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Final Exam Study Guide Some Conceptual Questions
1. Explain the prediction of the Bertrand duopoly model that both firms will price at marginal cost. How does
Cournot Duopoly differ? 2. Explain how first, second, and third degree price discrimination differ from the standard uniform pricing
problem of the monopolist. Give real world examples of both second and third degree price discrimination.
Explain why the examples accurately depict second and third degree price discrimination. 3. Briefly explain the logic behind backward induction, our approach to solving sequential (dynamic) games.
Use the difference between Cournot and Stackelberg duopoly as an example. 4. Briefly explain the difference in market structure between perfect competition, monopolistic competition,
oligopoly, duopoly, and monopoly, and explain why or why not we need to use game theory to understand
firms? decision-making. Some Perfect Competition and Monopoly Problems
1. (i) Suppose the market demand for your good is given by P=120-(1/5)Q. The cost function for each firm is
identical, and is given by c(q)=10+(1/10)q2. If all firms in this market have the same technology and are
price-takers, how many units should each firm produce if the market price for the firm?s product is $20?
Assuming the market clears, how many firms are in the market. If the market clears at P=20, with demand at 500 and each firm producing 100 units, there are 500/100=5
firms in the market. (ii) If there is only one firm in the market, what price should it charge and how many units should it sell to
maximize profits? Q=200, P=80 2. Suppose a monopolist having constant marginal cost equal to 1 sells to two consumers. Consumer 1
has an inverse demand given by 1 (1 ) = 3 ? 31 and Consumer 2 has an inverse demand of
2 (2 ) = 4 ? 42 . Resale between the two consumers is not allowed.
a) Assume the monopolist has to use uniform pricing. Calculate the monopolist?s profit maximizing
price and quantity. How much will each consumer consume at that price?
(Answer: = 31
14 = 17
24 1 = 11
42 2 = 25
56 = 289
336 b) Now assume the monopolist can engage in third degree price discrimination. Calculate the
monopolist?s profit maximizing prices and how much he will sell to each market.
1 3 43 (Answer: 13 = 2, 23 = 2.5, 13 = 3, 23 = 8, 3 = 48) c) Lastly, graphically show what the outcome will be if the monopolist can engage in first degree,
i.e. perfect, price discrimination. Identify the producer surplus, any consumer surplus, and any
deadweight welfare loss.
2 9 43 (Answer: CS=0, DWL=0, 11 = 3, 21 = 8, 1 = 1 = 24 ? draw a graph with all CS in
each market being expropriated by the producer) d) Graphically show the outcome in part (a) and the outcomes in part (b). Assuming there are no
fixed costs to production, compare the profits under each of the three different pricing schemes.
Which pricing scheme does the monopolist prefer and why?
(Answer: graphs should be straight-forward? we should have a ranking of 1 > 3 > , so
the monopolist prefers being able to engage in 1st degree, i.e. perfect, price discrimination) 1 3. A monopolist has a single production plant in California with a total cost function of () = 2 2 ,
where is total output. The good can be sold to two separate markets, those on the west coast and
those on the east coast. There is no resale between coasts. The inverse demand function for the east
coast is given by = 100 ? and the inverse demand function for the west coast is given by
1 = 80 ? 2 .
a) Calculate the monopolist?s profit maximizing prices if he can engage in third degree price
(Answer: Pe=76, Pw=66, Qe=24, Qw=28) b) Calculate the monopolist?s profits if he uses standard uniform pricing. How do these compare
with his profits under optimal third degree price discrimination that you solved in part (a)?
(Answer: profits under uniform pricing are 2253.33?. profits under 3rd degree are 2320) c) Suppose that we are back in the world of third degree price discrimination. The monopolist can
build a second plant in Massachusetts (i.e. an east coast plant) that is identical to the California
plant (i.e. west coast plant), and plans on producing output for each region at their respective
plant. If the cost of constructing the plant is $1200 should the monopolist build the second plant?
How much should the monopolist be willing to pay for the plant? Explain.
(Answer: Well, marginal costs are increasing in quantity, so intuitively producing at two plants might
be attractive. If we re-optimize our production levels and prices using two separate plants I get
profits of $3,266.66?. so your profits increase by $946.66? you would not expand if the fixed costs
of construction were more than that, so that is the most you are willing to pay and you will not
expand at a cost of $1200) 1
2 4. A monopolist with total cost function () = 10 + 2 sells to 10 identical consumers, each of
1 whom has demand function () = 10 ? 2 for the monopolist?s product.
a) Is this monopolist operating in the short run or the long run? Explain how you know.
(Answer: Short Run, and we know because there is a fixed component to the cost function, indicating
that there is some fixed input and, by definition, that we are in the short run.) b) Suppose the monopolist must charge a uniform price. Calculate the monopolist?s optimal price
and its profits.
(Answer: = 120
7 = 100
7 = 6,510
49 ? 132.86) c) Calculate any deadweight welfare loss associated with the monopoly in part (a) and depict it
(Answer: = 1,000
294 ? 3.40, just draw a standard monopoly diagram) d) Suppose the monopolist can now engage in second degree price discrimination. Calculate the
monopolist?s optimal two-part tariff and it?s total profit.
(Answer: 2 = 50
3 2 = 50
3 = 25
9 2 = 1,410
9 ? 156.67) Some Discounting Problems
1. Suppose you just graduated from UCR and have 2 offers. The first is a job offer at Jobs ?? US with a fixed
salary schedule which paid $40,000 this year and has a 2% pay raise every year. You will expect to hold
this position for 40 years and then retire. The second offer is acceptance to a 2 year master?s program
which will cost $20,000 a year. After you earn your master?s degree, you expect to earn $70,000/year for
the following 38 years, at which point you will retire. You can invest at 10%.
All salary payments are made at the end of the year while tuition is paid at the start of the year.
Compare the net present value of each option. Which should you take? = $$475,607 ?
= $524,864.14 2. Suppose you bought a 10-year bond when the interest rate was 5% per year. So, your bond would pay
$50 at the end of each of the 10 consecutive years and, after the tenth year, your $1000 is returned to
you. You just collected your seventh annual payment on the bond.
(a) Suppose that you decide to sell the bond now, assuming that the interest rate is still 5% and this
is expected to continue for the indefinite future. How much can you expect to sell the bond for?
(b) Suppose the interest rate has changed to 10%. Now how much can you expect to sell the bond
(Answer: $876.65) 3. Comparing Investment Options
Consider the following investment options:
Option 1: invest $100 now and get back $125 in one year
Option 2: invest $75 now and get back $100 in two years (a) Find the internal rate of return for both options.
(Answer: option 1, IRR is 25%. Option 2, IRR is 15.5%)
(b) If the interest rate is 5% which option would you prefer?
(Answer: NPV(option1)=$19.05, NPV(option2)=15.70) Some Game Theory Problems 1. Find all of the pure strategy Nash Equilibria of the following simultaneous move game. Solve it two
different ways: using IESDS and using the underlining method. When solving it with IESDS be sure to
clearly state the order of elimination of strategies and what your reasoning is. Column Row A B C D E a 9,4 12,10 15,7 2,8 15,5 b 14,8 3,10 12,18 4,7 20,12 c 7,8 6,8 20,10 3,12 15,9 d 15,0 7,4 14,2 5,3 9,1 e 20,18 2,9 10,14 3,7 19,20 (Answer: we should get (a,B) as the Nash Equilibrium) 2. Consider 2 firms selling fertilizer competing as Cournot duopolists. The inverse demand function
facing the fertilizer market is = 1 ? , where = + . For simplicity, assume that the longrun marginal cost for each firm is equal to ?, i.e. c(q)=?q for each firm. a) Find the Cournot Nash equilibrium where the firms choose output simultaneously.
(Answer: = = 1/4, = 1/2 b) Depict the outcome of (a) graphically with a diagram of the reaction functions. Identify the Nash
equilibrium, and clearly label your graph.
c) Find the Stackelberg Nash Equilibrium where firm A as the Stackelberg leader.
3 3 7 (Answer: = 8 , = 16, = 16) 3. Find all of the pure strategy Nash Equilibria of the following simultaneous move game. Solve it two
different ways: using IESDS and using the underlining method.
3,0 (Answer: (m,L) and (t,R)) 4. Consider the following entry or exit game. Recall that, if X is a random variable as payouts are here,
then the expected value of X is just the sum of the possible outcomes where each is weighted by the
probability that it happens. Assume that for any pair of strategies for the new and old cleaner there is
a 70% chance the economy is normal and a 30% chance there is a recession. Payouts are in the
order (New Cleaner, Old Cleaner).
a) Solve for the Subgame Perfect Nash Equilibrium of this sequential game.
(Answer: (Enter, High Price))
b) Suppose the order of play is reversed, i.e. the Old Cleaner first sets a price, then the New Cleaner
makes an entrance decision, and then we learn if we are in a recession or not and what our
payoffs are. Solve for the Nash Equilibrium now. Can the Old Cleaner deter market entry in this
case? What about in the first case?
(Answer: (Low Price, Stay Out) ? yes the entrant can now be deterred if the old cleaner takes
pre-emptive action. In part (a) the threat to lower prices and start a price war was an incredible
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