Benedict Economics - Consider a Bertrand duopoly with two firms

College of Saint Benedict Economics Questions

Midterm 2 Assignment

There are three questions and 32 points in total Show all steps of your work. Only writing the final answer is not enough.

Question 1 [8 points] Consider a Bertrand duopoly with two firms 1,2 who sell the same good that has demand curve Q = 16 − p if p < 16 and Q = 0 if p ≥ 16. Both firms have same constant unit cost 4. Firms 1,2 set prices p1, p2. If they set different prices, the firm which sets the minimum price receives the demand at that price while its rival receives zero demand. If both firms set the same price, they equally split the demand at that price.

(a) [4 points] Fix p2 = 3. Draw the profit of firm 1 as function of p1 and find all best responses of firm 1 to p2 = 3.

(b) [4 points] Fix p1 = 7. Draw the profit of firm 2 as function of p2 and find all best responses of firm 2 to p1 = 7.

Question 2 [16 points] Consider a Cournot duopoly with two firms 1,2. Let q1, q2 be the quantities produced by firms 1,2. The price p is given by the inverse demand p = 18 − Q where Q = q1 + q2. The unit cost of firm 1 is c1 = 5 and the unit cost of firm 2 is c2 = 12.

You can use the following result without proving it

For a Cournot duopoly with inverse demand p = a − Q, where firm 1 has unit cost c1 and firm 2 has unit cost c2, the best response functions are given as follows.

Best response of firm 1 (BR1) to q2 is: choose q1 = (a − c1 − q2)/2 if q2 < a − c1 and choose q1 = 0 if q2 ≥ a − c1.

Best response of firm 2 (BR2) to q1 is: choose q2 = (a − c2 − q1)/2 if q1 < a − c2 and choose q2 = 0 if q1 ≥ a − c2.

(a) [4 points] Drawing the best response functions in a diagram, identify Nash Equilibrium (NE) quantities of firms 1,2 and find their numerical values.

(b) [4 points] Suppose firm 2 is constrained by capacity K2 = 1, while firm 1 has no capacity constraint. Draw the best response functions and find NE quantities of this capacity constrained duopoly.

(c) [4 points] Suppose firm 1 is constrained by capacity K1 = 3, while firm 2 has no capacity constraint. Draw the best response functions and find NE quantities of this capacity constrained duopoly.

(d) [4 points] Suppose both firms 1,2 are capacity constrained. Firm 1 is constrained by capacity K1 = 3 and firm 2 is constrained by capacity K2 = 1. Draw the best response functions and find NE quantities of this capacity constrained duopoly.

Question 3 [8 points] Consider a variation of the linear city model of Hotelling. Buyers are uniformly distributed in a line of length one, where L is the left end and R the right end. Two firms 1,2 compete in prices. Firm 1 is located at distance 1/3 from the left end and firm 2 is located at distance 1/3 from the right end. Since the distance between L and R is 1, this means the distance between firms 1 and 2 is 1/3, as shown in the following diagram.

distance 1/3 distance 1/3 distance 1/3 L (left end) R (right end) Firm 1 Firm 2

Any buyer purchases one unit of the good from either firm 1 or firm 2. For any buyer, the good has benefit v > 0. To purchase the good from a firm, any buyer has to (i) pay the price set by the firm and (ii) pay the cost of transportation to travel to the firm. The unit transportation cost is 4. Let p1, p2 be the prices set by firms 1,2.

(a) [4 points] Consider a buyer b who has distance x from L. Suppose this buyer is on the right of L and left of firm 1, as shown below.

distance x L (left end) R (right end) buyer b Firm 1 Firm 2 In terms of p1, p2, x, determine when this buyer will buy from firm 1 and when it will buy from firm 2.

(b) [4 points] Consider a buyer b who has distance x from L. Suppose this buyer is on the right of firm 1, but on the left of firm 2, as shown below.

distance x L (left end) R (right end) Firm 1 buyer b Firm 2 In terms of p1, p2, x, determine when this buyer will buy from firm 1 and when it will buy from firm 2.