ECO380: Markets, Competition, and Strategy — Fall 2024
Graded Problem Set #2
Due: Nov 13, 2024, 11:59 PM
Please use Crowdmark to submit your answers as a PDF file. It is recommended that you use LATEX to write your solution. Read each question carefully and answer each part as best as you can. Partial credit will be given, so please show your work. Each problem contains several parts. Number each part of your answers.
Q1. (15 pts) The market is served by two firms, firm 1 and firm 2. The two firms supply identical products and compete by simultaneously setting prices. Inverse market de- mand is PD (Q) = 10 − Q, where Q is the sum of the firms’ sales. The marginal cost of either firm is $6.
(a) (5 pts) How many units of the good are sold in the Nash equilibrium?
(b) (5 pts) Suppose the marginal cost of firm 1 decreases to $4. Find the new Nash equilibrium and calculate firm 1’s equilibrium profit.
(c) (5 pts) Suppose the marginal cost of firm 1 decreases to $0. Find the new Nash equilibrium and calculate firm 1’s equilibrium profit.
Q2. (30 pts) Suppose that firm 1, firm 2, and firm 3 sell differentiated products. The demand for the product of firm j = 1, 2, 3 is
Q1 (p1 , p2 , p3 ) =12 − 2p1 + p2 + p3
Q2 (p1 , p2 , p3 ) =12 + p1 − 2p2 + p3
Q3 (p1 , p2 , p3 ) =12 + p1 + p2 − 2p3
respectively. Firm 1’s marginal cost is $2; firm 2’s marginal cost is $4; firm 3’s marginal cost is $2. There are no fixed costs. Firms compete by setting prices at the same time.
(a) (5 pts) Derive the best response function for each firm.
(b) (5 pts) Characterize the Nash equilibrium. Explain intuitively why p2 is larger than p1 and p3 .
(c) (5 pts) Verify that
in the Nash equilibrium.
Now suppose firm 1 and firm 2 merge, i.e., product 1 and product 2 are now sold by a single firm.
(d) (10 pts) Set up the profit maximization problem of the merged firm, compute the two FOCs (one for each price), and find the profit-maximizing price for either product as functions of p3 .
(e) (5 pts) Solve for the post-merger Nash equilibrium. Verify that all prices increase after the merger takes place.
Q3. (15 pts) A unit mass of consumers with unit demand search products 1, 2, 3, ... at random. The search cost is 0 < s < 1. A consumer values product j at Uj , where
Firm j sets the price pj of product j. There is no cost of supplying product j. In equilibrium, consumers believe that all prices are equal top* (and this belief is correct). Below, you’ll find useful information about exponential distributions.
f(U) = exp(-U) (probability density function)
F(U) = 1 - exp(-U) (cumulative distribution function)
E[U] = 1 (expected value)
E[U|U ≥ U*] = U* + 1 (expected value conditional on U ≥ U* )
(a) (5 pts) In the symmetric Nash equilibrium, consumers buy product j if Uj ≥ U* and continue searching otherwise. Find the reservation value U* .
(b) (5 pts) Set up firm j’s profit maximization problem and derive the first-order condition.
(c) (5 pts) Solve for p* in the symmetric Nash equilibrium.
Q4. (20 pts)
(a) (5 pts) Load “GPS1 data.csv” into your Python editor. Recall:
– The data includes weekly observations of the prices and sales in a duopoly market over a period of 52 weeks.
– Firm 1 and firm 2 sell products 1 and 2 respectively and compete in prices.
– The market size is one hundred consumers. (Recall that some of these con- sumers do not buy product 1 nor product 2 but take the “outside option”.)
Estimate the average marginal cost of firm 1 and firm 2 respectively if α = −0.23.
(b) (5 pts) Suppose the market share of firm j = 1, 2 is
where Vj = β0 + αpj . You have estimated that β0 = 0.3 and α = −0.1. You have also calculated that the marginal cost of firm 1 and firm 2 are $0.5 and $1 respectively. Use Python to determine the prices of both products in the Nash equilibrium. The following functions may help you do that:
def V(P, params):
alpha , beta0 = params
return beta0 + alpha*P
def s(P, params):
return np . exp(V(P, params ))/(1 + np . exp(V(P, params)) . sum ())
(c) (5 pts) Now suppose firm 1 and firm 2 merge. Write down the first-order condition that the merged entity solves for each product. Make sure to express the FOCs in terms of α, s1 , s2 , p1 , p2 , c1 , and c2 only. Remember that
Explain what upward pricing pressure (UPP) is and indicate the component of the first-order condition that represents UPP.
(d) (5 pts) In the lecture we talked about treating α as a random coefficient, i.e., α ~ N(¯(α),σ 2 ). Explain in your own words the issue we were able to address by adopting this approach.