Practice Problem Answers
1. You are a contestant on the famous game show "Let's Make a Deal" and Monty Hall has
selected you as a final contestant. You have a choice: you can take $64 in cash or you can gamble. If you
decide to gamble, there are three possible outcomes: door number one, door number two, and door
number three. Behind one of the doors is a prize valued at $100, another door has a prize valued at $81,
and one door has a prize worth $1.
a. What is the expected value of the gamble?
EV=(1/3)*(100) + (1/3)*(81) + (1/3)*(1) = 60 2/3
b. If you are risk neutral, which option would you choose?
The sure $64.
c. Suppose your utility function is of the form u = M1/2; which option would you choose?
You would choose the sure thing since you are risk averse and the expected value of the
sure thing exceeds the expected value of the gamble.
d. Suppose your utility function is of the form u = M1/2; what is the smallest amount of money
that Monty could offer you (with certainty) so that you would just be indifferent between the sure thing
and the gamble?
The utility you receive equals (1/3)*(100)1/2 + (1/3)*(81)1/2 + (1/3)*(1)1/2 = 6.66. Since U=Ml/2,
M=U2=44.36.
2. Ms Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from
the trip is a function of how much she actually spends on it (Y), given by U(Y) = ln Y
a. If there is a 25 percent probability that Ms. Fogg will lose $1000 of her cash on the trip, what is the
trip’s expected utility?
E(U) = .75ln(10,000) + .25ln(9,000) = 9.1840
b. Suppose the Ms. Fogg can buy insurance against losing the $1000 (say by purchasing traveler’s
checks) at an “actuarially fair” premium of $250. Show that her expected utility is higher is she
purchases this insurance than if she faces the chance of losing the $1000 without insurance.
E(U) = ln(9,750) = 9.1850
Because E(U) with insurance (9.185) is greater than without insurance (9.184), insurance is preferable.
c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her $1000?
ln(10,000 – p) = 9.1840
10,000 – p = e9.1840 = 9,740
p = 260
d. If you did part B. correctly you found that Ms Fogg was willing to buy insurance against a 25 percent
change of losing $1000 of her cash on her around-the-world trip. Suppose that people who buy such
insurance tend to become more careless with their cash and that their probability of losing $1000 rises to
30%. What is the actuarially fair insurance premium in this situation? What economic concept does this
situation relate to? Will Ms. Fogg buy insurance now?
The actuarially fair premium is now $300. If she buys insurance, spending is 9700, utility =
ln(9700) = 9.1799. This falls short of utility without insurance (9.1840), so here it is better to
forego insurance. The fact that people become more careless with insurance indicates that there
is a moral hazard in this case.
3. A farmer’s tomato crop is wilting, and he must decide whether or not to water it. If he waters the
tomatoes, or if it rains, the crop will yield $1000 in profits, but if the tomatoes do not get any water, the
crop will yield only $500. Operation of the farmer’s irrigation system costs $100. The farmer is risk
neutral and he seeks to maximize expected profits from tomato sales.
a) If the farmer believes that there is a 50 percent chance of rain, will he water the tomatoes?
Yes, because his expected profits would be higher if he watered (900 vs. 750)
b) What is the maximum amount the farmer would pay to obtain information from an itinerant weather
forecaster who can predict rain with 100 percent accuracy?
If he knows, his expected profit would be 950 (.5*1000 + .5*900); If he doesn’t know his expected
profit would be 900. So he would be will to pay 50.
c) How would your answer to part (b) change if the farmer were risk averse?
He would be willing to pay more than $50 to remove the uncertainty if he were risk averse.
4. According to the Kahneman-Tversky (hedonic framing model), which of the following options would
most people prefer to the other?
a.) Option 1: A gain of $5,000 or Option 2: a gain of $6,000 and a loss of $1,000?
Option 1 (Combine small losses with large gains).
b.) Option 1: A loss of $5,000 or Option 2: a loss of $4,000 and a loss of $1,000?
Option 1 (Combine losses).
c.) Option 1: A 50% chance of winning $1,000 and a 50% chance of winning $0 or
Option 2: A sure gain of $500?
Option 2 (People are risk averse with respect to gains).
5. A firm has the following production function Q=2K1/3 L1/3
a. What is the marginal product of labor if K is fixed at 27?
For K=27, we have Q=6L1/3, so MP= Q / L =2L-2/3.
b. Show whether the firm experiencing increasing, decreasing, or constant returns to scale.
This firm is experiencing decreasing returns to scale because 2(tK)1/3 (tL)1/3 is less than tQ.
c. Calculate the total cost as a function of Q.
TC = (Q3/216)*w + FC
d. What is the marginal cost if w=$10 and L=50?
MC=w/MPL=$10/.1474=$67.86
e. What is the profit maximizing amount of K and L if r=$50, w=$100 and the firm cannot spend more
than $2,000?
K*=20 and L*=10
6. When a firm is experiencing increasing returns to scale, what is happening to the long run total cost
function and what is happening to the long run average cost function?
The long run total cost function is increasing at a decreasing rate, while the long run average cost function
in decreasing.
7. When the average product of labor equals the marginal product of labor, how will marginal cost
compare with average variable cost?
When MP=AP, marginal cost and average variable cost will be the same. To see this, note first that MC=
 VC/  Q=w  L/  Q=w/MP. Similarly, AVC=wL/Q=w/AP. So when AP=MP, it follows that
MC=AVC.
8. A firm has access to two production processes with the following marginal cost curves: MC1=0.8Q
and MC2=10+0.6Q.
If the firm needs to produce 40 units of output, how much should it produce with each process?
Q2 = 15.7
Q1 = 24.3
9. Professor Smith and Professor Jones are going to produce a new introductory textbook. As true
economists, they have laid out the production function for the book as
Q= S0. 5 J0. 5
Where q equals the number of pages in the finished book, S equals the number of working hours spent by
Smith, and J equals the number of hours spent working by Jones.
Smith values his labor as $3 per working hour. He has spent 900 hours preparing the first draft. Jones,
whose labor is valued at $12 per working hour, will revise Smith’s draft to complete the book.
a. How many hours will Jones have to spend to produce a finished book of 150 pages? Of 300 pages?
Of 450 pages?
q  9000.5 J 0.5  30 J
J  25 q  150
J = 100 q = 300
J = 225 q = 450
b. What is the marginal cost of the 150th page of the finished book? Of the 300th page? Of the 450th
page?
Cost = 12 J = 12q2/900
MC 
dC 24q 2q


dq 900 75
q = 150
MC = 4
q = 300
MC = 8
q = 450
MC = 12
10. Given the following production function:
Q = 4L.8 K.8
a. Show the function has a diminishing rate of technical substitution.
MRTS =
Q / L 3.2 L0.2 K 0.8 K


Q / K 3.2L0.8 K 0.2 L
So obviously as L increases, the MRTS decreases. Or if the derivative of the MRTS with respect to L is
negative.
b. Assume the wage rate (w) is 25 and the rental rate of capital (v) is 100. Solve for the cost minimizing
levels of L and K to produce 1296 units of output. What is the total cost of producing in the long run?
w
25 K

  L  4K
v
100 L
1296  4(4 K ) 0.8 K 0.8
MRTS 
K *  18.54
L*  74.15
TC  74.15 * ($25)  18.54 * ($100)
TC  $3707
c. Assume in the short run that the level of capital is fixed at 10 units. What would be the short
run total cost of producing 1296 units of output? Compare the long run cost of producing 1296
units of output to the short run cost of producing 1296 units of output (which is greater and
why?)
1296  4L0.8 (10) 0.8
L*  137.5
TC  137.5 * ($25)  10 * ($100)
TC  $4436.25
In the short run (with a given amount of capital) the costs of producing a given amount will be higher than
it is in the long-run, for the simple reason that the firm does not have the ability to adjust it’s input use to
the cost minimizing allocation.
11. Doug wants to go into the donut business. For $500 per month he can rent a bakery complete with all
the equipment he needs to make a dozen different kinds of donuts (K=l). He must pay unionized donut
bakers a monthly salary of $400 each. He projects his production function to be Q = 5KL (where Q is
tons of donuts/month).
a. What is Doug's monthly total cost function, variable cost function, and marginal cost?
K=1 L=Q/5
TC = 500 + 400(Q/5) = 500 + 80Q
VC = 80Q
The marginal cost is equal to the slope of the total cost curve which is MC = 80.
b. How many bakers will Doug hire to make 25 tons of donuts?
L = Q/5 = 25/5 = 5
c. What will happen to Doug's total cost if his production function turns out to be Q=2KL?
K = 1, L = Q/2, TC = 500 + (400/2)Q = 500 + 200Q
12. An entrepreneur purchases two firms that produce the same good. Each firm has a production
function given by:
q  K i L i where i=1,2, referring to the amount of the input employed by either firm.
The firms differ in the amount of capital equipment they have. Firm 1 has K1 =25 and firm 2 has K 2
=100. For simplicity, the prices of K and L are normalized to $1 for each.
a. If the entrepreneur wishes to minimize short run total costs of production, how much should be
produced in each firm (find Q in firm 1 as a function of firm 2 or vice versa).
a.
q total = q1  q2 . q1 = 25l 1 = 5 l 1
2
SC1 = 25 + l1 = 25 + q1 /25
q2 = 10 l 2
S C 2 = 100 + q22 /100
2
SC total = SC1 + SC2 = 125 +
2
q1
q
+ 2
25 100
To minimize cost, set up Lagrangian: £  SC   ( q  q1  q2 ) .
 £ 2q1
=
  =0
 q1 25
 £ 2q 2
=
  =0
 q 2 100
Therefore q1  0.25q2 .
b. How much should be produced by each firm if the entrepreneur wishes to produce 100 units?
Q2 = 80; Q1=20
c. Given that output is optimally allocated, calculate the short-run total, average, and marginal cost
curves as a function of quantity produced.
c.
4 q1 = q2
q1 = 1/5 q
q2 = 4/5 q
2
SC = 125 +
q
125
SMC =
2q
125
SAC =
125
q
+
q
125
d. What is the marginal cost of the 100th unit? The 200th unit?
SMC (100) =
200
= $1.60
125
SMC(125) = $2.00
SMC(200) = $3.20
13. Suppose the long-run total cost function for a firm is given by:
TC = Qw2/3 v1/3
a. Calculate TC and ATC at Q = 100 and Q =150 if w = $15 and v = $27. Is this firm
experiencing increasing, decreasing, or constant returns to scale? Explain.
TC at Q = 100 = $1824.66
TC at Q = 150 = $2736.99
It’s constant returns to scale because TC change at the same proportion as output.
b. Use the cost function to calculate the contingent demand functions for K and L.
From Shephard's Lemma,
1
C 2  v  3
l
 q 
w 3  w 
2
C 1  w  3
k
 q  .
v 3  v 
14. Assume a single price firm is charging a price of $40, and is operating on a portion of the demand
where the elasticity equals -2. What is the marginal revenue? What is the marginal cost?
MR = P(1 + 1/e) = $20; If they are maximizing profit, MC would also equal $20, or
(P-MC)/P = -1/e, (40 – MC)/40 = ½, MC=20.
15. A perfectly competitive firm has a total cost function given by: 0.1q2 + 10q + 500 and the market
price is constant at P=$20 for all quantities produced.
a. What is the profit maximizing quantity of production for this firm?
MC  C q  0.2q  10 set MC = P = 20, yields q* = 50
b. What is their maximum attainable profit?
π = Pq – C = 1000 – 1250 = -250
c. Will this firm shutdown or continue to produce? How do you know?
This firm will continue to produce (even though their profit is negative) because their total revenue is
greater than their variable costs (TR=$1000; VC = $750).