Econ 460: Industrial Organization

0101, Fall 2008

Problem set 4

Due: Tuesday December8, 11am

1. Describe the distinction between pre-emption (as in our pre-emptive increase of plant size) and pre-commitment (as in our model of capacity commitment).

a) Describe the so-called “merger paradox”. What is the paradox here? How do we go about trying to solve it?
b) Describe why Cournot vs. Bertrand models of oligopoly can lead to very different conclusions about the incentives to merge.

c) Describe the difference between a horizontal merger, vertical merger and conglomerate merger. Describe a possible motivation for each type of merger.

3. Recall from lecture that in the n-firm Cournot model, where P = a – Q and C(q) = c, the unique Nash Equilibrium is for each firm to produce q* = (a – c)/(n+1), and for each firm’s profit to be (a – c)2/(n+1)2. (Make sure you see why this is true.)

Suppose that we have an industry with 5 firms, a demand function of P = 50 – Q, and a constant marginal cost of c – 4.

a)In the Cournot equilibrium, find the output of each firm, the aggregate industry output, the market price and the profits of each firm.

b)Suppose that 2 of the firms merge, leaving us with a 4-firm industry. After the merger, all firms are identical. Find the output of each firm, the aggregate industry output, the market price and the profits of each firm. Was the merger profitable for the firms that merged? For the firms that didn’t merge? What was the effect on consumers?

c)Suppose that instead of 2 firms merging, suppose that 3 firms had merged simultaneously, leaving us with a 3 firm industry.After the merger, all firms are identical. Find the output of each firm, the aggregate industry output, the market price and the profits of each firm. Was the merger profitable for the firms that merged? For the firms that didn’t merge? What was the effect on consumers?

4. Differentiated product Bertrand (long!)

Consider an industry with three firms, i=1, 2 and 3. Each firm is identical except that they produce differentiated product lines, goods 1, 2 and 3, respectively (1 good per firm). The demand for good each good is defined implicitly by:
p1 = 60 – 3q1 – 2(q2 + q3)

p2 = 60 – 3q2 – 2(q1 + q3)

p3 = 60 – 3q3 – 2(q2 + q1)

a) Find the demand functions for each firm by repeated substitution; ie express q1 solely as a function of prices p1, p2 and p3.

b) Suppose that all three firms simultaneously choose a price for the good they produce. Write down the profit function for each firm, find the FOC from each firm’s profit maximization problem, and solve these to find best response functions for each firm. Impose the fact that symmetry gives us p1 = p2 = p3. Solve to find the equilibrium price, output and profit for each firm.

c) Suppose that firms 1 and 2 merge to create a merged firm that produces both good 1 and good 2. Write down the profit maximization problem of the merged firm and the non-merged firm. Find the (three) FOCs, and solve these to find the post-merger equilibrium price and output level for each good, and the profit for each firm.

Was the merger profitable for the merging firms? For the non-merging firm?

Are consumers better off or worse off?

5. Consider a variant on the double marginalization model from class, where there are some positive costs to retailing.

As before, we have a monopolist upstream manufacturer, who produces output Q at a constant marginal cost c1, and sells this to a downstream monopolist retailer at a wholesale price r, who then sells this to final consumers at a retail price P. Unlike our model from class, the downstream retailer has a positive marginal retailing cost for each unit they sell, c2. As before, the final consumer demand for the product is P = A – BQ.

a) Write down the retailer’s profit maximization problem (with choice variable Q). Solve this problem to find the best response function, Q as a function of r. Find the retail price and the retailer’s profit in terms of r.

b) Find down the inverse demand curve faced by the upstream manufacturer as a function of the wholesale price (ie r as a function of Q). Write down the upstream manufacturer’s profit maximization problem (hint: use choice variable Q), and solve the profit maximization level to find the equilibrium level of output produced by the manufacturer (and purchased by the retailer). Use this to find the equilibrium wholesale price, equilibrium retail price, and equilibrium profits for the upstream and downstream firms.

c) Now suppose that the manufacturer and retailer merged, creating a single integrated monopolist. Write down the profit maximization problem for the merged firm (hint: what is the marginal cost for the merged firm?), and then find equilibrium output level Q, consumer price P and profit level πI.
Was the merger profitable? Are consumers better off? Why?

6. Explain the factors that determine optimal patent length. Describe what is meant by the term “patent race”.