1. 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

2. 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

3. 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

4. The Production Function

5. 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

6. The Demand for Stock of Capital

7. 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

8. 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

9. 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

10. The Labour Demand Curve

11. Labour Market Equilibrium

12. 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

13. A Production Table

14. 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

15. 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 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

16. One Variable Function

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. Contributions to Growth: Aust.

30. Contributions to Growth: Japan, UK, US

31. Contributions to Growth: Asia

32. 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