Production Functions

Students Should Be Able To

Use the Cobb-Douglas production function to calculate:

1. Output as a product of inputs

2. marginal and average factor products as a product of inputs or output and inputs

3. Total Factor Productivity Growth Construct input demand curve using

marginal products.

HK vs. USA

In 1998, USA’s real GDP per capita was about 1/3 greater than Hong Kong.

But average US growth rate over the preceding 50 years was about 2% per year. Average HK growth rate was 4.5% per year.

If these two growth performances continue, in 50 years HK GDP per Capita would be 2.5 times that in the USA.

Will this occur?

Sources of Growth

Because dividends are limited by capital income, dividend growth is determined by GDP growth.

Nominal GDP growth can be divided into two parts: 1) inflation; 2) real GDP growth.

Real GDP growth can be divided into two parts: 1) population growth; 2) growth in real GDP per capita.

Chinese GDP per Capita by Dynasty (1990 US$ per person)

Year Dynasty China Europe

50AD Han 400 450

960AD Tang 400 350

1280 Sung 600 450

1400 Ming 600 450

1820 Qing 600 1122

Industrial Age

In Britain in late 1700’s a new economic began to take shape

Key characteristic of this age was use of machinery (or capital) to augment labor.

Relatively large growth in output Population grows more slowly than output

GDP per capita, 1950

% of ACNZUS

GDP per capita, 1992

% of ACNZUS

ACNZUS 9255 100% 20,850 100%

W. EUROPE 5126 55.4 17,387 83.9

LATIN AMERICA

2487 26.8 4,820 23.1

ASIA 765 8.3 3,252 15.6

JAPAN 1873 20.4 19425 93.2

HONG KONG 1962 21.1 17,120 82.1

AFRICA 830 8.9 1284 6.1%

Post-War Facts

Large Income Differences Across Countries Convergence to World Leaders in Two Areas:

Europe and East Asia Low initial level of Japan and Europe due to destruction of

capital stock

Divergence from World Leaders in Africa and Latin America

Small Gains in Asia as Whole Interesting dynamics amongst East Asian economies.

Population Growth:Hong Kong and Singapore

Population

0100020003000400050006000

Tho

usan

ds

Hong Kong

Singapore

Population:China and India

Population

0200400600800

10001200

Mill

ions

China

India

GDP per Capita

GDP per Capita

0

2000

4000

6000

8000

1990

US$

1000

China

India

Burma

Philippines

Thailand

GDP per Capita pt. 2

GDP per Capita

0

5000

10000

15000

20000

25000

1990

US$

1000 Hong Kong

Singapore

S. Korea

Taiwan

Production Functions

Production Function

An economy’s value added is produced by its Stock of capital equipment denoted Kt

Labor force denoted Lt

Technology/Worker Efficiency denoted Zt

Cobb-Douglas production function

The parameter, a, is sometimes referred to as capital intensity, i.e., the greater is a, the more important capital is in production.

1( )a a

t t t tGDP K Z L

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 tGDP K Z L N GDP N K Z N L

1( )

a

at t tt

t t t

GDP K LZ

POP POP POP

Marginal Product

The marginal product of a factor is the extra output that results from the extra use of the factor relative to the size of the increase in factor use.

Marginal products of very small increases in factor use can be derived with derivatives

GDP GDPMPL MPK

L K

1(1 ) a a aGDPMPL a K X L

L

1 1( )a aGDPMPK a K XL

K

Production as a function of labor (holding capital fixed)

L

L

L

GDP

GDP

GDP

Marginal Product of Labor

L

GDP

L

Advantages of Cobb-Douglas Production Function Pt.2

Diminishing returns Holding capital & technology constant, the

marginal product of labor is a decreasing function of labor.

Holding labor & technology constant, the marginal product of capital is a decreasing function of capital.

Average Product

We define average productivity of a factor as the ratio of output to the level of factor use

Under Cobb-Douglas, the marginal product is proportional to average product.

GDP GDPAPL APK

L K

1

1 1

(1 ) (1 )

( )

a a a

a a

GDPMPL a K X L a

LGDP

MPK a K XL aK

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 flls as low as the real wage.

t

t

WMPL

P

Factor Shares

Under a Cobb-Douglas production function, labor compensation is a constant share of value added.

Labor compensation is the product of the wage rate and the quantity of labor WtLt.

Capital income is also a constant share of value added.

(1 ) (1 )t tt t t t

t t

W GDPa W L a PGDP

P L

t t t t t tEDIBTA PGDP W L aPGDP

Growth Rate Rules of Thumb

1. If Xt = Yt x Zt then

2. If then

3. If then

X Y Zt t tg g g

( )at tX YX Yt tg a g

tt

t

YX

Z

X Y Zt t tg g g

Productivity Growth

When economists study productivity, they often decompose output into two parts:

1. F: output due to the accumulation of the factors of production, capital and labor;

2. TFP: total factor producivity or output due to advances in technology.

Using Cobb-Douglas, it is easy to do thist t tGDP TFP F

1 1( )a a a

t t t t tF K L TFP Z

TFP Growth

Total factor productivity is implicitly defined as the ratio of output to a combination of the factors of production.

TFP growth is the difference between output growth and the growth of the combined factor.

tt

t

GDPTFP

F

TFP GDP Ft t tg g g

Measuring F

Measuring the growth in F has three parts1. Measuring a. Under Cobb-Douglas, we can

measure a, from labor’s share of income.

2. Measuring L. Government statistical bodies periodically measure the stock of labor using surveys of employers or households.

3. Measuring K: Perpetual Inventory Method. Guess at initial capital stock. Use constant dollar measures of investment and estimates of depreciation to recursively calculate investment.

1t t t tK K DPN I

1 t t

t

W La

PGDP

TFP Growth

The growth rate of factor is

TFP growth can be calculated as

Growth accounting attributes those parts of growth that are due to its different elements.

(1 )F K Lt t tg a g a g

(1 )TFP GDP K Lt t t tg g a g a g

Growth Due to

Capital (a) gk

Labor (1-a)gL

TFP gTFP

TFP Growth in HK & Singapore