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Productivity
Chap. 4, The Theory of
Aggregate Supply
Income depends on Output,
Output depends on productivity and
labor
GDP, Y, is value produced.
 GDP can be decomposed:

Yt
Yt
Lt
 
POPt Lt POPt
L = Labor is defined as hours worked.
 Main concern in this chapter is productivity

Income per person, 2003
Groningen Growth & Development Center http://www.ggdc.net
GDP per Person
160.00
140.00
HK = 100
120.00
100.00
80.00
60.00
40.00
20.00
0.00
Hong Kong Singapore
South
Korea
Taiwan
Japan
USA
France
Productivity
GDP per Hour Worked
200.00
180.00
160.00
HK = 100
140.00
120.00
100.00
80.00
60.00
40.00
20.00
0.00
Hong Kong Singapore
South
Korea
Taiwan
Japan
USA
France
Employment
Hours Worked per Capita
120.00
100.00
HK = 100
80.00
60.00
40.00
20.00
0.00
Hong Kong Singapore
South
Korea
Taiwan
Japan
USA
France
Production
Factors of Production
Capital
Output
Technology
Labor
etc.
Factors of Production: Capital
Capital (Kt) is the stock of durable goods
(machines, equipment, buildings, etc.)
used to produce other goods.
 Unit of measure is dollar-value.
 Difficult to measure directly, so it is defined
indirectly.

Stock vs. Flow
Stock: Some variable that accumulates. Flow:
Channel of increase or decrease of a stock.
 Example

 Stock:
Government Debt
 Flow: Government Revenue, Government
Expenditure

Example
 Capital
 Flow:
Investment (It), Depreciation (Dpnt)
Stocks and Flows
Figure 2.6
©2002 South-Western College Publishing
Capital is Defined Recursivel

Perpetual Inventory Method
Kt 1  Kt  I t  Dpnt
Method requires some initial guess at
capital stock. As original guess capital
depreciates, measure becomes more
accurate.
 Constant Depreciation Rate

Kt 1  Kt  It  dKt  (1  d ) Kt  I t
Hong Kong Investment to Capital
Ratio
Investment to Capital Ratio (d = .09)
0.25
0.2
0.15
0.1
0.05
20
02
#
20
00
19
98
19
96
19
94
19
92
19
90
19
88
19
86
19
84
19
82
19
80
0
Hong Kong capital stock
Capital
3500000
3000000
2500000
2000000
1500000
1000000
500000
0
80
9
1
82
9
1
84
9
1
86
9
1
88
9
1
90
9
1
92
9
1
94
9
1
96
9
1
98
9
1
00
0
2
2#
0
20
Productivity: Two Concepts

There are two basic measures of productivity.
1.
2.
Average Productivity: The average productivity of a
factor is output divided by amount of factor used.
Y Y
,
L K
Marginal Productivity: The extra output that would
be produced if an extra unit of a factor were used.
Y Y
,
L K
Capital Productivity
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Hong Kong
EU
USA
Aggregate Production Function
Assume aggregate output can be written
as an algebraic function of the aggregate
factors. Y  F ( K , L )
t
t
t
t
 Technological change over time is
represented as a scaling factor, Qt.
Yt  F (Qt , Kt , Lt )


Example: Cobb-Douglas
Yt   K t  (Qt Lt )1 a
a
Marginal Productivity of Labor

Holding capital constant, the effect on GDP of
increasing labor by a small amount. Y = F(L)
MPL = ΔY/ΔL
 The slope of the production function
 For very small increases in labor, can
be calculated
with first derivative of output with respect to labor.
MPL = F’(L)

Diminishing returns suggests that if you hold one
factor constant, marginal returns are a
diminishing function.
Production Function
(fixed K)
Y
ΔY
ΔY
ΔL
ΔL
L
Marginal Productivity Function
(fixed K)
MPL
MPL
L
Marginal Productivity Function
(fixed L)
MPK
MPK
K
Advantages of Cobb-Douglas Production Function
Constant Returns to Scale

If you increase both capital and labor by a
factor of N, then you will also increase
output by a factor of N
Yt   Kt  (Qt Lt )1 a  N  Yt   N  K t  (Qt N  Lt )1 a
a

a
Implications for Country Size: Output per
capita depends only on capital per capita
and labor per capita, not ona population
 Kt 
Yt
Lt 1 a
size itself.

(Q
)
POPt


POP
t 

t
POPt
Productivity Function

Labor productivity is a function of
technology and the capital-labor ratio.
Yt  Kt  (Qt Lt )
yt  
Lt
Lt
1 a
a
Kt 


Lat
a
1 a
(Qt Lt )

L1ta
Kt 


a
1 a
(Qt Lt )
Lat L1ta
a
 Kt  1a
   Qt  kta Qt1a
 Lt 
Advantages of Cobb-Douglas Production Function
Average Product & Marginal
Product

Under Cobb-Douglas, the marginal product is
proportional to average product.
Y
MPL  (1  a)  K Q L  (1  a )
L
Y
a 1
1 a
MPK  a  K (QL)  a
K
a

1 a  a
All intuition about things that change average
productivity carry-over 1-to-1 to marginal
productivity.
Advantages of Cobb-Douglas Production Function
Log-linear

Take natural log of output
 




Yt   K t  (Qt Lt )1 a  ln Yt  ln K t a  ln Qt1 a  ln Lt1 a 
a
 
 
ln Yt  a ln K t  (1  a) ln  Qt   (1  a) ln Lt

Growth rate of output is a linear function of
the growth rate of capital, labor, and
technology.
 
ln Yt  a ln K t  (1  a) ln  Qt   (1  a) ln  Lt 
ln Yt 1  a ln  Kt 1   (1  a) ln  Qt 1   (1  a) ln  Lt 1 
gtY  agtK  (1  a) gtQ  (1  a) gtL
Marginal Product =  A firm can raise its
profits by increasing
Marginal Cost
labor as long as the


Profit maximization
suggests that the
marginal product of a
factor should equal its
real cost.
The real cost of labor is
the real wage, the dollar
wage rate divided by the
price level.
Wt
MPL 
Pt
cost of the extra labor
is less than the extra
goods produced.
Since the extra goods
produced drops as
more labor is added,
firms will hire more
labor until the
marginal product falls
as low as the real
wage.
Labor Demand Schedule
(fixed K)
W/P
MPL
L
Advantages of Cobb-Douglas Production Function
Factor Shares

Labor compensation is the product of the
wage rate and the quantity of labor WtLt.
Wt
Yt
 (1  a)  Wt Lt  (1  a) PY
t t
Pt
Lt

Income left over to owners of capital is
also a constant share of output a∙Yt
Implications

Labor share of income (labor intensity) is
equal to the ratio of the marginal product
of labor to the average product.
Wt
Wt Lt
Pt MPLt Y L Y Y




 1 a
Yt
Y
L
PY
APLt
t t
L
L
Lt
Mar-98
Mar-96
Mar-94
Mar-92
Mar-90
Mar-88
Mar-86
Mar-84
Mar-82
Mar-80
Mar-78
Mar-76
Mar-74
Mar-72
Labor Intensity ≡1- a ≈2/3
Labor Share of Income
0.8
0.7
0.6
0.5
0.4
USA
JAPAN
0.3
0.2
0.1
0
Total Factor Productivity
Total factor productivity measures the total
effectiveness of an economy in applying
all of its factors of production.
 TFP is a geometrically weighted average
of capital and labor productivity with factor
WL
intensity, at and 1-at = PY used as weights.

[1 at ]
 Yt 
TFPt   
 Lt 
 Yt 
 
 Kt 
at
Wt Lt
[1  at ] 
PY
t t
TFP Growth
at
TFP
linear

[1 at ]
 Yt   Yt 
TFPt     
is log  Lt   Kt 
 ln TFPt  at ln
Yt
Y
 (1  at ) ln t
Lt
Kt
ln TFPt  at ln Yt  ln Lt  (1  at ) ln Yt  (1  at ) ln Kt
ln TFPt  ln Yt  ln Lt  (1  at ) ln K t

TFP growth rate is the gap between GDP growth
rate and the weighted average of the growth
rate of the factors of production.
ln TFPt  ln TFPt 1  ln Yt  ln Yt 1  (1  at )[ln Lt  ln Lt 1 ]  at [ln K t  ln K t 1 ]
tTFP  tY   at tL   (1  at ) tK 
TFP Growth Rates over time
1.4%
1.2%
1.0%
0.8%
1980-1995
0.6%
1995-2001
0.4%
0.2%
0.0%
USA
EU
Advantages of Cobb-Douglas Production Function
TFP equals technology

If production is according to CobbDouglas, then TFP directly measures
technology. at =a.
1 a
 Yt 
TFPt   
 Lt 
[a]
 Yt 
 
 Kt 
Yt
 1 a a 
Lt K t
K t a  Qt Lt 
1 a
Lt1 a K t a
 Qt1 a
Growth Accounting

When we measure growth, we might want
to determine if this is caused by capital
growth, labor growth or capital growth.
Growth caused by
gY
Capital
a×gK
Labor
(1-a)×gL
Technology
gTFP
The East Asian Miracle 1965-2001
Output Growth Rate
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
Hong Kong
Singapore
South Korea
Taiwan
USA
Myth of the East Asian Miracle
Alwyn Young, QJE 2001
TFP Growth Rates
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
Hong Kong
Singapore
South Korea
Taiwan
USA
EU
Criticisms



Critics of Young’s work that because of data
mismeasurement, they assumed that East Asian
production functions were different (greater
capital intensity) than developed economies.
Even using same production functions, most
East Asian growth differentials are due to factor
accumulation not TFP growth.
One key point, capital productivity was declining
in East Asia over this time period.
Growth Accounting: 1965-2000
12.0%
10.0%
8.0%
TFP
6.0%
Labor
Capital
4.0%
2.0%
0.0%
Hong Kong
Singapore
South Korea
Taiwan
USA
Capital Productivity
Capital Productivity
0
Hong Kong
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
Singapore
South Korea
Taiwan
USA