The Solow Growth Model (Part One)

1
The Solow Growth Model (Part One)

• The steady state level of capital and how savings
  affects output and economic growth.

2
Model Background

• Previous models such as the closed economy and
  small open economy models provide a static view
  of the economy at a given point in time. The
  Solow growth model allows us a dynamic view of
  how savings affects the economy over time.

3
Building the Model goods market supply

• We begin with a production function and assume
  constant returns. YF(K,L) so zYF(zK,zL)
• By setting z1/L we create a per worker
  function. Y/LF(K/L,1)
• So, output per worker is a function of capital
  per worker. We write this as, yf(k)

4
Building the Model goods market supply

• The slope of this function is the marginal
  product of capital per worker.MPK f(k1)f(k)

Change in y

• It tells us the change in output per worker that
  results when we increase the capital per worker
  by one.

Change in k
5
Building the Modelgoods market demand

• We begin with per worker consumption and
  investment. (Government purchases and net exports
  are not included in the Solow model). This gives
  us the following per worker national income
  accounting identity. y cI
• Given a savings rate (s) and a consumption rate
  (1s) we can generate a consumption function. c
  (1s)y which makes our identity, y (1s)y
  I rearranging, i sy so
  investment per worker equals savings per
  worker.

6
Steady State Equilibrium

• The Solow model long run equilibrium occurs at
  the point where both (y) and (k) are constant.
  These are the endogenous variables in the model.
• The exogenous variable is (s).

7
Steady State Equilibrium

• By substituting f(k) for (y), the investment per
  worker function (i sy) becomes a function of
  capital per worker (i sf(k)).
• To augment the model we define a depreciation
  rate (d).
• To see the impact of investment and depreciation
  on capital we develop the following (change in
  capital) formula,?k i dk substituting
  for (i) gives us,?k sf(k) dk

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Steady State Equilibrium

• If our initial allocation of (k) were too high,
  (k) would decrease because depreciation exceeds
  investment.

• If our initial allocation were too low, k would
  increase because investment exceeds depreciation.

• At the point where both (k) and (y) are constant
  it must be the case that, ?k sf(k) dk 0
  or, sf(k) dkthis occurs at our equilibrium
  point k.

• At k depreciation equals investment.

9
Steady State Equilibrium (getting there)

• Suppose our initial allocation of (k1) were too
  low.

k2k1?k
k3k2?k
k4k3?k
k5k4?k

This process continues until we converge to k
k1
k2
k3
k4
k5
K2 is still too low so
K3 is still too low so
K4 is still too low so
K5 is still too low so
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A Numerical Example

• Starting with the Cobb-Douglas production
  function we can arrive at our per worker
  production as follows, YK1/2L1/2 dividing by
  L, Y/L(K/L)1/2 or, yk1/2
• recall that (k) changes until, ?ksf(k)dk0
  ...i.e. until, sf(k)dk

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A Numerical Example

• Given s, d, and initial k, we can compute time
  paths for our variables as we approach the steady
  state.
• Lets assume s.4, d.09, and k4.
• To solve for equilibrium set sf(k)dk. This
  gives us .4k1/2.09k. Simplifying gives us
  k19.7531, so k19.7531.

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A Numerical Example

• But what it the time path toward k? To get this
  use the following algorithm for each period.
• k4, and yk1/2 , so y2.
• c(1s)y, and s.4, so c.6y1.2
• isy, so i.8
• dk .094.36
• ?ksydk so ?k.8.36.44
• so k4.444.44 for the next period.

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A Numerical Example

• Repeating the process gives

period k y c i dk ?k
1 4 2 1.2 .8 .36 .44
2 4.44 2.107... 1.264... .842 .399 .443
. . . . . . .
10 8.343... 2.888... 1.689... 1.126... .713 .412
. . . . . . .
8 19.75... 4.44 2.667... 1.777... 1.777... 0.000...
14
A Numerical Example

• Graphing our results in Mathematica gives us,

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Changing the exogenous variable - savings

• We know that steady state is at the point where
  sf(k)dk

sf(k)dk
sf(k)

• What happens if we increase savings?
• This would increase the slope of our investment
  function and cause the function to shift up.

k

• This would lead to a higher steady state level of
  capital.

• Similarly a lower savings rate leads to a lower
  steady state level of capital.

16
Conclusion

• The Solow Growth model is a dynamic model that
  allows us to see how our endogenous variables
  capital per worker and output per worker are
  affected by the exogenous variable savings. We
  also see how parameters such as depreciation
  enter the model, and finally the effects that
  initial capital allocations have on the time
  paths toward equilibrium.
• In the next section we augment this model to
  include changes in other exogenous variables
  population and technological growth.