Chapters 7 and 8 – In Class Group Problems
Simple Solow Growth Problem
1. Assume that production function is of Cobb-Douglas form
Y  F ( K , L)  20 K  L1
and the share of income that is attributed to capital is 30%. In this economy capital has an average
life of 30 years and the population is NOT growing.
a. Find the steady-state value of capital per worker if the rate of saving is currently at 10%.
 = 0.30 (capital’s share) and  = 1/30 = 3.3%
Y
K L
K
 F ( , )  20( )  y  20k 0.30
L
L L
L
steady-state k = 0 and i = sf(k)
k
1 0.70 10
k*
s

k

3
 
0.30
*
20
3.3
20k
f (k ) 
solving for k:
k (
200 10 7
)  (60.6)1.43  353
3.3
b. Given this value of steady-state capital, calculate output per worker (y), consumption per
worker (c), and investment per worker (i).
Plugging k* into the output per worker:
y  20k 0.30  20 * 3530.30  20 * 5.81  116.24
Plugging y into the consumption per worker function
c =(1-s)y  (1  .10) y  0.90 * 116.24  104.62
Plugging y and c into the consumption function, solved for i:
i = y - c = 11.62
c. What happens to the steady-state value of capital per worker if the rate of savings changes
from 10% to 5%?
steady-state k = 0 and i = sf(k)
k
1 0.70
5
k*
s

k

 1.52
 
0.30
*
20
3.3
20k
f (k ) 
solving for k:
100 10 7
k (
)  (30.3)1.43  131.36
3.3
d. Given this value of steady-state capital, calculate output per worker (y), consumption per
worker (c), and investment per worker (i).
Plugging k* into the output per worker:
y  20k 0.30  20 *131.36 0.30  20 * 4.32  86.4
Plugging y into the consumption per worker function
c =(1-s)y  (1  .05) y  0.95 * 86.4  82.08
Plugging y and c into the consumption function, solved for i:
i = y - c = 4.32
e.
Graph the original equilibrium (rate of savings 10%) and the new equilibrium (rate of savings
5%) and describe the process that moves the economy from the original steady-state to the
new steady-state.
 k = 3.3*k
ik
11.62
0.10*f(k)
4.32
0.05f(k)
131.36 353
k
Population Growth in the Solow Growth Model
2. Now suppose the rate of population growth is 1%.
a.
Find the steady-state value of capital per worker if the rate of saving is currently at 10%.
Y
K L
K
 F ( , )  20( )  y  20k 0.30
L
L L
L
steady-state k = 0 and i = k(+n)
k*
s
k
1 0.70
10



k

 2.3
*
0.30
20
3.3  1
f (k )   n
20k
solving for k:
k (
200 10 7
)  (46.51)1.43  242.43
4.3
b.
Given this value of steady-state capital, calculate output per worker (y), consumption per
worker (c), and investment per worker (i).
Plugging k* into the output per worker:
y  20k 0.30  20 * 242.430.30  20 * 5.19  103.8
Plugging y into the consumption per worker function
c =(1-s)y  (1  .10) y  0.90 *103.8  93.42
Plugging y and c into the consumption function, solved for i:
i = y - c = 10.38
c.
Graph the original equilibrium (population growth 0%) and the new equilibrium
(population growth 1%) and describe the process that moves the economy from the
original steady-state to the new steady-state.
k = 4.3*k
i,  k
11.62
10.38
 k = 3.3*k
0.10*f(k)
242.43 353
k