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Heather Krull
Econ 190
Homework 2
Due: Monday, March 27, 2006
Submit your answers on a separate sheet of paper. PLEASE staple your homework. I am not responsible
for any loose sheets of paper that may get lost as a result of not being attached to your work.
1. Suppose the government grants $2500 per child to households that have two or more children. Do
these child allowances influence the fertility behavior of households that had no children prior to
the government program? Graph (in the (N,X) space – see notes on fertility) and explain. You
may assume for simplicity that initially V = 0.
Assume the original budget line of a family that wishes to have no children is given by ABC. The budget
line will become ABDE when the child subsidy program is enacted. If a family is willing to have two
children, the government provides them with $5000, if they have three children, they will receive $7500, etc.
This system creates a new portion of the budget line (DE) which is flatter than the original since the subsidy
increases when the number of children increase.
The result, there are two changes to consider. The flattening
of the budget line is similar to a decrease in the cost of having
children. Theoretically, when the price of children decreases,
families have an incentive to substitute children for the
commodity good as relative prices have changed.
X
A
D
5000
7500
B
Additionally, the outward shift of the budget constraint serves
as an increase in the opportunity set. The income effect
suggests that with a high level of income, families will demand
C
E
higher quantities of all normal goods, children included.
N1
2 N2 3
N
Therefore, both the income and substitution effects suggest
that some families will demand more children. As drawn, the family will now have N 2 children and take
advantage of the child subsidy program.
2. A firm’s technology is such that is must combine 5 person-hours of labor with 3 machine-hours to
produce one unit of output. The firm has 15 machines in place and the wage rate rises from $10 per
hour to $20 per hour. What is the short-run elasticity of this firm’s labor demand?
If the firm must combine 3 machine-hours with 5 labor-hours, regardless of the level of output and quantity
of any one input, labor and capital are perfect complements in the production process. Thus, regardless of
how the price of one input changes, the firm will always continue to hire at a rate of 3 machines to 5
15
workers. If the quantity of capital hired is constant at 15, the firm will always hire
* 5 = 25 workers.
3
ΔE
% ΔE E 1
ΔE w 1
0
10
=
=
*
=
*
= 0. As
Therefore, the elasticity of labor demand is: elasticity =
Δw
% Δw
E 1 Δw E 1 10 - 20
w1
is the case with all perfect complements, this firm is completely unresponsive to wage changes (or changes
in the price of capital) in terms of the quantity of labor (capital) demanded in the short-run. The short-run
demand curve for labor is vertical and characterized as perfectly inelastic.
3. Consider initially a perfectly competitive firm initially in the short run with worker productivity
levels as provided. You may assume the firm sells its output at a price of $4. Complete the table
and use it to answer the questions that follow.
Labor (E)
Output
Marginal Product
of Labor
q
MPE 
E
-
Value of Marginal
Product of Labor
VMPE = MPE ∙ P
0
0
1
11
11  0
 11
10
11 ∙ 4 = 44
2
11 + 14 = 25
14
14 ∙ 4 = 56
3
25 + 16 = 41
16
16 ∙ 4 = 64
4
54
5
64
6
64 + 8 = 72
8
8 ∙ 4 = 32
7
11 ∙ 7 = 77
77  72
5
76
5 ∙ 4 = 20
54  41
 13
43
64  54
 10
54
-
13 ∙ 4 = 52
40
Average Product
of Labor
q
APE 
E
-
11
 11
1
25
 12.5
2
41
 13.6
3
54
 13.5
4
64
 12.8
5
72
 12
6
11
Value of Average
Product of Labor
VAPE = APE ∙ P
11 ∙ 4 = 44
12.5 ∙ 4 = 50
13.6 ∙ 4 = 54.6
13.5 ∙ 4 = 54
12.8 ∙ 4 = 51.2
48
11 ∙ 4 = 44
a. This firm will never hire fewer than 4 workers.
Firms hire up to the point were the cost of a worker (w) is equal to the revenue he generates
(VMP). E* will be maximize profits as long as marginal product is no longer increasing and the
value of marginal product is less than the value of average product. Marginal product is no longer
increasing after the third worker is hired, but it only four or more workers satisfy the condition that
the value of average product must exceed the value of marginal product. Therefore, the firm will
hire a minimum of four workers.
b. State and explain the profit-maximizing input rule for this firm in the short run.
w = VMPE: Firms hire workers as long as the additional revenue generated by the worker (VMP)
exceeds the cost of hiring him (w). In the event that there does not exist a worker for whom this
equality exists, the firm will hire workers as long as the value of marginal product exceeds the wage
rate.
c. Use the profit-maximizing input rule to determine how many workers the firm will hire at
the following daily wages. Additionally, construct this firm’s demand curve for labor for
wages of $50, $35, and $30.
w = $50  E* = 4: The fourth worker generates $52 in revenue, and it only costs the firm $50 to
hire him. Hiring a fifth worker would result in a loss, as she creates $40 worth of revenue, but it
would cost $50 to hire this worker. Therefore, the firm will hire 4 workers.
w = $35  E* = 5: The fifth worker generates $40 in revenue, and it only costs the firm $35 to
hire him. Hiring a sixth worker would result in a loss, as she creates $32 worth of revenue, but it
would cost $35 to hire this worker. Therefore, the firm will hire 5 workers.
w = $30  E* = 6: The same logic from above applies to this situation. The sixth worker
generates $32 in revenue, but it only costs $30 to hire her. However, the seventh worker’s wage,
$30, exceeds the $20 in revenue he can produce. The firm will hire six workers.
The short-run labor demand curve is the downwardsloping portion of the value of marginal product curve to
the right of the intersection of the value of marginal
product and value of average product curves (which is
where the value of average product is above the value of
marginal product). Alternatively, in this exercise, the
demand curve can be constructed using the quantity of
labor demanded at two different wages. Therefore, (5
workers, $35 wage) and (6 workers, $30 wage) define this
firm’s demand curve.
w
40
30
20
D = VMP
10
1
2 3
4 5 6
7
E
d. Calculate the elasticity of labor demand when the wage decreases from $35 to $30. Then
interpret this number and characterize the demand for labor as elastic or inelastic.
The
elasticity
of
labor
demand
is
calculated
using
the
formula
%E E w 1 6  5
35





 1.4. Its interpretation is: A 1% increase in the wage
%w E 1 w
5 30  35
rate will cause this firm to demand 1.4% fewer workers. Finally, because |δ| > 1, this is
considered elastic demand for labor.
4. Suppose a firm’s production technology is described by: q = 40K0.25E0.75. The corresponding
marginal product functions are MPK = 10K-0.75E0.75 and MPE = 30K0.25E-0.25. Suppose initially that
the firm faces input prices of r = $15 and w = $9.
a. State the profit-maximizing rule for a firm operating in the long run and use it to determine
the optimal capital-labor ratio. If the firm has hired 15 machines and 45 workers, is it
maximizing profits? If not, how should it adjust its input quantities?
In the long run, firms hire labor such that
1
1
3
3
30K 4 E
MPE w
 , which can be simplified in this problem as
MPK
r
9
3K 9
K 1

   . This firm profit maximizes by hiring 5 workers for each
E 15
E 5
10K 4 E 4 15
machine employed in the production process.
4

If the firm is currently utilizing 15 machines and 45 workers in the production process, it is not
K 1
K 15 1
profit-maximizing, as this is not a
 ratio (rather, it is a   relationship). 15 machines
E 45 3
E 5
require the use of 75 workers; alternatively, 45 workers would require only 9 machines in order to
profit maximize. Therefore, the firm must either hire more workers or fewer machines in order to
optimize.
b. If the firm faces input prices of r = $15 and w = $9, how many workers will the firm demand
if it has already decided to employ 10 units of capital? If the price of labor increases to $15
and the firm increases its capital stock to 12 units, does there exist a dominant scale or
substitution effect? How many workers will the firm choose to hire?
w = $9, K = 10  E* = 50
The information derived in part (a) can be used to solve this optimization problem. If the firm
requires 5 workers for every machine it employs, 10 machines necessitate the use of 50 workers in
 K 1 10 1

the production process      E  50 .
E 5
E 5

w = $15, K = 12  E* = 36
Similarly, this problem can be solved using the capital-labor ratio from part (a). Because
K 1
 utilizes input prices of w = $9 and r = $15, we must use a previous representation of this
E 5
ratio to incorporate the new wage.
Therefore, the new capital labor-ratio must be
3K w
3K 15
K 15
K 1
 
     , which suggests that the firm should hire 36 workers to
E
r
E 15
E 45
E 3
K
1
12
1


operate the 12 machines      E  36 .
E
3
E
3


Dominant Substitution Effect: The substitution effect dominates the scale effect in this
problem. When the wage rate increases, the firm will unambiguously decrease its use of labor.
However, the increase in the quantity of capital purchased suggests that the firm is substituting
capital for labor.
c. Finally, describe the scale and substitution effects associated with the wage increase from
$9 to $15 and decompose them graphically in the space provided (note: your graph should
correspond to your answers in part (b)).
The scale and substitution effects both suggest that the quantity of labor employed will decrease
when the firm experiences an increase in the wage rate.
The substitution effect further implies that the firm will increase its use of the now less expensive
input when the input price ratio changes (and it has, since w increased but r remained constant).
Therefore, the firm will substitute away from labor by utilizing more capital.
The scale effect predicts that the firm will use less of both inputs when the price of one input
increases. More specifically, an increase in the price of labor increases the overall marginal cost of
production. If the firm was initially profit-maximizing, marginal cost was equal to marginal revenue
(the profit-maximizing condition). Now, however, marginal cost exceeds marginal revenue, and the
firm has an incentive to reduce its production in order to again equate MR and MC. When
production decreases, the firm has a need for smaller quantities of all inputs. Therefore, it will
reduce its demand for both capital and labor.
In order to isolate the scale and substitution effects, the cost and production implications of a wage
increase must be considered.
Initially, assume the firm was producing q1 units of output
at a cost of C=600 (15*10 + 9*50). When the wage rate
increases, the firm is not only able to purchase less labor,
but it may also choose to incur less in total costs because of
the decrease in its production level (from q1 to q2). When
the wage rate increases from $9 to $15, total costs increase
to $720 (15*12 + 15*36), which produces a steeper isocost
which is closer to the origin over some regions of E, but
further from the origin over others (the two lines intersect
at E = 20, K = 28).
K
C = 720
28
12
10
R
P
Q
20 36
C=600
50
E
The scale effect is isolated by holding constant the ratio of
relative input prices. Therefore, a hypothetical budget line that is parallel to the original isocost and
tangent to the new isoquant will separate the two effects. P  Q is the scale effect (E and K both
decrease), and Q  R is the substitution effect (E decreases, K increases). Since K ultimately
increases, the final result must be a dominant substitution effect.