Reference no: EM13957648

PRoblem 1: (1) Two firms have technologies for producing identical paper clips. Assume that all paper clips are sold in boxes containing 100 paper clips. Firm A can produce each box at unit cost of cA = $6 whereas firm B (less efficient) at a unit cost of cB = $8.

(a) Suppose that the aggregate market demand for boxes of paper clips is p = 12 − Q/2, where p is the price per box and Q is the number of boxes sold. Solve for the Nash-Bertrand equilibrium prices p b A and p b B, and the equilibrium profits π b A and π b B. Explain your reasoning!

(b)  Answer the previous question assuming that firm A has developed a cheaper production technology so its unit cost is now given by cA = $2. 

Problem 2: Consider an infinitely-repeated price competition game between GM and FORD. Each firm can set a high price or a low price in each period t = 0, 1, 2, . . .. The profit of each outcome are given in the following matrix:

Suppose that each firm adopts a trigger-price strategy under which the firms may be able to implicitly collude on setting the high price. Let ρ (0 < ρ < 1) denote the time discount factor.

(2a) Compute the minimum threshold value of ρ which would ensure that GM sets p H in every period t. Show and explain your derivations

(2b) Compute the minimum threshold value of ρ which would ensure that FORD sets p H in every period t. Show and explain your derivations. 

Problem 3:

(3) Aike (Brand A) and Beebok (Brand B) are leading brand names of fitness shoes. The direct demand functions facing each producer are given by qA(pA, pB) = 180 − 2pA + pB and qB(pA, pB) = 120 − 2pB + pA. Assume zero production cost (cA = cB = 0).

(a) Derive the price best-response function of firm A as a function of the price set by firm B, pA = BRA(pB). Show your derivations, and draw the graph associated with this function. 

(b) Derive the price best-response function of firm B as a function of the price set by firm A, pB = BRB(pA). Show your derivations, and draw the graph associated with this function. 

(c)  Solve for the Nash-Bertrand equilibrium prices, hp b A, pb Bi. Then, compute the equilibrium output levels hq b A, qb Bi, the equilibrium profits hπ b A, πb Bi, and aggregate industry profit Πb = π b A + π b B. 

(d) Suppose now that the two producers hold secret meetings in which they discuss fixing the price of shoes to a uniform (brand-independent) level of p = pA = pB. Compute the price p which maximizes joint industry profit, πA + πB. Then, compute aggregate industry profit and compare it to the aggregate industry profit made under Bertrand competition which you computed in part (3c). 

Problem 4:  Ann Arbor and Ypsilanti are very similar cities, because each city has exactly one McDonald’s. Ann Arbor has NA = 200 residents and Ypsilanti has NY = 200 residents. Each resident demands one hamburger. A resident of Ann Arbor who wishes to buy a hamburger in Ypsilanti must bear a transportation cost of TA = $3. Similarly, a resident of Ypsilanti who wishes to buy a hamburger in Ann Arbor must bear a transportation cost of TY = $3. 

(4a) Solve for the undercut-proof equilibrium prices p U A and p U Y and profit levels π U A and π U Y assuming that McDonald’s has the technology for producing hamburgers at no cost. Show your derivation. 

(4b) Answer the previous question assuming now that McDonald’s in Ann Arbor bears a cost of $1 of producing each hamburger, whereas McDonald’s in Ypsilanti bears a cost of $4 of producing each hamburger. Show your derivation.