Homework Assignment 1
Economics 215
Intermediate Macroeconomics
1. Production Function. Assume that an economy has a Cobb-Douglas
production function with a = 12 . For simplicity, we will assume that the
technology level is constant (i.e. T = 1).
1
1
Q  F ( K , L)  K 2 L2  K  L
a.
Demonstrate that this production function demonstrates constant
returns to capital. Calculate the production level when K = 1 and L =
1. Calculate the level of output after the inputs double, K = 2 and L=
2. Calculate the level of output when inputs triple, K = 3, and L = 3.
If F ( K , L)  K  L , then
F (K , L)  K  L      K L    K L
When K = 1 and L = 1, Q=1. When K = 2 and L = 2, Q=2 When K = 3 and
L = 3, Q=3. Double all your inputs, double your output
Q
b.
Show that the average productivity of labor, APL = , is
L
decreasing in the amount of labor used if we hold the level of capital
constant. Assume that capital is constant at K = 1. Calculate the APL
when L = 1, L =4, L = 9, L = 16, L = 25. Is this decreasing as L
increases?
L
Q
Q
L
1
1
1
4
2
.5
9
3
1/3
16
4
.25
25
5
.2
The larger is L, the larger is Q, but the smaller is Q/L.
c.
Calculate the marginal product of labor for this production function
in two ways. Continue to assume that K = 1 throughout.
i.
Calculate
Q F ( K  1, L  L)  F ( K  1, L)
L  L  L


L
L
L
when L = 1 and L = 3 (i.e. Calculate the extra
production per extra worker as we move from 1 worker to
Q
4 workers). Calculate
, when L = 4 and L = 5.
L
Calculate the marginal product of labor when we move
from 9 workers to 16 workers and when we move from 16
workers to 25 workers. Is this amount decreasing as L
increases?
ii.
Calculus tells us that the marginal change in output from
an infinitessimal change in labor gives us the formula:
L
L
1
4
9
16
25
3
5
7
9
Q 1 Q

when L = 1, 4, 9, 16, and 25. Is this amount
L 2 L
decreasing as L increases?
Q 1 Q
Q
L  L  L
2

L
L
L
L
.5
1/3
.25
.2
1/6
1/7
.125
1/9=.111
.1
2. In our economy, we find that the income and substitution effects of real wage
changes always cancel out. Thus, the labor supply curve showing the
W
relationship between the real wage rate
, and labor supply is perfectly
P
vertical at L* = 4. The production function of firms is given by the CobbDouglas function Q  K  L . Assume that the capital stock, K = 1.
a.
How much output will be produced when L = L* = 16. Q = 4
Q 1 Q 1 1
b.
Calculate the marginal product of labor,
when L

 
L 2 L 2 L
= L*. Graph the labor market. Show, using the graph, that the
equilibrium real wage, w*, equals the marginal product of labor
calculated at L*. What is w* in numbers?
LS
w
w*
MPL=LD
L
L* = 4
w* = .25
c.
The workers must sign a dollar wage contract that specifies a wage
rate W and allows firms to choose the number of workers they want
to hire at that wage rate. Assume that workers specify a wage rate
that they expect will net them a real wage rate of w*. If workers
expect a price level of PE = 4, what wage rate, W , will they choose
in order to achieve the real wage rate, w*in section 2.a.
Workers would choose a wage rate W =1, to set the expected real wage = .25.
d.
The labor contracts have been signed at the wage rate solved for in
section 2.c. The workers have overestimated the price level. It
actually turns out that the wage rate will by P = 2. What is the ex
post real wage? How much labor will firms want to hire at the ex
post real wage? Answer this second question, by solving for the ex
post real wage, then solving for the level of labor L, that sets the
marginal product of labor equal to the ex post real wage. Is there a
worker shortage or a worker surplus?
The expost real wage turns out to be .5, which is larger than the equilibrium real
wage. At a real wage rate of .5, the firm will hire a level of labor equal to L=.5. There
is a worker surplus, since at the given real wage, workers would choose a labor
supply of L = 4
3. Real Consumption [Computer Assignment]. You are an analyst in the
department of statistics. You gather data on the level of consumer expenditure
in 2001 and are assigned to calculate the real consumption level for 2001. The
base year used by the department is 1990. You also have comparable data for
1990. Calculate consumption expenditure, PCC for 1990 and 2001. There are
N = 10 types of goods. Calculate weights w1…wN for each type of good based
on their share of overall consumption expenditure in the base year. Use these
weights to calculate the weighted average price level of consumer goods in
2001 (relative to the base year), P2001. Calculate real consumption spending in
2001 in 1990 dollars, C. [Download an Excel file from the Web Page].
Table 1
Consumer Goods
Hamburgers
Electricity
Cigarettes
Televisions
Shirts
Bus Tickets
Automobiles
Shoes
Haircuts
Doctor Visits
Total
Prices
10
4
40
1600
400
8
12000
250
100
400
Consumption Spending 1990
Consumption Spending 2001
Weighted Average of Relative Prices
Real Consumption Spending 2001
1990
Mil. HK$
Spending
5500
12000
8000
4000
6000
3000
17500
2000
9000
7000
74000
74000
90000
1.299775
69242.77
Weights
0.074324
0.162162
0.108108
0.054054
0.081081
0.040541
0.236486
0.027027
0.121622
0.094595
Prices
8
5
50
2000
500
10
16000
400
150
600
2001
Mil. HK$
Spending
8500
14500
6000
8000
7000
4500
22500
2500
9000
7500
90000
69242.77
Relative
Prices
0.8
1.25
1.25
1.25
1.25
1.25
1.333333
1.6
1.5
1.5
1.299775