Transcript Chapter 3

Chapter 3
PHYSICAL
CAPITAL
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The five features of capital
1. Capital is productive
2. Capital is itself produced through the process of savings and investment
• Not a natural resource
• Produced privately or publicly: buildings or computers vs. Messina bridge
3. Capital is rival in its use
• Only limited number of people can use one piece of capital at a time
• Are ideas a form of capital?
4. If capital privately produced, rate of return (ror) motivates its production; if
publicly produced, other criteria matter. But “social” ror important as well
5. Capital wears out
• Capital gradually depreciates. Depreciation for buildings is slower than for
computers
• Depreciation may be physical, economic, fiscal – different types of depreciation
need not coincide
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Capital’s role in production
• Given that capital is productive, it enables
workers to produce more output
• Study relation between K and output (GDP)
through production function
• Focus on capital accumulation is crucial in
the Solow model
Robert Solow
Nobel Prize, 1978
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The main properties of the
production function
Y=F(K,L)
• Constant returns to scale
– F(zK,zL) = z F(K,L)
– If z=1/L:
• (1/L)Y = (1/L) F(K,L) = F(K/L,L/L)= F(K/L,1)
• Y/L = F(K/L) or y=F(k,1) or, ignoring 1, y=f(k)
– So we convert output into output per worker and obtain a
function that relates output per worker with capital per
worker only (L disappears)
• Positive but diminishing returns on each factor K, L
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Figure 3.1 GDP per worker and Capital
per Worker, 2009
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The main properties of the
production function
Second property of the production function
• Positive but diminishing returns on each factor K, L
• Or Diminishing marginal product of capital
– Marginal product of capital = extra output from using one
additional unit of K, holding L fixed
– MPK = f(k+1) – f(k)
– Aka (also known as) the partial derivative of y with respect to
k
– ∂F/∂K = f’(k)
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Figure 3.2 A Production Function with
Diminishing Marginal Product of Capital
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Decreasing Marginal Product of Capital
Example: the Cobb Douglas
production function
• Y=AKαL1-α
– A, a measure of productivity (Why?)
– α says how exactly K and L combine to produce output
– Empirically, α not too far from 1/3 (see box in next slide)
• In per worker terms: y = Akα
– where y = Y/L and k = K/L
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Capital’s Share of Income in a CrossSection of Countries: average not too far
from 1/3
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The Solow model
Constant population (hence L is constant)
Capital accumulation
• ΔK = I – D
– I investment, D depreciation
In per worker terms: Δk = i – d
• Investment is constant fraction of output
– i = γy = γf(k)
• Depreciation constant fraction of capital
– D = δk
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Combining the three equations here is
the equation of capital accumulation
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Figure 3.4 The Steady State of the
Solow Model
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Figure 3.6 Effect of Increasing the
Investment Rate on the Steady State
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The Solow model in a Cobb-Douglas
world – capital accumulation first
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The steady state level of capital per
worker
If Δk = 0 (investment net of depreciation = 0)
• kss = (γA/δ)1/(1-α)
The steady state capital stock is higher
• The higher the propensity to save and invest
• The lower the depreciation rate
• The higher the efficiency of the economy
The same holds for the level of output (Gdp) per worker
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The Solow model in a Cobb-Douglas
world – output per worker in the steady
state equilibrium second
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The Solow model as a theory of
income differences
Using the Solow model, we have obtained the steady state
equilibrium levels of Gdp per worker
Hence we have a theory to explain income differences across
countries
• This was our purpose to start with!
Question: is the Solow model a good theory of income differences?
Figure 3.7 contrasts the predicted values of cross-country income
gaps with respect to the US with the actual values of these gaps
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Predicted Versus Actual GDP per
Worker
Question: what would be the picture like if the Solow model
were an exact predictor of the data we see in the real
world?
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The Solow model as a theory of
relative growth rates
Can the Solow model explain cross-country differences in growth
rates as well?
First of all, so far we have a model of income differences in steay
states where by definition growth is zero. So as such the model
cannot say anything about growth
BUT The model may still have something to say as to relative
growth rates, hence why some countries grow faster than others
This is the case when we think of countries AWAY from their
steady states.
In steady state growth is zero, but away from the steady state some
transitional growth will occur (exactly to take that country to the
steady state!). Hence the Solow model predicts that countries
converge - by growing fast or slow - to their steady state
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Transitional growth in the Solow
model (Cobb-Douglas version)
Dividing by k we obtain the growth rate of k:
K hat =Δk/k = γAkα-1 – δ
This equation can be nicely graphed as follows:
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Growth in the Solow model
This picture is very useful to learn immediately about the
growth rate of k (k with the hat in the picture) and the
speed of convergence to the steady state (diminishing as
the steady state equilibrium is approached)
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Three predictions on transitional
growth rates
1. If two countries have the same rate of investment (as well as
depreciation and productivity) but different levels of income, the
country with lower income will experience faster growth
– Same investment rate means same steady state level of income.
So the initially rich country will grow less fast than the initially
poor country
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Three predictions on transitional
growth rates
2. If two countries have the same level of income (as well as
depreciation and productivity) but different rates of investment,
the country with a higher rate of investment will experience faster
growth
– Higher investment means higher steady state level of output,
and thus faster growth along the way
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Three predictions on transitional
growth rates
3. A country that raises its investment rate will experience faster
growth
– Higher investment raises the steady state level of income. So
even if a country starts from a steady state equilibrium, it will
start growing to reach the new, higher, steady state equilibrium
implied by the higher investment rate
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