Productivity Chap. 4, The Theory of Aggregate Supply

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



HK

= 1

00

Employment

Hours Worked per Capita

0.00

20.00

40.00

60.00

80.00

100.00

120.00



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%


TFP

Labor

Capital



-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0


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Productivity Chap. 4, The Theory of Aggregate Supply