Production Functions

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Production Functions
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Students Should Be Able To

• Use the Cobb-Douglas production function to
  calculate
• Output as a product of inputs
• marginal and average factor products as a product
  of inputs or output and inputs
• Total Factor Productivity Growth
• Construct input demand curve using marginal
  products.

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HK vs. USA

• In 1998, USAs 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?

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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.

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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
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Industrial Age

• In Britain in late 1700s 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

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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

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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.

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Population GrowthHong Kong and Singapore
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PopulationChina and India
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GDP per Capita
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GDP per Capita pt. 2
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Production Functions
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Production Function

• An economys 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.

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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.

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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

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Production as a function of labor (holding
capital fixed)
GDP
L
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Marginal Product of Labor
L
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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.

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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.

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Marginal Product Marginal Cost

• 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.

• 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.

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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.

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Growth Rate Rules of Thumb

1. If Xt Yt x Zt then
2. If then
3. If then

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Productivity Growth

• When economists study productivity, they often
  decompose output into two parts
• F output due to the accumulation of the factors
  of production, capital and labor
• TFP total factor producivity or output due to
  advances in technology.
• Using Cobb-Douglas, it is easy to do this

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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.

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Measuring F

• Measuring the growth in F has three parts
• Measuring a. Under Cobb-Douglas, we can measure
  a, from labors share of income.
• Measuring L. Government statistical bodies
  periodically measure the stock of labor using
  surveys of employers or households.
• Measuring K Perpetual Inventory Method. Guess
  at initial capital stock. Use constant dollar
  measures of investment and estimates of
  depreciation to recursively calculate investment.

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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.

Growth Due to
Capital (a) gk
Labor (1-a)gL
TFP gTFP
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TFP Growth in HK Singapore