500 billion kr
1.00
2500
2001
2600 billion kr
1.02
2600/1.02=2549
2002
2700 billion kr
1.04
2700/1.04=2596
2003
2800
1.07
2800/1.07=2616
2004
2900
1.10
2900/1.10=2636
2005
3000
1.12
3000/1.12=2678
2006
3100
1.14
3100/1.14=2719
Note that Q can not be observed as there are many goods in the economy.
Nominal GDP increases by (3100-2500)/2500=0.24; that is by 24 percent
The price level increases by (1.14-1)/1=0.14; that is by 14 percent between 2000 and 2006.
The real GDP increases by (2719-2500)/2500=0.088; that is, by 8.8 percent.
Real GDP (Y) = Nominal GDP (P*Y)/Price level (P)
Calculating the growth rate of real GDP using an approximate formula:

realGDP no m GDP GDPdeflator



rea l GDP
nomGDP
deflator
3100  2500  114  100  0.24  0.14  0.10
2500
100
1.2 percent difference between the exact and the approximate formula.
Real wage = nominal wage / price level
If the nominal wage in percentage terms increases more than the price level increases, then
the real wage increases, which means that you can buy more
goods and services: your purchasing power increases.
realwage no m wage GDPdeflator


rea l wage
nomwage
deflator
More rules for computing GDP:
2) used goods are not includes in the calculation of GDP.
3) If newly produced final goods is stored, it is inventory investment which is part of private
investment. When the goods are finally sold, they are considered used goods.
4) Some goods are not sold in the market place and do therefore not have market prices. We
must use their imputed value as an estimate of their value.
For example, home ownership and government services.
Price indexes provide an overall measure of the price level in the economy. We have to
choose a base year. CPI = Consumer price index.
In Swedish: KPI = konsumentprisindex
Assume CPI (2000) = 100. Assume 2 goods in the economy:
CPI (2001) 
1
1
2
2
P2001
 Q2000
 P2001
 Q2000
1
1
2
2
P2000
 Q2000
 P2000
 Q2000
When we use the original basket of consumption to calculate the price index, it is a Laspereys
index. If CPI (2001) > CPI (2000) then the overall price level has increased.
Alternatively: CPI (2001) 
1
1
2
2
P2001
 Q2001
 P2001
 Q2001
1
1
2
2
P2000
 Q2001
 P2000
 Q2001
When we use the current basket of consumption to calculate the price index, it is a Paasche
index.
CPI versus the GDP deflator (BNP deflatorn)
The GDP deflator measures the development of the prices of all goods and services produced.
The CPI measures the development of prices only of the goods and services bought by
consumers, including imported goods.
GDP-deflator (2001) = Nom. GPD in 2001 (P01*Y01)/real GDP in 2001 in 2000 year prices
GPD  deflator P(2001) * Y (2001) / P(2000) * Y (2001)
The GDP-deflator is a Paasche index.
In macroeconomic models real GDP is denoted by Y (instead by Q that is used for quantity in
microeconomics) and the price level is denoted by P.
Comparing standard of living across countries/regions
Who has a higher material standard; that is, is richer in terms of goods and services they can
consume?
2 methods to compare GDP across countries:
1. The Exchange Rate Method uses the current exchange rate to convert GDP in domestic
currency to GDP expressed in dollars:
Formula for calculating GDP from spending when assuming 2 goods in the economy:
GDP (kr) = Pnt(kr)*Qnt + Pt(kr)*Qt
Pnt = price of the non-tradable good, e.g. hair-cuts, housing services, etc
Qnt = quantity of the non-tradable good,
Pt= price tradable good, Qt=quantity of the tradable good
GDP in dollars:
GDP ($) = e ($/kronor)*GDP(kr) =
e ($/kronor)*Pnt(kr)*Qnt + e ($/kronor)*Pt(kr)*Qt
e ($/kronor) is the nominal exchange rate: In 2007: e ($/kronor) =1/6.
2. Purchasing Power Parity Method controls for the fact that prices differ across
countries. The method replaces domestic prices with average prices across countries
(in $).
GDP in dollars according to the PPP-method:
GDP ($) = e ($/kronor)*GDP(kr) =
Average Pnt ($)*Qnt + Average Pt ($)*Qt
We want to use the same prices because we want our measure of income to reflect the
quantity of goods that are available in one country during a year. Using different prices for
different countries distorts the picture.
If the law of one price holds for tradable goods, then
average Pt ($) = e ($/kronor)*Pt(kr)
That is, the law of one price says that the price of a tradable good expressed in the same
currency should be the same in all countries. We expect to hold if transportation costs and
differences in VAT are zero across countries.
Assuming zero transportation costs and no country-differences in VAT, arbitrage tends to
equalize prices across countries: It is profitable to buy the good where the price is low, and
ship it to the country where the price is high. Higher demand in the country in which the price
was initially low tends to put an upward pressure on the price in this country, and a higher
supply in the country where the price was initially high tends to put a downward pressure on
the price in this country. Thus, the market mechanism tends to equalize prices across
countries for tradable goods.
The law of one price does not hold for non-tradable goods and services:
In a poor country, the price of a non-tradable good tends to be lower because of lower labor
costs than in a richer country.
Ex: e($/Etiopian currency)* Pnt(Etiopia) < Average Pnt($)
 PPP-adjusted GDP ($) for Etiopia > Not PPP-adjusted GDP ($) for Etiopia= GDP ($)
according to the exchange rate method.
Thus, the exchange rate method understates GDP ($) in poor countries.
Because domestic prices (in $) are lower in poor countries.
In news reports you hear that in this country they earn 400 per capita and year. I have often
thought: How can they survive? The explanation is that this income figure is not PPPadjusted.
Purchasing Power Parity means that prices expressed in the same currency are the same
across countries and regions. PPP does empirically not hold as prices on non-tradable goods
tend to be lower in poor countries.
Application on Regions within a country: If nominal income per capita in the Stockholm
region is twice as high as the nominal income per capita in the Karlstad region, the purchasing
power of income per capita in the Stockholm region might be less than twice as high due to a
higher price level; e.g. on non-tradable goods and services such as housing.
The BIG MAC INDEX:
The law of one price implies: e($/kronor) P(kronor) = P*(dollars)
If e($/kronor) P(kronor) > P*(dollars)
Then the Swedish currency is too strong; “ overvalued” according to the journal Economist.
Critique: The law of one price should not hold for Big Mac as it is not a tradable good.
Measuring GDP through expenditures on final goods and services (method 1) in the real
world: that is, with a public sector and internatinal trade.
Availability of newly produced goods
And services
GDP = 3000 billions kronor
Imports = 1500 billions kronor
Sum: 4500 billions kronor
In Swedish: FÖRSÖRJNINGSBALANS
Tillgång på nyproducerade varor och
tjänsterr
BNP = 3000 miljarder kronor
Import = 1500 miljarder kronor
Summa: 4500 billions kronor
Use of newly produced goods and services
Private consumption = 1500
Private investment = 400
Public consumption = 700
Public investment =300
Exports = 1600
Sum: 4500
Användning
Privat konsumtion = 1500
Privata investeringar = 400
Offentlig konsumtion = 700
Offentliga investeringar =300
Export = 1600
Summa: 4500
In other words, GDP+imports = private consumption+private investment +
Public consumption and public investment + exports
Rearranging:
GDP = private consumption+private investment +
Public consumption and public investment + exports - imports
Expressing this NATIONAL INCOME IDENTITY in real terms (in constant prices):
Real GDP (Y) = real private consumption (C)+real private investment (I) +
Real Public consumption (GC) and public investment (GI) + exports – imports (NX)
Thus,
Real GDP (Y) = C+ I+ G+NX
Where G = GC+ GI. Note that P*Y= GDP in current prices.
C = real expenditures of households on final goods and services. Goods are sometimes
categorized into nondurable and durable goods.
I = real expenditures on new machines, new buildings, and inventory build-up by firms. I is
gross private investment.
G: real government spending on (/purchases of) final goods and services.
Often (but not in the Mankiw textbook) G is split up into government consumption (GC) and
government investment (GI).
Examples of GC are expenditures on teachers’ and doctors’ salaries.
Examples of GI is expenditures on new roads and government buildings.
Note: Government transfers to households such as unemployment benefits etc. are not
included in G: Total government expenditures = G + government transfers to households
(unemployments benefits, transfers to poor, to kids, to retired people) and to firms.
NX = real net exports = trade balance = value of exports of final goods and services – value of
imports of final goods and services. Thus, NX is net expenditure from abroad on our goods
and services.
Note: Capital stock this year = capital stock last year + I – depreciation (= reduction of the
value of the capital stock). Net investment = I – depreciation.
One way to think about real variables. Assume that only one good is produced in the
economy; e.g. corn. Then production, Y, is measured in tonnes of corn. Some of this corn is
used for private consumption (C), some for private investment (it is planted in the ground to
yield production next year) (I), some for government consumption and investment (G), and
some of the production is shipped abroad (Export) and some corn is imported (Imports).
(NX=exports-imports).
Ex.: Y=C+I+G+NX: 100 = 50 + 20 + 20 + 10
One aim of this section is to show that GDP is roughly the income a country’s citizens.
Other measures of income:
Gross national product (GNP) = GDP – (wages and capital income of Swedish workers and
firms operating abroad – wages and capital income of foreign workers and firms operating in
Sweden).
By Swedish worker I mean a Swedish resident.
Gross National Product (GNP) = GDP + net factor income from abroad (NFI)
In Swedish: Bruttonationalinkomsten till marknadspris (BNI) = Bruttonationalprodukten till
marknadspris (BNP) +
faktorinkomster från utlandet, netto (NFI).
GNP counts all final output produced by domestically owned inputs (workers and capital), no
matter where those inputs are situated in the world.
GDP counts all final output produced within the country regardless of who owns the inputs
(foreign or domestic citizens) involved.
GNP is a income measure whereas GDP is a production measure.
In a closed economy: GNP = GDP
In Stockholm: GDP > GNP as many workers compute to Stockholm but live in other regions
e.g. Uppsala. That is, they contribute to GDP in Stockholm but their incomes are not part of
the Stockholm GNP.
GNP (Stockholm) = GDP (Stockholm) – wages of workers that live in other regions than
Stockholm.
Another concept is disposable GNP (DGNP)
DGNP = GNP + (transfers from foreign countries – transfers to foreign countries) = GNP +
net transfers from abroad (NFTr).
For Sweden: DGNP < GNP as net transfers from abroad are negative.
From now on we assume that net transfers from abroad = 0.
In Swedish: Disponibel bruttonationalinkomst (DBNI) = bruttonationalprodukt till
marknadspris (BNP) + faktorinkomster från utlandet, netto (NFI) + transfereringar från
utlandet, netto (NFTr).
Other income measures continued:
Net national product (NNP) = GNP – value of depreciation of capital
In Swedish: Nettonationalinkomst till marknadspris (NNI) =
Bruttonationalinkomst till marknadspris (BNI) – kapitalförslitning
National Income = NNP – Indirect taxes (such as the value added tax)
In Swedish: Nettonationalinkomst till faktorpris =
nettonationalinkomst till marknadspris (NNI) – indirekta skatter
National income = compensation to employees (70 %) +
income to noncorporate business + corporate profits + net interest of firms.
Note: compensation to employees (workers) is total labor cost = gross income + social
insurance contributions, which is around 2/3 of national income.
Personal Income = National Income
- corporate profits - social insurance contributions
- net interest received by the business sector + dividends
+ government transfers to individuals + personal interest income
In Swedish: Hushållens inkomster
Disposable Personal Income = Personal Income – individual tax payments
In Swedish: Hushållens disponibla inkomst
Summary:
Gross National Product (GNP) = Gross Domestic Product (GDP) + net factor income from
abroad (NFI)
Net National Product (NNP) = GNP – depreciation of physical capital.
National Income = NNP – indirect taxes (e.g. value added tax) = labor income including
direct labor income taxes and social security contribution + capital income.
Dispsable Personal Income = National Income – (direct taxes and social security
contributions) + transfers to households.
In Swedish:
Bruttonationalinkomst till marknadspris (BNI) = Bruttonationalprodukt (BNP) +
faktorinkomster från utlandet, netto.
Nettonationalinkomst till marknadspris (NNP) = BNI – kapitalförslitning
Nettonationalinkomst till faktorpris = NNP – indirekta skatter (t ex moms) = arbetsinkomster
inklusive direkta skatter på arbete och arbetsgivaravgifter + kapitalinkomster före skatt.
Hushållens disponibla inkomst = nettonationalinkomst till faktorpris – direkta skatter på
arbete och kapital – arbetsgivaravgifter + transfereringar till hushållen (barnbidrag,
studiebidrag, A-kassa).
Problem med att använda BNP per capita som ett mått på levnadsstandard eller
välfärd:
* BNP-måttet tar inte hänsyn till produktion som inte prissätts på marknader eller utförs av
den offentliga sektorn.
Det gäller t ex produktion som utförs i hemmet. Tidigare var de flesta kvinnor hemma och
utförde där en rad sysslor som numera utförs av den offentliga sektorn.
T ex dagis. Värdet av den offentliga sektorns produktion räknas in i BNP från inkomstsidan.
Dvs. lönekostnaderna för dagis-personalen räknas in i BNP.
Är det så att för U-länder så sker en större andel av produktionen utanför marknaden och den
offentliga sektorn? Vad innebär detta ifall man använder BNP per capita som ett mått på
levnadsstandard? Över- eller underskattas u-ländernas BNP per capita i relation till iländernas BNP-per capita?
* USA har högre BNP per capita än Europa. Men amerikanen jobbar i genomsnitt fler timmar
än europeen…Ifall vi gillar fritid så är BNP per capita ett dåligt mått på lycka.
* En stor svart sektor räknas inte in i BNP-måttet.
*En högre BNP innebär en högre materiell standard men kan även innebära större
miljöföroreningar. En ekonomi som inte producerar något förorenar inte heller.
Simplifying assumptions in Classical and Keynesian models:
NFI=depreciation of capital=0
Y =GDP=GNP=NNP= National Income + indirect taxes= indirect taxes +
labor income (including direct taxes and social security contributions) + capital income
Assume: T = net taxes = tax payments – transfers to households.
Households’ disposable income = Y - T
In Swedish:Y= BNP till marknadspris =BNI till marknadspris = NNI till marknadspris=NNI
till faktorpris + indirekta skatter = indirekta skatter + Löneinkomster+direkta skatter + sociala
avgifter + kapitalinkomster.
ARBETSLÖSHET och något om inflation
Arbetslöshetsmått etc:
I arbetskraften är sysselsatta och arbetslösa.
Arbetslöshetsgraden = antalet arbetslösa/antalet i arbetskraften.
Arbetskraftsdeltagandegraden = arbetskraften/befolkningen 19-65
Sysselsättningsgraden = sysselsatta/befolkningen, 19-65
Detta mått visar hur stor andel av dom som kan jobba som faktiskt gör det.
Teori: Typer av arbetslöshet
Vid full sysselsättning är arbetslösheten lika med den naturliga arbetslösheten.
Arbetslösheten = den naturliga arbetslösheten + cyklisk arbetslöshet.
Den cykliska arbetslösheten > 0 eller < 0 eller = 0.
In recessions both cyclical unemployment is above 0. Also machines and factories are not
fully employed (they are idle) during a recession.
Den naturliga arbetslösheten = friktionsarbetslöshet och strukturell arbetslöshet
The unemployment caused by the time it takes workers to search for a job is called frictional
unemployment. Because changes in the composition of demand among industries or regions
are always occurring, and because it takes time for workers to change sectors, there is always
frictional unemployment. The fact that the European unemployment rate is higher than the
unemployment rate in the USA is often explained by a more generous unemployment
insurance: The replacement rate = Unemployment benefits/previous labor earnings.
Strukturell arbetslöshet uppstår pga. att nya branscher uppstår och expanderar medan andra
stagnerar. Kompentensen för att jobba i de expanderande branscherna inte är desamma som
kompetenserna för att jobba i de gamla sektorerna. Därför måste folk utbilda sig på nytt vilket
tar tid.
Strukturell arbetslöshet uppstår också när lönerna är för höga.
Ibland kallas arbetslöshet pga. för höga löner klassisk arbetslöset. I så fall:
Den naturliga arbetslösheten = friktionsarbetslöshet + strukturell arbetslöshet + klassisk
arbetslöshet
Real
wage
Wage rigidity is the failure of
wages to adjust until labor
supply equals labor demand.
S
U
Rigid
real
wage
D
Labor
If the real wage is stuck above the
equilibrium level, then the supply
of labor exceeds the demand.
Result: unemployment U.
The unemployment resulting
from wage rigidity and job
rationing is called structural
unemployment. Workers are
unemployed not because they
can’t find a job that best suits
their skills, but rather, at the
going wage, the supply of labor
exceeds the demand. These
workers are simply waiting for
jobs to become available.
Arbetslöshetens kostnader:
Produktionsbortfall.
Vid ofrivillig arbetslöshet ohälsa etc
The costs of inflation:
If inflation is expected the people have adjusted, so the only cost is shoe-leather inflation as
you go to the money machine more often because of the higher nominal interest rate. Also
restaurants want to reprint menus more often, which is a social cost (=samhällsekonomisk
kostnad).
If actual inflation is higher than the extected then the real actual interest rate is lower than the
expected at a given bank interest rate, which means that savers loose and borrowers gain.
Realräntan (räntan korrigerad för inflation) = bankräntan (som antas vara fixt) – inflation.
När man bestämmer sig för att sätta in pengar på bank så gör man en bedömning av
inflationen. Dvs. man beräknar den förväntade inflationen.
Den förväntade reala avkastningen (=realräntan) = fix bankränta – förväntad inflation.
Faktisk real avkastning = fix bankränta – faktisk inflation.
Also your real wage becomes lower than expected.
Hyperinflation is more than 50 percent per month.
AGGREGATE SUPPLY: FACTOR MARKETS: CAPITAL AND LABOR MARKETS
Assume an aggregate production of Cobb-Douglas type:
real GDPt  Yt  F ( At , Kt , Lt )  At  G(Kt , Lt )  At  Kt  L1t , where 0 <  < 1.
Ex.:   0.5 , easy to calculate with… Y  A  K 0.5  L0.5
If there is one good in the economy, then Y is the quantity of this good.
In kilograms, or liters depending on what the good is.
K = aggregate physical capital (machines and buildings)
Should be measured by machine-hours and by hours buildings are used per year. But in reality
K is measured by the real dollar value of the aggregate physical capital. It is then implicitly
assumed that the real dollar value of K is proportional to the number of machine hours and
hours buildings are used.
L = aggregate labor input is measured by total hours worked (or by number of workers if
every worker works the same number of hours).
A = totalfactorproductivity, captures the effect on Y of all factors apart from K and L that
impacts Y. For example, technological progress (innovations) or increased education of
workers increases Y at given levels of K and L. Higher energy prices should decrease the use
of energy and thereby also A and Y at given levels of K and L. Empirically  is estimated to
be around 1/3, which is the typical value of the share of capital income of national income.
The aggregate production function:
Assume that K and A are constant: A=1, K=9, and  =0.5.
 Y  A0  K00.5 L0.5  1 90.5  L0.5  3  L0.5
L
Y  3  L0.5 MPL  Y  Y1  Y0 Y/L
L L 1  L0
0
0
0
1
3
3
3
2
4.2
4.2-3=1.2
4.2/2=2.1
3
5.2
1
5.2/3=1.7
5
6.7
(6.7-5.2)/2=0.75
6.7/5=1.34
8
8.5
(8.5-6.7)/3=0.6
8.5/8=1.1
From the table we see that:
When L increases, Y increases but at a diminishing rate; because the capital-labor ratio
decreases when L increases. Each worker gets less capital to work with.
*MPL and APL(=Y/L) falls when L increases and K and A are constant.
When MPL is below APL (=labor productivity), APL is decreasing.
Plotting the production function for A=1, K=9 and alpha = 0.5:
14
12
10
8
Y
Serie1
6
4
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
L
MPL is the slope of the production function.
1.6
1.4
1.2
MPL
1
Serie1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
L
If the stock of physical capital is higher:
 Y  A0  K00.5 L0.5  1160.5  L0.5  4  L0.5
Parameters:
A=1, K=16,
A=1, K=16,
A=1, K=16,
 =0.5.
 =0.5.
 =0.5.
0.5
L
Y/L
Y

Y

Y
Y 4 L
0
MPL 
 1
L L 1  L0
0
0
1
4
4
4
2
5.6
1.6
5.6/2=2.8
3
6.9
1.3
6.9/3=2.3
5
8.9
1
8.9/5=1.8
8
11.3
0.8
11.3/8=1.4
* MPL and APL increases when K (or A) increases and L is constant.
This is shown by comparing the 2 tables above.
16
17
DO NOT READ:If A Y and MPL and APL at a given L (and K):
Assume that A=1 and A=2 , K=9, and  =0.5.
A=1 (curve below) and A=2 (curve above)
30
25
Y
20
Serie1
15
Serie2
10
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
L
A=1 (curve below), A=2 (curve above)
3.5
3
MPL
2.5
2
Serie1
1.5
Serie2
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
L
Summary:
MPL (=
Y
) and Y/L decrease when L increases and K and A are constant.
L
MPL and Y/L increase when K (or A) increases and L is constant.
By symmetry, the same holds for MPK and K: that is,
MPK (=
Y
) and Y/K decrease when K increases and L and A are constant.
K
MPK and APK(=Y/K) increase when L (or A) increases and K is constant.
Review of exponents
Definition:
x n  x  x  ...  x .
E.g.
22  2  2  4
n terms
m
 x n
Ex.:
x2  x3  x5
Rule 1: x n  x m
Rule 2:
Rule 3:
Rule 4:
Y
L
x n

1
xn  xn m
xm
x0  1
 A K
0.5
 L0.5
L
Or more generally,
Y A  K   L1
L

Ex.:
xn
L
x 4 
and
x2  x  x
1
x4
x 2  x 23  x 1  1
x
x3
Ex.: 20  1
Ex.:
 A  K 0.5  L0.5  L1  A  K 0.5  L0.51 
A  K 0.5
L0.5
 A  K   L1  L1  A  K   L 
A K 
L

K
 A 
L
Production per worker, Y/L, increases if technology improves (A) or if every worker gets
more capital (K/L)
Deriving the mathematical expression for MPL:
Review of the derivative:
(1) If y  x 2
(2) If y  b  x 2 where b is a constant
(3) More generally, y  b  x n
then
then
then
dy
 2  x 21  2  x
dx
dy
 2  b  x 21  2  b  x
dx
dy
 2  b  x n 1
dx
(1   )  A0  K0
dY d ( A0  K0  L1 )


MPL 

 (1   )  A0  K0  L 
dL
dK
L
Thus, MPL = (1-alpha)*(Y/L). That is, MPL is always lower than Y/L
2. Y  A  K   L1 exhibits constant returns to scale.
If you e.g. increase each factor of production by 10 percent, then production also increases by
10 percent.
If production increases by more, increasing returns to scale (IRS), and if production increases
by less, decreasing returns to scale (DRS). CRS imply constant long-run average costs, IRS
imply decreasing long-run average costs and DRS imply increasing long-run average costs
when production increases.
THE DEMAND OF K AND L IN THE LONG RUN BY FIRMS
In macro models households/individuals supply L and K (through saving as saving equals
investment in a closed economy). Firms demand L and K.
Assumptions: There exist many identical firms that produce an identical good.  Perfect
competition is assumed  The firms maximize profits.
The problem of the representative firm is to choose K and L and thereby output (Y) so that
profits are maximized. Assuming two factors of production (K, L):
Profits ($) = Total revenue ($) – capital cost ($) – labor costs ($)
 Profits ($) = P  Y – R  K – W  L
where Y  A  K   L1 , P=price of the good, Y = quantity of the good,
R= rental price of one unit of capital per period of time,
W= nominal wage per worker per period of time.
Problem of the firm: Choose K and L (and thereby output) to maximize profits:
Profits ($) = P  A  K   L1 – R  K – W  L
Assuming perfect competition implies that the individual firm cannot influence prices: P, R
and W are exogenous from the point of view of the firm. We also assume that A and  are
exogenous from the point of view of the firm as they are assumed to be given by the
technology. To maximize profits, the firm should choose K and L so that:
R  P  MPK and W  P  MPL
W
 MPL
P
R
Y
W
Y
 MPK   
 MPL  (1   ) 

and
P
K
P
L
or
R
 MPK
P
and
These 2 conditions should hold simultaneously:
If we combine them we find the profit-maximizing firm’s capital-labor ratio, K/L, which is
independent of the level of production. In other words, when assuming a Cobb-Douglas
production function the optimal (=cost-minizing) capital-labor ratio is identical for a low level
and for a high level of Y.
Divide by W/P on both sides of the first condition:
R / P   (Y / K )
  (Y / K )
Y
1
L
 L


 
 
W /P
(W / P )
(1   )  (Y / L)
K (1   ) Y (1   )  K
W (1   )  K
R
L
K

 
 

R
 L
W (1   )  K
L
demand


W

(1   ) R
Thus, a higher Wage relative to the Rental price of capital makes optimal to use more capital
relative to labor at a given level of production.
Determining Equilibrium Factor Prices:
They are found where Supply = Demand
Assume that the supply of K and L is fixed: K , L

demand

W
K K
W / P (1   ) K






L L
(1   ) R
R/P

L
   
Thus, a higher capital labor ratio increases the equilbrium real wage relative to the
equilibrium real rental rate.
Example: Assume: Y  A  K   L1 , and A=1,  = 0.5 , K =9, L =9:
L supply increases from 9 to 16
2.5
MPK, R/P
2
1.5
Serie1
1
Serie2
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
K (Kbar = 9)
L supply increases from 9 to 16
1.6
1.4
MPL, W/P
1.2
1
0.8
Serie1
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
L (Lbar = 9 or 16)
R / P  MPK  0.5  ( K / L )0.5  0.5  (9 / 9) 0.5  0.5
W / P  MPL  0.5  ( K / L)0.5  0.5  (9 / 9)0.5  0.5
Assume now that L =16 due to labor immigration
R / P  MPK  0.5  ( K / L)0.5  0.5  (9 /16) 0.5  0.6667
W / P  MPL  0.5  ( K / L )0.5  0.5  (9 /16)0.5  0.375
15
16
17
Thus, If L  (due to labor immigration) and K constant 
R / P  MPK  and W/P=MPL .
 Domestic workers loose in terms of W/P from labor immigration, whereas domestic capital
owners gain as R / P  MPK .
2 other examples:
(2) If L  (due to black death) and K constant  R / P  MPK , W/P=MPL 
 Workers gain and capital owners loose income.
(3) If K  and L constant  R / P  MPK  and W/P 
 Workers gain and capital owners loose.
Conclusion: w/p  and R / P  MPK  if ( K / L ) 
The distribution of income between workers and capital-owners
Pr ofits($)  P  Y  R  K  W  L
Assuming perfect competition in the rental market for capital:
 Pr ofits($)  P  Y  (r   )  P  K  W  L , where r=real return to capital
Note that: if   0 : PY=GDP=GNP=National income.
(net of taxes; that is, taxes are ignored.)
If   0 , PY=GDP=GNP= national income +   P  K (depreciation)
where (r   )  P  K = nominal capital income
Real Profits =
Pr ofits($)
W
 Y  (r   )  K   L
P
P
where (r   )  K = real capital income, W/P= real wage
Assuming perfect competition in the goods market means that the profits are zero and that
firms maximizes profits by choosing K and L so that W/P=MPL and
R / P  (r   )  MPK .
 Y  MPK  K  MPL  L  Y   

Y
Y
 K  (1   )   L  Y    Y  (1   )  Y  0
K
L
Y    Y  (1   )  Y
Thus, the share of GDP (net of taxes) that goes to capital owners is  . The share of GDP (net
of taxes) that goes to workers is 1-  .
We have data on labor and capital income. Thereby, we get an estimate of  , which is
around 1/3 for both developing and developed countries. Why?
In poor (rich) countries: K is low (high), r is high (low), L is high (low) and W/P is low (high)
Elaborating on the production function
A in the production function depends on business climate, tax levels, knowledge of workers,
energy used, etc. If we particularly want to focus on energy used we might want to include
this variable specifically in the production function:
Yt  At  Energyt Kt   L1t  , where 0 <  ,  < 1  0<1-  -  <1.
Knowledge of the work force measured by educational level

In the basic formulation of the pf: Yt  At  Kt  L1t , where 0 <  < 1, an increase of the
knowledge of the workforce e.g. measured by the educational level increases A. In an
alternative formulation:
Yt  At  Kt  H t  L1t  , where 0 <  ,  < 1  0<1-  -  <1.
Where H = number of workers with higher education, L = number of workers with low
education. With this formulation of the production function, A does not capture the
knowledge or the educational level of the work force.
THE LABOR MARKET FROM A MACRO PERSPECTIVE
In macro models households/individuals supply L and K (through saving as saving equals
investment in a closed economy). Firms demand L and K.
Assumptions: There exist many identical firms that produce an identical good.  Perfect
competition is assumed  The firms maximize profits.
The problem of the representative firm is to choose K and L and thereby output (Y) so that
profits are maximized. Assuming two factors of production (K, L):
Labor Demand in the short run (=when the capital stock, K, is fixed)
The firms are assumed to be price-takers in output and input markets; that is, P (=product
price), W (nominal wage per period of time), and R (rental price per unit of capital per period
of time) are exogenous from the individual firms’ point of view; that is, an individual cannot
impact P, W, and R. Assume also that the capital stock (K) is fixed; in other words, assume
short-run analysis. The firm has to pay for a fixed factor of production regardless of its chosen
level of production.
If the representative firm maximizes profits it should choose L (and thereby the level of
production, Y) so that: P*MPL = W  W/P = MPL
MPL,W/P
MPL and Employment
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
1
2 3 4
5 6 7
8 9 10 11 12 13 14 15 16
Employment (L)
Production (Y)
Production and employment
14
12
10
8
6
4
Serie1
2
0
1
2
3 4
5 6
7
8 9 10 11 12 13 14 15 16
Employment (L)
In the graphs above we assumed:
Y  A K  L1 , and if A=1,
K  9 , and   0.5
 Y  3 L
The profit-maximizing level of employment, L* , is given by:
MPL 1.5 L*1/2  W / P
Solving for L* as a function of the exogenous variables:
1/2
*
L
 1.5 


W / P 
2
 If e.g. W/P=0.75, then L* =4.
In top figure above: where is Y/L-curve?
The producer surplus in real terms (= real profits + real fixed costs = real profits + ( R / P)  K )
is the area between the MPL-curve and the real wage, W/P=0.75.
More generally:
The problem of the representative firm is to choose L (and thereby Y) in order to maximize
Profits ($) = P($)  A K   L1  W ($)  L  R($)  K
Where P = Product price, W = nominal wage, R= nominal rental price of capital, and
Y  A K  L1
The profit-maximizing level of employment, L* , is given by:
P  MPL  (1 )  P($)  A K   L*  W
 (1 )  A K   L*  W / P
 This is the inverse demand curve for L.
Solving for L* : that is, deriving the demand curve for L:


L*   (1   )  A  K
W /P

 If W/P  L* ; If
1/



K  L* 
Equilibrium in the labor market and taxes
If there is perfect competition, the equilibrium real wage is where labor demand = labor
supply. See graph below.
Labor Supply: 3 possibilities:
(1) Labor supply is unrelated to the real wage; vertical labor supply-curve.
(2) Labor supply increases when the real wage increases; a positively sloped labor
supply-curve.
(3) Labor supply decreases when the real wage increases: when the real wage increases
the individual can afford to take more leisure, which she likes.
Factors that increase aggregate labor supply at a given real wage:
1. Labor immigration. 2. Lower unemployment benefits should increase the labor supply of
the domestic population.
Introducing labor taxes that creates a difference between the labor costs of the firms and
what the worker receives:
MPL,W/P
Employment and taxes
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
LS
Eqb W/P:t=0
LD
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Employment (L)
Introducing a tax creates a wedge between what the firm pays (producer real wage) and what
the worker receives (consumer real wage). A tax decreases employment. L decreases in figure
from 5 to about 3.5. The producer real wage increases to 0.8 (from 0.7 when tax is zero) and
the consumer real wage drops to below 0.4. Taxes on labor include social insurance- and
income taxes.
The consumer real wage drops more than the producer real wage increases as the labor supply
is more inelastic than labor demand in the figure above. The inelastic side of the market bears
the most of the tax burden.
Flows on the labor market
The average rate of unemployment around which the economy fluctuates is called the natural
rate of unemployment.
THE POPULATION Between 19-64 years are either employed (work or study) or
unemployed or outside the labor force. The labor force is employed + unemployed.
Also people below 19 or above 64 may work, but we ignore this.
NOTE: This model is a simplification: some that get jobs or become are initially outside the
labor force; they are students that finish school, or house-wife or husbands that return to the
labor market after having raised kids.
(1) Labor force (L)=Employment (E)+number of unemployed (U)
(2) Number of people finding jobs per period of time (f*U) =
Number of people loosing jobs per period of time (s*E)
f = the job finding rate (e.g. 0.1), which tells us what proportion of the unemployed that finds
jobs per period of time (e.g. 10 percent).
s = the job separation rate (e.g. 0.05 = 5 percent) tells of what proportion of the employed that
looses their jobs per period of time.
Solving for the natural rate of unemployment (U/L) as a function of the exogenous
parameters(s,f): Insert equation (1) into (2): f*U=s*(L-U)
Divide by 1/L:  f*U/L=s*(L-U)/L  f*U/L=s-s*U/L (f+s)*U/L=s
 U/L=s/(s+f)
If f  U/L ; If s  U/L . Plug in numbers see that this is true.
But s and f tend to be positively correlated.
Relative to old EU-countries, the US have a higher f and a higher s. A high f and a high s can
theoretically yield the same natural unemployment rate as a low f and a low s. However, the
natural unemployment rate is higher in the old EU-countries than in the USA. This model
relates the long-run unemployment rate to job finding and to job separation rate. It does not
explain why there is unemployment.
The reasons of unemployment are often divided into job search and too high real wages which
lead to too few jobs.
The natural rate of unemployment = frictional and structural unemployment.
The unemployment rate = the natural rate + cyclical unemployment.
Cyclical unemployment > 0 or < 0 or = 0.
In a recessions both cyclical unemployment is above 0. Also machines and factories are
unemployed during a recession, which means that K used in production is less that the K that
is available in the economy.
The unemployment caused by the time it takes workers to search for a job is called frictional
unemployment. Because changes in the composition of demand among industries or regions
are always occurring, and because it takes time for workers to change sectors, there is always
frictional unemployment. The fact that the European unemployment rate is higher than the
unemployment rate in the USA is often explained by a more generous unemployment
insurance: The replacement rate = Unemployment benefits/previous labor earnings.
Real
wage
Wage rigidity is the failure of
wages to adjust until labor
supply equals labor demand.
S
U
Rigid
real
wage
D
Labor
If the real wage is stuck above the
equilibrium level, then the supply
of labor exceeds the demand.
Result: unemployment U.
The unemployment resulting
from wage rigidity and job
rationing is called structural
unemployment. Workers are
unemployed not because they
can’t find a job that best suits
their skills, but rather, at the
going wage, the supply of labor
exceeds the demand. These
workers are simply waiting for
jobs to become available.
3 reasons for structural unemployment:
1. Minimum wages.
2. Trade unions
Unions and collective bargaining between unions and employer organizations may lead the
actual real wage to be higher than the equilibrium real wage. Unions cares about the average
union member who has a job.
3. Efficiency wages. Efficiency wage theories imply that firms voluntarily (that is, even in the
case when unions do not exist) pay a higher real wage than they have to pay to attract workers
(=equilibrium real wage). Why?
Higher real wages than the equilibrium real wage may induce higher labor productivity (as
workers value their job and do not want to loose it) and is therefore consistent with profitmaximization and perfect competition.
Moreover, paying higher real wages than they have to attract workers is a way to lower labor
turnover and a way to attract high quality workers. Labor turnover might be costly for the
firms as it may be costly to recruit labor.
One main lesson: A higher real wage decreases labor demand;
A larger stock of physical capital increases labor demand;
decreased unemployment benefits increases labor supply.
Labor supply and unions
The model assumes that the union organizes all workers, and it is also assumed that the union
acts as a monopoly. The good it sells is labor. We also assume (to start with) that the union
faces unorganized competitive firms that demand labor.
Review: Maximization of Total Revenue
Assume a monopoly that faces a downward-sloping demand curve.
Assume that the inverse demand-curve is: P=10-Q
 MR = 10 – Q*2
Demand and Marginal Revenue
10
P,MR
5
0
-5
Serie1
1
2
3
4
5
6
7
8
9
10
Serie2
-10
-15
Quantity(Q)
Total revenue, P*Q, is maximized along the Demand-curve at the quantity where MR=0.
This occurs when 10  Q*  2  0  Q*  5
 Optimal price: P*  10  5  5 , TR*  P*  Q*  5  5  25
The union faces the aggregate demand curve for labor. We assume that the firms are not
organized into an employment organization that bargains over wages with the union. Firms
are assumed to be perfectly competitive.
The assumed problem of the union is to choose the point (real wage and the employment) on
the aggregate demand curve for labor that maximizes labor income: real wage*employment.
(An somewhat modified union objective would be to maximize labor income subject to the
constraint that more employment implies that leisure are foregone. The cost of foregone
leisure is reflected by the labor supply curve.)
In wage negotiations in Sweden there is an employer organization (who may act as an
monopsonist) and a union.
OPTIONAL (=not necessary to read) Numerical Example:
I The competitive outcome: Competitive firms and no union.
Assume: LS  50  (W / P)
Assume a perfectly competitive firm in the short run (=K is fixed):
Re al TR(L)  70  L  0.05  L2
Re al TC (L)  (W / P)  L
 Real profits=TR-TC
The profit-maximizing level of employment, Ld , is given by:
MR( Ld )=MC( Ld )
 70  0.1 Ld  W / P  700 10 W / P  Ld
In the competitive equilibrium: Ld  LS
 700 10  (W / P)*  50  (W / P)*
 700  60  (W / P)*  (W / P)*  11.6666.. ,  L*  50 11.666  583.333
 Re al labor income  (W / P)*  L*  11.666  583.33  6805
Re al profits  TR(L)  TC(L)  70  L  0.05 L2  (W / P)  L
 Re al profits  70  583  0.05  (583)2  (11.666)  583 
 40810 16994  6802  17014
II A union facing competitive demand for labor
The union wants to maximize (W/P)*L subject to the constraint that the union must choose a
point on the (inverse) labor demand curve: W / P  70  0.1 Ld
 Thus, the union chooses L (and thereby W/P) to maximize W / P  L  (70  0.1 L)  L .
The optimal labor supply is given by MR  70  0.2  L  0 :  L*  350 ,
 (W / P)*  70  0.1 350  35 , (W / P)*  L*  35  350  12250
 Re al profits  70  350  0.05  (350)2  (35)  350 
 24500  6125 12250  6125
 Labor income is higher and profits are lower relative to the competitive case (I). (Note: A
competitive firm can make a profit in the short run.)
Exercise: Calculate the (structural) unemployment. That is, the number of people that would
like to work when the real wage = 35 minus the actual employment.
III A single buyer of labor (monopsonist) facing unorganized workers
A monoponist chooses the point on the labor supply curve that maximizes profits. The real
wage the monopsonist has to pay is dependent on the number of workers it hires.
The problem of the monopsonist is to choose L (and thereby Y) and W/P in order to maximize
real profits: Re al Pr ofits  TR( L)  TC(L)  70  L  0.05  L2  (W / P)  L
The constraint is that the monopsonist must choose a point on the (inverse) labor supply
curve: W / P  (1/ 50)  Ls
Inserting the constraint into the profit function:
L
Re al Pr ofits  70  L  0.05  L2  ( )  L
50
The profit-maximizing level of employment, L* , is found by the equation
d (real profits )
 0.
dL
2  L*
0
50
Solving for L* : L*  500 , the optimal real wage: W / P  (1/ 50)  500  10
 70  0.05  L* 
W/P*L=5000,
Re al Pr ofits  70  500  0.05  (500)2 10 500 =35000-12500-5000=17500.
 Highest profits and lowest labor income of the cases I-III.
LABOR DEMAND AND SUPPLY
80
60
40
W/P
20
0
-20
1
2
3
4
5
6
7
Serie1
Serie2
Serie3
-40
-60
-80
Employment in 100s (L)
Case II and III are depicted in figure above.
The graphs are:
1.The inverse competitive demand curve for labor:
W/P=70-0.1*L
2. The marginal revenue curve associated to this demand curve:
MR=70-0.2*L
3. The inverse competitive labor supply curve: W/P=L/50
In wage negotiations between a monopsonist (an employer organization) and a union we
expect an outcome somewhere in between case II and case III. Where exactly should depend
on the relative strengths of these organizations.
Recall that unions in Europe are blamed for high unemployment.
GROWTH ACCOUNTING (TILLVÄXTBOKFÖRING)
Mathematics: Percentage Changes in Economics.
Expressing levels into growth rates; that is, into percentage changes:
y
x z

 .
y
x
z
Rule 1. If y(t) = x(t)*z(t), then
Ex.: Total Revenue (TR) = Price(P)*Quantity(Q)
 If P is raised by 10 % and sold quantity (Q) thereby decreases by 5%, then TR increases by
5 %.
Rule 2. If y(t) = x(t)/z(t), then
y
x z
.


y
x
z
Ex.: GNP per capita (y) = GNP(Y)/Population(Pop)
 If GNP (Y) increases by 5 % and the population increases by 3 % , then GNP per capita
increases by 2 %.
y
x
 a .
y
x
Rule 3. If
y(t)  x(t)a , then
Rule 4. If
Yt  At  Kt  Lt1 ,
then
Y
A
K
L

 
 (1   ) 
Y
A
K
L
(1)
(2)
(3)
thus the growth rate of Y equals:
(1)
The growth rate of totalfactorproductivity.
(2)
The contribution of physical capital.
(3)
The contribution of labor.
Question addressed by so-called growth accounting (Mankiw, ch. 8, app.)
growth accounting:
How big share of the growth rate of the GDP can be attributed to changes in capital, to
changes in the labor input and to changes in total factor productivity?
For developed countries we have good data on
We have not direct data on
A
A
Y  K
,
Y
K
and
L
L
,
as A captures the influence on Y of many different factors
on Y. E.g. taxes, climate for business, educational level of work force, infrastructure, social
capital etc.
Under perfect competion,  , is the share of national income that is capital income, and (1 ) is the share of national income that is labor income. We have data on labor income and
national income. Thereby, we get an estimate of  .
Example:
Year
2005
2006
Y
100
103
A
?
?
K
300
306
L
1000
1010
Y
K
L
 0.03,
 0.02 and
 0.01
Y
K
L
A
 0.03 =
+ 0.3*0.02 + 0.7*0.01
A
A

= 0.03 – 0.006 – 0.007 = 0.017.
A
0.017/0.03 = 0.57 : 57 percent of the growth rate of Y can be attributed to an increase in A.
0.006/0.03 = 0.2: 20 percent can be attributed to an increase in K.
0.007/0.03 = 0.23: 23 procent can be attributed to an increase in L.
We have not explained why K, L and A changes over time.
We have only been engaged in accounting.
The neoclassical growth model explains why K and thereby Y increase.
(A and L are exogenously given in this model; that is, they are determined outside the model.)
The growth rate of output equals the growth rate of aggregate demand, which can be
split up as follows:
Y C C I I G G NX NX
 
 
 


Y
Y C Y I
Y G
Y
NX
NATIONAL INCOME ACCOUNTING II: National Saving, Domestic Investment, the
trade balance and the current account balance.
Swedish: Nationalräkenskaper II: Nationellt finansiellt sparande, inhemska investeringar,
handelsbalansen och bytesbalansen.
Assume a closed economy: GNP = GDP = C + I + G
“income measure” = “production measure” = C + G + I
 S = GNP – C – G = I
 National saving (= income – (private consumption + government spending)) = gross
private investment
In Swedish: Nationellt finansiellt sparande.
Most often but not in the Mankiw textbook G is split up into GC and GI; that is, G = GC + GI.
GNP = C + I + GC + GI
 S = GNP – C – GC = I + GI
 National saving (= income – consumption (private and public)) = private and public gross
investment.
From now on we use the definition of national saving by Mankiw.
Disaggregating national saving into private and public saving:
Model assumption: T = net taxes (= government tax revenues – government transfer
expenditures). Thus, although in the real world there are many taxes, we lump them all
together. If T > 0, then government tax revenues > government transfer expenditures. This is
typically the case in the real world as taxes apart from paying for transfers also pays for G.
 GNP – T = private disposable income.
Often it is assumed that net taxes are positively related to income: T is procyclical; tax
revenues increases and transfer expenditures decreases in a boom (when Y increases relative
to potential GDP where the unemployment rate is at its natural rate). In a recession, when Y is
below potential Y, tax revenues fall and transfer expenditures increases due to higher
expenditures for the unemployed.
The fact that net taxes tend to fall in a recession and rise in booms is an so-called automatic
stabilizator of the market economy. Lower net taxes in a recession implies that disposable
income fall less than income which may mean that private consumption which constitute half
of aggregate demand fall less.
It is often assumed Taxes = t*GDP, where t is a proportional tax rate, e.g. 0.4 (that is, 40
percent). With this formulation government tax revenues increase when GDP increases. In a
recession tax revenues tend to decrease and transfer expenditures tend to increase (e.g.
unemployment benefits), decreasing public saving. In a boom the opposite happens, tax
revenues high and transfer expenditures low, increasing public saving.
S = GNP – C – G = I
 National Saving (S) = I
 S = GNP – C – G + T – T = I
 S = (GNP – T – C) + (T – G) = I
 National Saving (S) = private saving + public saving = I
In Swedish: Privat finansiellt sparande.
Public saving = government income (government tax revenues) - total government
expenditures (= government transfer expenditures + government spending on final goods and
services, GC+GI) = government budget surplus
In Swedish: Offentliga sektorns finansiella sparande =offentliga sektorns budgetöverskott.
The open economy: THE TRADE BALANCE AND THE CURRENT ACCOUNT
GDP (Y) = C + I + G + NX (=Exports – Imports)
 NX = GDP – (C+I+G) = Domestic output – domestic spending/absorbtion
 If domestic output exceeds domestic spending, exports are larger than imports. In other
words, the trade balance, NX, is positive.
In Swedish: NX = handelsbalans
National Saving:
 GNP = GDP + net factor income from abroad (NFI) = C + I + G + NX + NFI
 S = GNP – C – G = I + NX + NFI
where NX + NFI = current account balance
 national saving = private domestic investment + current account balance
In Swedish: Nationellt finansiellt sparande = inhemska privata investeringar + bytesbalansen.
The current account balance is often called net foreign investment. Saving is either invested
domestically or abroad. Assume that NFI = 0, and NX>0. This means that the value of exports
is bigger than the value of imports, which means that foreign countries become indebted
towards us and that our foreign assets thereby are increased. Hence, the term net foreign
investment is explained.
To be exact, one should also account for net transfers from abroad when calculating national
saving:
 GNP + net transfers from abroad (NFTr) = C + I + G + NX + NFI + NFTr
 S = GNP + NFTr – C – G = I + NX + NFI + NFTr
where NX + NFI + NFTr = current account balance (net foreign investment)
 national saving (S) = private domestic investment + current account balance
If NFI and NFTr are relatively stable over time, changes in national saving (S) and in
domestic investment (I) are reflected in the trade balance (NX).
If national saving exceeds private domestic investment, the current account is positive.
As mentioned most often (but not in Mankiw) G is split up into GC and GI:
 S = GNP + NFTr – C – GC = I + GI + NX + NFI + NFTr
 national saving (S) = private and public domestic investment + current account balance
(net foreign investment)
Back to Mankiw’s formulation and disaggregating national saving (S) into private and public
saving:
 S = GNP + NFTr – C – G + T – T = I + NX + NFI + NFTr
 (GNP + NFTr – T – C) + (T – G) = I + NX + NFI + NFTr
 (private saving) + (public saving) = private domestic investment + current account balance
(net foreign investment)
Application: “Twin Deficits”.
A government budget deficit, G-T>0, generates a current account deficit if private saving and
domestic private investment (I) are constant.
E.g. if military spending increases due to the Iraqi war, G (T-G)  NX 
if private saving, I, NFI, and NFTr are constant.
Summary:
National Saving = GDP (Y) + net factor income from abroad (NFI) + net transfers from
abroad (NFTr)
 National Saving – (C+G) = I + NX + NFI + NFTr
where NX + NFI + NTr = current account balance
ECONOMIC GROWTH
Aim to explain why the standard of living (GDP/GNP per capita) changes over time. Main
text: Mankiw, Ch. 7-8.
Math: Growth rate = Percentage Change
y y
y
, e.g. r1 = 0.02, that is, 2 % .
 1 0  r1
y
y0
where y 0 = income per capita year 0, y1 = income per capita year 1.
r1 = growth rate/percentage change between year 0 and year1.
y1  y0  r1  y0  y1  r1  y0  y0  y0  (1  r1 )

Analogously:
y2  y1  (1  r2 ) , y3  y2  (1  r3 )

y3  y0  (1  r1 )  (1  r2 )  (1  r3 )
y1
y2
At a constant yearly percentage change (growth rate) income year 3 is:
y3  y0  (1  r )  (1  r )  (1  r )  y0  (1  r )3
where r = constant yearly growth rate/percentage change.
After t years and a constant growth rate income per capita equals:
yt  y0  (1  r )t
,
where t = number of years.
Exercise: If GDP per capita (in 1995 prices) in 1995 and in 2000 was 194 and 222 thousands,
what is the average annual growth rate during this 5-year period?
Graphical representation of the exponential function:
yt  y0  (1  r )t . Let y0  1 and r = 0.03: yt  (1  0.03)t
4.5
4
3.5
3
2.5
y
Serie1
2
1.5
1
0.5
0
0
10
20
30
40
time
If r increases, steeper slope. If y0 increases, the curve shifts upwards.
Students do not have to know logarithms:
Alternative graphical representation of the function:
yt  y0  (1  r )t
50
t
 ln( yt ) = ln( y 0 ) + ln( (1  r ) )
 (1  r )t )
ln( y 0 ) + t  ln(1  r )
 ln yt  ln y0  ln(1  r )  t
 ln( yt ) = ln( y0
 ln( yt ) =
int ercept
This is the equation for a straight line: y = a + b  x
If r is a small number < 0.1  ln(1 r )  r 
slope coefficient
ln yt  ln y0  r  t
The logarithm function (r=0.03)
1.6
1.4
1.2
ln y
1
0.8
Serie1
0.6
0.4
0.2
0
0
10
20
30
40
time
Formula: yt  y0  (1  r )
How many years does it take to double y at different growth rates?
t

2 y0  y0  (1  r )t
2  (1  r )t
t
 ln(2) = ln( (1  r ) )
 ln(2) = ln(1  r )  t  r  t

 ln(2)/r  t
 t  ln(2)/r
If r = 0.05  t  14 years.
If r = 0.015  t  46 years.
50
THE SOLOW GROWTH MODEL:
Aim to explain the development over time of K and Y, k=K/L, and y=Y/L.
In the model the growth rate of the technological level, A / A  g , and the growth rate of
the numbers of workers, L / L  n , are exogenous variables.
In the model everyone is a worker. Thus, the number of workers = population.
Thus, they are determined outside the model.
As A / A , and L / L are assumed to be exogenous variables, the model is about the
accumulation of physical capital, and its effects on k, Y, and y.
To simplify we start by assuming: A / A  L / L  0 .
Thus, the level of A and L are assumed to be constant over time.
Assumption (A1): The production function:
where 0 <
Yt  A  Kt  L1 ,
 < 1.
Expressing production in terms of per worker (labor productivity):
Yt At  Kt   Lt1

Lt
Lt

 At  Kt  Lt   At  Kt   Lt 
Labor productivity depends on:
* Totalfaktorproductivity, A. If A   Y/L 
* Physical capital per worker, K/L. If K/L   Y/L 
Note: (1-  )* Labor productivity (Y/L) = MPL (= W/P)
 y  A  k
Thus, the level of A and L are assumed to be constant over time.
As L is assumed to be constant we can assume L=1.
Small letters indicate that variables are expressed in terms of per worker.
K 
 At  t 
 Lt 

Labor productivity
y=Y/L
9
8
7
6
5
4
3
2
1
0
Y/L=A(K/L)
A=2
A=2
Serie1
Serie2
A=1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k=K/L
In figure: A=1,2, and
 =0.5 .
Slope of the curve above is MPK=dY/dK.
Note: L is assumed to be a constant; for example, L=1  K=k
More complicated proof which is optional:
dy
   A  k  1 , which is MPK:
dk
 1
dY
K
 ( 1)
 1 1
 1
   A K  L    A K  L
   A   ]
MPK=
dK
L
[
Assumption (A2): A constant share of income is saved
(= a constant share of production is invested).
Goods market equilibrium condition: Y  C  I  G  NX
We assume a closed economy without a government sector:
 G=NX=0
 S=Y-C=I
 National saving equals gross investment.
It is easy to augment the model so it includes a government sector as well as exports and
imports we do this on the C-level.
(A2):
S  s Y ,
where s is the share of income that is saved.
Note1: Y=GDP=GNP
Note2: s is not saving per worker even though s is a small letter.

s Y  I
 s
Y I

L L
 s y i
Y CI C I

   ci
L
L
L L
 y  c  i  (1  s)  y  s  y  (1  s )  A  k   s  A  k 
Moreover,
s S
Y
y=Y/L,i
production and investment per worker
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
y=Ak

i=sy=sAk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Serie1
Serie2

16
17
k=K/L
In figure: s=0.3, A=1, and  =0.5. The vertical distance between the curves for production
per worker, and investment per worker is consumption per worker.
Assumption (A3): K  I    K
where K is net investment, I = gross investment, and   K = depreciation of capital per
period.  is the depreciation rate, which is between 0 and 1; e.g. 0.05. (That is, 5 %).  If
I    K , then K  0
If I    K , then K  0 ;
K  0 .
k  i    k
If I    K , then
Expressing (A3) in terms of per worker: 
K I
I L
I /L
i
     
  
K
K
K L
K/L
k
L
k K L
Using k = K/L 
, where
=0 by assumption.


L
k
K
L
k  i    k  i    k ]
k K



k
K
k k
Derivation optional:[
d*k
Depreciation of capital per worker
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
1
2
3
4
5
6
7
8
9
k
In figure:
  0.1
10
11
12
13
14
15
16
17
The whole model:
(A1): y  A  k  ,
(A2): i  s  y
constant share of income.
(A3): k  i    k ,
the production function
investment = saving (equilibrium condition) where saving is
The time path of the capital stock per worker
A-level students need not know mathematical derivation below:
The whole model can be reduced to one equation:
Inserting (A1) and (A2) into (A3):
k  i    k  s  y    k  s  A k     k
The long-run equilibrium (steady state) value for k, k * , occurs when k
That is, when gross investment equals depreciation

s y   k

s  A k     k
Solving for k in equilibrium:

k
k

1
1

A 
s  A  k 1

1
1

A 
 s
 k*  


  
  
What is the long-run equilibrium value of y, y* ?

1
1 1
k 
 s
s A 
 0.


 s  A 1
y *  A  (k * )  A  
 If s  or A  

  
k *  and y* .
If the economy is not in its equilibrium, it converges over time towards the equilibrium
because if k< k * , then i>   k  k * , and
if k> k * , then i<   k  k * . See figure below.
Showing the equilibrium in the Solow diagram:
y, i, dk
The SOLOW MODEL
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0

y=Ak
y=A*k*exp(alfa)
Serie1
Serie2
Serie3
dk
dk
i=sy=sAk
1
2
3
4
5
6
7
8
9
k
In figure: A=1, s=0.3,
  0.1 , and  =0.5.
10
11
12
13
14
15

16
17
The transition to equilibrium: a numerical example
Starting below the equilibrium: The initial value of k: k(year=1)=4.00.
Assume also: A=1, s=0.3,   0.1 , and  =0.5.
year k
i
 k k
 y
y  k 0.5 c 
(1  0.3)  y 0.3  y
1
2
3
4
5
…
4.00
4.2
4.395
4.584
4.768
2.00
2.049
2.096
2.141
2.184
1.4
1.435
1.467
1.499
1.529
0.6
0.615
0.629
0.642
0.655
0.4
0.420
0.440
0.458
0.477
0.2
0.195
0.189
0.184
0.178
0.049
0.047
0.045
0.043
9
3
2.1
0.9
0.9
0
0

The equilibrium values of k and y are calculated by using the formulas:
1
 s  A 1
k*  
 ,
  
k
k
 y
0.05
0.046
0.043
0.040
0.0245
0.0229
0.021
0.020
0
0
y

 s  A 1
y *  A  (k * )  A  

  
How to fill out the Table based on an initial value and assumed parameter values: A=1,
s=0.3,   0.1 , and  =0.5.
Start by filling out the column for k based on the formula:
k  i    k  s  y    k  s  A k     k
k2  k1  s  A k1    k1
 k2  k1  s  A k1    k1  (1   )  k1  s  A  k1
If k(year=1)=4, A=1, s=0.3,   0.1 , and  =0.5.
k2  0.9  k1  0.3  k10.5 , k3  0.9  k2  0.3  k20.5 , etc.

After the values of k has been filled out, all other values of other variables (columns) can be
calculated.
Graphical description of transition to equilibrium when economy start below and above the
equilibrium ln( y*  3 )=1.1:
(1) k(t=1)=4, y (t=1)=2 , and  ln(y (t=1)=2)=0.69
(2) k(t=1)=14, y (t=1)=3.74 , and  ln(y (t=1)=2)=1.32
Transition to equilibrium
1.4
1.2
ln (Y/L)
1
0.8
Serie1
Serie2
0.6
0.4
0.2
0
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
time
According to model:
The growth rates of k and y are higher the lower k and y are. This explains why the slope of
the curves for lny becomes flatter and flatter when lny approaches its equilibrium. Recall that
the slope of lny is the growth rate of y.

If two economies share the same equilibrium; that is, have the same parameter values on A, s
(as well as on  and  ) but differ with respect to initial values, then the economy with lower
k and y experience higher growth rates of k and y than the economy with higher k and y.

the model says that y (and k) of these two economies converge over time. In other words, the
model says that y over time converge across economies if the economies share the same
equilibrium value of y).
Main lesson of empirical work on growth:
Real per capita income tends over time to converge across economies, which are similar with
respect to “institutions”.

An economy with an initially relatively low real income per capita has on average a higher
growth rate of real income per capita than an economy with an initially relatively high real
income per capita if “institutions” are similar. Ex.: EU-countries and regions within countries.
Evidence from the OECD-countries (the currently rich countries)
Growth rate of GDP p.c.
Average annual growth rate of GDP p.c., 1960-2000, and GDP
p.c. in 1960
0.05
0.04
0.03
Serie1
0.02
0.01
0
0
2000
4000
6000
8000
10000
12000
14000
16000
Real GDP per capita 1960
Sample includes: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece,
Iceland, Ireland, Italy, Japan, Netherlands, New Zealand, Norway Portugal, Spain, Sweden,
Switzerland, United Kingdom and USA.
Initially poor countries grow faster in terms of real GDP per capita during the period 19602000 than initially rich countries. The correlation between the average annual growth rate of
real GDP per capita between 1960 and 2000 and real GDP per capita in 1960 = - 0.89
Evidence from the 24 Swedish Regions, 1911-1993
Regions that were relatively poor in terms of real income per capita in 1911, on average had a
higher growth rate of real income per capita.Higher growth rates in poor regions caused
relative differences in real per capita income to diminish across the Swedish Regions between
1
911 and 1993.
The dispersion is lower for real per capita income when it is adjusted for regional differences
in cost of living as counties with high unadjusted real per capita incomes tend to have cost of
living.
Per capita Income adjusted and unadjusted for cost of living
The empirical evidence on convergence in real per capita income across the Swedish regions
is consistent with the predications of the textbook model:
Low real per capita income 
Little capital (physical + human) per worker,
low wages, high rates of return to capital capital per worker 
 production per worker   income per capita 
Also factor mobility tends to contribute to convergence:
Low wages and high returns to capital out-migration, and foreign investment
 capital per worker 
 production per worker 
Evidence from the countries of the world
Growth rate of GDP per
capita
Average annual growth rate growth rate of GDP p.c., 19602000, and GDP p.c. in 1960
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01 0
-0.02
S
2000
4000
6000
8000
10000
12000
14000
Real GDP per capita in 1960
Sample includes 80 countries.
 No convergence in real GDP per capita across the countries of the world. The correlation
between the average annual growth rate of real GDP per capita between 1960 and 2000 and
real GDP per capita in 1960 = + 0.14.
Is lack of convergence in GDP per capita for the countries of the world, evidence against the
model? NO!
The model says that if countries have the same equilibrium, the poorer country should grow
faster in terms of y and k than the country that is richer in terms of y and k.
But if countries differ with respect to equilibrium, that is, with respect to values on A, s (as
well as on  and  ), the poorer economy need, according to model, not grow faster than the
initially richer economy.
Africa is poor because it has a low equilibrium.
Example that a rich country can grow faster than a poor country
Country A (Poor Country): A=1, s=0.2,   0.1 , and  =0.5  : k *  4 , y *  2
Assumed initial values of k and y: 4 and 2.
 Country A’s growth rate of y=0
Country B (Rich Country): A=1, s=0.3,   0.1 , and
Assumed initial values of k and y: 5 and 2.24.
 Country B’s growth rate of y is positive.
 =0.5  : k *  9 , y*  3
16000
y, i, dk
Countries with different saving rates
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
y=k**0.5
i=0.3*y
i=0.2*y
0.1*k
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
Time path of y of two countries
3.5
3
Y/L
2.5
2
Country B
1.5
Country A
1
0.5
0
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93
time
The short and long run effects of an increase of L (e.g. due to immigration)
y, i, dk
The SOLOW MODEL
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0

y=A*k*exp(alfa)
y=Ak
Serie1
dk
dk
Serie2
Serie3
i=sy=sAk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
A one-time increase of L:L(t=0)<L(t=1)= L(t=2)= L(t=3)= L(t=4)
 At time 1: K/L and Y/L, At time 2 and onwards: K/L  and Y/L
If the economy initially is in its equilibrium, it will over time revert to the initial equilibrium
as gross investment exceeds depreciation of capital.
K/L
Time path K/L
10
9
8
7
6
5
4
3
2
1
0
Serie1
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
time
In figure: A=1, s=0.3,   0.1 , and  =0.5, K(0)=900, L(0)=100 and L(1)=200.
The long run values of k and y are unchanged. However, adjustment takes a long time so
migration plays a role for y during a long time according to model.
What happens to the long run values of Y and K?
Y *  A  K   L1  A  k *  L  Y *  A  k *  L(1)  Y *  A  k *  L(0)
K *  k *  L(1)  K *  A  k *  L(0)
 Size of economy increases when L increases.
Example: Y * increases from 3*100= 300 to 3*200=600,
and K * increases from 9*100=900 to 9*200=1800.
In case of a pandemic, L decreases, the results are the opposite.
The effect of an increase in A
y, i, dk
The SOLOW MODEL
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0

y=A*k*exp(alfa)
y=Ak
Serie1
dk
dk
Serie2
Serie3
i=sy=sAk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
k
  0.1 , and  =0.5  k *  9 and y*  3 .
y*  6.75
New value: A=1.5  k *  20.25 ,
Old value: A=1, s=0.3,
The transition to the new long run equilibrium
Transition to new equilibrium
8
7
6
Y/L
5
Serie1
4
Serie2
3
2
1
0
1
7
13
19
25
31
37
43
49
time
55
61
67
73
79
85
91
97
Long-run growth of the number of workers
Before we assumed: Lt 1  Lt
 L(0)=L(1)=L(2)=L(3)
Now we assume : Lt 1  (1  n)  Lt
where n is the constant growth rate of the number of workers; e.g. 0.01.
L
L
L
L L
L
 t 1  (1  n)
 t 1  1  n  t 1  t  n  t 1
n
Lt
Lt

n
Lt
Lt
Lt
Lt
Lt
 L(0)<L(1)<L(2)<L(3)
Assumption A4.
To keep k constant gross investment (I) now needs to compensate not only for depreciation of
capital to keep k constant but also for the fact that the number of workers increases over time:
(A3):
K  I    K
Derivation below optional:
[
K I
I L
I /L
i
     
  
K
K
K L
K/L
k
L
k K L
 n by assumption.
, where


L
k
K
L
k  n  i   ]  k  i  (n   )  k
k
K


n 
k
k
K
k
Using k = K/L 
The whole model:
(A1): y  A  k  ,
the production function
(A2): i  s  y
investment = saving (equilibrium condition) where saving is
constant share of income.
(A3)+(A4): k  i  (n   )  k , The time path of the capital stock per worker
A-level students need not know mathematical derivation below:
The whole model can be reduced to one equation:
Inserting (A1) and (A2) into (A3)+(A4):
k  i  (n   )  k  s  y  (n   )  k  s  A k   (n   )  k
The long-run equilibrium
The long-run equilibrium (steady state) value for k, k * , occurs when k
That is, when gross investment equals “depreciation”
 s  y  (n   )  k
 s  A  k   (n   )  k
Solving for k in equilibrium:

1
1 1
k 
 s A

 n 
1
1

 


s A 
n 

k
k

s  A  k1
n 
 s A
k*  
1
1

 

 n  
 0.
What is the long-run equilibrium value of y, y* ?

 s  A 1
y *  A  (k * )  A  
 If n 

 n  

k *  and y* .
The transition to the equilibrium
If the economy initially is in equilibrium and n  the economy moves over time to the new
lower equilibrium because when i< (n   )  k  k:
y, i,(n+d)k
Growth rate of L increases
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
y=k**0.5
i=0.3*y
(0+0.1)*k
(0.05+0.1)*k
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17
k
In figure: A=1, s=0.3,   0.1 ,  =0.5 and n=0 and 0.05.
(1) when n=0 
k *  9 and y*  3 .
k *  4 and y *  2 .
(2) when n=0.05 
Transition to new equilibrium
3.5
3
Y/L
2.5
Serie1
2
Serie2
1.5
Serie3
1
0.5
0
1
7
13
19
25
31
37
43
49
55
Time
The growth rates of aggregate variables:
K k L k
   n
K
k
L
k
Y y L y
 

n

Y
y
L
y
K  kL 
Y  yL
61
67
73
79
85
91
97
In the steady-state:
k * y*
 * 0
k*
y
K * k * L
 *   0n  n
K  k L 
L
K*
k
*
*
y L
Y
Y *  y*  L
 *  * 
 0n  n
L
Y
y
*
*
Factor prices: In the model:
C + I = Y = capital income + labor income
= MPL*L+MPK*K= (W/P)*L + (r+  )K
r= real return on physical capital.
Note: There is only one good in the Solow model, which is consumed or invested. If it is
invested it is an asset which yields a return. K is the only asset in the economy. There exist no
bonds, shares or money in the model.
Expressing the equilibrium condition above in terms of per worker:
 c + i=y=(W/P) + (r+  )k
(1) W/P=MPL= (1   )  A  k   (1   )  y
(2) r      A  k  1    y / k  If k   W/P  and r 
In poor and rich countries K/L is low and high, respectively.
Factor mobility across economies
If the value of A is the same in poor and rich countries, the real return on capital is higher in
poor countries. As a result, we expect capital to move from rich to poor countries, increasing
K/L in poor countries and lowering K/L in rich countries. Thereby, mobility of capital
contributes to convergence in K/L between rich and poor economies. We expect L to move
the opposite way because W/P is higher in rich countries. Mobility of L increases K/L in poor
countries and decreases K/L in rich countries. Thereby, it also contributes to convergence in
K/L between rich and poor countries.
Why do capital not flow to Africa? In other words, why
are not large investments taking place in some African countries?
Answer: Because A is low, which means that MPK=r+d is not so high.
This can be seen in Solow-diagram. (Allow countries to differ w.r.t. A.)
In other words, if A is the same across countries (which it is not). (assume g=0).
K will move from richer countries with low r (due to high k) to poorer countries where r is
high due to low k. Thereby k and y will tend to equalize across countries.
Nowadays, rich EU-countries invest capital or move production to new EU-countries or to
CHINA or India. L will move the opposite way, from low-wage countries to high wage
countries, which also contributes to equalize real wages, r and k across countries.
Workers move from new EU-countries to old EU-countries where real wages are higher.
Specific example: assume two countries that are the same with respect to the
parameters: A, n, d, and alfa, but one country has a higher saving rate than the other. Assume
that these countries are in their respective equilibria. Allowing for factor mobility across
countries equalizes the real wage, the real return to capital, k and y across countries.
The new equilibrium will be joint for the two countries and is determined by a weighted
average of the saving rates in the 2 countries, where the weights are given by the size of the
populations in the two countries.
Capital to labor (k) ratio is low in developing countries. As a result, one would
expect a high real rate of return on investment in those countries. Why then do not a lot of
investment (construction of new factories, etc.) take place in many of these countries?
Answer: There is a lot of corruption, which makes the actual rate of return much lower; that
is, after the investor have paid off a lot of government official, there might not be so much
money left. A is low. There might also be a political risk. Investors might risk that some
bandits take over the factories, like in Zimbabve.
Important Exercise: Derive the equilibrium expressions for the real wage and for the real
return on capital; that is, express the real wage and the real return to capital as functions of the
exogenous variables: s, A, n, the depreciation rate and alfa.
Golden rule is optional reading for A-level students:
The golden rule level of capital:
The level of capital that maximizes consumption per worker in equilibrium
Consumption per worker is the distance between the curve for labor productivity
( y  A  k  ) and the curve for depreciation of capital per worker: (n+d)k. This distance is
maximized at the level of k where the slopes of these two curves are the same:
dy
   A  k  1  n  
dk
Solving the equation   A  k  1  n   for k yields the answer.
MPK 
A government that wants to maximize consumption per worker should choose the saving rate
(s) so that this level of capital is achieved.
An economy can save too much. That is, by decreasing the saving rate per capita consumption
can increase in the steady state.
Adding realism in the model: continuing technological progress
There is technological progress if new production techniques arise due to innovations such as
the computer, engine, electricity, etc.
A  dA / dt  g
A
A
gt
[optional reading: A(t )  A(0)  e ]
Model assumption: (A5):
,
where g =rate of technological progress is exogenously assumed.
[Optional reading: The model is here formulated in continuous time which means that time
changes continuously. Previously the model was in discrete time which means that the time is
in periods. If the model were in discrete time: At  A0  (1  g )t .]
Only technological progress can explain long run increases in the living standard= GDP
per capita = Y/L=y
Growth rates in the long-run equilibrium:
Before:
k * y*
k * y* w*
,
Now:


0
 * 
g
w
k*
y*
k*
y
K *
Y *
 g n,
 g  n , r*  d    ( y / k ) ,
*
*
K
Y
y* k *
r*
  ( *  * )  0
r
y
k
Technological progress is exogenous
As the rate of technological progress is unexplained by the SOLOW model (that is,
exogenous), adding g to the model does not add any more economic insights than the version
of the model with g=0.
For this reason and because it is simpler we will focus on the version of the model where g=0.
Keep however in mind that technological progress makes the model more realistic because in
the real world y typically increases over time due to new production techniques; that is, due to
innovations.
What happens if the economy is off its equilibrium growth path?
ln(Y/L)
Transition to equilibrium growth path
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
Serie2
Serie3
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
Time
When the economy approaches its equilibrium growth path, the growth rate of y deviates from
the long-run growth rate (g). If an economy starts out below (above) the equilibrium growth
path, the growth rate of y is higher (lower) than g. Holding constant the equilibrium growth
path that is holding constant A(0), s, n, g, d and alfa, a lower y means a higher growth rate of
y.
What happens to the growth rate and to the equilibrium growth path if the saving rate
increases (or institutions improve or population growth )?
Transition to higher equilibrium growth path
2.5
ln(Y/L)
2
Serie1
1.5
Serie2
1
Serie3
0.5
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
time
If s increases, the equilibrium shifts upwards, and the growth rate of y is higher than the long
run growth rate during to the transition to the new equilibrium growth path.
Factors that impact GDP per capita in the real world:
GDP
GDP
Hours worked Employment POP 19  64



POP Hours worked Employment POP 19  64
POP
POP = Population. If GDP per hour (=labor productivity) increases or the hours worked per
employed increases or the number of employed as a share of population increases, then GDP
per person increases.
In other words, get each worker to produce more or get more people in production, then GDP
per person increases.
Production per employed (= the first 2 terms on the left hand side of the equation above) is in
macromodels is GDP per worker, Y/L.
GDP (or GNP) per capita as a measure of the standard of living
The income distribution
GDP per capita (=average income) can be a poor indicator of the income of the average
citizen; that is, of the median income.
The median is the person in the middle of the income distribution.
Typical income distribution
Number of income earners
60
50
40
30
Serie1
20
10
0
Income classes of equal size
The income distribution is typically assymmetric:
 median income < average income
 the more unequal income distribution the greater difference between median and average
income and a larger proportion of the population tends to have an income below the average
income.
An extreme example:
Country Equal. 10 individuals each with an income of 5000.
The median and mean income is 5000.
Country Unequal. 9 individuals each has an income of 2000.
One individual has an income of 32000.
Average income is 5000. Median income is 2000.
GDP per capita as an indicator of “human development/happiness”
We have concluded that average income per capita may be a poor indicator of the income of
the average person; that is, of the median person.
What is the relationship between income per capita and other indicators of
“welfare/happiness”? We want (but cannot) measure is happiness/utility:
U = U (y, x1, x2, x3, x4,…)
Where y = income per capita, x1=literacy rate, x2=assess to clean water, x3= infant mortality
rate, x4=life expectancy, etc.
2 views:
1. The correlation between income per capita and other variables (x1,x2,x3,x4,..) which we
believe impact the welfare of people is high.
Therefore, it is sufficient to study determinants to income per capita.
2.The correlation is not necessarily high.
The UN (UNDP’s) “Human Development Index” has 3 components:
1. Life expectancy. 2. Educational level (e.g. literacy rate).
2. Income per capita.
According to this index Sweden’s is a top 5 whereas with respect to income per capita
Sweden is only top 20.
Problem of Household surveys that ask “Are you happy?” is that the meaning of the word
happy may differ across cultures.
We are rich now but are we happier?
The importance of relative position.
Harvard-students were asked what alternative they preferred:
a) USD 50000/year whereas others get half.
b) USD 100000/year whereas others get the double.
Source: The economist, Aug. 9, 2003.
Some characteristics of poor countries
Large agricultural sector.
They have a comparative advantage with respect to labor-intensive production as they have a
lot of labor but only a little capital (physical and human).
Demography: Young populations, many kids per woman.
In the Solow-model there is one type of asset; physical capital, in this chapter and in the
Keynesian model there are also financial assets: money, bonds, and equities or stocks.
Chapter 4 (and 18): MONEY SUPPLY AND INFLATION IN THE LONG RUN
4.1There are 4 main kinds of assets in the economy: money, bonds, equities orstocks, and real
assets.
MONEY
The money stock consists of assets that can be immediately used for making payments
Money includes currency (notes and coins) and also deposits on which checks can be written.
The Functions of money are to serve as : 1. A Medium of exchange 2. A Unit of account. 3. A
Store of value (=An asset).
There are different measures of money that vary with respect to liquidity:
By liquidity we mean how easy it is to transform money into goods and services.
M1= currency (=notes and coins), salary- and check accounts, travellers’ checks.
M2 = M1 + saving deposits. Etc. M2 is less liquid than M1.
Notes and coins are sometimes called the monetary base (MB).
Without a medium of exchange we have a barter economy (=bytesekonomi).
Today that would be inefficient.
BONDS (from Dornbusch and Fischer)
A bond is a promise by a borrower to pay the lender a certain amount (the principal) at a
specified date (the maturity date of the bond) and in the meantime to pay a given amount of
interest per year. Thus we might have a bond, issued by the Swedish Debt Office, that pays
10000 on June 1, 2012, and until that time pays 8 percent per year, or 800 kronor per year.
Bonds are issued by many types of borrowers – the government, municipalities, corporations.
The interest rates on bonds issued by different borrowers reflect the differing risk of defaults.
Default occurs when a borrower is unable to meet the commitment to pay interest or principal.
EQUITIES OR STOCKS (from Dornbusch and Fischer)
Equities or stocks are claims to a share of the profits in an enterprise. For example, a share in
Ericsson entitles the owner to a share of the profits of that corporation. The shareholder or
stockholder receives the return on equity in two forms. Most firms pay regular dividends,
which means that stockholders receive a certain amount of dollars for each share they own.
Firms may also decide not to distribute profits to the stockholders, but rather retain them and
reinvest them and reinvest these profits by adding to their stock machines and
buildings/structures. When this occurs, the shares become more valuable since they now
represent claims on the profits from a larger capital stock. Therefore, the price of the stock in
the market will rise, and stockholders make capital gains. A capital gain is an increase, per
period of time, in the price of an asset. Of course, when the outlook for a corporation turns
sour, stock prices can fall and stockholders make capital losses. Thus the return on stocks or
the yield to a holder of a stock is equal to the dividend (as a percentage of price) plus the
capital gain.
Example: Suppose that a stock in 2006 trades for 15 kronor. In 2007 the stock pays a
dividend of 0.75 kronor and the stock price increases to 16.50 kronor. What is the yield on the
stock? The yield per year is equal to 15 percent, which is the dividend as a percent of initial
price (5 percent = (0.75/15)*100) plus 10 percent, which is the 1.50 kronor capital gain as
percent of initial price. Alternatively, 2.25/15*100 = 15 percent.
REAL ASSETS
Real assets, or tangible assets are machines, land, and structures owned by corporations, and
the consumer durables (cars, washing machines, stereos etc) and houses owned by
households. These assets carry a return that differs from one asset to another. Owner-occupied
houses provide a return to owners who enjoy living in them and not paying monthly rent; the
machines a firm owns contribute to producing output and thus making profits. The assets are
called real to distinguish them from financial assets (money, stocks, bonds).
The value of equities and bonds held by individuals cannot be added to tangible wealth
to get the total wealth of individuals. The reason is that the equities and bonds they hold are
claims on part of the tangible wealth, that part held by corporations. The equity share gives an
individual a part ownership in the factory and machinery.
4.2 IN MACROMODELS:
In macroeconomics, to make things manageble, we lump financial assets into two categories.
On one side we have money, with the specific characteristic that it is the only asset that serves
as a means of payment. On the other hand we have all other financial assets. Because money
offers the convenience of being a means of payment, it carries a lower return than other
financial assets, but that differential depends on the relative supplies of financial assets. As we
soon will see, when the central bank reduces the money stock and increases the supply of
other financial assets (we say “bonds”), the yield on other financial assets increases.
To sum up: 2 types of financial assets: money and interest-bearing assets (“bonds”).
A bond is a promise to pay to its holder certain agreed-upon amounts of money at specified
dates in the future. For example, a borrower sells a bond in exchange for a given amount of
money today, say 100 kronor, and promises to pay a fixed amount, say, 6 percent, each year
to the person who owns the bond; an to repay the full 100 kronor (the principal) after some
fixedd period of time, such as 3 years, or perhaps longer. In this example, the interest rate is 6
percent, for that is the percentage of the amount borrowed that the borrower pays each year.
4.3 THE WEALTH CONSTRAINT
At any given point in time, an individual has to decide how to allocate his or her financial
wealth between alternative assets. The wealth budget constraint in the asset markets states that
the real money demand, the quantity of nominal money divided by the price level, which we
denote L, plus the demand for real money bond holdings, which we denote DB, must add up
to the real financial wealth of the individual. Real financial wealth is, of course, simply
nominal wealth WN divided by the price level, P:
L+DB=WN/P.
Note that the wealth budget constraint implies, given an individual’s real wealth, that
a decision to hold more real balances is also a decision to hold less real wealth in the form of
bonds.
The total amount of real financial wealth in the economy consists of the real money
supply and of real bonds in existence. Thus, total real financial wealth is equal to:
WN/P=M/P+SB
Where M is the nominal money supply and SB is the real value of the supply of bonds.
Thus, L+DB=WN/P=M/P+SB
(L-M/P)+(DB-SB)=0.
Thus, the wealth budget constrain implies that when the money market is in equilibrium
(L=M/P), the bond market, too, is in equilibrium (DB=SB). Similarly, when there is excess
demand in the money market, so that L>M/P, there is an excess supply of bonds; DB<SB. We
can therefore fully discuss the assets markets by concentrating on the money market.
SIMPLIFICATION IN MODELS
In our macromodels the central bank controls the money supply.
The central bank directly controls the amount of currency and indirectly the amount of bank
deposits (M2) through reserve requirements, and through the interest rate bank pays to the
central bank if they borrow from the central bank.
The Bank Sector creates money
Assume that Money supply (M) = monetary base (MB) + bank deposits.
Assume that the central bank increases the currency by 100 millions, through an open-market
operation; that is, the central bank buys government bonds and pays with newly printed notes
and coins. We assume that the public increases their bank deposits by 100 as a result. If the
reserve-requirement for banks is 10 %, the banks can increase their lending by 90 million to
people who needs to make investments, or buy houses. If we assume that the seller of the
investment goods (for example new machines) or the sellers of the houses deposits the 90
millions in the bank system, the money supply increases by an additional 90 millions. The
banks can now increases their lending further; by 0.9*90=81 millions to people that needs to
money to invest in stocks, physical capital, home-owned houses, etc. The sellers of these
goods and assets therefore receive 81 millions, which we assume are deposited in the banks.
Thus, bank deposits increase by an additional 81 millions, and the banks can therefore
increase their lending further; by 0.9*81 millions to people that needs to money to invest in
stocks, physical capital, home-owned houses, etc.
We summarize:
Period 0 The bank deposits increase by 100 and banks increase their lending by 0.9*100=90
Period 1: The bank deposits increase by 90, and banks increase their lending by 0.9*90=81.
Period 2: The bank deposits increase by 81, and banks increase their lending by 0.9*81.
Period 3: The bank deposits increase by 0.9*81, and banks increase their lending by
0.9*0.9*81. Etc.
Thus ∆M = ∆MB + ∆bank deposits = 100 + 90 + 81 + 0.9*81 + …
More generally, ∆M = ∆MB(1+ (1-rr) + (1-rr)*(1-rr) + (1-rr)*(1-rr)*(1-rr) + ….
Where r is reserve requirement, which is determined by central bank.
As profit-maximizing banks should not want to have excess reserves, the central banks
indirectly controls the money supply.
Using the formula for the sum of a geometric series (see below):
M=MB/rr.
Thus, if reserve-requirements is 0.1, the money supply is 10 times higher than MB.
What is a bank run?
The central bank is the bank of the banks. Private banks can borrow from the central bank.
The central bank sets the interest rate the private banks have to pay on their loans. This
interest rate together with the currency and reserve requirement are factors that impact the
money supply.
When the central bank increases the supply of money it usually buys government bonds from
the public and pays with newly printed currency.
If the money supply increases through an open-market operation the supply of bonds
decreases, then the interest rate goes down, and the price of outstanding bonds increases.
An open market operation implies that the government buys outstanding bonds with newly
printed money.
When the interest rate goes down also stock prices tend to increase.
The quantity theory for money:
The quantity equation:
M*V=P*Y
M=nominal money supply
V= income velocity of money. It is the number of times a dollar is used for purchases of
newly produced goods and services during a year.
P = price level. Y= real GDP. Note: P*Y= GDP in current prices.
Note that this equation is an identity, which means that it holds for sure.
The quantity theory for money:
Assumptions:
1.The factors of production and the production function determine real GDP:
Yt  At  Kt  Lt1
The quantity theory for money is a theory for the long run as it assumes that the level of Y is
independent of M.
2. Velocity is constant.
Thus, changes in M translate into changes in P.
We can write the quantity equation in percentage terms:
∆M/M + ∆V/V = ∆P/P + ∆Y/Y
Ex. 1: If ∆V/V and ∆Y/Y are zero, and money supply increases by 10 percent, what is the
rate of inflation: 10 + 0 = = ∆P/P + 0 .
Thus, inflation is 10 percent as a result.
Ex. 2:
If ∆V/V = 0 and ∆Y/Y=3, and money supply increases by 10 percent, what is the rate of
inflation:
10 % + 0 = inflation + 3
Thus, inflation is 7 percent according to this theory.
Ex. 3:
If ∆V/V = 0 and ∆Y/Y=3, and the central bank has an inflation target of 2 percent by how
much should it increase the money supply to obtain this target?
∆M/M + 0 = 2 + 3 = 5 percent.
The demand for money in real terms:
The demand for money in terms of how many goods and services that your money can buy:
d
M 
 P   k Y


,where k is positive constant.
If k is large people hold a lot of dollars in relation to income.
The real money demand function shows that when real income increases, the real money
demand increases as well. This is because when your income increases then your
consumption increases which means that you need more money to pay for your increased
purchases.
Relation between money market equilibrium and the quantity theory of money:
Equilibrium in the money (and bond) market:
Real Money Supply = Real money demand
d
M   M   k Y , k>0


P  P 
If k=1/V, then

M  (1/ V ) Y
P
M V  P Y , which is the quantity equation.
Thus, the quantity equation shows the equilibrium in the money market when real money
demand = k*Y.
The Fischer effect:
In the Keynesian model in the long run the real interest rate (r) is determined by
Saving (r)=Investment (r) for a closed economy.
For a small open economy, the real interest rate (r) is determined by
World Saving (r)=World Investment (r)
The real interest rate, r= rate the bank pays (nominal interest rate, i)–Inflation.
 The Fischer equation: i=r+inflation.
The Fisher effect says as the real interest rate is determined by real factors in the long
run(S=I(r)), inflation increases the nominal interest rate one-for-one.
From the quantity theory of money we find out the inflation rate in the long run,
And from the Fisher equation we find out the nominal interest rate.
Ex.: If Velocity is constant, and the growth rate of real GDP is 2 percent,
And the growth rate of the money supply is 8 percent, and the real interest rate is 3 percent,
what is the inflation rate according to the quantity theory of money and what is the nominal
interest rate according to the fischer equation?
Modifying the Fischer equation. Making it more realistic.
Assume: nominal interest rate is set before actual inflation for the period is known. Thus, i is
assumed to be a fixed rate, when you take the bank loan or when you deposit your money in
the bank.
 i= expected r + expected inflation
Thus, the expected real interest rate r is determined by S(r)=I(r).
The nominal interest rate moves one-for-one with changes in expected inflation.
Making the real money demand function more realistic:
When you have bills and coins in your pocket you give up interest that this money could have
generated. The nominal interest rate is the cost of holding money, which should impact your
real money demand. Thus:
d
M 
 P   L(i, Y )


r
M/P
r2
r1
L(r,Y1
L ))'
((r,Y)
M/p
M/p
,PP
PPP
P negatively on the nominal interest
/PP for money depends
The figure shows that the real demand
rate and positively on the real income, Y.
More on this demand function for real money in Chapter 10.
[If real money demand depends on the nominal interest rate = r + inflation,
Then velocity should depend on inflation as well.
A higher inflation and thereby a higher nominal interest rate, a higher velocity.
Example: Assume L(i, Y) = f(i)*Y,
if nominal interest rate  f(i) 
Money market equilibrium implies: M/P=f(i)*Y
M*(1/f(i)) = P*Y. M*V(i) = P*Y.
if nominal interest rate (i)  f(i) , V(i) .]
The costs of inflation:
If inflation is expected the people have adjusted, so the only cost is shoe-leather inflation as
you go to the money machine more often because of the higher nominal interest rate. Also
restaurants want to reprint menus more often, which is a cost.
If actual inflation is higher than the extected then the real actual interest rate is lower than the
expected at a given bank interest rate, which means that savers loose and borrowers gain. Also
your real wage becomes lower than expected.
Hyperinflation is more than 50 percent per month.
Summary:
In the classical model = Keynesian model in the long run (=flexible prices) changes in
nominal money supply do not impact real variables such as real GDP and the real interest rate,
and employment.
Bonds In more detail (from Varian, 6th edition): Perhaps use explanation from dornbusch.
Bonds are basically a way to borrow money. The borrower – the agent who issues the bond –
promises to pay a fixed number of dollars x (the coupon) each period until a certain date T
(the maturity date), at which point the borrower will pay an amount F (the face value) to the
holder of the bond. Thus, the payment stream of a bond looks like (x,x,x,x,x,,,,F). If the
interest rate (i) is constant, the present value of such a bond is easy to compute. It is given by
x
x
xF
PV 

 ... 
2
(1  i ) (1  i )
(1  i )T
For someone to be willing to buy a bond the rate of return (interest rate) must be at least as
high as the rates of returns (interest rates) on assets of similar risk. Governments bonds for
countries like Sweden are relatively safe. Assume the coupon is 10 kr and the face value is
1000 kr, then the interest rate is 10 percent:
Note that the present value (PV) of an issued bond will decline if the interest rate
increases. Why? When the interest rate goes up the price now for 1 dollar delivered in the
future goes down. So the future payments of the bond will be worth less now. There is a large
and developed secondary market for bonds. That is, a market where outstanding bonds are
traded daily. The market value of outstanding bonds will fluctuate as the interest rate
fluctuates since the present value of the stream of payments represented by the bond will
change.
An interesting special kind of a bond is a bond that makes payments forever. These are
called concols or perpetuities. Suppose that we consider a consol that promises to pay $x
dollars a year forever. To compute the value of this consol we have to compute the infinite
sum:
x
x
PV 

 ...
(1  i ) (1  i )2
The trick to computing this is to factor out 1/(1+i) to get

1 
x
x
PV 
x

 ...
2

1 i 
(1  i ) (1  i )

But the term in the brackets is just x plus the present value! Substituting and solving for PV:
1
PV 
 x  PV 
1 i
PV=x/i.
For a consol it is easy to see how increasing the interest rate reduces the value of a bond.
Suppose, for example, that a consol is issued when the interest rate is 10 percent. Then if it
promises to pay 10 dollars a year, forever, it will be worth 100 dollars now – since 100 dollars
would generate 10 dollars a year in interest income. NOW, suppose that the interest rate goes
up to 20 percent. The value of the consol must fall to 50 dollars, since it only takes 50 dollars
to earn 10 dollars a year at a 20 percent interest rate. The formula for the consol can be used
to calculate an approximate value of a long term bond.
THE CLASSICAL MODEL = KEYNESIAN MODEL IN THE LONG RUN.
The closed economy with its own currency.
In the model changes in nominal money supply does not impact real variables such as the real
GDP in the long run.
In the Keynesian Model wages and prices are flexible in the long run but not in the short run.
In the long run flexible wages and prices ensure that factors of production are fully employed:
that the demand for K and L become equal to the fixed supply of K and L.
Because labor and capital are fixed in supply, the supply of goods and services, Y, is fixed as
well in the long run.
Y  F ( K , L)  Y
What determines aggregate demand (AD) for goods and services?
AD = C+I+G
C = C(Y
C(Y-- T)
depends
on
consumption
spending by
households
C
disposable
income
T))
T
Y
((Y
C
C
==
CC
Y-T
The slope of the consumption function is
the MPC.
C  C  MPC  (Y  T )  C  MPC T  MPC Y
MPC = marginal propensity to consume shows by how much C increases when the
households’ disposable income, Y-T, increases by one unit.
We assume that
0  MPC  C(Y  T )  1
(Y  T )
Thus, if Y-T increases by 1 dollar, C increases by less than 1 dollar as some of the increase in
disposable income is saved.
The saving function
Households use their disposable income to C and to saving (S) :
Y-T= C+S.  S  C  (1  MPC )  (Y  T )
A richer private consumption function
In many textbooks the Keynesian private consumption function is:
C  C  MPC  (1  t )  (Y  TR ) , where t=proportional tax rate; eg. t=0.4.
TR=transfers to households, unemployment benefits, child allowances, sick benefits, etc.
This consumption which is more realistic, the households’ tax payments, t*Y, is related to
income, Y.
Realistically, current consumption may depend not only on Current disposable income but
also on expected future disposable income, Wealth, and on the interest rate.
For example, we expect C  if wealth .
Many textbooks assume that a higher real interest rate increases private saving, S, and lowers
private consumption at a given level of Y-T.
The investment function
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The investment function relates the quantity of investment I to the real
interest rate r. Investment depends on the real interest rate because the
interest rate is the cost of borrowing. The investment function slopes
downward; when the interest rate rises, fewer investment projects are
profitable.
Real
interest
rate, r
Investment function, I(r)
Quantity of investment, I
Real interest rate (r)  nominal interest rate (i)– inflation rate (
P
 )
P
The nominal interest rate is the bank interest rate.
The Exact formula:
(1  r)  1  i
1 

ln(1 r)  ln(1 i)  ln(1   )
Often we assume a linear investment function:
I (r)  I  d  r , where d>0
- We expect the intercept,
I , increases if firms’ expectations about the future improves.
The government sector:
T=net taxes = tax revenues – transfers to households.
G = GC + GI
T and G are assumed to be fixed; determined outside the model:
T  T, G  G
How does equilibrium in the goods market occur?
Equilibrium in the goods market:
Aggregate supply of output (Y) = aggregate demand for output (AD)
Y  AD  C  I  G  C (Y  T )  I (r )  G
where Y=GDP=GNP=national income (as depreciation and indirect taxes are 0).
Goods market equilibrium in the long run; that is, when Y  F ( K , L) :
Y  AD  C (Y  T )  I (r )  G
Note: r is the only variable not already determined in the last equation. Thus, r plays a key
role: it must adjust to ensure that the demand for goods and services equals the supply of
newly produced goods and services. The greater the interest rate, the lower level of
investment, and thus the lower the aggregate demand for goods and services, and vice versa.
The real interest rate brings financial markets into equilibrium, which also leads to
equilibrium in the goods market:
Real
interest
rate, r
S'
Saving, S
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If G
S  (Y  C(Y  T )  G )  ,
S  (Y  T  C(Y  T ))  (T  G ) 
Private saving, Y  T  C (Y  T ) , is unchanged.
Public saving, (T  G)  , decreases.
(Y  T )   C(Y  T )  , S  (Y  C(Y  T )  G)  ,
S  (Y  T  C(Y  T ) )  (T  G)  
If T 


Private saving Y  T  C(Y  T )  increases because Y-T increases more than C
increases when T is lowered because 0<MPC <1.
Public saving,
T  G , decreases.
Summary: Lower taxes (T) or higher public spending (G) decreases public saving and
national saving. Thereby, r increases and I decreases.
S(r)
Real
interest
rate, r
B
I2
A
I1
Investment, Saving, I, S
When saving is positively related to the interest rate, as shown by
the upward-sloping S(r) curve, a rightward shift in the investment
schedule I(r), increases the interest rate and the amount of
investment. The higher interest rate induces people to increase
saving, which in turn allows investment to increase.
If private saving increases when the real interest rate increases, this means that private
consumption decreases when the real interest rate increases as private saving plus private
consumption always equals disposable income; Y  T , which is unrelated to the real interest
rate in the classical model.
If national saving is unrelated to the real interest rate, a shift in the investment-curve only
increases the real interest rate.
A higher intercept, I  , which shifts the investment-curve,
happens if the firms’ expectations about the future improves.
I  I  d  r , outwards,
Summary: If G  and T   I(r)
Y  C (Y  T )  I (r)  G
Main lesson: Increased government spending or lower net taxes (which implies a higher
government budget deficit) increases the real interest rate and thereby lowers private
investment to the same extent so that aggregate demand and output are unchanged.
THE CLASSICAL MODEL = THE KEYNESIAN MODEL IN THE LONG RUN
The small open economy with its own currency
Simplify by assuming that NFI=NTr=0:
 GDP (Y) = GNP, trade balance = current account balance
Equilibrium in the goods market: Y  C(Y  T )  I (r )  G  NX
 S = GNP – C – G = I + NX
The model below is for a small open economy:
For a small open economy: its real interest rate, r, equals the world interest rate, r *
The economy’s output Y is fixed by the
factors of production and the production
function.
Consumption is positively related to
C = C (Y-T)
disposable income (Y-T).
Investment is negatively related to the
I = I (r)
real interest rate.
The national income accounts identity,
NX = (Y-C-G) - I
expressed in terms of saving and investment.
or NX = S - I
Now substitute our three assumptions from Chapter 3 and the condition
that the interest rate equals the world interest rate, r*.
Y = Y = F(K,L)
NX = (Y-C(Y-T) - G) - I (r*)
NX = S
- I (r*)
This equation suggests that the trade balance is determined by the
difference between saving and investment at the world interest rate.
S  I  (Y  C (Y  T )  G)  I (r * )
THE FOREIGN EXCHANGE MARKET
Suppose that there is an increase in the demand for U.S. goods and
services. How will this affect the nominal exchange rate?
S$
e
e1
e0
A
B
D$
$
Dollar Value of Transactions
D$ shifts rightward and increases
the nominal exchange rate, e.
This is known as appreciation
of the dollar.
Events which decrease the
demand for the dollar, and thus
D$  decrease e would be a
depreciation of the dollar.
When e(Yen/$) increases, the dollar becomes stronger, appreciates, which implies that
american goods become more expensive relative Japanese goods.
When e(Yen/$) increases American goods become more expensive in Yen: e(Yen / $)  P($) ,
where P($) is the domestic price level.
When
e(Yen/$)
increases
Japanese
goods
become
cheaper
in
dollars:
*
*
(1/ e(Yen / $))  P (Yen)  e($ / Yen)  P (Yen) .
Explaining slopes of D- and S-curves: More expensive american goods means a lower foreign
demand for american goods, and thereby a lower demand for US dollars, and higher domestic
(=American)e demand for japanese goods, and therefore a higher supply of US dollars.
Note: The exchange rate is to a large extent determined by financial flows due to financial
investments rather than due to the trade of goods and services. That is, the demand and supply
of foreign currency are determined mainly by interest differentials on financial investments.
The real exchange rate is often called terms of trade.
The relation between the real (  ) and nominal exchange rate (e):
  e(Yen*/$)  P($)  Yen
P (Yen)
Yen
 A higher real exchange rate means that domestic goods and services become more
expensive relative to foreign goods and services in the same currency.
Example: Nominal exchange rate (e)=120 yen/$. Price of car in the us,
Ford = 10,000 $, Price of car in Japan, Toyota = 2,400,000 yen
Real exchange rate =
e( yen / $)  P(dollar) 120*10,000 price ford ( yen)


 0.5
P( yen)
2400000
pricetoyota( yen)
We get one half of a Japanese car per American car.
If e(yen/$) increases to 240, we get one Japanese car per American car.
The American car has become more expensive!
Model assumption:
  Exports , Imports  NX=(Exports-Imports) 
Purchasing power parity (PPP) implies:
PPP and the law of one price implies: 1   
Real
exchange
rate, 
S-I
e(Yen / $)  P($) Yen

Yen
P* (Yen)
The law of one price applied to the
international marketplace suggests that
net exports are highly sensitive to small
movements in the real exchange rate.
This high sensitivity is reflected here
with a very flat net-exports schedule.
NX()
Net Exports, NX
Real
exchange
rate, 
S2-I
2
1
NX2
Real
exchange
rate, 
S1-I Expansionary fiscal policy at home, such as an
increase in government purchases G or a cut in
taxes, reduces national saving.
The fall in saving reduces the supply of dollars
to be exchanged into foreign currency, from
S1-I to S2-I. This shift raises the equilibrium real
exchange rate from 1 to 2.
NX() A reduction in saving reduces
NX1 Net Exports, NXthe supply of dollars which
causes the real exchange rate
to rise and causes net exports
to fall.
S-I(r1*) S-I (r2*) Expansionary fiscal policy abroad reduces
world saving and raises the world interest
rate from r1* to r2*.
1
2
NX1
The increase in the world interest rate reduces
investment at home, which in turn raises the
supply of dollars to be exchanged into foreign
currencies.
As a result, the equilibrium
NX() real exchange rate falls from
1 to 2.
NX2 Net Exports, NX
Summary: If G  and T  
 , NX(  ),
Y  C (Y  T )  I (r*)  G  NX ( )
Main lesson: Increased government spending or lower net taxes (which implies a higher
government budget deficit) increases the real exchange rate and thereby lowers net exports
(=exports – imports) to the same extent so that aggregate demand and output are unchanged.
The inflation and depreciation of the nominal exchange rate:
  e(Yen*/$)  P($)
P (Yen)
If
  0 
  e  P($)  P*( yen)

e
P($) P*( yen)

e( yen /$)  P*( yen)  P($)
e( yen /$) P*( yen) P($)

 If inflation is higher in Japan than it is in the US, then the dollar appreciates
,e(yen/dollar), and the yen depreciates.
The law of one price applied on financial returns in different countries
Bond yields in different countries must have the same expected rate of return in the same
currency because of arbitrage.
Uncovered interest rate parity:
(1 i)  et ( yen / $)  (1  i* ) 
1
ete1 ( yen / $)
The left-hand side of the equation tells you how much you have after one year if you invest
one dollar in a bank in your home country USA. et ( yen / $)  (1  i* ) tells you how much you
have in Yen after one year if you invest the dollar in a bank in Japan. The right-hand side of
the equation shows the expected return of saving in Japan after one year in $. ete1 ($ / yen ) is
the expected exchange rate in a year. Students need not know derivation below:

(1 i)  (1  i
*
)
 (1  i) /(1  i* ) 
ete1 ($ / yen)
et ($ / yen)
e
t 1
e ($ / yen)
et ($ / yen)
. Divide by (1  i* ) :
. Subtract 1 from both sides:
(1  i) (1  i* )
ete1 ($ / yen)
e ($ / yen)


 t
*
*
e t ($ / yen)
e t ($ / yen)
(1  i ) (1  i )
(i  i* )
ete1 ($ / yen )  e t ($ / yen )


. For small values of i* : (1  i* )  1
*
e t ($ / yen)
(1  i )

ete1($ / yen)  et ($ / yen)
*
 i i 
.
et ($ / yen)
If nominal interest rate = 1percent in the U.S.A. And 5 percent in Japan. Do you expect the
dollar to appreciate or depreciate? We expect et 1 ($ / yen )
To decrease by 4 %. That is, we the dollar to appreciate by 4 percent.
INTRODUCTION TO AND SUMMARY OF THE KEYNESIAN MODEL IN THE
SHORT AND THE LONG RUN
In the short run actual GDP, Y, may be lower or higher or equal to potential GDP, Y .
Aim of the Keynesian model in the short run is to explain short-run fluctuations in
production around a long-run trend.
The short-run supply curve is assumed to be horizontal. Thus, in this model aggregate demand
determines output at a fixed price level.
Actual Output can be below or above the full-employment GDP, Y . K is fixed more or less.
By more or less I mean that some part of the fixed stock of physical capital can be unused in a
recession.
In the long run nominal variables such as the nominal money supply only impact nominal
variables such as the price level. Nominal money supply do not impact real variables such as
real GDP, employment, unemployment, the real interest rate, etc.
In the short run, however, nominal money supply may impact real variables such as real GDP.
In the short run the money and goods market interact and jointly determine the level of real
variables such as real GDP.
P
LRAS
C
P0
B
A
SRAS
AD=C(Y-T)+I(r)+G2
AD=C(Y-T)+I(r)+G1
Y
Y
Y = F (K,L)
The closed economy with its own currency:
Aggregate demand, AD= C(Y-T) + I(r)+G is measured on the horizontal axis, and the price
level (P) on the vertical axis:
Why does aggregate demand, AD= C(Y-T) + I(r)+G, depends negatively on P?
When P M/Pr I(r), AD= (C(Y-T) + I(r)+G) 
In many textbooks, it is assumed that C(Y-T) also depends negatively on the real interest rate,
r, which provides one additonal reason why the AD-curve depends negatively on P.
What factors increases AD at a given price level; that is, shifts the AD-curve to the right?
*Expansionary monetary policy by the Central Bank:
If M  (M/P)   nominal interest (=real interest rate when P is constant)   I(r) , AD=
(C(Y-T) + I(r)+G) at a given level of P.
When the short-run supply curve is horizontal, when AD   Y C(Y-T) .
In the new short run equilibrium: r is lower, I,Y, and C are higher than before.
*Expansionary fiscal policy by the political authorities:
If G (or T)  AD= (C(Y-T) + I(r)+G)  at a given price level.
When the short-run supply curve is horizontal, when AD   Y C(Y-T) , the demand
for real money, L(r,Y)  r I.
In the new short run equilibrium:
r is higher, I is lower, G, C, and Y higher than before. See figure above.
Other factors that shifts the AD-curve to the right:
If C at a given Y-T due to e.g. higher expected future disposable income, Y-T.
If I at a given r due to e.g. improved expectations about the future.
(If L(r,Y)  (= velocity) at a given i and Y, lowers the nominal interest rate and thereby
increases private investment, I.)
Conclusion: An increase in aggregate demand increases income, Y in the short run.
The small open economy (in which r=r*) with its own currency:
Why does aggregate demand, AD = (C(Y-T)+I(r*) + G + NX(real exchange rate)), depends
negatively on P?
If P  the real exchange rate: (e  P) / P*  which lowers exports and increases imports: NX
, AD = (C(Y-T)+I(r*) + G + NX(real exchange rate)) 
Also: If P M/P   r so that r>r* instantaneously  financial investments in domestic
country turns more profitable  demand for the domestic currency   the nominal
exchange rate, e, appreciates (e) (e  P) / P*  NX,
AD = (C(Y-T)+I(r*) + G + NX(( (e  P) / P* )) 
Factors that increase AD at a given price level; that is, shifts the AD-curve to the right:
If M   (M/P)  and [if L(r,Y)  (= velocity) for given r and Y] 
 r so that r<r* financial investments in domestic country turns less profitable 
demand for the domestic currency   the nominal exchange rate, e, depreciates (e)
(e  P) / P*  NX,
AD = (C(Y-T)+I(r*) + G + NX(( (e  P) / P* ))
Fiscal policy (G, T) does not shift the AD-curve for a small open economy:
If G (or T or C(Y-T) at a given Y-T or I  at a given r=r*)
 r so that r>r* financial investments in domestic country turns more profitable 
demand for the domestic currency   the nominal exchange rate, e, appreciates (e)
(e  P) / P*  NX, AD = (C(Y-T)+I(r*) + G + NX(( (e  P) / P* )) ]
In the long run:When actual Y>full employment (=potential) Y; that is; when actual
employment is above full employment, and actual unemployment is above the natural rate of
unemployment at point B in diagram, nominal wages and prices increase, which shifts the
short-run aggregate supply-curve upwards. (The SRAS-curve is equivalent to the MC-curve
in microeconomics.) A higher price level decreases the real money supply (M/P), which
increases the interest rate and thereby lowers private investment, I, for the closed economy.
For the small open economy a higher domestic price level, increases the real exchange rate,
and thereby lowers net exports, NX.
In the long run the economy is back at the full-employment output but at a higher price level.
Example: A fall in aggregate demand
LRAS
P
B
A
SRAS
AD
AD'
C
Y
Y
A reduction in aggregate demand, due e.g. to an increase of private saving, C(Y-T) at a given
Y-T, results in unemployment above the natural rate (=employment below full employment)
in the short run. The new short-run equilibrium is at point B. Over time employment below
full employment decreases nominal wages and prices, which increases aggregate demand
because a lower price level increases real money supply (M/P), which lowers the interest rate
and thereby stimulates private investment, I, for the closed economy. For the small open
economy a lower domestic price level, decreases the real exchange rate, and thereby
stimulates net exports, NX.
In the long run the economy back at full employment.
At the new long-run equilibrium (point C) the price level is lower than at the old long-run
equilibrium.
Keynes said that it may take a long time for nominal wages and prices to fall. As a result, it
might be motivated with expansionary monetary (M) or fiscal policy (G, T) to increase
employment and production.
Example: An adverse aggregate supply shock
P
LRAS
B
A
Y
SRAS'
SRAS
AD'
AD
Y
Think of the short-run aggregate supply-curve (SRAS) as being equivalent to the marginal
cost-curve, MC-curve, in microeconomics:
Increases in input prices, e.g. higher nominal wages, or higher oil prices, increase production
cost at a given level of production; that is, shifts the SRAS-curve upwards.
The new short-run equilibrium is at point B.
The price level is higher and Y is lower than before.
An event with higher prices and lower production is called stagflation.
In the long run:
Over time employment below the full-employment level decreases nominal wages and prices,
which shifts the SRAS-curve shifts back.
In the long run the economy is back in its old long-run equilibrium.
Keynes said that it may take a long time for nominal wages and prices to fall. As a result, it
might be motivated with expansionary monetary (M) or fiscal policy (G, T) to increase
employment and production.
THE KEYNESIAN MODEL MORE IN DEPTH:
The simple Keynesian model for a closed economy witout its own currency:
Assumption: The short-run supply curve is horizontal (= P is fixed), which implies that
aggregate demand alone determines output.
The model also assumes that the real interest rate is fixed; and that planned investment is an
exogenous variable. The money market plays no explicit role here.
The model is thus relevant for small economies without monetary authorities that are not or
far from full-employment. If economies are at full-employment government spending cannot
increase aggregate production. For example: the model is relevant for small municipalities
within Sweden or small countries within the Euro-area or small states within the US states.
If they are big, expansionary fiscal policies might increase the real interest rate.
The model how income Y is determined for given levels of the exogenous variables:
government purchases (G), net taxes (T) and planned investment, I (planned). Actual
expenditure (Y) is the amount households, firms, and the government spends on newly
produced goods and services (GDP).
Planned expeditures is the amount households, firms and the government would like to spend
on newly produced goods and services. The economy is in equilibrium when Actual
expenditures (Y) = Planned expeditures (E)
Actual Expenditure, Y=E
Y, E
Planned Expenditure,
E = C(Y-T)+ I(pl.) + G
Y2 Y* Y1
Income, Output, Y
Actual expenditures (Y) = Planned expenditures (E) + involuntary inventory investment
(positive or negative).
Actual investment = I(planned) + involuntary inventory investment
(which is zero, negative or positive).
Planned Expenditure (E) =
C(Y  T )  I ( pl.)  G  C  MPC  (Y  T )  I ( pl.)  G
where Y=GDP=GNP= households’ income before tax and transfers
(as we assume that NFI=VAT=depreciation of capital=0).
Note that: I ( pl.)  I ( planned ) .
If we rewrite the equation above:
E
 C  MPC  T  I ( pl.)  G  MPC Y , where 0<MPC<1.
Slope
Intercept
How is the equilibrium achieved?
At Y2 planned expenditures (E) > actual expenditures (Y)
 involuntary depletion of inventories,
I ( pl.) > I (actual)
 firms increase production, Y
At Y1 planned expenditures (E) < actual expenditures (Y)
 involuntary inventory build-up,
I ( pl.) < I (actual)
 firms decrease production, Y
The goods market Equilibrium implies: Y = E
Y
 E  C  MPC  T  I ( pl.)  G  MPC Y
The effect on Y of increased government spending (G):
Actual Expenditure, Y=E
Y, E
B
G
A
Y
Y1
Planned Expenditure,
E = C(Y-T) + I(pl.) + G0
Income, Output, Y
E  C  MPC  T  I ( pl.)  G  MPC Y
Slope
Intercept
An increase in government purchases of G raises planned expenditures (E) by that amount
for any given level of income (Y): the E-schedule shifts up. The equilibrium moves from A to
B and income rises. Note that the increase in income Y exceeds the increase in government
purchases G. Thus, fiscal policy has a multiplied effect on income. This is because a higher
income increases private consumption, C(Y-T).
Y=C(Y-T) + G.
The multiplier process:
If government spending (G) increase by 1 $, you might expect equilibrium output (Y*) to also
rise by 1$. But it does not:
Initially planned expenditures (E) increases by G, and income increases.
A higher income increases consumption (and thereby planned expenditures) by MPC*G,
which raises Y again.
This second increase in income of MPC*G again raises consumption, this time by
MPC*(MPC*G), which again raises income and so on.
The change in equilibrium Y is:
Y *  G (1 MPC  MPC2  MPC3  ....)
Using the expression for the sum of an infinite series:
 Y * 
 Y
*
G
1
 G >1 because 0<MPC<1. E.g., MPC=0.8
1  MPC
1
 1  MPC
1
This is the government spending multiplier, which is 5 if MPC equals 0.8.
The multiplier effect operates fully if the factors of production are not fully utilized; that is,
when the short-run aggregate supply curve is horizontal.
The tax multiplier: The effect of lower net taxes on Y
Actual Expenditure, Y=E
Y, E
B
A
Planned Expenditure,
E = C(Y-T0) + I(pl.) + G
-MPC*T
Y
Y1
Income, Output, Y
E  C  MPC  T  I ( pl.)  G  MPC Y
Slope
Intercept
A decrease in net taxes, T<0, raises planned expenditures (E) by that amount for any given
level of income (Y). The equilibrium moves from A to B and income rises. Note that the
decrease in income Y exceeds the initial decrease in net taxes T. Thus, tax policies have a
multiplied effect on income when the short-run supply curve is horizontal.
The multiplier process:
If net taxes (T) are decreased by 1 $, you might expect equilibrium output (Y*) to rise by
1$*MPC. But it does not:
Initially private consumption and planned expenditures (E) increases by –MPC*T, and
income increases by this amount.
A higher income increases consumption (and thereby planned expenditures) by MPC*(MPC*T) in the next round, which raises Y again.
This second increase in income of MPC*(- MPC*T) again raises consumption, this time by
MPC* MPC*(- MPC*T) , which again raises income and so on.
The change in equilibrium Y is:
Y *  T  (MPC  MPC * MPC  MPC * MPC * MPC  ....)
Using the expression for the sum of an infinite series:
 Y *   MPC T >1 because 0<MPC<1.
1  MPC

Y *   MPC  1. This is the tax multiplier.
T 1  MPC
NOTE: A tax reduction of 1 dollar increases Y less than a dollar increase of G because the
consumer uses the increase in disposable income both to consumption (which increases
aggregate demand and therefore Y) and to saving (which does not have an effect on aggregate
demand and hence no effect on Y).
Solving for equilibrium Y mathematically and deriving multipliers mathematically:
Goods market Equilibrium: Y = E 
Y  E  C (Y  T )  I ( pl.)  G  C  MPC  (Y  T )  I ( pl.)  G
 Y  E  C  MPC  (Y
Rewriting somewhat:
Y
 T )  I ( pl.)  G
 E  C  MPC  T  I ( pl.)  G  MPC Y
This is one equation with one unknown variable (one endogenous variable): Y.
Y
 MPC Y  C  MPC  T  I ( pl.)  G
 Y  (1  MPC )  C  MPC  T
 Y* 
 I ( pl.)  G
C  MPC  T  I ( pl.)  G
1  MPC
Y * is equilibrium Y, the value of Y which implies equilibrium in the goods market.
The solution implies that the unknown/endogenous variable, Y, is expressed as a function of
the exogenous variables.
What happens to equilibrium Y when the exogenous variables change?
 Y * 
C  MPC T  I ( pl.)  G
1  MPC
The government spending multiplier
If


C  T  I ( pl.)  0
G
Y *  1 MPC
Y *  1  1
G 1  MPC
T  I ( pl.)  G
Y *  C  MPC 
1  MPC
The tax multiplier is:
C  G  I ( pl.)  0
 Y *   MPC T
1  MPC

Y *   MPC  1
T 1  MPC
The balanced budget multiplier:
Question: How much does Y increase if G and T increases by the same amount?
If T is increased by the same amount that G increases, the government budget does not
worsen.
T  I ( pl.)  G
Y *  C  MPC 
1  MPC
 Y *   MPC T  G
1  MPC
A balanced budget implies: T  G
 Y *   MPC G  G
1  MPC

Y *   MPC 1  1  MPC  1
G 1 MPC 1  MPC
Thus, the balanced-budget multiplier equals one.
Accounting for a foreign sector:
Some of the demand of the production of a municipality comes from abroad.
Here we assume that X is an exogenous variable, whereas imports depends on the economy’s
income. (By the same reasoning exports should depend on the income in the rest of the
world.) Moreover, both exports and imports should depend on the real exchange rate.
If the economy has no currency of its own: exports and imports should depend on the relative
prices. In sum:
Exports = F(world income, Y * ;   (e  P) / P* ))
Note: if the economy lacks its own currency: e=1.
A higher “world” income should increase the exports of newly produced goods and services.
A higher price level relative to the price level in the rest of the world should decrease exports.
Imports = G(Y;   (e  P) / P* )
A higher income, Y, should increase imports.
A higher real exchange rate should increase imports.
Simplifying assumptions:
Exports, X , is assumed to be exogenously given.
Imports is assumed to depend on income: M=m*Y, where m is assumed to be a positive
constant, e.g. m=0.1. m is called the marginal propensity to import, which says if income, Y,
increases by 10 million, then imports increase by m millions. If m=0.1 then imports increase
by 1 million.
E  C (Y  T )  I ( pl.)  G  X  m Y
Equilibrium in the goods market implies that Y=E
 Y  E  C (Y  T )  I ( pl.)  G  X
Assuming our specific consumption function:
 m Y
Y  E  C  MPC  (Y  T )  I ( pl.)  G  X  m Y
This is one equation with one unknown variable (one endogenous variable): Y.
Y
 MPC Y  m Y  C  MPC T  I ( pl.)  G  X
 Y (1 MPC  m)  C  MPC T  I ( pl.)  G  X
 Y *  C  MPC T  I ( pl.)  G
1 MPC  m
Y * is equilibrium Y, the value of Y which implies equilibrium in the goods market.
The solution implies that the unknown/endogenous variable, Y, is expressed as a function of
the exogenous variables.
What happens to equilibrium Y when the exogenous variables change?
 Y *  C  MPC T  I ( pl.) G
1 MPC  m
The government spending multiplier
If

C  T  I ( pl.)  0
G
Y *  1  MPC
m

Y * 
1
1
G 1 MPC  m
Note: The multiplier is lower as some of the increased demand for goods and services when Y
increases falls on imported goods and services.
Deriving multipliers mathematically for a more elaborate private consumption function:
Assume now the following private consumption function:
C  C  MPC  (1  t )  (Y  TR )
E  C (Y  T )  I ( pl.)  G  X  m Y
Equilibrium in the goods market implies that Y=E, when we assume our new specific
consumption function:
Y  E  C  MPC  (1  t )  (Y  TR)  I ( pl.)  G  X  m Y
This is one equation with one unknown variable (one endogenous variable): Y.
Y
 MPC (1 t ) Y  m Y  C  MPC TR  I ( pl.)  G  X
 Y (1 MPC  (1 t )  m)  C  MPC TR  I ( pl.)  G  X
 Y *  C  MPC TR  I ( pl.)  G
1 MPC  (1 t )  m
Y * is equilibrium Y, the value of Y which implies equilibrium in the goods market.
The solution implies that the unknown/endogenous variable, Y, is expressed as a function of
the exogenous variables.
What happens to equilibrium Y when the exogenous variables change?
 Y *  C  MPC TR  I ( pl.) G
1 MPC  (1 t )  m
The government spending multiplier
If


C  T  I ( pl.)  0
Y * 
G
1  MPC  (1  t )  m
Y * 
1
1
1

MPC

(1 t )  m
G
Note: The multiplier is lower as some of the increased income, Y, increases taxes.
Before a million increases of income, Y, increased disposable income, Y-T, by 1 million.
Now a million increase of income, Y, increases disposable income by (1-t)*(1 million).
BACK TO OUR ORIGINAL SPECIFICATION OF CONSUMPTION FUNCTION:
Summary: Factors that increase the equilibrium income (Y):
E  C (Y  T )  I ( pl.)  G  C  MPC  (Y  T )  I ( pl.)  G
E  C  MPC  T  I ( pl.)  G  MPC  Y
Slope
Intercept
Planned expenditures(E)  at a given level of Y:
If C(Y-T)  at a given level of Y, e.g. if T .
Also if wealth (the stock market), or expected future income or preferences become more
impatient, we expect C(Y-T)  at a given level of Y-T.
If
I (pl.) due to improved expectations about the future, and if G.
If assuming a linear consumption function:
C(Y  T )  C  MPC  (Y  T )
If
C , or MPC  (when Y-T>0) , or T , C(Y-T)  at a given level of Y.
Thus: If C , or MPC (when Y-T>0), or T , or
Y so that E>Y at the old equilibrium Y  Y .
I (pl.), or G   E  at a given level of
The effect of Y on private saving:
Households use their disposable income to saving and consumption:
Y  T  S  C  S  C  MPC  (Y  T )
Thus, the private-saving function (S) is
S  C  (1  MPC )  (Y  T )
Because with this saving function S+C equals Y-T.
MAIN LESSON IN THIS MODEL: If government spending increases by 1 dollar,
aggregate demand and output may increase by more than 1 dollar.
Leftist parties talk often about multipliers, when they want a larger government sector.
They rarely mention that increased G may decrease I and NX.
THE AD-CURVE in the PY-diagram is drawn under the assumption that Y=E, which
means that actual investment = I (planned).
Actual Expenditure, Y=E
Y, E
B

A
Planned Expenditure,
E = C(Y-T) + I(r0) + G
Y* Y1
Income, Output, Y
P
AD
Output (Y)
Why is aggregate demand, AD= C(Y-T) + I(r)+G, higher when P is lower?
For Closed economy with its own currency:
when P M/Pr I(r) ,AD= (C(Y-T) + I(r)+G).
For small open economy (r=r*) with its own currency: If P  the real exchange rate:
(e  P) / P*   NX , AD = (C(Y-T)+I(r*) + G + NX(real exchange rate)) 
Also: If P M/P   r so that r<r* instantaneously  financial investments in domestic
country turns less profitable  demand for the domestic currency   the nominal exchange
rate, e, depreciates (e) (e  P) / P*  NX, AD = (C(Y-T)+I(r*) + G +
NX(( (e  P) / P* ))
THE KEYNESIAN MODEL MORE IN DEPTH CONTINUED:
INTRODUCING THE MONEY AND BOND MARKET
If the closed economy has its own currency and a central bank, the model below is relevant.
Now we have 2 markets: the goods market and the money (and bond) market, which interact:
A higher income (Y) increases the real money demand which results in a higher interest rate
(= real interest rate when M/P is fixed), which lowers I(planned): Our equilibrium requires
both these markets to be in equilibrium.
Assumptions: Short run: P is fixed. K is fixed more or less.
2 markets:
Equilibrium in the goods market: Y=E=C(Y-T)+I(r)+G..
Equilibrium in the money market: M/P = L(r,Y)
Real interest rate (r)=nominal interest rate (i) – inflation rate = nominal interest rate
Note1: since P is assumed to be constant, the inflation rate is 0, and the real interest rate
equals the nominal interest rate.
Note2: We now assume that planned investment depends on the real interest rate: I(r).
Equation system with the endogenous variables: Y, r, I(r), C(Y-T).
Exogenous variables determined outside the model:
M determined by the central bank.
G, T are determined by the government.
P is fixed in the short run.
In the long run actual Y = full employment Y.
P increases as long as actual Y>full-employment Y: the short-run aggregate supply curve
(SRAS) shifts upwards.
P decreases as long as actual Y<full-employment: the SRAS-curve shifts downwards.
Analysis of expansionary fiscal policy: that is, a higher G or a lower T.
The short and long-run effects on increased government spending (G)
Actual Expenditure, Y=E
Y, E
B
G+
A
Planned Expenditure,
E = C(Y-T) + I(r0) + G0
Y* Y1
Income, Output, Y
If G  AD= (C(Y-T) + I(r)+G)  at given level of Y
When AD   Y the real money demand, L(r,Y)  r I.
In the new short run equilibrium (B):
r is higher, I is lower, G, C, and Y higher than before.
Y increases by less than
G
Y  1 MPC
because private investment falls (“is crowded
out”) due to a higher interest rate.
P
LRAS
C
P0
B
A
SRAS
AD=C(Y-T)+I(r)+G1
AD=C(Y-T)+I(r)+G0
Y
Y
Y = F (K,L)
i
M/P
i
i
M/p
PPP
/PP
Variable
G
Y
Interest rate
I
C
P
National saving
L(i,Y1
L(i,Y0
))'
)
((r,Y)
M/p
PPP
Old short-and long-run
Equilibrium (A)
G0
Y0
R0
I0
C(Y0-T)
P0
S0=I0
New short-run
Equilibrium (B)
<G1
< Y1
<R1
>I1
<C(Y1-T)
=P1
>S1=I1
New
long-run
equilibrium (C)
= G1
Y2=Y0
<r2
>I2
C(Y0-T)
<P2
>S2=I2
The long run effect .
In the long run: the price level increases when Y is larger than the full-employment Y. A
higher price lever lowers M/P which increases the real interest rate, and lowers private
investment even more.
NOTE that in the long run I decreases by the same amount that G increased because
in the long run actual output equals full employment or potential output.
The short and long-run effects of lower net taxes (T): T  T1  T 0  0
Actual Expenditure, Y=E
Y, E
B
MPC*+
A
Planned Expenditure,
E = C(Y-T0) + I(r0) + G
Y* Y1
Income, Output, Y
If T  AD= (C(Y-T) + I(r)+G)  at given level of Y
When AD   Y the real money demand, L(r,Y)  r I.
In the new short run equilibrium (B):
r is higher, I is lower, C, and Y are higher than before.
Y increases by less than
 T
Y  1MPC
 MPC
because private investment falls (“is crowded
out”) due to a higher interest rate.
P
LRAS
C
P0
B
A
SRAS
AD=C(Y-T1)+I(r)+G
AD=C(Y-T0)+I(r)+G
Y
Y
Y = F (K,L)
i
M/P
i
i
L(i,Y1
L(i,Y0
))'
)
((r,Y)
M/p
PPP
Variable
M/p
PPP
/PP
Old short-and long-run
T
Y
Interest rate
I
C
P
National saving
Equilibrium (A)
T0
Y0
R0
I0
C(Y0-T0)
P0
S0=I0
New short-run
Equilibrium (B)
>T1
< Y1
<R1
>I1
<C(Y1-T1)
=P1
>S1=I1
New
long-run
equilibrium (C)
= T1
Y2=Y0
<r2
>I2
C(Y0-T1)
<P2
>S2=I2
The long run effect .
In the long run: the price level increases when Y is larger than the full-employment Y. A
higher price lever lowers M/P which increases the real interest rate, and lowers private
investment even more.
NOTE that in the long run I decreases by the same amount that C increased because
in the long run actual output equals full employment output.
Shortand long run effects of expansionary monetary policy
Assume that nominal money supply (M) is increased:
Note that the picture shows the opposite, a lower nominal money supply.
Recall that the price level (P) is assumed to be constant in the short run.
r
Supply' Supply
Demand, L (r,Y)
M/P
M/P
If M  (M/P)   nominal interest (=real interest rate when P is constant)   I(r) , AD=
(C(Y-T) + I(r)+G) at a given level of P, which increases Y, which increases real money
demand; which pushes up the interest somewhat.
Actual Expenditure, Y=E
Y, E
B
A
Planned Expenditure,
E = C(Y-T) + I(r0) + G
Y* Y1
Income, Output, Y
+
P
LRAS
C
P0
B
A
SRAS
AD=C(Y-T)+I(r)+G
AD=C(Y-T)+I(r)+G
Y
Y
Y = F (K,L)
Variable
Y
Interest rate
I
C
P
M/P
Old short-and long-run
Equilibrium (A)
Y0
R0
I0
C(Y0-T)
P0
M0/P0
New short-run
Equilibrium (B)
< Y1
>R1
<I1
<C(Y1-T)
=P1
<M1/P0
New
long-run
equilibrium (C)
Y2=Y0
r2=r0
I2=I0
C(Y0-T)
<P2
M0/P0
In the long run P increases until M/P becomes identical to M0/P0. Thus,
An increased nominal money supply has no effect on real variables in the long run.
MATHEMATICAL TREATMENT OF KEYNESIAN MODEL IN SHORT RUN
(and of the AD-curve in the PY-diagram.)
Assumption: Short run: P is fixed and K is fixed more or less.
2 markets:
Equilibrium in the goods market: Y=C(Y-T)+I(r)+G
Equilibrium in the money market: M/P=L(i,Y)
Real interest rate(r) =nominal interest rate (i)-inflation rate=i
Note: since P is assumed to be constant, the inflation rate is 0, and the real interest rate equals
the nominal interest rate.
Assume now: planned investment depends on the real interest rate: I(r).
Thus, 3 equations and 3 endogenous variables: Y,r,i.
If reducing the equation system to 2 equations:
Equilibrium in the goods market: Y=C(Y-T)+I(r)+G
Equilibrium in the money market: M/P=L(r,Y)
Thus, 2 equations and 2 endogenous variables: Y, r.
Section not required for A-level students:
Equilibrium in the goods market:
Before:
Y  E  C  MPC  (Y  T )  I ( pl.)  G
If I depends on r:
Y  E  C  MPC  (Y  T )  I  d  r  G
where d>0.
 I  d  r  G  C  MPC  T  I  G 
d
Y *  C  MPC1TMPC
r
1  MPC
1  MPC
Equilibrium in the money market (and bond market):
M / P  L(r,Y )  e Y  f  r

r   1  ( M / P)  e  Y
f
f
Solving for equilibrium Y and r means that the endogenous variables Y and r are expressed as
functions of the exogenous variables and parameters; that is, as functions of autonomous
spending: C , I , G , T , and of real money supply, M / P , and of the parameters: MPC, d,
e, and f (which all are assumed to be positive).
To do so, we substitute the equation for money market equilibrium into the equation for goods
market equilibrium:


Y 1

d e
(1 MPC)  f
 C  MPC  T  I


1  MPC

G
d M /P
(1  MPC )  f


 T  I  G  d  ( M / P)
Y  f  (1  MPC )  d  e   C  MPC
1  MPC
f  (1  MPC )

(1  MPC )  f


Y  f  (C  MPC  T  I  G) 
f  (1  MPC )  d  e
 If
d M /P
f  (1  MPC )  d  e
C , or MPC, or T , or I , or G , or M / P 
equilibrium Y increases.
NOTE: That the equation above also is the equation for the AD-curve: If P increases AD goes
down.
Solving for equilibrium r by inserting the expression for equilibrium Y into the equation for
the equation that shows equilibrium in the money market:
r   1  M / P  e  f  (C  MPC  T  I  G) 

f
f


f  (1  MPC )  d  e

d M /P

f  (1  MPC )  d  e 
which can be simplified:
r   ( f  (1  MPC )  d  e)  M / P 
f  ( f  (1  MPC )  d  e)

ed  M / P

f  ( f  (1  MPC )  d  e)
e  (C  MPC  T  I  G)
f  (1  MPC )  d  e

r

r
e  (C  MPC  T  I  G)
 f  (1  MPC )
M /P
f  ( f  (1  MPC )  d  e)
f  (1  MPC )  d  e
e  (C  MPC  T  I  G)
(1  MPC )
M /P
f  (1  MPC )  d  e
f  (1  MPC )  d  e
 If C , or I , or G , or T , equilibrium r increases.
If M / P , equilibrium r decreases.
THE SMALL OPEN ECONOMY (r= r * ) with its own currency:
The short and long run effect of monetary policy
If M   (M/P)  and [if L(r,Y)  (= velocity) for given r and Y] 
 r so that r<r* financial investments in domestic country turns less profitable 
demand for the domestic currency   the nominal exchange rate, e, depreciates (e)
(e  P) / P*  NX,
AD = (C(Y-T)+I(r*) + G + NX(( (e  P) / P* ))
Variable
Old short-and long-run
Equilibrium (A)
Y0
R0
I0
E0
C(Y0-T)
NX0
P0
M0/P0
Y
Interest rate
I
e
C
NX
P
M/P
New short-run
Equilibrium (B)
< Y1
=R1
=I1
>e1
<C(Y1-T)
<nx1
=P1
<M1/P0
New
long-run
equilibrium (C)
Y2=Y0
=r2=r0
=I2=i0
e2=e0
C(Y0-T)
Nx2=nx0
<P2
M0/P0
Fiscal policy (G, T) does not shift the AD-curve for a small open economy:
If G (or T)  r so that r>r* financial investments in domestic country turns more
profitable  demand for the domestic currency   the nominal exchange rate, e,
appreciates (e) (e  P) / P*  NX,
AD = (C(Y-T)+I(r*) + G + NX(( (e  P) / P* ))
Increased G or increased C(Y-T) due to lower net taxes crowds out net exports completely
even in the short run.
More in-depth analysis of the short-run effect for the small open economy:
2 equations: Equilibrium in the goods market: Y = C(Y-T) + I(r*) + G + NX (e)
Equilibrium in the money market: M/P = L(r*,Y)
Assumption 1: r = r*
Assumption 2: P($) and
changes in e change
P*(Yen)
are assumed to be constant, which means that only
  e(Yen* / $)  P($)  Yen
P (Yen)
Yen
Thus, e   , e  
‘e=yen/dollar:  e ()  appreciation (depreciation) of dollar 
Domestic goods become more expensive (cheaper) in foreign currency (Yen) and foreign
goods become cheaper (more expensive) in domestic currency (dollar).  Exports,
Imports NX
The money market equilibrium condition:
level of output (Y).
M , P and r *
M / P  L(r* ,Y ) , determines the equilibrium
are exogenously given in the model. The only endogenous
variable in the LM-equation is Y, which then is determined by the exogenous variables
M / P , and r * : M / P  L(r* ,Yeq )
M / P  L(r* ,Y )
L(r* ,Y )  Y 100  r* , where r * =0.04
Ex.: The money market equilibrium condition:
If M=1000 and P=1, and
 1000=Y-100*0.04  Y = 1000 + 4 = 1004
If M=1100  Y = 1100 + 4 = 1104
 Given P and r * , M determines Y.
Since M determines Y, fiscal policy (changes in G and T) has no effect on Y.
A monetary expansion in the standard model above:
If M / P r : r < r*  not profitable with financial investments in our country:  the
demand for our currency   the currency depreciates: e NX
 Monetary policy has a larger impact on Y in small open economy compared to the effect in
a closed economy because there is no crowding-out of private investment.
Fiscal Policy(G,T):
Changes in G and T do not impact equilibrium Y
(because eqb. Y is determined by the equation: M s / P  L(r * ,Yeq ) :
Yeq  C(Yeq  T )  I (r* )  G  NX ( )
Aggregate Demand
If e.g. G   aggregate demand  at a given level of Y.
But AD must be unchanged as Yeq is fixed.
C(Yeq  T ) and I ( r* ) also are fixed, e must increase so that NX  by the same
amount that G . That is, AD  G  NX  0
As
Why does a fiscal expansion increase e?
If G  aggregate demand  at a given exchange rate.
When aggregate demand   Y 
real money demand  domestic interest rate, r :
r > r*  profitable with financial investments in our country  the demand for our currency
  the currency appreciates  NX 
(In a richer model, fiscal policy impacts Y in the short run:
A currency appreciation (e) tends to make the price on imported goods and services cheaper
in the domestic currency P because prices on imported goods are included in the
consumer price index  M / P : Fiscal policy has an effect on Y.
*Fixed exchange rates are not included in the course
*The large open economy is an average of the closed economy and the small open economy.
To find how any policy will impact any variable, find the answer in the 2 extreme cases and
take an average.
THE SHORT-RUN AGGREGATE SUPPLY CURVE WITH A POSITIVE SLOPE:
Y = Y + (P-Pe)
P
P2
P1
P0
SRAS (Pe=P2)
Start at point A; the economy is at full employment Y and the
LRAS*
e
SRAS (P =P0)actual price level is P0. Here the actual price level equals the
expected price level. Now let’s suppose we increase the price
B
level to P1.
A'
Since P (the actual price level) is now greater than Pe (the
expected price level) Y will rise above the natural rate, and we
A
AD' slide along the SRAS (Pe=P0) curve to A' .
AD
Y Y' Output
Remember that our new SRAS (Pe=P0) curve is defined by the
presence of fixed expectations (in this case at P0). So in terms
of the SRAS equation, when P rises to P1, holding Pe constant
at P0, Y must rise.
Y = Y + (P-Pe)
The “long-run” will be defined when the expected price level equals the actual price level. So, as price level
expectations adjust, PeP2, we’ll end up on a new short-run aggregate supply curve, SRAS (Pe=P2) at point
B.
Hooray! We made it back to LRAS, a situation characterized by perfect information where the actual price
level (now P2) equals the expected price level (also, P2).
In terms of the SRAS equation, we can see that as Pe catches up with P, that entire “expectations gap”
disappears and we end up on the long run aggregate supply curve at full employment where Y = Y.
Y = Y + (P-Pe)
Assume: P0=expected P0.
Assume that the AD-curve shifts out:
Y  if W is fixed,
or if the increase in aggregate demand is unexpected.
If W in the next period is flexible, workers demand higher nominal wages to compensate for
the unexpected fall in the real wage due to the unexpected P.
In the new long-run equilibrium: P2 and Y , where the
real wage is identical to the real wage in the old long-run equilibrium.
In this long-run equilibrium the actual real wage is equal to the expected real wage.
In the short-run aggregate demand can impact Y if:
*Nominal wages are flexible but workers are surprised by P  so that W/P L. If W
is flexible and workers are not surprised W adjust when P so that Y remains at Y .
In figure above: the economy moves from A to B.
* Nominal wages are determined by contracts. This means that even if workers are not
surprised when the change in aggregate demand occurs, their real wage changes when P
changes.
The SRAS-curve equals the Marginal Cost-curve (MC-curve) in micro when labor is the
variable factor of production:
MC  W L  W
Y MPL
If L for given values of A and K MPL   MC 
Factors that shifts the SRAS-curve?
1. Higher nominal wages increases MC at given level of Y:
If W   Marginal cost (MC) increases at a given level of Y
SRAS-curve shift inwards as .
The new short-run eqb. implies a higher P and a lower Y.
In the long-run W will go down to the original level.
P
LRAS
B
A
SRAS'
SRAS
AD'
AD
Y
Y
adverse supply shock pushes up costs and prices. If AD is held
constant, the economy moves from point A to point B, leading to
stagflation-- a combination of increasing prices and declining output.
Eventually, as prices fall, the economy returns to the natural rate at
point A.
*An
In response to an adverse supply shock, the monetary authorities can increase aggregate
demand by increasing the nominal money supply to prevent reduction in output. The cost of
this policy is a higher price level permanently. In this diagram SRAS is horizontal because I
do not have the appropriate figure.
2. A higher labor productivity decreases MC at given level of Y:
If A or K at given W  Marginal Cost (MC)  at a given level of Y because MPL=(1alpha)*(Y/L)  at given L:  SRAS-curve and LRAS outwards.
Maybe the SRAS-curve shifts more to the right than LRAS because in the long run W adjust
upwards.
Ex.1: A permanent increase in oil prices impacts energy use and thereby A.
Thereby, both the SRAS and the LRAS-curves shift out.
A temporary increase in oil prices only shifts the SRAS-curve.
Ex.2:Globalisation increased the Y/L because of increased competition.
Deriving the Phillips-curve from the Aggregate Supply Curve:
A more realistic formulation of the SRAS-curve:

 P 
Y  Y  e 
P 
Thus, if actual P > than expected P, then actual real wage < expected real wage
employment > full-employment, unemployment < the natural rate of unemployment, actual
output > potential output.

 P 
Y  Y   e 
P 
Rewrite this equation in percentage terms and relate potential output to the natural rate of
unemployment:
 t   te  beta *(u  unatural )  v ,
We have added a term, v, which is a supply chock:
v is positive if oil prices go up, and is negative if oil prices go down.
There exists a short-run trade-off between inflation and unemployment.
*If inflationary expectations are not rational, e.g. adaptive. Then an increase in aggregate
demand and thereby a higher P can lower the real wage and thereby lower unemployment.
*If inflation expectations are rational but nominal wages are fixed by nominal wage contracts.
The rational expectations school says that monetary and fiscal authorities cannot
systematically surprise workers. That is, they cannot systematically use the short-run trade-off
between unemployment and inflation.
 = -1  n) + 
One example of adaptive expectations is that expected inflation equals last years inflation
rate.
If the monetary policies can surprise workers and firms, e.g. if inflation expectations are
adaptive, unemployment can be lowered below the natural rate in the short run.
Rational expectation school argues that the short-run Phillips curve does not accurately
represent the options that policymakers have available. If policy makers are credibly
committed to reducing inflation, rational people will understand the commitment and lower
their expectations of inflation. Inflation can then come down without a rise in unemployment
and fall in output.
Hysteris means that a recession has permanent or long-lasting effect on unemployment. That
is, the long-run supply curve is not relevant.
Unemployed never come back they become drunks so they will be useless in production in the
future.
SHOULD STABILIZATION POLICIES BE ACTIVE OR PASSIVE?
Stabilization policies consist of fiscal policy (T, G) and by monetary policy (M). The aim is to
lower the variation in employment and output that arise in a market economy.
Should the government try to shift AD-curve back with e.g. fiscal policy?
The issue at stake:
Should policy-makers use monetary and fiscal policy to stabilize the economy when shocks
occur to aggregate demand and to aggregate supply to avoid recessions?
Some say Yes, other say No.
The view on this issue depends on ones belief on whether the politicians have the capacity or
the will to stabilize the economy.
Or do they tend to destabilize the economy?:
Do politicians increase the economic fluctuations rather than decrease them?
Problems with an active stabilization policy
Problem1: Lags in the implementation and effects of stabilization policies.
The inside lag is the time between a shock to the economy and the policy action responding to
that shock. This lag arises because it takes time for policymakers first to recognize that a
shock has occurred and then to put appropriate policies into effect.
Monetary policy has a short inside lag whereas fiscal policy has a long lag.
The outside lag is the time between a policy action and its influence on the economy. This lag
arises because policies do not immediately influence spending, income, and employment.
Automatic stabilizers are fiscal policy without any inside lag.
For example, in recessions taxes become lower because they are related to income. Similarly,
the unemployment insurance and welfare systems automatically raise transfer payments in a
recession.
Thus, if YT (= tax revenues  - transfer expenditures)
And if YT (= tax revenues  - transfer expenditures)
Thus, public saving = public budget surplus = (T- G )  when Y.
Are fiscal policies expansionary or not?
As public saving increase with Y, to find out whether fiscal policy are expansionary one
calculate public saving at full-employment Y.
If public saving is negative at full-employment Y, fiscal policies are said to be expansionary.
Problem2: Economic Forecasting is difficult and even experts have imperfect knowledge
about the functioning of the economy
Problem3: “Established” Historical economic relations can change over time due to changes
in the way people form expectations.
Thus, it is hard to predict the future on the basis on past empirical relations.
Should monetary and fiscal policies be governed by rules or by discretion?
Policy is conducted by rule if policymakers announce in advance how policy will respond to
various situations and commit themselves to this.
Policy is conducted by discretion if policymakers are free to act in the way they feel
appropriate at the time an event occurs.
The debate over rules versus discretion is distinct from the debate over passive versus active
policy. Policy can be conducted by rule yet be either passive or active. Ex.: One rule that is an
active stabilization policy is the automatic stabilizers. Another rule that is an active
stabilization policy is the Taylor rule.
Nominal Official Interest Rate =
Inflation + 2.0 + 0.5(inflation -2.0)- 0.5*GDP gap
where the GDP gap = (potential output – actual output)/potential output
The view against discretionary policy is that politicians are incompetent and/or are driven by
their interest to be reelected, which might cause political business cycles.
Even if politicians are driven by a desire to stabilize the economy rather than favour their own
interest, e.g. to be reelected, an argument against discretionary policies is the TIME
INCONSISTENCY PROBLEM:
Policy makers announce in advance the policy they will follow in order to influence the
expectations of the private decision makers.
Later, after the private decision makers have acted on the basis of their expectations,
policymakers may be tempted to renege on their announcement.
EX.:1. To encourage research, the government announces that it will give a temporary
monopoly to companies that discover new drugs. But, after the drugs have been discovered,
the government is tempted to revoke the patent to increase consumer surplus. Of course if it
does, next time the government promises a patent researchers might not believe it, and devote
less effort to find new drugs than they otherwise would have done.
The time inconsistency problem is the reason behind an inflation target and an
independent central bank. Show in the phillipsdiagram.
If the central bank does not have a target it may be tempted to exploit the short-run trade-off
between unemployment and inflation in order to lower the unemployment rate in the short
run. Firms and workers understand this, which leads to higher inflation expectations than
would be the case with an inflation target. The end result is higher inflation, and no lower
unemployment than the natural rate. If the central bank were not independent but instead run
by Politicians the temptation to increase the growth rate of nominal money supply to lower
the unemployment rate would be higher because politicians want to be reelected. In the long
run the unemployment is at the natural rate.
But, the inflation rate will be higher with politicians in charge.
MICROECONOMIC FOUNDATIONS OF CONSUMPTION
According to Keynesian theory the private consumption function is:
C (Y  T )  C  MPC  (Y  T ) , Where T=net taxes=taxes – transfers
Y-T=disposable income
Current private consumption depends on current disposable income.
More elaborate theories say that current consumption depends not also on expected future
disposable income, Wealth, and on the interest rate. For example, we expect an increase in
wealth to increase current consumption at a given level of Y-T, which would increase C in
the equation above.
A Simple Model of Intertemporal Choice over the life-cycle.
Assumptions: The individual lives 2 periods.
The individual consumes in both periods and also receives incomes in both periods. The
incomes are exogenously given.
We assume a perfect capital market, which means that the individual can borrow and lend as
much as she wants at a given interest rate.
The individual receives and leaves no bequest.
Preferences are represented by the utility function: U(C1,C2)
where C1=consumption in first period of life, and C2=consumption in second and last period
of life. The individual values both goods (C1 and C2). The marginal utility of C1 is
diminishing when C1 increases (and C2 is constant), and the marginal utility of C2 is
diminishing when C2 increases (and C1 is constant).
Diminishing marginal utilities implies that the individual wants to “smooth” consumption
rather than consuming a lot in one period and little in the other period. The perfect capital
market, which implies that the individual can lend and borrow at a given interest rate, makes
consumption “smoothing” possible.
A consequence of diminishing marginal utilities is that if income only is received in one
period of life, the individual wants to spread this income over both periods of life. If income
increases only in one period, the individual wants to spread this increase of income over both
periods.
Secondperiod
consumption
Here are the combinations of first-period and second-period consumption
the consumer can choose. If he chooses a point between A and B, he
consumes less than his income in the first period and saves the rest for
the second period. If he chooses between A and C, he consumes more that
his income in the first period and borrows to make up the difference.
Consumer’s
Consumer’sbudget
budgetconstraint
constraint
B
Saving
Vertical
Verticalintercept
interceptisis
(1+r)Y
(1+r)Y11++YY22
A
Y2
Borrowing
Horizontal
Horizontalintercept
interceptisis
YY11++YY22/(1+r)
/(1+r)
Y1
First-period consumption
C
The constraints: (1) S=Y1-C1, (2) C2=(1+r)S+Y2
where S = Saving in first period of life (can be negative), r=interest rate, Y1 and Y2 income
net of taxes received in period 1 and in period 2. r is the real interest rate net of taxes as
before. If capital or interest income is taxed, real interest rate after tax, r= (1-t)*nominal
interest rate –inflation= (1-t)*(real interest rate before tax). The proportional tax rate, t, is 0.3
in Sweden.
Combining the constraints (1) and (2):
 C2=(1+r)*(Y1-C1) + Y2
The budget constraint in figure above:
 C 2  (1  r ) Y1  Y 2  (1  r )  C1
Slope of the constraint:
dC2  (1  r)
dC1
Giving up one unit of C1 means more than one unit of C2 can be consumed because of
positive return (interest) on saving. Thus, C1 is “more expensive” than C2.
The budget constraint can also be written:
 1 C1 
1
1
 C 2  Y1 
Y 2
1 r
1 r
Present value of life-time consumption = Present value of life-time income
What is a present value? If r=0.05, the present value, x(t), of a value next year, x(t+1): e.g.
105 dollars, is the value you have to deposit in a bank today to receive 105 dollars next year.
Thus, x(t)*(1+r)=x(t+1). If there is no uncertainty, and there is a perfect capital market, the
individual should be indifferent between receiving 100 dollars today and receiving 105 dollars
next year if the interest rate is 5 %.
The intertemporal budget constraint above corresponds to our usual budget constraint that has
prices in front of the quantities:
 P1 C1  P2  C 2  Y1 
1
Y 2
1 r
where P1= price of current consumption=1, P2=1/(1+r)=the price of future consumption.
P1>P2. Because if giving up one unit of C1, positive interest on savings means more than one
unit of C2 can be consumed. Thus, C1 is more expensive than C2.
If either Y1  or Y2 , the budget constraint shifts outwards.
 C1*  and C 2*  because of diminishing marginal utilities. The optimal levels of C1 and
C2 depends on the present value of life-time income, Y1 
Y1 
1
Y 2 :
1 r
1
Y 2   C1*  and C 2* 
1 r
Regardless of whether Y1  or Y2  increase, the consumer spread the increase in
1
Y 2 over both periods.
1 r
If Y1 S *  (Y1  C1* )  ,
Y1 
as the consumer wants to increase consumption in both periods.
If Y2 S *  (Y1  C1* )  , for the same reason.
 Thus, if Y2 C1* . This result does not happen in the Keynesian model.
Secondperiod
consumption
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, and more of the other good, C 2* .
For a saver: If r , a saver becomes richer: the income effect: C1*  and C 2* .
The net effect (substitution + income effect): C 2* , C1* ? , S=(Y1- C1* )?
For a borrower:
If r , a borrower becomes poorer: the income effect: C1*  and C 2* .
The net effect (substitution + income effect): C1*  , S=(Y1- C1* ), C 2* ?
In aggregate an economy typically saves: r   S=(Y1- C1* )?
It is often assumed that an increase in r has no or a positive effect on S.
Borrowing constraints: C1  Y1
Consider an individual that consumes less than she would like in period 1:
If Y1   C1* , C 2*  0 . That is, she uses all of the increase in Y1 for C1.
Borrowing constraints are facts of life: They should increase aggregate saving in the
economy, but may be an obstacle for small-business that may have profitable investment
projects that the banks might not want to lend money to because of imperfect information.
The motive for saving in the intertemporal choice model is that the individual wants to
smooth consumption over the life-time. If we add uncertainty to the model, people also save
because future income may be uncertain or because the individual might live longer than
expected. This is called precautionary saving.
Modigliani’s life-cycle model, and Friedman’s permanent income hypothesis builds on the
microeconomic intertemporal choice model above.
GOVERNMENT DEBT
When a government spends more than it collects in taxes, it borrows from the private sector to
finance the budget deficit:
Debtt 1  Debtt (1 rt 1)  (Gt 1 Tt 1)
Pr imary Deficit
Government Deficit  Debtt 1  Debtt  Debtt rt 1  (Gt 1 Tt 1)
Size of government debt and “fiscal sustainability”:
To compare the government debt across countries or over time it should be related to income;
that is, to GDP or to GNP.
When does Debt/GDP increase?
If G-T =0, Debt/GDP increases if the r > growth rate of GDP


  Debt 
 GDP   Debt  GDP  r  GDP
Debt
GDP
Debt
GDP
GDP
Problems in measurement:
1.Debt should be measured in real terms
2. Government not only has a debt but also assets.
The budget procedure that accounts for assets as well as liabilities is called capital budgeting.
Under capital budgeting, government borrowing to finance the purchase of a capital good
would not raise the deficit.
3. The government budgets deficit is countercyclical. It increases in a recession and decreases
in a boom. To see whether fiscal policies are expansionary economists often calculate a
cyclically-adjusted budget deficit:
the government budget deficit at the full-employment output.
The intertemporal budget constraint of the government:
3
2
Debt  11r 

 .... 
(1

r
)

(1
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)
(1
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
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
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1
1
2
1
2
3
G
G
G
0

T3
T1
T2


 ...
1 r1 (1 r1) (1 r2 ) (1 r1) (1 r2) (1 r3)
This intertemporal budget constraint implies that the present discounted value of net taxes
must cover the present discounted value of government spending plus the initial government
debt.
Lower taxes today means higher taxes in the future.
Thus, an increased tax burden on future generations.
Do government budget deficits imply a tax burden on future generations?
The intertemporal budget constraint of the government shows that the government can
redistribute across generations. It shows e.g. that if the government lowers T1,which benefits
the current generation, the government must raise future taxes, which hurts “the future
generations”. Also spending :G1, G2, benefits different generations.
Some redistribution from present to future generation is optimal if the real incomes
Of the future generations will be higher if there is technological growth in the economy;
That is, growth in A. If there is growth in A, then Y/L will increase by the same rate in the
steady state in the Solow model. Y will increase by the same rate that A increases by
If G are constant over time (they won’t), and Ts are constant over the time periods, this would
mean that the tax rate of future generations will fall:
T=t1*Y1=t2*Y2=t2*Y3=
where Y1<Y2<Y3<Y4, implies t1>t2>t3>t4 if T should be constant over time.
Lower-letter t is the tax rate.
From utilitarian point of view it would be socially optimal if the tax payments were higher for
future generations if their incomes are higher as the utility of income is assumed to be
diminshing. Redistribution of income may thus be socially optimal.
The Ricardian view on whether a tax cut can stimulate the economy:
A tax cut today will mean higher future taxes for the households so that their present value of
life-time income will be unchanged, so private consumption is unchanged. Households only
increase their saving in response to a tax cut today. Thus, no effect on Y.
Also an increase in G will have a lower effect on Y as people realize that a
higher G will imply higher taxes in the future, so households’ life-time income will go down
so households will cut their consumption and increase their saving. Thus, lower effect on Y of
an increase of G as people at the same time cut their consumption. Y=C+I+G.
Proponents of the Ricardian view assume that people are rational when making
decisions such as choosing how much of their income to consume and how much to save.
When the government borrows to pay for current spending, rational consumers look ahead to
anticipate the future taxes required to support this debt.
Why the ricardian view might not hold? That is, why a tax cut today may increase
current consumption even though future taxes must be higher, leaving the life-time income in
present value of the consumers unchanged.
1. Short-sighted consumers.
2. Borrowing constraints: The theory says that people want a smooth consumption over time
to diminishing marginal utility of consumption at a particular point of time. The Ricardian
view is that consumers base their spending not only on current but on their lifetime income,
which includes both current and expected future income. Advocates of the traditional view of
government debt argue that current consumption is more important than lifetime income for
those consumers who face borrowing constraints, which are limits on how much an
individual can borrow from financial institutions. A person who wants to consume more than
his current income must borrow, for example because his current disposable income is low,
e.g. for students. If he can’t borrow to finance his current consumption, his current income
determines what he can consume, regardless of his future income. In this case, a debt-financed
tax cut raises current income and thus consumption, even though future income is lower.
3. Also generations may not be altruistic towards their kids so they do not compensate their
kids with a higher bequests to compensate for higher future taxes.
Other questions:
Should the government balance the budget in every period?
A budget deficit or surplus can help stabilize the economy. A balanced
budget rule would revoke the automatic stabilizing powers of the system of taxes and
transfers. When the economy goes into a recession, tax receipts fall, and transfers
automatically rise. Although these automatic responses help stabilize the economy, they push
the budget into deficit. A strict balanced budget rule would require that the government raise
taxes or reduce spending in a recession, but these actions would further depress aggregate
demand.
Money supply and budget deficits
One way to finance a budget deficit is to print more money. Money-financed budget deficits
cause inflation, particularly in developing countries.
The Laffer-curve: Tax rates and total tax revenues:
Government tax revenues=Tax rate*Y(L(tax rate))
If the tax rate increases tax revenues increases ceteris paribus.
However, higher taxes might lower labor supply and thereby production.
Laffer pointed out that government revenues actually can actually fall when tax rates are
increased.
Government bonds that are indexed with inflation:
The return is based on CPI, and when the principal (the price of the bond when it is issued) is
repaid by government it is increased by rate of inflation.
The interest rate paid on the bonds, therefore, is a real interest rate.
Indexed bonds reduce the government’s incentive to produce surprise inflation to reduce the
real value of its debt.
Optional reading:
2-period model to show the Ricardian view:
Assumptions:
Debt1 G1 T1
(1)
if G>T, the government borrows from the public.
0  Debt2  Debt1  (1 r)  G2 T2
(2)
The government is assumed to “live” 2 periods and balance its budget over these 2 periods.
Combining equations (1) and (2):
0  (1 r)  (G1  T1)  G2 T2
 T1 
T2  G  G2
1
(1  r )
(1  r )
 Present value of income equals present value of consumption.
The consumer maximizes U  C1  C21 by choosing C1, (and thereby S and C2) subject
to the constraints: (1) S=Y1-T1-C1, (2) C2=(1+r)S+Y2-T2
where S = Saving in first period of life (can be negative), r=interest rate, Y1-T1 and Y2-T2
net income received in period 1 and in period 2.
Combining the constraints (1) and (2):
 C2=(1+r)*(Y1-T2-C1) + Y2-T2 
1 C1
1
Y 2 T 2
 C 2  Y1  T1 
1 r
1 r
If the government decreases T1 to stimulate aggregate demand it must increase T2 due to the
government’s intertemporal budget constraint:
G
T1  T2  G1  2
(1  r )
 T1 
T2
(1  r )
 T1 
T2  G  G2
1
(1  r )
(1  r )
because G1 and G2 are assumed to be constant.
(1  r )
How is private consumption (C1) impacted by the decrease of T1?
The present value of the life-time income of the consumer is not affected by T1. As a result,
C1 is unchanged, and Y1 is unchanged.
What is the effect on private saving of T1?
S *  Y1 T1 C1*
 S *  Y1T1C1*  0 T1 0  T1  0
 Private saving increases when T1
What is the effect on public saving?
S public  T1  G1 
S public  T1  G1
 T1
 When current net taxes are lowered, public saving decreases.
There is no effect on aggregate saving, the decrease of public saving is fully compensated by
an increase in private saving.
S  S private  S public  T1  T1  0
As aggregate saving is unchanged, the interest rate is unchanged, and
the domestic investment is unchanged. Also C1 and C2 are unchanged
which means that aggregate demand is not impacted.
INVESTMENT AND THE COST OF CAPITAL. Not included in the course.
In this course we have thus far implicitly assumed that there is one good in the economy
which means that the price of one unit of capital is equal to the general price level, P. Now we
abandon this assumption:
Let’s consider the benefit and cost of owning capital. For each period of time that a firm
rents out a unit of capital, the rental firm bears 3 costs:
1.Interest on their loans, which equals the market price of a unit of capital, PK , times the
interest rate, i, so
i  PK
2.The loss on the market price of a unit of capital: - PK
If market price of capital increases (capital gain), it impacts cost negatively.
Thus, the minus sign is because we are measuring costs, not benefits.
3.Depreciation, the fraction of value lost per period of time because of wear and tear:
where  is the depreciation rate, for example 10 % = 0.1
The cost of one unit of capital per period of time, R =
i  PK  PK    PK  PK  (i 
  PK ,
PK
)
PK
PK
  of an apartment is higher than i, then the capital cost is negative
PK
Assume you buy one machine 1/1 2006, what is the cost of this unit of
capital during the year of 2006.
1 million kronor
PK January 1 2006
If
PK
December 31, 2006
i=0.04,  = 0.02
Cost of 1 machine during 2006:
1. i  PK
1.25 million kronor
0.04*1,000,000 = 40,000 kronor
2. - PK
-(1250000 – 100000) = - 250 000
3.
0.02*1,000,000 = 20,000 kronor
  PK
Total cost
PK
If
equals the overall rate of inflation,
PK
where the real interest rate: r  i   
The cost of one unit of capital per period
- 190,000
P
  :  R = PK  (i     )
P
R = PK  (i     ) = PK  ( r   )
of time in terms of the economy’s output: R/P =
PK
 (r   )
P
A common assumption in macro models is that there exist one good in the economy (e.g.
corn):  PK  P  R  P  (r   )  R/P = ( r   )
Thus, the real cost of one unit of capital (R/P) accounts both for the real return on capital (the
real opportunity cost of funds) and for the fact that using capital depreciates it: it is worth less
at the end of the year because of wear and tear.
AN ALTERNTATIVE GRAPHICAL REPRESENTATION OF THE KEYNESIAN
MODEL IN THE SHORT RUN: THE IS-LM-MODEL.
THE IS-LM-model provides a more complete derivation of the
AD-curve in the PY-diagram for a closed economy.
It shows the short-run determination of Y when P is fixed.
Assumption: Short run: P is fixed and K is fixed more or less.
2 markets:
Equilibrium in the goods market: Y=C(Y-T)+I(r)+G
Equilibrium in the money market: M/P=L(i,Y)
Real interest rate(r) =nominal interest rate (i)-inflation rate=i
Note: since P is assumed to be constant, the inflation rate is 0, and the real interest rate equals
the nominal interest rate.
Assume now: planned investment depends on the real interest rate: I(r).
An increase in the interest
rate (in graph a), lowers
planned investment,
which shifts planned
expenditure downward (in
graph b) and lowers
income (in graph c).
(a)
r
(b)
E
Y=E
Planned Expenditure,
E=C+I+G
Income, Output, Y
(c)
r
I(r)
The IS curve shows the equilibriums in the goods market.
Investment, I
IS
Income, Output, Y
If r I(r)  so that AD (= planned expenditures)  at the old equilibrium Y AD<Y at the
old equilibrium  Y
Factors that increase aggregate demand (=planned expenditures) at a given level of Y and
thereby shifts the IS-curve to the right:
1. If C(Y-T)  at a given level of Y, e.g. if
T .
Also if wealth (the stock market), or expected future income or preferences become more
impatient, we expect C(Y-T)  at a given level of Y.
If assuming a linear consumption function:
C(Y  T )  C  MPC  (Y  T )
If
C  or MPC (when Y-T>0), or T , then C(Y-T)  at a given level of Y.
2. If firms’ expectations about the future improves so that I(r) increases at a given level of r. If
assuming a linear investment function: I (r)  I  d  r
I , then I(r) increases at a given level of r.
3. If G 
If
If assuming linear functions:
If C , or MPC  (when Y-T>0), or T , or I , or
level of Y  Y : The IS-curve shifts to the right.
G , aggregate demand  at a given
Optional reading:
[Mathematical Derivation of IS-curve:
Equilibrium in the goods market:
Before:
Y  E  C  MPC  (Y  T )  I ( pl.)  G
If I depends on r:
Y  E  C  MPC  (Y  T )  I  d  r  G
where d>0.
 I  d  r  G  C  MPC  T  I  G 
d
Y *  C  MPC1TMPC
r
1  MPC
1  MPC
If we like we can instead solve for r: The equation for the IS-curve:
r  C  MPC  T  I  G   (1  MPC )  Y

d
Intercept
If


d

C , or T , or I , or G , the intercept increases.]
Real money demand:
( M / P)d  L(r,Y )
The quantity of real money demanded is negatively related to the nominal interest rate
(because it is the opportunity cost of holding money) and positively related to income
(because of transactions demand).
Along a LM-curve real money supply,
M / P  L(r,Y )
M / P , equals real money demand:
The LM-curve has a positive slope as an increase in Y increases Real money demand, which
means that the interest rate has to increase to lower real money demand to the same extent so
that it continues to be equal to the real money supply ( M / P ), which is fixed along a given
LM-curve.
Nominal money supply ( M ) is assumed to be exogenous as it is determined by the Central
Bank. The price level is also assumed to be fixed. Thus, the supply of real money balances,
M / P , is assumed to be exogenously given; that is, determined outside the model. The cost
of holding real money is actually the nominal interest rate and not the real interest rate.
However, as the price level is fixed in the model, the real interest rate equals the nominal
interest rate.
P
Note: i = r (=real interest rate) as
=0 as P is assumed to be fixed.
P
If
M / P , LM-curve shifts to the right.
Mathematical derivation of the LM-curve:
Assume: L(r,Y )  e Y  f  r , where e> 0 and f > 0.
In equilibrium; that is, along the LM-curve:
M / P  L(r,Y )  e Y  f  r
Solving for r:
r   1  ( M / P)  e Y
f
Intercept
If
f
Slope
M / P  , the intercept becomes more negative. Thus, the LM-curve shifts to the right.
r IS
LM(P0)
r0
Y0
Y
The
Theintersection
intersectionof
ofthe
theIS
IScurve/equation,
curve/equation,Y=
Y=CC(Y-T)
(Y-T)++I(r)
I(r)++GGand
and
the
LM
curve/equation
M/P
=
L(r,
Y)
determines
the
level
of
the LM curve/equation M/P = L(r, Y) determines the level of
aggregate
aggregatedemand.
demand. The
Theintersection
intersectionof
ofthe
theIS
ISand
andLM
LMcurves
curves
If C(Y-T)  at a given level of Y, e.g. if T , or if I(r)  at a given r, or if G 
represents
representssimultaneous
simultaneousequilibrium
equilibriumin
inthe
themarket
marketfor
forgoods
goodsand
and
services
servicesand
andin
inthe
themarket
marketfor
forreal
realmoney
moneybalances
balancesfor
forgiven
givenvalues
valuesof
of
AD
at
a
given
level
of
Y:
the
IS-curve
shifts
to
the
right.
government
governmentspending,
spending,taxes,
taxes,the
themoney
moneysupply,
supply,and
andthe
theprice
pricelevel.
level.
If assuming linear functions:
If C  or MPC  (when Y-T>0), or T , or
level of Y: The IS-curve shifts to the right.
The new equilibrium r and Y are higher.
I , or G , aggregate demand  at a given
If M / P : the LM-curve shifts to the right  the equilibrium r decreases, and equilibrium
Y increases.
I(r) increases. As a result, national saving increases as well.
[If L(r,Y)  (= velocity )  at a given r and Y due to a higher use of credit cards:
the LM-curve shifts to the right.]
+G
Consider an increase in government purchases.
This will raise the level of income by G/(1- MPC)
r
IS IS´
A
LM
B
Y
The IS curve shifts to the right by G/(1- MPC) which raises income
and the interest rate.
Y increases by less than
G
Y  1 MPC
because private investment falls (“is crowded
out”) due to a higher interest rate.
The economic mechanism:
If G  aggregate demand  at given level of Y  Y  real money demand  r 
investment 
Comparing equilibrium A with equilibrium B:
Y0 < Y1
C(Y0-T) < C(Y1-T)
R0 < r1
I(r0) > I(r1)
National saving0 > national saving1
+M
Consider an increase in the money supply.
r IS
LM
LM
A
B
Y
The LM curve shifts downward and lowers the interest rate which raises
income. Why? Because when the Fed increases the supply of money, people
have more money than they want to hold at the prevailing interest rate. As a
result, they start depositing this extra money in banks or use it to buy bonds.
The interest rate r then falls until people are willing to hold all the extra
money that the Fed has created; this brings the money market to a new
equilibrium. The lower interest rate, in turn has ramifications for the goods
market. A lower interest rate stimulates planned investment, which increases
planned expenditure, production, and income Y.
If M / P  r  so that real money demand  and becomes equal to
aggregate demand  at a given level of Y Y ..
Comparing equilibrium A with equilibrium B:
Y0 < Y1;
C(Y0-T) < C(Y1-T)
R0 > r1;
I(r0) < I(r1)
National saving0 < national saving1
M /P
 I(r) ,
THE KEYNESIAN MODEL IN THE SHORT RUN: FROM IS-LM-curve to the ADcurve:
You probably noticed from the IS and LM diagrams that r and Y were on
the two axes. Now we’re going to bring a third variable, the price level
(P) into the analysis. We can accomplish this by linking both twodimensional graphs.
LM(P2)
To derive AD, start at point A in the top
r IS
LM(P1) graph. Now increase the price level from P1
to P2.
B
An increase in P lowers the value of real money
A
balances, and Y, shifting LM leftward to point B.
Notice that r increased. Since r increased, we know
Y
P
that investment will decrease as it just got more
costly to take on various investment projects. This
B
P2
sets off a multiplier process since -I causes a –Y.
A
P1
The - Y triggers -C as we move up the IS curve.
AD The +P triggers a sequence of events that end
Y with a -Y, the inverse relationship that defines
the downward slope of AD.
Why has the AD-curve a negative slope in the PY-diagram?
Closed economy:
If P  ( P1  P2 ) M
/ P  r 
 I(r) ,
AD = (C(Y-T)+I(r) + G)  at a given level of Y
Small open economy:If P  the real exchange rate: (e*P)/(P*)  which lowers exports and
increases imports. Hence, NX = (Exports – Imports) ,
AD = (C(Y-T)+I(r*) + G + NX(real exchange rate)) 
Also:
If P M/P   r so that r>r* financial investments in domestic country turns more
profitable  demand for the domestic currency   nominal exchange rate, e, appreciates
(e) NX,
AD = (C(Y-T)+I(r*) + G + NX(real exchange rate)) 
Factors that shift the AD-curve to the right at a given P:
= factors that increases AD at a given Price level.
For the Closed economy:
*Factors that shift the IS-curve to the right.
*Factors that shift the LM-curve to the right apart from changes in P.
If assuming linear functions:
If C , or MPC  (when Y-T>0), or T , or I , or
the AD-curve shifts to the right in the PY-diagram.
G , or M ,
For the Small open-economy standard model:
Only factors that shift the LM-curve in the r/Y-diagram apart from changes in P:
If M  and [if L(r,Y)  (= velocity) at a given r and Y].
For long-run effects of fiscal policy see chapter 5.
2 examples of short and long run in the AD/AS-model with horizontal SRAS-curve and
using the IS-LM-framework.
Short and long run effects of
Closed economy
C , or T :
Note: Long-run effects have been covered earlier.
A decrease in T lowers private, and national saving = I(r)
r
IS
LM(P2)
LM(P0)
IS'
C 


Short
Run:
Y
P
r
C
I
Y
P
P2
C 
P0
 
LRAS
SRAS
AD'
Long
Run:
+
0
+
+
-
0
+
++
+
--
Short and long run effects of monetary policy
If M : in the short run: movement from A to B. CLOSED ECONOMY.
Y, C(Y-T), and I(r)=S increase, and r decreases.
Remember that LR is the movement from A to C.
For the variables Y, P and r, you can read the effects right off the diagrams.
Y 0, because rising P shifts LM to left, returning
r
Y to Y* as required by LRAS.
P +, in order to eliminate the excess demand at P .
0
r 0, reflecting the leftward shift in LM due
to +P, restoring r to its original level.
C 0, since both Y and T are back to their initial
levels (C=C(Y-T)).
I 0, since Y or r has not changed.
P
Notice that the only LR impact of an
increase in the money supply was an
increase in the price level.
P2
P0
IS
A= C
LM(P0)
LM
B
LRAS Y
C
A
B SRAS
AD´
AD
Y* Y´ Y
THE MICROECONOMICS OF LABOR SUPPLY
The individual or household faces the choice between consumption and leisure. More
consumption requires more hours worked and hence less leisure.
The problem of the individual is to maximize:
U = U(C,R)
where
C= consumption during a period of time, e.g. a day.
R = hours of leisure enjoyed during a day.
If C U(.), and if R U(.). Also diminishing marginal utilities.
The two constraints the individual faces are:
(1) The time constraint:
LR  L
where L = labor supply in hours,
L is the time endowment which is 24 hours per day.
(2)
P C  W  L  M
where P= Price of the consumption good
W = Nominal Hourly Wage
M= non-labor income, e.g. government transfers
Let M  P  C
In other words, C is the quantity of goods that the individual receives that is not related to
hours worked.
 P  C W  L  P  C
 P  C W  L  P  C
 P  C W  L  W  L  P  C  W  L
 P  C  W  (L  L)  P  C  W  L
 P  C W  R  P  C W  L
Now we have combined the two constraints that the individual faces, and the result is similar
to the usual budget constraint: px  x  p y  y  I
Thus, the goods that the individual derives utility from (C and R) are on the left-hand-side of
the equation. And in front of the quantities of these goods are the respective prices of these
goods  W is the price of leisure: it is what the individual gives up by taking one hour of
leisure. P  C W  L is called full or potential income. If R=0, then P  C  P  C W  L .
The constraint can be rewritten in real terms:
1 C  (W / P)  R  C  (W / P)  L
where 1 = real price of consumption, W/P is the real price of leisure = the quantity of goods
the individual gives up by consuming one more unit of leisure.
Graphical illustration of the choice possibilities of the individual:
Let C  0 ,  C  (W / P)  L  (W / P)  R
Intercept
Slopecoefficient
An Increase of the Real Wage
Consumption
20
15
Serie1
10
Serie2
5
0
1
2
3
4
5
6
7
8
9
10
Leisure (0.0-1.0)
Note: The choice constraint cuts the x-axis where R= L .
In the figure we assume that L =1, and that W/P increases from 10 to 20.
The numbers on the x-axis are 0.0, 0.1, 0.2,…, 1.0.
Note also that labor supply (L) = L - R: When R=0, then L= L .
If W/P  the intercept increases, and the slope becomes more negative.
If W/P , the individual can afford more of both C and R. On the other hand, when W/P , R
becomes more expensive in terms of the quantity of consumption goods the individual gives
up by consuming one more unit (hour) of leisure.
3 hypothetical possibilities on demand for leisure and on labor supply (L= L -R) when W/P :
1.No effect on the demand for leisure and on the labor supply if the substitution (price) effect
= income effect. The substitution effect is negative for the demand of leisure when the price
of leisure (that is, the real wage) increases. The income effect for the demand of leisure is
positive as a higher real wage means that the individual can afford and wants more leisure
when income increases.
2.Negative effect on the demand for leisure (= positive effect on labor supply) if the
substitution effect > income effect.
3. Positive effect on the demand for leisure (= negative effect on labor supply) if the
substitution effect < income effect.
The optimal choice with positive non-labor income ( C  0 )
C  (W / P)  R  C  (W / P)  L
 C  C  (W / P)  L  (W / P)  R
Intercept
Slopecoefficient
An Increase of Non-Labor Income
Consumption
20
15
Serie1
10
Serie2
5
0
1
2
3
4
5
6
7
8
9
10
LEISURE (0.0-1.0)
In the figure we assume that L =1, W/P=10, and that C increases from 5 to 10. The numbers
on the x-axis are 0.0, 0.1, 0.2,…, 1.0.
When C increases the individual wants more of both goods as they are assumed to be socalled normal goods. You want more of normal goods when your income increases.
An increase of C does not change the opportunity cost of enjoying leisure, and constitutes
therefore a pure income effect.
Summary:
The effect of changes in the exogenous variables on optimal demand for C and R, and on
optimal labor supply:
If C  C * , R * , L*  L  R* 
If W/P   C * , R * ?, L*  L  R* ?
A mathematical note on how to derive optimal demand-functions in case of a CobbDouglas (or a logarithmic) utility function:
If the individual maximizes U ( x, y)  x  y 
subject to the budget constraint: px  x  p y  y  I
where x = quantity of good x, y= quantity of good y, p x = price of good x, p y = price of good
y, and I= income.
The optimal demand for x and y are such that the consumer chooses to spend a constant
fraction of its income on these goods:
p x  x*


I

p y  y*


I

Note if
  1 



I
   px

I
y* 

   py
 x* 

x*  (1  ) 
I , y*    I
py
px
A mathematical example on the optimal choice of leisure (optimal labor supply):
Assume that the individual has the following utility function: U  C1/ 2  R1/ 2
The constraints of the individual are: (1) L  R  L  1
(2) C  W / P  L  C
Note: W, P and C can not be affected by the individual. Thus, they are exogenous from the
point of view of the individual.
Combining the constraints yields:
1 C  (W / P)  R  C  (W / P)  L

1 C  (W / P)  R  C  (W / P)
The result is similar to the usual budget constraint: px  x  p y  y  I
Optimal demands for C and R, and optimal labor supply are:
(W / P  C)  0.5  (W / P  C)
C*  0.5  I  0.5 
pc
1
(W / P  C )  0.5  0.5  C
R*  0.5  I  0.5 
pR
W /P
W /P
L*  1 R*  0.5  0.5  C
W /P
When C  0 :
If C  C * , R * , L*  L  R* 
If W/P   C * , R * , L*  L  R* : More labor is supplied when W/P .
When C  0 :
If W/P   C * , R * =0.5 and L*  L  R* =0.5. That is, labor supply and optimal leisure
are unrelated to W/P.
Thus, the substitution effect equals the income effect.
A MATHEMATICAL EXAMPLE OF THE LIFE-CYCLE MODEL WITH COBBDOUGLAS UTILITY
The individual/household chooses C1 (and thereby S and C2) to maximize
U  C1  C21 , where 0    1
If the individual is impatient which is a common assumption:
The budget constraint of the individual is:
1 C1
  1/ 2
1
1
 C 2  Y1 
Y 2
1 r
1 r
where r, Y1 and Y2 cannot be affected by the individual (are exogenous).

C1* 
 I
C 2* 
PC1

  (Y1  Y 2 )
(1   )  I

PC 2
(1  r )
1
   (Y1 
(1   )  (Y1 
1
(1  r )
Y2
)
(1  r )
Y2
)
(1  r )
 (1  r )  (1   )  (Y1 
Y2
)
(1  r )
S *  Y1 C1*  Y1   (Y1 Y 2 )  (1  ) Y1  Y 2
(1 r )
(1 r )
Note: The solution to the mathematical problem is such that the endogenous (the choice)
variables are expressed as functions of the exogenous variables.
 If Y1 
1
Y 2   C1* , C 2* ; S *  if Y1, S *  if Y2.
1 r
If r   C1* , S *  (Y1  C1* ) , C 2*  (1 r)  (1  ) Y1 (1  ) Y 2 
THE LIFE-CYCLE MODEL WITH ENDOGENOUS LABOR SUPPLY
In other words, the individual chooses how much to work and consume in both periods
of life.
Exercise: Assume that Y1 and Y2 are not exogenous from the point of view of the individual.
Assume that Y1=W1*L1, where L1=1-R1, where L1 is hours worked in period 1, and R1 is
hours of leisure in period 1. 1=L1+R1 equals time endowment (total number of hours
available) in period 1 that is normalized to 1. Assume also that Y2=W2*L2, where L2=1-R2,
where L2 is hours worked in period 2, and R2 is hours of leisure in period 2 of life. 1=L2+R2
equals total number of hours available in period 2 that are normalized to 1. We also assume
that W1 and W2 are exogenous from the point of view of the individual.
Assume: U  C1  C2  R1  R21  
Where the preference parameters,  ,  ,  , 1       , all are assumed to be between
zero and 1.Write up the intertemporal budget constraint of the individual. Derive the optimal
levels of C1, C2, R1, R2, L1, and L2 as functions of the exogenous variables.What happens to
the optimal levels of C1, C2,
R1,R2, L1, and L2 if W2 increases?
• Real business cycle theory emphasizes the
idea that the quantity of labor supplied at any
given time depends on the incentives that
workers face.
• The willingness to reallocate hours of work
over time is called the intertemporal
substitution of labor.
Consider this example:
Let W1 be the real wage in the first period.
Let W2 be the real wage in the second period.
Let r be the real interest rate.
If you work in the first period, and save your earnings, you will have
(1 + r)W1 a year later. If you work in period 2, you will have W2.
A MATHEMATICAL TREATMENT:
THE SOLOW MODEL WITH CONTINUING TECHNOLOGICAL PROGRESS:
Mathematical model (no need to understand details focus on figures). We do the model in
continuous time instead of in discrete time:
(A1): Y (t )  K (t )   ( A(t )  L(t ))1 ,
A(t)  A(0)  egt
Where A(0) is a shift variable that increases if human capital of workers increases, if
business climate improves, g is the rate of technological progress.
To get nice mathematical expressions we assume that technological progress is laboraugmenting which means that A is multiplied with L.
(In case of a Cobb-Douglas pf labor-augmenting technological progress is equivalent to
neutral technological progress that we thus far have assumed:
 Y (t )  A(t )1  K (t )   L(t )1 Let B(t )  A(t )1
 Y (t )  B(t )  K (t )   L(t )1
 Neutral technological progress.)
Model continued:
We express all variables per effective worker, AL:
(A1): y  k 
where y  Y / AL
(A2): s  y  i
(A3):

dK / dt  I    K
dK / dt I
I AL
I / AL
i
   
 
  
K
K
K AL
K / AL
k
Using k  K / AL 
dk / dt dK / dt dA / dt dL / dt



K
A
L
k
 inserting (A4) and (A5):
dk / dt dK / dt

gn
K
k

dk / dt
dK / dt
ng 
K
k
dk / dt  g  n  i    dk / dt  i  ( g  n   )
k
k
k
k
 dk / dt  i  ( g  n   )  k  s  y  ( g  n   )  k  s  k   ( g  n   )  k
In equilibrium: dk / dt  0

1


1
s
,
k*  

 g  n  


1
s
y*  

 g  n  
Multiply both sides by A(t):

1

1

1
s
s
*
 k*  
,

A
(
t
)
y
(
t
)

 A(t )



g

n


g

n






 The equilibrium growth paths of k(t) and y(t).
In other words, the equilibrium is no longer a point but a time path.
1


1
s
k * (t )  
 A(0)  e gt

 g  n  


1
s
 y* (t )  
 A(0)  e gt

 g  n  


1
s
 ln k * (t ) 
 ln 
  ln A(0)  g  t
 g  n  
1





s
 ln y* (t ) 
 ln 
  ln A(0)  g  t
 g  n  
1


These are the equilibrium growth paths of lnk(t) and lny(t):
If the saving rate (s) increases, or the growth rate of the labor force (n) decreases ,or if A(0)
increases due to e.g. a higher educational level among the workers, then the equilibrium
growth paths of y and k shifts upwards. As the economy moves towards its new equilibrium
growth path, the growth rate of y is higher than the long run growth rate (g).
Transition to the equilibrium growth path
If the economy is not on its equilibrium growth path, it will over time move to its equilibrium
growth path.
Assume: A(0)=1, s=0.3, and (n+g+d)=0.1, g=0.015, n=0.015, d=0.07.
 k * =9, y * =3, ln( y * =3)=1.1
We assume two different starting values:
(1) k (1) =4, y (1) =2, ln( y (1) =2)=0.69, (2) k (1) =16, y (1) =4, ln( y (1) =4)=1.39
To find out how K/AL develops over time, we use the transitional equation:
dk / dt  s  k   ( g  n   )  k
 k  s  k   ( g  n   )  k
 k2  k1  s  k1  ( g  n   )  k1
 s  k1  (1  ( g  n   ))  k1  0.3  k1  (1  0.1)  k1
Multiplying k (t ) with A(t) gives k(t), then it is trivial to find y(t) and lny(t):
 k21
ln(Y/L)
Transition to equilibrium growth path
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Serie1
Serie2
Serie3
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
Time
When the economy approaches its equilibrium growth path, the growth rate of y deviates from
the long-run growth rate (g). If an economy starts out below (above) the equilibrium growth
path, the growth rate of y is higher (lower) than g. Holding constant the equilibrium growth
path that is holding constant A(0), s, n, g, d and alfa, a lower y means a higher growth rate of
y.
What happens to the growth rate and to the equilibrium growth path if the saving rate
increases?
Initially assume that the economy is on its equilibrium growth path and that: A(0)=1, s=0.3,
and (n+g+d)=0.1, g=0.015, n=0.015, d=0.07.
 k * =9,
y * =3, ln( y * =3)=1.1
We assume that s increases to 0.4  k * =16,
y * =4, ln( y * =3)=1.39
Transition to higher equilibrium growth path
2.5
ln(Y/L)
2
Serie1
1.5
Serie2
1
Serie3
0.5
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
time
If s increases, the equilibrium shifts upwards, and the growth rate of y is higher than the long
run growth rate during to the transition to the new equilibrium growth path.
Summing up the results:
The long run growth rate of y is g; that is, the growth rate that occurs along the equilibrium
growth path.
The growth rate of y = g + growth that occurs during the adjustment to equilibrium growth
path.
y
y

y
Holding constant the variables that determine the equilibrium growth path constant; that is,
A(0), s, and n, a lower y means a higher growth rate.
A(0)  , s, or n 

y

y
Increasing A(0), s or decreasing n shifts the equilibrium growth path upwards, and thereby
induces transitional growth.
The growth rates of other variables in the model:
K k A L k
  

gn
K
A
L
k
k
Y y A L y
  

gn

Y
y
A
L
y
K  k  A L 
Y  y  A L
In the steady-state:
k * y*
 *  0,
y
k*
k * y* w*
 * 
g
w
k*
y
K * k * A L
 *  
 0 g n  g n
K*  k*  A L 
A
L
K*
k
Y * y* A L
Y *  y*  A  L

 *    0 g n  g n
A
L
Y*
y
y* k *
r*
r*  d    ( y / k ) ,
  ( *  * )  0
r
y
k
Testing the model empirically by regression analysis on a cross-section of countries:
i
i
i
i
i
y2000
 y1960
/ y1960
   1  y1960
 2  (S / Y )i  3  n19602000
 etc.
where i = swe, Norway, finland, usa, etc.
According to model
1, 3 should be negative. And  2 should be positive.
The dependent variable is typically the average annual growth rate of GDP per capita:
y2000  y1960  (1  g )40




y2000
 (1  g )40
y1960
1/40
y2000 
y1960 
1  g
Alternatively an approximate formula,
(it is approximate because in the real world data is discrete).
y(2000)  y(1960)  e g40  ln y(2000)  ln y(1960)  g  40
  ln y (2000)  ln y (1960)  / 40  g
Note: gr for small r and g=ln(1+r)>r .
Other variables than the standard variables in the SOLOW model
In empirical analysis often more variables than initial income per capita (or initial income per
employed), the investment rate, the population growth rate are included. For example,
educational level, variables measuring tax rates, corruption, openness to trade, population age
structure, population density.
To make the empirical analysis fully consistent with the Solow model, it is typically assumed
that these variables impact the level of technology (A) and thereby the steady state level of
production per worker.
Thus, A is assumed to depend on a host of variables.
If the model is tested in terms of per capita, often age structure variables; the share of the
population below 15 and above 65 years, are included as explanatory variables in the
regressions to account for the fact that people of these age groups typically do not work.
ENDOGENOUS GROWTH MODELS
Endogenous growth models rejects the assumption of the Solow model that technological
progess is exogenous
The Basic Model
Start with a simple production function: Y(t)=A*K(t), where Y is output, K is the capital
stock, and A is a constant measuring the amount of output produced for each unit of capital
(noticing this production function does not have diminishing returns to capita). One extra unit
of capital produced A extra units of output regardless of how much capital there is. This
absence of dimishing returns to capital is the key difference between this endogenous growth
model and the Solow model.
Let’s describe capital accumulation with an equation similar to those we have been using:
K  s Y    K . This equation states that the change in the capital stock K equals
investment s*Y minus depreciation. We combine this equation with the production function,
Y K

 s A
do som rearranging, and we get:
Y
K
Y
. Notice that as long as
Y
sY>  (depreciation rate), the economy’s income grows forever, even without the
assumption of exogenous technological progress. In the Solow model, saving leads to growth
temporarily, but diminishing returns to capital eventually force the economy to approach a
steady state in which the growth rate of output per worker equals the exogenous rate of
technological progress. In contrast, in this endogenous growth model, saving and investment
can generate persistent growth.
This equation shows what determines the growth rate of output,
A second and more realistic class of endogenous growth models are complementary to
the Solow model builds on Paul Romer (1990).
In the model above MPK does not fall when K increases. A second and more realistic class of
endogenous growth models are complementary to the Solow model. They try to explain the
long-run growth rate, g, which is exogenous in the Solow model, by the number of scientists,
etc.
Read 2-sector model in Mankiw, ch. 8
THE FERTILITY CHOICE
Assume that a household derives utility from a consumption good and from having kids.
Assume the following utility function:
U (C , K )  C  alfa * ln K
where C is quantity of consumption goods and K is number of kids.
(You may use the greek notation for alfa.)
Alfa is assumed to have a positive value.
Assume that W is the wage income that the household receives if the household works full
time. Note that if y=lnx, then dy/dx=1/x
(Assume also that the household lack other sources of income than labor income.)
Assume that the price of the consumption good is 1.
Assume that the price of (the cost of) children is related to the wage income. This is the case
because when the household have kids, it is assumed that the household no longer can work
full time because the have to look after/raise the kids. Thus, the household give up part of the
wage income, which constitutes the price (cost) of kids. Assume that the price per kid is
W*beta, where beta is the proportion of the household’s full time that the each kid require.
For example, if beta=0.2 and the household has one kid, then 80 percent of the time of the
household is devoted to work and 20 percent is devoted to the kid. If the household has 2 kids,
60 percent of the time is devoted to work and 40 percent is devoted to raising these two kids.
a.Write up the budget restriction of the household. 2p
b. Derive the optimal levels of C and K as functions of the exogenous variables. 4p
Assume that the wage income is exogenous from the point of view of the household.
Also show the optimal choice graphically with C on the vertical axis and K on the horizontal
axis. c.What happens to the optimal choice of C and K if W increases. Show mathematically
by using the derivative.
What happens to the utility level of the household. Explain why! 2p
d. Is this theoretical effect consistent with empirical observations from the real world? 1p
e. What happens to the optimal choice of C and K if beta increases? Show mathematically.
What happens to the utility level of the household. Explain why. 1p