Transcript Chapter 12

Chapter 12
The Production Function Approach
to Understanding Growth
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12–1
Chapter 12: The Production Function
Approach to Understanding Growth
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Economists and economic growth
The production function
The Cobb–Douglas production function
Growth accounting
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12–2
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
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The Production Function
12–4
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
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12–5
The Demand for Stock of Capital
12–6
Demand for Stock of Capital
• Change real interest rate – move along demand
curve
• Change MRPK – shift demand curve
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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
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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
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12–9
The Labour Demand Curve
12–10
Labour Market Equilibrium
12–11
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
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12–12
A Production Table
12–13
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
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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
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12–15
One Variable Function
12–16
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)
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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)
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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
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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
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Properties: Labour Income
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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%
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Properties: Capital Income
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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%
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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
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Contribution to Additional GDP
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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
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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
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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
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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?
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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%
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Contributions to Growth: Aust.
12–29
Contributions to Growth: Japan, UK, US
12–30
Contributions to Growth: Asia
12–31
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?
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