Chapter 12 The Production Function Approach to Understanding Growth Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Chapter 12: The Production Function Approach to Understanding Growth Economists and economic growth The production function The Cobb–Douglas production function Growth accounting Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Introduction In this chapter, growth refers to growth in total GDP, not GDP per capita, as in the previous chapter Growth in labour productivity, the foundation of growth in living standards, is equal to the rate of growth of GDP minus the rate of growth of the labour force This is achieved through increasing the capital labour ratio by saving and investment, and by promoting total factor productivity through research, innovation and better management Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

The Production Function

Employment of Capital Assume fixed labour force Diminishing returns and diminishing marginal product of capital (MPK) Value of the MPK or marginal revenue product of capital (MRPK) = P . MPK Opportunity cost of capital – the real interest rate is what you give up by holding capital instead of interest-bearing assets Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

The Demand for Stock of Capital

Demand for Stock of Capital Change real interest rate – move along demand curve Change MRPK – shift demand curve Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Labour Supply Flow supply of labour measured in person-hours of work Opportunity cost of labour supplied is leisure given up This must be compensated for with real wage At some point the opportunity cost of labour supplied rises The labour supply curve slopes up Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Labour Demand Assume fixed stock of capital Diminishing returns and diminishing marginal product of labour (MPL) Marginal benefit of hiring is MRPL = P . MPL Marginal cost of hiring is money wage, W In equilibrium P . MPL = W So in equilibrium MPL = W/P = real wage We measure the MPL by the real wage Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

The Labour Demand Curve

Labour Market Equilibrium

The Production Function Labour and capital are primary factors of production Secondary factors of production are all the other relevant factors, such as managerial expertise, which are assumed to be constant The production function shows the level of production associated with different combinations of capital and labour, holding secondary factors constant Likewise, with inputs of labour and capital constant, it shows the level of production with different inputs of secondary factors Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

A Production Table

Measuring the MPL Hold the capital input constant at, say, 3 units and increase labour input progressively by 1 unit Total output goes from, say, 0 to 3.46 to 4.9 to 6.0 … The change in total output measures the marginal product of labour, 3.46, 1.43, 1.1 … The MPL declines as labour input rises Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Measuring the MPK Hold labour input constant at, say, 6 units and increase capital input progressively by 1 unit Total output rises from, say, 0 to 4.0 to 5.66 to 6.93 … The change in total output measures the marginal product of capital, 4.0, 1.66, 1.27… The MPK declines as capital input rises Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

One Variable Function

Algebraic Version Production in period t (Yt) depends on the employment of secondary factors in period t (At) multiplied by some function (f) of the combination of the employment of capital and labour in period t (Kt and Lt) Yt = At . f(Kt, Lt) Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Cobb–Douglas Function Founded on the observation that the shares of capital income (α) and labour income (1 – α) in US national income have been more or less constant over time Assumes constant returns to scale Yt = AKtα . Lt(1 – α) Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Properties: the MPK and K/L The average product of capital, Y/K = A(L/K)(1 - α) With constant A and α, this must fall as the capital–labour ratio rises (as L/K falls) Differentiating the production function with respect to K gives MPK = α Y/K This must fall as the capital–labour ratio rises and Y/K falls Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Properties: the MPL and L/K The average product of labour, Y/L = A(K/L)α With constant A and α, this must fall as the labour–capital ratio rises (as K/L falls) Differentiating the production function with respect to L gives the MPL = (1 – α)Y/L This must fall as the labour–capital ratio rises and Y/L falls Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Properties: Labour Income The real wage is the MPL Total labour income is MPL . L The national income is Y The share of labour income is MPL . L/Y MPL = (1 – α)Y/L So labour share = MPL . L/Y = (1 – α)Y/L . L/Y = (1 – α) In Australia about 75% Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Properties: Capital Income By the same reasoning, capital income is MPK.K Capital’s share in national income is MPK . K/Y MPK = αY/K So capital’s share is = α In Australia about 25% Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Factor Shares and CRTS CRTS = constant returns to scale, meaning that increasing capital and labour inputs by a factor of x will increase output by that factor Let’s show that if shares of capital and labour add to unity, then CRTS applies Shares add to unity of capital’s share is α and labour’s share is (1 – α) With Y = AKα L(1- α), then scaling up labour and capital inputs by a factor of x gives Y’ = A(xK)α (xL)(1-α) = xY Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Contribution to Additional GDP Capital’s share = α = MPK . K/Y Let MPK = dY/dK Then α = [dY.Y]/[dK/K] Capital’s share, α, measures the proportional change in output following a proportional change in capital input Likewise, labour’s share, (1 – α), measures the proportional change in output following a proportional change in labour input With α and (1 – α) < 1, this change is less than proportional to the change in the primary input Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Secondary Factors’ Contribution Yt = At . F(Kt, Lt) With constant inputs of labour and capital, dY/Y = dA/A With constant primary inputs, output changes in the same proportion as secondary inputs Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Total Factor Productivity (TFP) TFP measures the output which is obtained from given inputs of capital and labour It measures the productivity of the secondary factors whose input is measured by A Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Accounting for Growth Growth is measured by the rate of change of GDP over time = [dY/Y]/dt [dY/Y]/dt = [dA/A]/dt + α[dK/K]/dt + (1 – α)[dL/L]/dt So there are 3 causes of growth: Growth in secondary inputs (A) growth in capital stock (K) growth in labour force (L) What are their respective contributions? Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Contributions to Growth Capital’s contribution is [dY/Y]/[dK/K] = α = .25 Labour’s contribution is [dY/Y]/[dL/L] = (1 - α) = .75 Secondary factors’ contribution is the residual, dY/Y – .25[dK/K] – .75[dL/L] So if output rises by 10% when capital and labour inputs both rise by 1%, the rise in TFP is 9% Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank

Contributions to Growth: Aust.

Contributions to Growth: Japan, UK, US

Contributions to Growth: Asia

Significance? Does an economy get a bigger growth pay-off from growth in the capital stock and the capital–labour ratio by promoting saving and investment, or from growth in TFP through promoting better management, research and innovation? Copyright  2005 McGraw-Hill Australia Pty Ltd PowerPoint® Slides t/a Principles of Macroeconomics by Bernanke, Olekalns and Frank