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Production Functions. 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
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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.
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.
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
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.
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.
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.
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
Production as a function of labor (holding capital fixed) 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.
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. Marginal Product = Marginal Cost
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.
Growth Rate Rules of Thumb • If Xt = Yt x Ztthen • If then • If then
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
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.
Measuring F • Measuring the growth in F has three parts • Measuring a. Under Cobb-Douglas, we can measure a, from labor’s 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.
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.
