Economics 580 Assignment #3: Differentiated Products, Bertrand Price Competition, and Mergers

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Economics 580

Assignment #3:  Differentiated Products, Bertrand Price Competition, and Mergers

(Due by Feb 21, 2023, 1 PM Pacific Time)

I. (Representative Consumer Utility Maximization with Quasi-linear Quadratic Utility (and Linear Demand System) Consider a representative consumer with the following quasi-linear utility function over three goods:

U (q0 , ql , q2 ) = q0 + αl ql + α2 q2 _ [βl q l(2) + 2γql q2 + β2 q2(2)]/2,

with αi   > 0, βi   > 0, and βl β2  _ γ 2   > 0, where q0  is the numeraire good.  (Note: If the parameters make your calculations too tedious, you may assume some specific parameter values for simplicity. For instance, you may set αi  = βi  = 1 but keep γ .)

1. Derive the consumer’s demands for goods  1 and 2  (both direct and inverse demand systems).

2. How should we parameterize the demand system in order to measure the degree of substitution/complementarity between the two goods (or the degree of prod- uct differentiation)?  For instance, we can use γ as an independent parameter, or set βl  = β2  = 1 _ γ . Discuss your reasoning.

3. Derive the consumer’s surplus (or indirect utility as a function of prices).

4.  Suppose two firms produce these two goods at constant marginal cost cl  and c2 , respectively. Compute the Bertrand-Nash equilibrium when the firms compete by setting prices simultaneously.   Determine their own and cross cost-pass- through (CPT) rates (i.e., how the equilibrium prices change with respect to costs, respectively). Explain.

5.  Suppose the two firms merge to become a monopolist over the two products (or they collude in setting prices).  Compute the joint-profit-maximizing prices as well as own and cross cost-pass-through (CPT) rates. Explain.

6.  Compare the prices from (d) and (e) and explain the result.

II. (Asymmetric Marginal Costs) Consider two firms, 1 and 2, producing differ-

entiated products. The demands for the two products are symmetric and given by

qi  = 10 _ 2pi + pj ,                                                (1)

for j  i, i = 1, 2. The marginal cost of production for firm 1 is $1 and for firm 2 is 0. Answer the following three questions.

1.  Suppose that the two firms compete by simultaneously choosing their prices. Determine the Bertrand-Nash equilibrium prices, and firm profits.

2.  Suppose that the two firms choose their prices to maximize their joint profits (through price-fixing collusive agreements or through a merger). Determine the optimal prices, and firm profits.

3.  Compare the outcomes from the above two settings and discuss the implications.

III. (Asymmetric Demands) Consider two firms, 1 and 2, producing differentiated

products. The demands for the two products are asymmetric and given by

q1  = 10 _ 2p1 + p2 ,                                                (2)

q2  = 14 _ 2p2 + p1 ,                                                (3)

The marginal cost of production for both firms are 0.  Answer the following three questions.

1.  Suppose that the two firms compete by simultaneously choosing their prices. Determine the Bertrand-Nash equilibrium prices, and firm profits.

2.  Suppose that the two firms, jointly behaving as a single monopolist, choose their prices to maximize their joint profits (through price-fixing collusive agreements or through a merger). Determine the optimal prices, and firm profits.

3.  Compare and the outcomes from the above two settings and discuss the impli- cations.

IV. (Symmetry with an Arbitrary Degree of Substitution/Complements) Consider

two firms,  1 and 2, producing differentiated products.   The demands for the two products are symmetric and given by

qi  = 10 _ 2pi + γpj ,                                               (4)

for j  i, i = 1, 2, γ is a demand parameter measuring diversion ratio and the degree of product differentiation, and γ e [_2, 2). The marginal costs of production for both firms are $1. Answer the following three questions for each of the following values of γ = 0, 1, _1, and _2, respectively.

1.  Suppose that the two firms compete by simultaneously choosing their prices. Determine the Bertrand-Nash equilibrium prices, and firm profits.

2.  Suppose that the two firms choose their prices to maximize their joint profits (through price-fixing collusive agreements or through a merger). Determine the optimal prices, and firm profits.

3.  Compare the outcomes from the above two settings and discuss the implications.


2023-02-10