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1. Bundling with correlated values
A monopolist sells both products A and B at zero cost. Consumers’ valuations for product A, vA, is uniformly distributed on [0,1]. Consumers’ valuations for product B, vB, is correlated with vA. If a consumer buys both products, his gross utility is vA+ vB . First, assume vB is perfectly and negatively correlated with vA, which impliesvB =1−vA.
a) Suppose the monopolist sells the two products separately. For price pA and pB, what are the demands for products A and B?
b) Under separate selling, what are the optimal prices p∗A and p∗B?
c) What is the equilibrium profit under separate selling?
d) Now suppose the monopolist sells the two products as a bundle with a bundle price pAB. What is the demand for the bundle under price pAB ?
e) Under bundling, what is the optimal bundle prices p∗AB?
f) What is the equilibrium profit under bundling?
Next assume vB is perfectly and positively correlated with vA. This implies vB = vA.
g) Suppose the monopolist sells the two products separately. For price pA and pB, what are the demands for products A and B?
h) Under separate selling, what are the optimal prices p∗A and p∗B?
i) What is the equilibrium profit under separate selling?
j) Now suppose the monopolist sells the two products as a bundle with a bundle price pAB. What is the demand for the bundle under price pAB ?
k) Under bundling, what is the optimal bundle prices p∗AB?
l) What is the equilibrium profit under bundling?
m) Explain your intuition why bundling does not help increase profits when valuations are positively correlated.
2. Merger with horizontally differentiated products
Three firms produce horizontally differentiated products and compete in price. The demand function for each firm is as the following
q1 = 1 − p1 + p2 + p3 , 2
q2 = 1 − p2 + p1 + p3 , 2
q3 = 1 − p3 + p1 + p1 2
The production cost is assumed to be zero. Each firm i maximizes its profit πi = piqi by choosing pi, taking other firms’ prices as given.
a) Use the first-order condition to derive each firm’s best response (note the best responses are in price).
b) Use symmetry, p∗1 = p∗2 = p∗3 = p∗ and the best responses derived in [a.] to solve for the symmetric equilibrium price p∗.
Now suppose firm 1 and firm 2 merge. Suppose the merged firm is called firm m. Firm m sells both products 1 and 2 by setting p1 and p2 (since products are horizontally differentiated, selling two varieties is more profitable for firm m). It competes with firm 3 in price. The production costs remain zero for firms m and 3. The demand functions for each product are unchanged.
c) Write down firm m’s profit maximization problem.
d) Holding p3 fixed, use two first-order conditions (w.r.t. p1 and p2) to derive firm m’s best response of p1 and p2.
e) Holding p1 and p2 fixed, use first-order condition to derive firm 3’s best response of p3.
f) Apply symmetry p∗∗ = p∗∗ = p∗∗ (note that symmetry does not apply to p3∗∗). Solve for the equilibrium post-merger prices pm∗∗ and p3∗∗.
g) Compare p∗, pm∗∗ and p3∗∗ and give some intuition why they are ordered in that way.
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