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Ch3: National Income:
Where it Comes From
and Where it Goes
Mankiw: Macro Ch 3
Mankiw: Econ Ch24, Ch26
Varian: Ch10, Ch19, Ch29
Williamson: Ch 7
Learning Objectives
what determines the economy’s total
output/income
 how the prices of the factors of production
are determined
 how total income is distributed
 what determines the demand for goods
and services
 how equilibrium in the goods market is
achieved

Outline of model
A closed economy, market-clearing model
Three markets
 Labor market: MPL=W/P
 Goods market:YS=Yd
 (Financial market) Loanable funds market: S=I
Aggregate Supply side
 AS=YS=Production function
Aggregate Demand side
 AD=Yd=C+ I + G
Aggregate Supply Side:
The production function




describe relationship between inputs and output.
Real Output (Y，這章定義為大寫)
Inputs: factors of production 生產要素
Y = F(K, L)
K = capital: tools, machines, and structures
L = labor: physical and mental efforts of workers


F( ) reflects the economy’s level of technology
Assumes constant returns to scale
●
Returns to scale:
Initially
Y1 = F (K1 , L1 )
Scale all inputs by the same factor z:
K2 = zK1 and L2 = zL1
(e.g., if z = 1.25, then all inputs are increased by
25%)
What happens to output, Y2 = F (K2, L2 )?
 If constant returns to scale, Y2 = zY1
 If increasing returns to scale, Y2 > zY1
 If decreasing returns to scale, Y2 < zY1
Examples
F (K , L)  K  L : CRS
2
2
F (K , L)  K  L : IRS
F (K , L) 
KL : CRS
F (K , L) 
K  L : DRS
2
K
: CRS
F (K , L) 
L
The distribution of national income

determined by factor prices,
the prices per unit that firms pay for the
factors of production

wage = price of L

rental rate = price of K
Notation
W = nominal wage
Re = nominal rental rate
P = price of output
W /P = real wage
(measured in units of output)
Re /P= real rental rate
 = change in a variable

X = “the change in X ”
Marginal Product of Labor:
Y Y
MPL 

L L
Diminishing marginal returns: diminishing MPL
Suppose L while holding K fixed
 fewer machines per worker
 lower worker productivity
Marginal Product of Capital:
Y Y
MPK 

K K
Fig 3.3: MPL( K fixed )
Diminishing marginal returns
Y
output
F (K , L )
1
MPL
MPL
As more labor is
added, MPL 
1
MPL
1
Slope of the production
function equals MPL
L
labor
Eg,

Which of these production functions have
diminishing marginal returns to labor?
a) F (K , L)  2K  15L
b) F (K , L) 
KL
c) F (K , L)  2 K  15 L
Determination of factor prices
Varian: 19.7-19.9 and Appendix
Factor prices are determined by supply and
demand in factor markets.
 Assume: Supply of each factor is fixed.
 Assume markets are competitive:
each firm takes W, Re, and P as given.

Max   PF ( K , L)  Re K  WL
FOC wrt K : P  MPK  Re
FOC wrt L : P  MPL  W
Demand for labor
P  MPL  W , MPL 
W
P
Profit Mazimization( FOC wrt Y) : P 
W
 MC
MPL
benefit = MPL cost = real wage
A firm hires each unit of labor
if the cost does not exceed the benefit.

W
P
W
MPL 
: demand for labor
P
MPL 
Fig 3.4:
MPL = Demand for labor
Units of
output
Each firm hires labor
up to the point where
MPL = W/P.
Real
wage
MPL, Labor
demand
Units of labor, L
Quantity of labor
demanded
Labor Market:
the equilibrium real wage
Units of
output
Labor
supply
equilibrium
real wage
MPL, Labor
demand
L
Units of labor, L
Determining the rental rate
MPK = Re/P :
diminishing returns to capital:
MPK  as K 
Firms maximize profits by choosing K
such that P．MPK = Re .
MPK curve is firm’s demand curve
for renting capital.
(for one-time decision)
Neoclassical Theory of Distribution
W
total labor income = L  MPL  L
P
Re
K  MPK  
total capital income =
P
If production function has CRS,
Y  MPL  L  MPK  
national
income
labor
income
capital
income
The ratio of labor income to total income in
the U.S.
1
Labor’s
share
of total 0.8
income
0.6
Labor’s share of income
is approximately constant over time.
(Hence, capital’s share is, too.)
0.4
0.2
0
1960
1970
1980
1990
2000
Taiwan data: labor share
薪資報酬佔所得比例(%)
100
80
%
60
40
20
0
1997
1998
1999
2000
2002
2001
年度
2003
2004
2005
2006
薪資報酬佔所得比例
Cobb-Douglas Production Function

1
Y  AK L
A: the level of technology. (A is exogenous)
Each factor’s marginal product is proportional to
its average product.
MPK   AK
 1 1
L
Y

K
Y
MPL  (1   ) AK L  (1   )
L


C-D Production Function

C-D production function has
constant factor shares:
capital income = MPK x K =  Y
labor income = MPL x L = (1 –  )Y
 = capital’s share of total income
1- = labor’s share of total income
C-D Production Function
Proof that
Y  MPL  L  MPK  
--- Exhaustion of the product
--- imply zero profits for competitive
firms in the LR.
--- Since π=0 for all periods,
can ignore intertemporal analysis:
profit maximization over-time
Aggregate Demand Side:
Demand for goods & services
Components of aggregate demand:
AD = C+I+G+NX = C+I+G
C = consumer demand for goods & services
I = demand for investment goods
G = government demand for goods & services
(closed economy: no NX )
Consumption, C

Disposable income
total income minus total taxes:
Yd=Y – T.

Consumption function: C = C(Yd)
assume Yd   C 

Marginal propensity to consume (MPC)
C
MPC 
Yd
Fig 3.6: Consumption function
C
C (Y –T )
MPC
1
The slope of the
consumption function
is the MPC.
Y–T
Alternative (具個體化更完整的設定):
Household’s intertemporal analysis:
utility maximization over-time

With micro-foundation:
refer to Ch16 and Varian: Ch10
r＝the real rate of return for saving
(the payment to compensate the deferment of C)
= the real cost of borrowing
C = C(lifetime wealth, real interest rate)
= C (current income, future income,
real interest rate)
Investment, I
function: I = I (r ),
r :real interest rate,
the nominal interest rate corrected
for inflation.
 Investment
 Real
interest rate
近似= Nominal interest rate – Inflation rate
Real and Nominal Interest Rates
You lend out $100 for one year. P=$1/candy
 Nominal interest rate was 10%.
 During the year inflation was 6%.
 Present $100/$1= 100 units of candies
 $100*(1+10%)/[$1*(1+6%)]=103.7 units of
candies
 Real return= 3.7 units, real rate of return= 3.7%
 Real interest rate （real rate of return）
近似= Nominal interest rate – Inflation rate

= 10% - 6% = 4%
( r=R-π)
Investment function
 Real
interest rate is

the cost of borrowing

the opportunity cost of using one’s own
funds to finance investment spending.

(for over-time decision)
So, r  I 
Fig 3.7: The investment function
r
I (r )
I
Firm’s intertemporal analysis:
profit maximization over-time

With micro-foundation:
refer to Ch17 and Williamson Ch7
I t  Kt 1  Yt 1
MPK ,t 1  r  
--- over-time decision rule: I=I(r)
Cost of investment= r+δ
Benefit of investment = MPK
 Earlier: one-time period decision rule:
Re
MPK 
P
Government spending, G
G = govt spending on goods and services.
 G excludes transfer payments
(e.g., social security benefits,
unemployment insurance benefits).
 Assume government spending and total
taxes are exogenous:

G G
and
T T
Goods market equilibrium

Aggregate demand: Y d  C (Y  T )  I (r )  G

Aggregate supply:

Equilibrium:

The real interest rate adjusts
to equate demand with supply.
Y s  F (K , L )
Y = C (Y  T )  I (r )  G
Saving and Investment
in the National Income Accounts
(1) GDP= Y=total expenditure
Y = C + I + G + NX
 A closed economy: no international trade
Y = C + I + G → Y – C – G =I
(2) GDP = Y = national income
 Y= C + G + S
→ National Saving ：S= Y – C - G
→ (3) For the economy as a whole, S = I
The Meaning of Saving and Investment

National Saving 國民儲蓄S = Y –C – G


the total income that remains after paying for
consumption and government purchases.
Private Saving 私部門儲蓄Sp ≡ Y – T – C
the amount of income that households have left
after paying their taxes and paying for their
consumption.

Public Saving 公部門儲蓄Sg ≡ T –G
the amount of tax revenue that the government
has left after paying for its spending.
Gov’t budget constraint (Gov’t BC):

Surplus and Deficit （預算盈餘與赤字）
Sg≡ T –G >0 if T > G (budget surplus )
Sg≡ T –G <0 if G > T (budget deficit)
Budget Deficit ≡ D ≡ G-T = - Sg
Gov’t Bond（公債）：Bg
Gov’t issue new bonds to finance Budget Deficit
△Bg = D ≡ G-T ，
亦即 Gov’t BC: G= T + △Bg
Flow vs. Stock
The accumulation of past budget deficits
=Public Debt （Gov’t Bond）
Flow 流量：I, education, S, D
vs.
Stock存量：K, H, wealth, Bg
Market for Loanable Funds

Financial markets
coordinate economy’s saving and
investment
in the market for loanable funds.
(可貸資金市場，LF)
Supply and Demand for Loanable Funds
supply of loanable funds (SLF）＝S (net)
The demand for loanable funds （DLF）＝I
 The
 The
price (of loan) in the LF market is
real interest rate. r=R-π
r＝the real rate of return for saving
= the real cost of borrowing
Supply and Demand for Loanable Funds
SLF =S： (+)vely-slpoed， QsLF↑ =S ↑ as r ↑
其實不一定是正斜率 Varian:Ch10
As r ↑，S ↑：if Substitution effect >Income effect
As r ↑，S ↓ ：if SE <IE
Here we assume SE >IE, so r ↑→ S ↑
(課本忽略r對C與對S的影響，故設SLF為垂直線)



DLF=I： (-) vely-slpoed ，QdLF↓= I↓ as r ↑
r as the opportunity cost of investment.
LF Market equilibrium:
the intersection of SLF and DLF determines r*.
Fig 3.12 Market for Loanable Funds
Interest
Rate
Supply
5%
Demand
0
$1,200
Loanable Funds
(in billions of dollars)
Copyright©2004 South-Western
Alternative setup (FYI)
In equilibrium: Sp + Sg = I
You can set up the model as
in equilibrium: Sp= I-Sg = I+ △Bg
SLF=Sp
DLF=I+△Bg=I+(G-T)


The following comparative static analysis
(△SLF，△DLF) would be different,
but the result still be the same.
Comparative Statics (比較靜態)
St = Yt-Ct-Gt = It
 Kt+1=(1-δ)Kt+It δ: depreciation rate(折舊率)
St↑ → It↑ → Kt+1↑ → Yt+1↑


Government Policies that affect S and I：
1. Taxes on saving （利息所得稅）
2. Tax credits on investment (投資抵減)
3. Government budget deficits（預算赤字）
Policy 1: Saving Incentives
Taxes on interest income：
real rate of return = (1-t）r
 t↑ reduce future payoff from current saving
→reduce the incentive to save.
 t↓ → increase the incentive to save:
QsLF↑ =S ↑ at any given r.
 SLF curve shifts to the right.
 r* ↓， QdLF ↑= I ↑
→ lower interest rates and greater investment.

Fig 3.9b
An Increase in Supply of Loanable Funds
Interest
Rate
Supply, S1
S2
1. Tax incentives for
saving increase the
supply of loanable
funds . . .
5%
4%
2. . . . which
reduces the
equilibrium
interest rate . . .
Demand
0
$1,200
$1,600
Loanable Funds
(in billions of dollars)
3. . . . and raises the equilibrium
quantity of loanable funds.
Copyright©2004 South-Western
Policy 2: Investment Incentives
An investment tax credit （1-κ）r
lowers the cost of borrowing
and increases the incentive to borrow.
 κ↑： QdLF↑ =I ↑ at any given r.
 DLF curve shifts to the right.
→ higher interest rates
and greater saving/investment.

Fig 3.11
An Increase in Demand for Loanable Funds
Interest
Rate
Supply
1. An investment
tax credit
increases the
demand for
loanable funds . . .
6%
5%
2. . . . which
raises the
equilibrium
interest rate . . .
0
D2
Demand, D1
$1,200
$1,400
Loanable Funds
(in billions of dollars)
3. . . . and raises the equilibrium
quantity of loanable funds.
Copyright©2004 South-Western
Policy 3: Government Budget Deficits
and Surpluses
I= S = (Y – T – C) + (T – G) = Sp + Sg
 Gov’t budget deficit： T< G， Sg<0
at any given r, QsLF↓ =S↓ < Sp when Sg<0
SLF curve shifts to the left.
 →r*↑，LF*↓
higher interest rate and lower investment.
referred to as crowding out effect (排擠效果):
The deficit borrowing crowds out private investments.
Figure 3.9:
The Effect of a Government Budget Deficit
Interest
Rate
S2
Supply, S1
1. A budget deficit
decreases the
supply of loanable
funds . . .
6%
5%
2. . . . which
raises the
equilibrium
interest rate . . .
Demand
0
$800
$1,200
Loanable Funds
(in billions of dollars)
3. . . . and reduces the equilibrium
quantity of loanable funds.
Copyright©2004 South-Western
U.S. Federal Government
Surplus/Deficit, 1940-2004
5%
0%
(% of GDP)
-5%
-10%
-15%
-20%
-25%
-30%
1940
1950
1960
1970
1980
1990
2000
U.S. Federal Government Debt,
1940-2004
Fact: In the early 1990s,
about 18 cents of every tax
dollar went to pay interest on
the debt.
(Today it’s about 9 cents.)
120%
(% of GDP)
100%
80%
60%
40%
20%
0%
1940
1950
1960
1970
1980
1990
2000
Taiwan data:台灣政府收支淨額
Fig 26.5a台灣政府收支淨額
3,500,000
3,000,000
2,000,000
政府收入淨額
政府支出淨額
1,500,000
1,000,000
500,000
年度
20
06
20
04
20
02
20
00
19
98
19
96
19
94
19
92
19
90
19
88
19
86
19
84
19
82
0
19
80
百萬元
2,500,000
Taiwan data:
台灣政府預算盈餘赤字與新增公債
Fig26.5b台灣政府收支餘絀（預算盈餘赤字）與新增公債
600,000
500,000
400,000
300,000
100,000
政府收支餘絀
公債收入
-200,000
-300,000
-400,000
-500,000
年度
20
06
20
04
20
02
20
00
19
98
19
96
19
94
19
92
19
90
19
88
19
86
19
84
-100,000
19
82
0
19
80
百萬元
200,000
Taiwan data：公債餘額
Fig 26.5c 政府公債餘額
35,000
30,000
25,000
億元
20,000
政府公債餘額
15,000
10,000
5,000
0
1995
1996
1997
1998
1999
2000
年度
2001
2002
2003
2004
2005
The special role of r
r adjusts to equilibrate the goods market and
the loanable funds market simultaneously:
If LF market is in equilibrium,
S=I
→Y–C–G =I
→Y=C+I+G
goods market is in equilibrium
Thus,
LF market
Goods market
equilibrium

equilibrium
Walras’ Law (Varian: Ch 29)

If there are markets for k goods,
then we only need to find a set of prices
where k-1 of markets are in equilibrium.

Here in our macro model,
we have 3 markets, we only need 2 prices
to assure general equilibrium:
real wage and real interest rate
Chapter Summary

Total output is determined by


the economy’s quantities of capital and labor
the level of technology
Competitive firms hire each factor until its
marginal product equals its price.
 If the production function has constant
returns to scale, then labor income plus
capital income equals total income
(output).

CHAPTER 3
National Income
slide 57
Chapter Summary

A closed economy’s output is used for




consumption
investment
government spending
The real interest rate adjusts to equate
the demand for and supply of


goods and services
loanable funds
CHAPTER 3
National Income
slide 58
Chapter Summary

A decrease in national saving causes the
interest rate to rise and investment to fall.
CHAPTER 3
National Income
slide 59