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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:
L = Labor is defined as hours worked. Main concern in this chapter is productivity
t t t
t t t
Y Y L
POP L POP
Income per person, 2003Groningen Growth & Development Center http://www.ggdc.net
GDP per Person
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
Hong Kong Singapore SouthKorea
Taiwan Japan USA France
HK
= 1
00
ProductivityGDP per Hour Worked
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
200.00
Hong Kong Singapore SouthKorea
Taiwan Japan USA France
HK
= 1
00
Employment
Hours Worked per Capita
0.00
20.00
40.00
60.00
80.00
100.00
120.00
Hong Kong Singapore SouthKorea
Taiwan Japan USA France
HK
= 1
00
Capital
Labor
Factors of Production
etc.
TechnologyOutput
Production
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.
ExampleStock: Government DebtFlow: Government Revenue, Government
Expenditure Example
CapitalFlow: Investment (It), Depreciation (Dpnt)
©2002 South-Western College PublishingFigure 2.6
Stocks and Flows
Capital is Defined Recursivel
Perpetual Inventory Method
Method requires some initial guess at capital stock. As original guess capital depreciates, measure becomes more accurate.
Constant Depreciation Rate
1t t t tK K I Dpn
1 (1 )t t t t t tK K I dK d K I
Hong Kong Investment to Capital Ratio
Investment to Capital Ratio (d = .09)
0
0.05
0.1
0.15
0.2
0.25
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
#
Hong Kong capital stock
Capital
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
#
Productivity: Two Concepts
There are two basic measures of productivity.1. Average Productivity: The average productivity of a
factor is output divided by amount of factor used.
2. Marginal Productivity: The extra output that would be produced if an extra unit of a factor were used.
,Y Y
L K
,Y Y
L K
Capital Productivity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Hong Kong EU USA
Aggregate Production Function
Assume aggregate output can be written as an algebraic function of the aggregate factors.
Technological change over time is represented as a scaling factor, Qt.
Example: Cobb-Douglas
( , )t t t tY F K L
( , , )t t t tY F Q K L
1( )a a
t t t tY K Q L
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)
L
Y
ΔL
ΔYΔL
ΔY
Marginal Productivity Function (fixed K)
L
MPL
MPL
Marginal Productivity Function (fixed L)
K
MPK
MPK
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
Implications for Country Size: Output per capita depends only on capital per capita and labor per capita, not on population size itself.
1 1( ) ( )a aa a
t t t t t t t tY K Q L N Y N K Q N L
1( )a
at t tt
t t t
Y K LQ
POP POP POP
Productivity Function
Labor productivity is a function of technology and the capital-labor ratio.
11
1 1
1
11
( ) ( )
( )
a aa at t t t t tt
a at t t t
aa at at t t
ta at t t
t
a at t
K Q L K Q LY
L L L L
K Q L
y
QK
QL L
kL
Advantages of Cobb-Douglas Production Function Average Product & Marginal Product Under Cobb-Douglas, the marginal product is
proportional to average product.
All intuition about things that change average productivity carry-over 1-to-1 to marginal productivity.
1
1 1
(1 ) (1 )
( )
a a a
a a
YMPL a K Q L a
LY
MPK a K QL aK
Advantages of Cobb-Douglas Production Function Log-linear Take natural log of output
Growth rate of output is a linear function of the growth rate of capital, labor, and technology.
1 1 1( ) ln ln ln ln
ln ln (1 ) ln (1 ) ln
a a a a at t t t t t t t
t t t t
Y K Q L Y K Q L
Y a K a Q a L
1 1 1 1
ln ln (1 ) ln (1 ) ln
ln ln (1 ) ln (1 ) ln
(1 ) (1 )
t t t t
t t t t
Y K Q Lt t t t
Y a K a Q a L
Y a K a Q a L
g ag a g a g
Marginal Product = Marginal Cost 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.
A firm can raise its profits by increasing labor as long as the 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.
t
t
WMPL
P
Labor Demand Schedule (fixed K)
L
W/P
MPL
Advantages of Cobb-Douglas Production Function Factor Shares Labor compensation is the product of the
wage rate and the quantity of labor WtLt.
Income left over to owners of capital is also a constant share of output a∙Yt
(1 ) (1 )t tt t t t
t t
W Ya W L a PY
P L
Implications
Labor share of income (labor intensity) is equal to the ratio of the marginal product of labor to the average product.
1
t
t t t t
tt t t
t
W Y YW L P MPL L Y aY Y LPY APL
L LL
Labor Intensity ≡1- a ≈2/3Labor Share of Income
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mar
-72
Mar
-74
Mar
-76
Mar
-78
Mar
-80
Mar
-82
Mar
-84
Mar
-86
Mar
-88
Mar
-90
Mar
-92
Mar
-94
Mar
-96
Mar
-98
USA
JAPAN
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 intensity, at and 1-at = used as weights.
[1 ]
[1 ]t ta a
t t t tt t
t t t t
Y Y W LTFP a
L K PY
WLPY
TFP Growth
TFP is log linear
TFP growth rate is the gap between GDP growth rate and the weighted average of the growth rate of the factors of production.
[1 ]
ln ln
ln ln (1 ) ln
ln ln ln (1 ) ln (1 ) l
ln (1 ) ln
n
t t
t
a a
t t t tt t t t
t t t t
t t
t t t
t
t
t t
t
t t
Y Y Y YTFP TFP a a
L K L K
TFP a Y L a
TFP
Y a
Y L a K
K
1 1 1 1ln ln ln ln (1 )[ln l
(1
n ] [ln ln
)
]
TFP Y L
t t t t t t t t
Kt t t t t
t t
t
TFP TFP Y Y a L L a K
a a
K
TFP Growth Rates over time
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
USA EU
1980-1995
1995-2001
Advantages of Cobb-Douglas Production FunctionTFP equals technology
If production is according to Cobb-Douglas, then TFP directly measures technology. at =a.
1 [ ] 1
11 1
a a aat t t at t t
t ta a a at t t t t t
K Q LY Y YTFP Q
L K L K L K
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 1965-2001Output Growth Rate
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Hong Kong Singapore South Korea Taiwan USA
Myth of the East Asian MiracleAlwyn Young, QJE 2001
TFP Growth Rates
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
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
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Hong Kong Singapore South Korea Taiwan USA
TFP
Labor
Capital
Capital Productivity
Capital Productivity
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
Hong Kong Singapore South Korea Taiwan USA