Questions -

Q1. Two firms produce close substitute products. They face interdependent demand curves:

q_i = 1 - p_i + θp_j,

where i = 1, 2; i ≠ j, 0 < θ < 1. The firms can produce at zero cost.

A. What are the equilibrium prices and outputs under Bertrand price competition?

B. Suppose firm 1 discovers firm 2's reaction function through industrial espionage.

Find the equilibrium prices when firm 1 acts as a price leader in the market.

Q2. Now suppose the owner of firm 1 delegates pricing decisions to a manager. The owner writes a contract that provides the manager with returns of

π_i^M = β(p₁- t)q₁,

where q_i =1- p_i + θp_j i = 1, 2; i ≠ j; 0 < θ, β < 1. As before, the firms have zero cost.

A. Solve for the equilibrium prices as a function of t: p₁*(t) and p₂*(t).

B. Show that the optimal managerial contract over-emphasizes cost (t* > 0) and that the equilibrium prices under the optimal contract are Stackelberg prices.

Q3. A manufacturer distributes her product through duopoly retailers who compete in a retail Cournot market facing (inverse) demand:

P(Q) = p(q₁ + q₂) = a - b(q₁ + q₂),

where q₁ and q₂ are the quantities sold by each retailer. There are no other retail-specific costs apart from the wholesale price, w. The firms procure the goods from a single, monopoly manufacturer who has the cost function c(Q) = cQ + F.

A. Suppose the manufacturer sets only wholesale prices to retailers, w*. Solve for the equilibrium price (p*) and equilibrium quantity Q*. Calculate the profit level for retailers, π_R*, and the wholesaler, π_w*.

B. Now suppose the manufacturer sells to the retailers through a two-part tariff arrangement at the wholesale price, w_m, and franchise fee F_R. Assume that retailers must be left with at least π_R* in retail profit to sign the franchise agreement. Solve for the optimal 2-pail tariff for the monopoly manufacturer, (w_m*, F_R*). Verify that wholesaler profit is larger under the 2-part tariff arrangement.