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ECONOMIC GROWTH COURSE
Main questions:
Why are some countries so poor while other countries are rich in terms of income per capita?
Why are some countries growing faster than others?
Readings:
Weil, ch. 1-2.
Weil, ch. 3: The Solow Model without population growth and technological progress.
Weil, ch. 4.2: The Solow Model with population growth but without technological progress.
Weil, skim appendix of ch. 8: Incorporating technological progress in the Solow model
Note: There will be a lot more reading.
Review: Read by yourselves: NATIONAL INCOME ACCOUNTING
MEASURING PRODUCTION AND INCOME
Main lesson:
The value of the production of a country (GDP) approximately equals the income its
citizens.
Gross domestic product (GDP) is the market value of all final goods and services produced
within an economy during a year.
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
Comparing GDP over time
Nominal GDP = GDP in current prices
real GDP = GDP in constant prices
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 according to method 1.
Assume two goods in the economy: Q1 , Q 2
Nominal GDP in 2000:
1
1
2
2
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.
In macromodels:
real GDP (=GDP in constant prices) is denoted by Y.
This means that P is assumed to equal 1 in the base year as 1*Y=Y
Nominal GDP is denoted by P*Y.
It may be useful to think that there is one good in the economy; e.g. potatoes.
Then Y is quantity of potatoes in terms of kilogrammes, and P is the price of a kg potatoes.
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:
1
1
2
2
P2001
 Q2000
 P2001
 Q2000
CPI (2001)  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.
1
1
2
2
P2001
 Q2001
 P2001
 Q2001
Alternatively: CPI (2001)  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/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.
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
Use of newly produced goods and services
Private consumption = 1500
Private investment = 400
Public consumption = 700
Public investment =300
Exports = 1600
Sum: 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.
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)
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 a closed economy macro model:
Y = C+I+G
Then Y=GDP=GNP=DGDP. DGNP is the income used by government (T) and households,
Y-T, which is household disposable income. (Households earn both labor and capital income.)
Thus, T + (Y-T) = Y = C+I+G+NX.
T = net taxes = tax payments (indirect and direct taxes plus social security contributions) –
transfers to households and firms.
Typical simplifying assumptions in an open economy macro model:
The basic identity in national income accounting: Y = C+I+G+NX
Assume that Net Foreign Income (NFI) =net transfers from abroad (NFTr)=0
Then Y=GDP=GNP=DGDP.
Comparing Standard-of-Living across Countries
2 methods to determine which country has the highest material standard: the amount of goods
and services:
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 in domestic currency 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 (PPP) 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 ($)*Qt
We want to use the same prices because we want our measure of income to reflect the
quantity of goods that are available in one country during a year. Using different prices for
different countries distorts the picture when we want a measure of material standard.
If the law of one price holds for tradable goods, then
average Pt ($) = e ($/kronor)*Pt(kr)
That is, the law of one price says that the price of a tradable good expressed in the same
currency should be the same in all countries. We expect the law of one price to hold if
transportation costs and differences in VAT are zero across countries because the market
mechanism tends to equalize prices across countries: If there are price differences (in the
same currency) then it is profitable to buy the good where the price is low, and ship it to the
country where the price is high. This leads to higher demand in the country in which the price
was initially low which tends to put an upward pressure on the price in this country.
Moreover, a higher supply in the country where the price was initially high tends to put a
downward pressure on the price in this country. Thus, the market mechanism tends to
equalize prices across countries for tradable goods.
The law of one price does not hold for non-tradable goods and services:
In a poor country, the price of a non-tradable good tends to be lower because of lower labor
costs than in a richer country.
Ex: e($/Etiopian currency)* Pnt(Etiopian currency) < 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 for non-tradable goods; they 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.
Read by yourselves:
AGGREGATE SUPPLY: FACTOR MARKETS: CAPITAL AND LABOR MARKETS
Assume an aggregate production function of Cobb-Douglas type:
real GDPt  Yt  F ( At , Kt , Lt )  At  G( Kt , Lt )  At  Kt  L1t , where 0 <  < 1.
Real GDP = GDP in constant prices.
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.
MPL is below APL (=labor productivity=Y/L).
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
16
17
L
If the stock of physical capital increases:
 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 Y1  Y0 Y/L
Y 4 L
MPL 

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 (=Y/L) increases when K (or A) increases and L is constant.
This is shown by comparing the 2 tables above.
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
 A K  L  L  A K  L 
L

1

1


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
MPL 
then
then
then
dy
 2  x 21  2  x
dx
dy
 2  b  x 21  2  b  x
dx
dy
 n  b  x n 1
dx
(1   )  A0  K0
dY d ( A0  K0  L1 )

 (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. Firms demand L and K.
Assumptions: There exist many identical firms that produce an identical good.  Perfect
competition is assumed  Firms are price-takers and they are assumed to 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 for profit-maximization 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-minimizing) 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
R
L
W (1   )  K
K

 
 

R
 L
W (1   )  K
L
demand


W

(1   ) R
Thus, a higher Wage relative to the Rental price of capital makes it optimal to use more
capital relative to labor at a given level of production. You may know the concept of
isoquants.
Determining Equilibrium Factor Prices:
They are found where Supply = Demand
Assume that the supply of K and L is fixed: K , L

demand
W / P (1   ) K

W
K 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 ) 
Do exercise: Labor market!!
The distribution of income (Y) between workers and capital-owners.
Assume no government sector.
Pr ofits($)  P  Y  R  K  W  L
Assuming perfect competition in the rental market for capital:
 Pr ofits($)  P  Y  (r   )  P  K  W  L , where r=real return to capital
Note that: if   0 : PY=GDP=GNP
(taxes are ignored.)
If   0 , PY=GDP=GNP
where (r   )  P  K = nominal capital income
Real Profits =
Pr ofits($)
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)
If   0 :
Y=r*K+(W/P)*L,
Y, K, (W/P) are in real terms; eg. In kilo of potatoes or in constant dollars.
r is real interest rate. For example in r=0,03. This means that rK is in kg potatoes.
rK is capital income in kg of potatoes and (W/P)*L is labor income in kg of potatoes.
Equation in nominal terms (in kronor):
PY=r*PK+W*L, P is the price of a kg of potatoes in kronor.
PY, r*PK= capital income, and WL= labor income are in kronor.
Real wage and labor productivity:
When perfect competition: MPL  (1 )*Y / L  W / P
If the share of income (Y) that goes to workers: (1-alpha) is constant,
Then the real wage grows at the same rate that labor productivity, Y/L, grows.
What nominal wage demands, keep the income distribution between capital owners and
workers unchanged? Trade union strategy:
“We need to forecast future development of labor productivity and inflation, when we make
our wage demands”:
W/P = MPL=(1-alfa)* (Y/L), when assuming perfect competition and a cobb-Douglas pf.
This means:ΔW/W - ΔP/P = Δ(1-alfa)/(1-alfa) + Δ(Y/L)/(Y/L)
If the distribution of income between firm owners and workers are constant:
Δ(1-alfa)/(1-alfa) = 0.
Thus: ΔW/W = Δ(Y/L)/(Y/L) + ΔP/P
Read by yourselves:
Elaborating on the production function:
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  Ht  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.
Do the exercise with human capital in the production function.
LECTURE: ECONOMIC GROWTH
How to calculate growth rates: Weil, ch. 1.
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
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
Where does this approximate formula come from?
dy = (dy/dx)*dx + (dy/dz)*dz= z*dx + x*dz.
dy/y=(1/y)*(z*dx+x*dz)=dx/x+dz/z
Ex1.: Total Revenue (TR) = Price(P)*Quantity(Q).
 If P increases by 10 % and (Q thereby decreases by 5%, then TR increases by 5 %.
Ex.1: real wage = MPL = (1-alpha)*Y/L:
The percentage change of real wage = percentage change of (1-alpha) + percentage change of
Y/L. Note: (1-alpha) = share of labor income
Rule 2. If y(t) = x(t)/z(t), then
y
x z
.


y
x
z
Ex.1: GDP 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 %.
Ex. 2: Debt-income-ratio (y)= nominal debt(D)/nominal GDP. In case no amortizations are
done, the growth rate of nominal debt is the interest rate. Thus:
Percentage change of y = interest rate – percentage change of nominal GDP.
Ex3: Real wage(w) = W/P.
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
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
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.
To explain this model it is best to start to work with K and Y.
Instead of expressing variables in terms of per worker, to start with.
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
MPK=
   A K  L    A K  L
   A   ]
dK
L
[
If you increase saving, K increases, the rate of return is dY/dK-d, which falls.
Assumption (A2): A constant share of income is saved: S=s*Y, 0<s<1.
(= a constant share of production is invested) and a constant share of income is consumed:
C=(1-s)*Y.
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
s S
Y
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,
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
k K L
L
Using k = K/L 
, where
=0 by assumption.


L
k
K
L
k K
k  i    k  i    k ]



k
K
k k
Derivation optional:[
Depreciation of capital per worker
d*k
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
10
11
12
13
14
15
16
17
k
In figure:
  0.1
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
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,
That is, when gross investment equals depreciation
 s y   k
 s  A k     k
Solving for k in equilibrium:
s A 

k
k

k * , occurs when k  0 .
s  A  k 1

1
1
A 1
 s
 s  A 1
*
 k


k


  
  


*
What is the long-run equilibrium value of y, y ?

1
1 1

 s
y *  A  (k * )  A  



A 1


 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
10
11
12
13
14
15

16
17
k
In figure: A=1, s=0.3,
  0.1 , and  =0.5.
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.
16000
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 *
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.
 4 , y*  2
 =0.5  : k*  9 , y*  3
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
k K L
L
 n by assumption.
, where


L
k
K
L
k
K
k  n  i   ]  k  i  (n   )  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 
1
1
 s A


 n  
s A 
n 
k
k

s  A  k 1
n 
1

 s  A 1
k*  

 n  
 0.
What is the long-run equilibrium value of y, y* ?

 s  A 1
 If n 
y *  A  (k * )  A  

 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 k L k
   n
K
k
L
k
61
67
73
79
85
91
97
Y  yL

Y y L y
 

n
Y
y
L
y
In the steady-state:
k * y*
 * 0
k*
y
K * k * L
 *   0n  n
L
K*
k
*
*
y L
Y
 *  * 
 0n  n
L
Y
y
K*  k*  L 
Y *  y*  L
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:
k * y*
k * y* w*
,
Now:


0
 * 
g
w
k*
y*
k*
y
y* k *
r*
K *
Y *
*
  ( *  * )  0
 g n,
 g  n , r  d    ( y / k) ,
r
y
k
K*
Y*
Before:
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.
READ BY YOURSELVES: 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.