Solow

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Solows Model

• (Modeling economic growth)

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Solow model I Constant productivity

• Assumptions of the model
• Population grows at rate n
• L (1 n)L
• Population equals labor force
• No productivity growth
• Capital depreciates at rate ?

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1. Per-capital Income

• Production function Y F(K, L)
• In per worker terms y f(k)
• Relationship between variables

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From the above we can get

• Per-person or per-capita income level (y)
  depends on each workers capital equipment(k).
• yf(k) shows DMR.
• Can you draw the graph with y and k?

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• Growth rate is measured by the slope of the
  tangent line of the y or f(k) curve.
• Growth rate decreases as the per-capita capital
  stock rises. It is true for all countries-
  Convergence
• Countries that start further away from the steady
  state grow faster

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2. Actual Supply of Capital

• Assume FIXED SAVINGS RATE or APS s S/N/Y/N
  savings /income
• Given an income of y
• Actual savings s y s f(k)

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• EXAMPLE
• Savings rate of 40
• s .4 (you save a fraction of your income)
• Can you draw the actual savings curve in the
  previous graph you have drawn?

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Minimum Capital Requirement to just keep up for
each work is proportional to population growth
rate(n) and capital depreciation rate(d)
3. Required Capital for Just Keep-Up
if you do not replenish the economy with the
minimum requirement of capital, then the level of
capital and thus the level of production or
income fall.
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• Example)
• Y 100 L 20 K 10
• y Y/L 5
• k K/L 10/20 0.5
• n 3 d 5
• Then you need 8 of capital every year to keep
  constant each workers capital equipment.

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4. Equilibrium or Not

• The Change in capital per worker is the actual
  supply of capital over the minimum required
  capital

We may call this net investment.
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• Thus
• If ?k gt 0 economy accumulates capital per worker
• If ?k lt 0 economy reduces capital per worker
• If ?k 0 constant capital per worker steady
  state

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Graphically
f(k)
(?n)k
?k lt 0
s f(k)
?k gt 0
k
k
k0
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• Steady-state Per-capita Income or y Y/N is
  determined where s f(k) (?n)k.

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Implications of the model

• The economy converges, over time, to its steady
  state.
• If the economy starts BELOW the steady state, it
  accumulates capital until it reaches the steady
  state.
• If the economy starts ABOVE the steady state, it
  reduces capital until it reaches the steady state.

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• Growth rates
• Capital per worker grows at rate 0
• Output per worker grows at rate 0
• Total capital K k L grows at rate n
• Total output Y y L grows at rate n

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Comparative statistics

• Parameters of the model s, n, ?
• Predictions of the model
• In steady state
• Higher savings rate implies higher income per
  worker
• Higher population growth implies lower income per
  worker

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Savings rate and growth
(?n)k
s2f(k)
s1f(k)
k
kss
kss2
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• Note that an increase in savings rate do increase
  the level of income, but not the rate of growth
  of income.

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Population growth rate and growth
(?n2)k
(?n1)k
sf(k)
k
kss2
kss
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Technical Innovations

• How is this different for the y curve from an
  increase in savings rate?