Heather Krull
Econ 190
Final Exam Solution
December 16, 2005
Instructions:
1. Write your name: ____________________________________
2. Write your answers in the space provided. If you need additional space, you may continue answers on extra paper
I have provided. Be sure to indicate that answers are continued and carefully identify any work that carries over.
3. After you have completed the exam, sign the Honor Code statement below.
HONOR CODE PLEDGE: I affirm that I have neither given nor received aid on this exam:
___________________________________________
Multiple Choice (2 points each)
1. A When wages increase, the income and substitution effects will lead to a(n) ambiguous change in the quantity
of leisure a worker consumes and a(n) increase in the amount of money the individual spends on consumption
goods.
2. D Theory suggests that the supply curve of labor is downward-sloping for some wage levels, and upwardsloping for others.
3. C If the wage increases from below Larry's reservation wage to above Larry's reservation wage, we know that
Larry will enter the labor force.
4. B The intertemporal substitution hypothesis suggests that labor force participation rates and hours worked
increase when wages are higher as people substitute time over the life cycle to take advantage of changes
in the price of leisure.
5. C According to the household production model, if a married couple decides to have one spouse stay at home, it
should be the spouse for whom the difference between the wage and the household marginal productivity
is lowest.
6. C In the context of the retirement model, an increase in pension benefits will unambiguously increase the years
a worker is retired by both the income and substitution effects.
7. A The short run differs from the long run in that at least one input is fixed in the short run.
8. C A firm is more responsive to wage changes in the long-run, which suggests the long-run labor demand curve is
flatter than the short-run labor demand curve.
9. A There exists a positive relationship between the demand for labor and the price charged for the firm’s output.
10. D Marshallian's rules of derived demand state that the demand for labor will be more elastic if labor is a large
share of the firm’s production costs.
11. B A payroll tax: is borne entirely by workers if the supply of labor is perfectly inelastic.
12. C Firms with monopoly power hire workers up to the point where MRMPE = w.
13. B Non-discriminating monopsonists hire fewer workers than a competitive firm would and pay all workers the
same wage.
Page 1 of 8
14. A According to the optimal stopping rule, an individual should continue to acquire additional schooling up to the
point where the marginal rate of return to schooling is equal to the discount rate.
15. A If ability and schooling are positively correlated then ability bias will tend to overstate the return to schooling.
Short Answer (60 total points). Show your work in the space provided.
1. (22 points) The utility function of a worker is represented by U(C,L)=12C3/4L1/4. The marginal utility of
leisure is then given by MUL=3C3/4L-3/4, and the marginal utility of consumption is given by MUC=9C1/4L1/4. Suppose there are 16 hours in a day during which this individual can work or leisure.
a.
(4 points) State and interpret the utility maximization rule that is used to determine L*. Then use
the rule to determine the optimal number of hours of leisure that should be consumed and the
number of hours that should be worked if w = $8 and consumption = $192.
Rule: The optimal level of consumption and leisure occurs at the tangency of the budget constraint and
indifference curve. At the tangency, the slopes of the two lines is equal, so
MUL
MUL
w
 MUC
MUC
w
Interpretation: The condition can be interpreted as follows: at the tangency, the additional utility
enjoyed from spending an additional dollar on leisure is exactly equal to the additional utility enjoyed from
spending an additional dollar on consumption. In other words, the individual is indifferent between
consuming more leisure or consumption.
Using the above rule, we can solve for the optimal amount of leisure, and from that, hours of work:
3
3
MUL
3C 4 L 4
C
 w  1 1  8 
 8  192  24 L  L*  8, and H *  T  L*  H *  16  8  H *  8
4
4
MUC
3L
9C L
b. (3 points) Calculate this worker’s reservation wage.
This problem requires that you solve the wage offer that makes the worker indifferent between working
and not working. Thus, it is the slope of his indifference curve at the endowment point, V. The slope of
his indifference curve is given by:
C ~
 w,
3L
where L = 16 at the endowment point, and consumption must be calculated from part (a). Specifically, if
the worker is able to consume $192 when working 8 hours (part a), he must have $192-(8)($8) = $128 in
non-labor income that he can consume even if not working. Therefore,
128 ~ ~
 w  w  $2.67.
3  16
c. (2 points) Suppose now that wages increase to $10 per hour, and consumption increases to $270.
Calculate the new optimal levels of leisure and hours of work.
Using the same equations as above,
MUL
C
w
 10  270  30L  L*  9, and H *  T  L*  H *  16  9  H *  7
MUC
3L
Page 2 of 8
d. (3 points) Use your answers from parts (a) and (c) to calculate this individual’s elasticity of labor
supply. Interpret and characterize as elastic or inelastic.
H
%H H 1 H w 1
The elasticity of labor supply is given by the formula:  
and is calculated here



%w w H 1 w
w1
H w 1 7  8
8
as  



 0.5 This suggests that when wages increase by 1%, hours of work
H 1 w
8 10  8
will decrease by 0.5%. Because |σ| < 1, this is considered inelastic labor supply, which means that this
worker is relatively unresponsive in terms of hours worked to wage changes.
e. (10 points) Does your answer in part (c) reflect a dominant income or substitution effect?
Explain both, and give your reasoning for which is dominant. Finally, graph the changes in this
problem. In other words, draw the original budget constraint and optimal consumption point,
(L1, C1), with an appropriate indifference curve. Then, draw in the new budget constraint that
results from the increase in wages. Draw the new optimal consumption bundle, (L2, C2), and
decompose the change into income and substitution effects.
A negative elasticity of labor supply indicates that the income effect dominates in this example. The
substitution effect states that a change in wages is a chance in the opportunity cost of not working, or of
leisure. When the wage rate increased from $8 to $10, there was an incentive to work more because the
cost of not doing so had increased. The income effect suggests that once the wage rate increased, each
hour worked added more to this individual’s wealth (income). An increase in wealth will result in fewer
hours worked if leisure is a normal good, since more leisure will be demanded. In this case, both effects
were present, but there was an overwhelming income effect, as fewer hours were worked when wages
increased.
C
2.
P  Q: Income Effect (H↓)
Q  R: Substitution Effect (H↑)
U1
288
U2
256
slope = -w
= -8
R
P
Q
slope = -10
128
16
Page 3 of 8
L
2. (9 points) Consider a perfectly competitive firm with production function q = 30K1/3E2/3. As such, the
marginal productivity equations are given by MPE = 20K1/3E-1/3 and MPK = 10K-2/3E2/3. Assume capital
rents at a price of $7.50 per unit.
a. (3 points) Suppose initially that output sells at a price of $2 and workers are paid a competitive
wage of $20 per hour. Use this information to state the firms’ short-run profit-maximization
condition, and use it to determine the optimal number of employee-hours to hire if K = 5.
Rule: w  VMP  P  MP
Interpretation: Firms should continue hiring workers as long as the additional revenue generated by that
worker (VMP = P·MP) exceeds the cost of hiring the worker (w). Optimally, hire E* such that w = VMP.
This rule can be used to determine E* in the short run. Specifically:
P  20K
1
3E
1
3
 w  2  20 
K
E
w  40K
 2K
 2K

1
1
1
3
3
 20E
E
1
1
3

3
3
3
3
3
1
1
  E 1 3 



3
8K  E  E  40
b. (2 points) If the firm is instead in the long-run, what is the profit-maximizing condition it should
use to determine E* and K*? Use the rule, then, to determine how many employees will be hired
if the competitive wage decreases to $10, and the quantity of capital used increases to 10.
Rule:
MPE w

MPK
r
Using this rule,
1
1
2
2
20K 3 E
10K
3
E
3
3

w
2K 10
20 10




 E *  15
r
E 7.50
E 7.50
c. (4 points) Using your answers to parts (a) and (b), what can you conclude about the
complementarity/substitutability of capital and labor in the production process? Additionally,
did the wage decrease cause a dominant substitution or scale effect? Explain.
The decrease in wages from $20 to $10 ultimately resulted in an increase in capital from 5 units to 10,
which means that labor and capital are complements in the production process (note that if the LR
condition were used to solve for the original quantity of labor, E* would not have been 40, as in #1 -2K
20
10 20

 
 E *  3.75 -- we would have seen an increase in labor from 3.75 (well, four
E 7.50
E 7.50
workers) to 15, which is what should happen as wages decrease). The increase in the quantity of capital
suggests that the scale effect is dominant. When wages decreased, the marginal cost of production
decreased, so total output produced will increase, which means more of both inputs will be used. If the
substitution effect were dominant, less capital would have been used as the firm substituted labor (now
less expensive) for capital.
Page 4 of 8
3. (7 points) Use your answers to questions 1 and 2 to determine the market equilibrium. Specifically, you
know from #1 how many hours that particular worker is willing to work each day. You know from #2
how many employee hours the firm is willing to hire. Use this information to determine the market
equilibrium (i.e. wage and employee hours) if there are 750 workers that are identical to the worker in #1,
and there are 350 firms just like the firm in #2. [Hint: I call the employee hours in #1 H, and the
employee hours in #2 are labeled E, but in words and concept, they represent the same quantity. If it
helps, explicitly write (H1,w1) and (H2,w2) from #1 and (E1,w1) and (E2,w2) from #2 to construct the
market supply of and demand for labor.] Draw the supply and demand curves, label the equilibrium E*
and w* (with actual numbers), and describe (in terms of characteristics like unemployment, the wage
each worker is paid, etc).
The only wage that showed up in the labor supply and labor demand questions was $10. When w = $10, E* = 15
for an individual firm, and H* = 7 for an individual worker. Thus, if there are 350 firms, the total quantity of
employee-hours demanded will be 15*350 = 5250. Similarly, if 750 workers want to work 7 hours, that will be
7*750 = 5250 total labor hours supplied. Since the income effect is dominant for the typical worker in this
problem, the equilibrium occurs on the downward-sloping portion of the labor supply curve. Also, problem 2
looks as though the quantity of labor demanded decreases when the wage decreases, but as the note in 2(c)
suggests, it really isn’t. Thus, equilibrium looks as follows:
w
dominant
income
effect
10
D
dominant
subsitution
effect
S
E
5250
Equilibrium in this competitive market is characterized by no unemployment. Exactly 5250 employee hours are
demanded by firms, and workers are willing to supply exactly 5250 hours of work, all at the market-clearing wage
of $10 per hour. Additionally, each worker is paid $10, which is not true in at least one of the market described in
the following question.
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4. (9 points) Describe and illustrate graphically the hiring decisions of the following non-competitive labor
markets. How does each outcome (E* and w*) compare to competitive labor markets you illustrated and
described in #3?
w
w
MC
w
S
wC
S
w
wC
wM
D
E
EC
(a) Discriminating
Monopsonist
a.
D
E
EM EC
(b) Non-Discriminating
Monopsonist
MRP VMP
E
EM EC
(c) Monopolist
Discriminating Monopsonist: Discriminating monopsonists are able to hire workers according to their
reservation wages. Therefore, they hire the first worker at the wage it requires to get him to enter the
labor market. The next worker hired demands a higher wage, and the discriminating monopsonist hires
him at that rate. This continues up to the point where the demand and supply curves of labor intersect.
The discriminating monopsonist hires the same number of workers as a competitive firm, but only the last
worker hired is paid the competitive wage. All workers hired previously are paid according to their (lower)
reservation wages.
b. Non-Discriminating Monopsonist: Non-discriminating monopsonists cannot identify workers’
reservation wages, so they pay all workers the same rate. However, as a result of this, the marginal cost of
hiring the last worker is not the same as the wage paid to that worker (see example in class notes). The
non-discriminating monopsonist hires where the marginal cost of hiring another worker (MC) is equal to
the revenue generated by hiring that worker (D = VMP), not where supply equal demand. Thus, this type
of firm hires fewer workers than a competitive firm. Additionally, since workers are willing to work at
wages depicted by the supply curve, the firm pays all workers according to the supply curve, not demand
(which would measure the worker’s worth to the firm). Therefore, a non-discriminating monopsonist
hires fewer workers, and pays them less, than a competitive firm.
c. Monopolist: Since demand and marginal revenue are not the same for a monopolist, the additional
revenue generated by the last worker hired (VMP for competitive firms and monopsonists) is MR∙MP =
MRP (Marginal Revenue Product). The monopolist has no control over wages, and like a competitive
firm, takes the market-clearing wage as given. Thus, a monopolist hires where MRP = w (similar to VMP
= w for competitive firms), but since MRP < VMP (since MR < P), this profit-maximizing condition
results in fewer workers hired, relative to competitive markets. Monopolists pay the same wage as
competitive firms, but they hire fewer workers, as denoted above.
Page 6 of 8
5.
(13 points) Consider the figure to the right which PV (in hundreds of thousands)
illustrates the present value of lifetime earnings (PV) firms
24
22
are willing to offer to workers who have attained different
e
20
levels of education. Those workers who have completed 16
18
years or more of education (4 or more years after high
16
school) will be offered a wage leading to a present value of
14
lifetime income of $2,000,000 (PV2), while those
12
d
10 b
PV1
completing less than 16 years of education will be offered a
8
c
wage leading to a present value of lifetime income equal to
6
$1,000,000 (PV1). Lines CA and CB show the total cost of
4
different levels of education for two types of workers.
2
0 a
Type A workers are low-productivity workers while type B
12 13 14 15 16 17 18
workers are high-productivity workers. Note: 12 years of
Years of School
school in this problem is the equivalent of “S=0” in class;
we’re assuming here that all workers obtain at least 12 years of schooling.
CA
PV2
CB
19
20
a. (3 points) What is the net benefit (i.e., the difference between the PV and the cost of education) a
type A person would derive from attaining 16 years of education? What would be the net benefit
for A from 12 years of education? What is the optimal level of education for a type A person?
Net benefit of 16 years: 800,000
Net benefit of 12 years: 1,000,000
2,000,000 – 1,200,000 = 800,000
1,000,000 – 0 = 1,000,000
Optimal level of education: 12 years
b. (3 points) What is the net benefit a type B person would derive from attaining 16 years of
education? What would be the net benefit for B from 12 years of education? What is the optimal
level of education for a type B person?
Net benefit of 16 years: 1,200,000
Net benefit of 12 years: 1,000,000
2,000,000 – 800,000 = 1,200,000
1,000,000 – 0 = 1,000,000
Optimal level of education: 16 years
c. (1 point) Is 16 years of education an effective way to distinguish low-productivity workers from
high-productivity workers?
Yes, because workers accurately sort themselves according to ability. Low-ability workers (B types) will
never attend school beyond high school, and high-ability workers (A types) find it optimal to bear the
costs of schooling to attend for 16 years.
d. (3 points) Suppose firms lowered the cutoff for the higher wage job to 14 years of education.
What level of education would type A workers attain? What level of education would type B
workers attain? Would the level of education be an effective signal of worker productivity?
From the graph, it looks like 14 years of education costs worker A roughly $600,000, and worker B
$400,000.
Worker A education: 2,000,000 – 600,000 = 1,400,000
Since 1,400,000 is more than the 1,000,000 earned from not attending school, these workers will go to
school for 14 years to signal themselves as high ability.
Page 7 of 8
Worker B education: 2,000,000 – 400,000 = 1,600,000
Clearly, if B workers were willing to spend money to attend 16 years of school, they will also attend 14,
which is clear here as 1,600,000 is better than the 1,000,000 they could earn by not going to school.
Effective signal? No, since both workers obtain 14 years of schooling, and thus the firm will not be able
to distinguish high-ability workers from low-ability workers.
e. (3 points) Derive the schooling range the firm can use to determine that an individual is of highability that will lead to the proper sorting of workers. [Hint: Use points a, c, and d to determine
the cost of one year of schooling for each type of worker.]
Cost per year for A-type workers (using points a and d):
1,200,000 - 0
 300,000
16 - 12
Cost per year for B-type workers (using points a and c):
800,000 - 0
 200,000
16 - 12
B-type workers will obtain the schooling threshold if:
w H  CH S  w L
2,000,000  200,000S  1,000,000
5S
A-type workers will obtain 12 years of schooling:
w H  CL S  w L
2,000,000  300,000S  1,000,000
10
S
3
Therefore, if
10
 S  5, workers will sort themselves according by the quantity of schooling obtained.
3
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