The Solow Growth Model (Part Two)

1
The Solow Growth Model (Part Two)

• The golden rule level of capital, maximizing
  consumption per worker.

2
Model Background

• As mentioned in part I, the Solow growth model
  allows us a dynamic view of how savings affects
  the economy over time. We also learned about the
  steady state level of capital.
• Now, we assume policy makers can set the savings
  rate to determine a steady state level of capital
  that maximizes consumption per worker. This is
  known as the golden rule level of capital (kgold)

3
Building the Model

• We begin by finding the steady state consumption
  per worker.From the national income accounts
  identity, y c iwe get c y i
• We want steady state c so we substitute steady
  state values for both output (f(k)) and
  investment which equals depreciation in steady
  state (dk) giving us cf(k) dk

f(k),dk

• Because, consumption per worker is the difference
  between output and investment per worker we want
  to choose k so that this distance is maximized.
• This is the golden rule level of capital kgold

cgold
k
kgold
Above kgold, increasing k reduces c
Below kgold, increasing k increases c

• A condition that characterizes the golden rule
  level of capital is MPK d

4
Building the Model

• While the economy moves toward a steady state it
  is not necessarily the golden rule steady state.
• Any increase or decrease in savings would shift
  the sf(k) curve and would result in a steady
  state with a lower level of consumption.

f(k),dk
dk
f(k)
sgoldf(k)
sgoldf(k)
k
kgold
To reach the golden rule steady state
The economy needs the right savings rate.
5
A Numerical Example

• Starting with the Cobb-Douglas production
  function from part I, (1) yk1/2 recall that
  the following condition holds in steady
  state, (2) s/d k/f(k)
• assume depreciation is 10 and the policy maker
  chooses the savings rate and thus the economys
  steady state. Equation (2) becomes, s/.1
  k/vkSquaring both sides yields, k 100s2
• With this we can compute steady state capital for
  any savings rate.

6
A Numerical Example

• Using the functions from the previous slide and
  solving for a range of savings rates
• We can see that at s.5 we get c2.5 so at
  savings rate of .5 consumption per worker is
  maximized. Also note that at that level MPKd0
  and k25.

s k y dk c MPK MPK-d
0 0 0 0 0 8 8
.1 1 1 .1 .9 .5 .4
.2 4 2 .4 1.6 .25 .15
.3 9 3 .9 2.1 .167 .067
.4 16 4 1.6 2.4 .125 .025
.5 25 5 2.5 2.5 .1 0
.6 36 6 3.6 2.4 .083 .017
.7 49 7 4.9 2.1 .071 .029
.8 64 8 6.4 1.6 .062 .038
.9 81 9 8.1 .9 .056 .044
1.0 100 10 10 0 .05 .05
7
A Numerical Example

• Another way to identify the golden rule steady
  state is to choose the level of capital stock
  where MPK d 0
• In this example MPK 1/(2vk) .1 0so 1
  .1(2vk) and 5 vkand 25 k

8
A Numerical Example

• But what is the time path toward k? To get this
  use the following algorithm for each period.
• k 4, and y k1/2 so, y 2.
• c (1 s)y, and s .5 so c .5y 1.0
• i sy, so i 1.0
• dk .14 .4
• ?k sy dk so ?k 1.0 .4 .6
• so k 4.6 4.6 for the next period.

9
A Numerical Example

• Repeating the process gives

period k y c i dk ?k
1 4 2 1.0 1.0 .4 .6
2 4.6 2.144... 1.072... .536 .46 .612
. . . . . . .
10 10.12... 3.087... 1.543... 1.543... .953 .590
. . . . . . .
8 25 5 2.5 2.5 2.5 0.0
And we converge to k25
10
The Transition to the Golden Rule Steady State

• Suppose an economy starts with more capital than
  in the golden rule steady state.

• This causes an immediate increase in consumption
  and an equal decrease in investment.

Output, y

• Over time, as the capital stock falls, output,
  consumption, and investment fall.

Consumption, c
Investment, i

• The new steady state has a higher level of
  consumption than the initial steady state.

t0
Time
At t0, the savings rate is reduced.
11
The Transition to the Golden Rule Steady State

• Suppose an economy starts with less capital than
  in the golden rule steady state.

• This causes an immediate decrease in consumption
  and an equal increase in investment.

Output, y
Consumption, c

• Over time, as the capital stock grows, output,
  consumption, and investment increase.

Investment, i

• The new steady state has a higher level of
  consumption than the initial steady state.

t0
Time
At t0, the savings rate is increased.
12
Conclusion

• In this section we used our knowledge that
  savings affects the steady state and chose the
  savings rate to maximize consumption per worker.
  This is known as the golden rule level of capital
  (kgold)
• In the next section we augment this model to
  include changes in other exogenous variables
  population and technological growth.