Econ 460: Industrial Organization
0101, Fall 2008
Problem set 3
Due: Tuesday December 8, 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).
The difference is subtle. In our model of pre-emption, we have distinct time periods with different time valuations of money. Our story for pre-emption works only because it is more costly to invest in the new plant in the first period than in the second period; if we invest now we must pay the full costs up front (despite the fact that the extra plant is useless now) whereas if we invested in the future this would have a lower cost, because we would discount the investment back to the present. The presence of actual time is important.
In contrast, in our capacity expansion model, no real time needed to pass. All we needed was a mechanism that would allow a firm to credibly commit to producing larger output. Though we used a dynamic model where the incumbent firm acted “first” in order to deter entry, we did not need any actual chronological time to pass between the decision made by the incumbent and the entrant’s entry decision – these decisions could be 2 seconds apart and the model would still work, as long as the capacity decision was in fact irreversible.
2.
a) Describe the so-called “merger paradox”. What is the paradox here? How do we go about trying to solve it?
The “merger paradox” is the point that it is nontrivial to design a model where a merger is rational for firms to make (unless the merger is a merger to monopoly). In many simple models, the profits of a merged firm are lower than the sum of the pre-merger firms’ profits, while profits of non-merging firms are increased by the merger. So there are weak incentives to merge, and strong incentives to “free-ride” on other firms merging.
We solve this by realizing that many features of the simple models are unrealistic. In particular, the integrated firm after a merger is not the same as other firms that did not merge. We can explain this through models that give the merged firm a strategic advantage (such as through a commitment to higher output), or product differentiation models where the merged firm produces multiple product lines.
b) Describe why Cournot vs. Bertrand models of oligopoly can lead to very different conclusions about the incentives to merge.
In Cournot models, firms choose quantities, which are strategic substitutes. This means they have downward sloping best response functions; when one firm increases its output, this causes other firms to want to decrease their output. In contrast, in Bertrand models firms choose prices, which are strategic complements. This means that they have upward sloping best response functions; when one firm increases its price, this causes other firms to want to increase their price.
When two or more firms merge, they have more market power, and so decrease their output and increase their prices. In a Cournot world, this causes other firms to increase their output, which decreases the profits of the merged firm, which means the firms have less incentive to merge. But in a Bertrand world, other firms response by increasing their prices, which increases the profits of the merged firm, which means the firms have more incentive to merge.
c) Describe the difference between a horizontal merger, vertical merger and conglomerate merger. Describe a possible motivation for each type of merger.
A horizontal merger is a merger between firms that produce products that are substitutes, typically in the same industry. Horizontal mergers may be motivated by a desire to increase market power in the industry in order to increase prices, or it may be motivated by economies of scale that mean the merged firm has lower costs.
A vertical merger is a merger between firms that produce products that are complements, particularly between firms that are in the same supply chain (eg. a firm merging with its supplier). A vertical merger might be motivated by a desire to reduce the problem of multiple marginalization, or it might be motivated by a desire to foreclose on rival firms in either the upstream or downstream markets.
A conglomerate merger is a merger between firms that produce unrelated products. Conglomerate mergers might be motivated by economies of scope; production costs may be lowered by the merger if the products share common inputs. Alternatively, conglomerate mergers may not be in the interests of firm owners at all; they may be undertaken by managers who have their own preferences and interests which are not perfectly aligned with those of shareholders.
3. Recall from lecture that in the n-firm Cournot model, where P = a – Q and C(q) = cq, 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.
Here n=5, A = 50, c = 4, so q* = (46/6) = 7.76, Q* = (5)(46)/6 = 38.33, P* = 50 – 38.33 = 11.67, and π* = (46/6)2= 58.78.
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?
If 2 firms merged, we are left with n=4.
As before, A = 50, c = 4, so q* = (46/5) = 9.2, Q* = (4)(46)/5 = 36.8, P* = 50 – 36.8 = 13.2, and π* = (46/5)2 = 84.64.
The firms that merged are worse off; they used to earn (2)(58.78) = 117.56, but now they only earn 84.64. The firms that didn’t merge are better off; they used to earn 58.78, now they earn 84.64. Consumers are worse off; aggregate output is lower, and prices are higher.
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?
If 3 firms merged, we are left with n=3.
As before, A = 50, c = 4, so q* = (46/4) = 11.5, Q* = (3)(46)/4 = 34.5, P* = 50 – 34.5 = 15.5, and π* = (46/4)2 = 132.25.
The firms that merged are worse off; they used to earn (3)(58.78) = 176.33, but now they only earn 132.25. The firms that didn’t merge are better off; they used to earn 58.78, now they earn 132.25. Consumers are worse off; aggregate output is lower, and prices are higher.
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)
Each firm has a constant marginal cost of c.
Note that this problem is exactly the same as the one from class and the text, as described in Appendix A page 426, except we have specific values for A, B, and s.
a) Find the demand functions for each firm by repeated substitution; ie express q1 solely as a function of prices p1, p2 and p3.
Start with demand for good 1.
3q1 = 60 – p1 – 2(q2 + q3)
q1 = (60 – p1 – 2(q2 + q3))/3 (1)
By symmetry,
q2 = (60 – p2 – 2(q1 + q3))/3 (2)
q3 = (60 – p3 – 2(q2 + q1))/3 (3)
Solve simultaneously. Substitute (2) into (1):
q1 = (60 – p1 – 2[(60 – p2 – 2q1 - 2q3)/3] - 2q3)/3
= [60 – p1 – 120/3 + 2p2/3 + 4q1/3 + 4q3/3 - 2q3]/3
q1 – 4q1/9 = [20 – p1 + 2p2/3 -2q3/3]/3
5q1/9 =[20 – p1 + 2p2/3 -2q3/3]/3
5q1 =3[20 – p1 + 2p2/3 -2q3/3]
q1 =[60 – 3p1 + 2p2 -2q3]/5 (4)
By symmetry, substituting (2) into (3) gives:
q3 =[60 – 3p3 + 2p2 -2q1]/5 (5)
Substituting (5) into (4) gives:
q1 =[60 – 3p1 + 2p2 -2{[60 – 3p3 + 2p2 -2q1]/5}]/5
= [60 – 3p1 + 2p2 – 24 + 6p3/5 – 4p2/5 +4q1/5]/5
5q1 – 4q1/5 = 36 – 3p1 + 6p2/5 + 6p3/5
21q1 = 180 – 15p1 + 6p2 + 6p3
7q1 = 60 – 5p1 + 2p2 + 2p3
q1 = (60 – 5p1 + 2p2 + 2p3)/7
and by symmetry:
q2 = (60 – 5p2 + 2p1 + 2p3)/7
q3 = (60 – 5p3 + 2p2 + 2p1)/7
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.
Firm 1 solves:
Maxp1 : q1(p1 – c)
Maxp1 :[(60 – 5p1 + 2p2 + 2p3)/7](p1 – c)
FOC: (60 – 10p1 + 2p2 + 2p3)/7 + 5c/7 = 0
(60 + 2p2 + 2p3 +5c)/7 =10p1/7
BR1: p1 = (60 + 2p2 + 2p3 +5c)/10
By symmetry, we have:
BR2: p2 = (60 + 2p1 + 2p3 +5c)/10
BR3: p3 = (60 + 2p2 + 2p1 +5c)/10
Imposing a symmetrical solution p1 = p2 = p3 = p* gives us:
p* = (60 + 2p* + 2p* +5c)/10
10p* - 2p* - 2p* = 60 + 5c
p* = (60 + 5c)/6
Substituting back the demand functions from a) gives us
q* = (60 – 5p* + 2p*+ 2p*)/7
= (60 – p*)/7
q* = (60 – [(60 + 5c)/6])/7
q* = (50 – 5c/6)/7
q* = 5(60 – c)/42
Substituting into the profit function gives:
π* = q*(p* – c)
= 5(60 – c)/42[(60 + 5c)/6 – c]
= 5(60 – c)/42[(10 – c/6]
= 5(60 – c)/42[(60 – c]/6
π* = 5(60 – c)2/252
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?
The non-merged firm profit maximization problem and best responses remains the same as before:
Maxp3 : q3(p3 – c)
Maxp3 :[(60 – 5p3 + 2p2 + 2p1)/7](p3 – c)
FOC: (60 – 10p3 + 2p2 + 2p1)/7 + 5c/7 = 0
(60 + 2p2 + 2p1 +5c)/7 =10p3/7
BR3: p3 = (60 + 2p2 + 2p1 +5c)/10 (6)
But the merged firm has two choice variables, p1 and p2. It solves:
Maxp1,p2 : q1(p1 – c) + q2(p2 – c)
Maxp1,p2 :[(60 – 5p1 + 2p2 + 2p3)/7](p1 – c) + [(60 – 5p2 + 2p1 + 2p3)/7](p2 – c)
The firm has two FOCs, one for each choice variable.
FOC1: (60 – 10p1 + 2p2 + 2p3)/7 + 5c/7 + 2(p2 – c)/7 = 0
Ie: (60 + 2p2 + 2p3) + 5c + 2(p2 – c) = 10p1
p1 = (60 + 4p2 + 2p3 +3c)/10 (7)
FOC2: 2(p1 – c)/7 + (60 – 10p2 + 2p1 + 2p3)/7 + 5c/7 = 0
p2 = (60 + 4p1 + 2p3 +3c)/10 (8)
Imposing that p1 = p2 = pm gives:
pm = (60 + 4pm + 2p3 +3c)/10
(10 – 4)pm = (60 + 2p3 +3c)
p1 = p2 = pm = (60 + 2p3 +3c)/6 (9)
Then, we can solve equations (6) and (9) simultaneously to
p3 = (60 + 4pm +5c)/10
p3 = (60 + 4[(60 + 2p3 +3c)/6] +5c)/10
10p3 = (60 + 240/6 +8p3/6 + 12c/6 + 5c)
52p3/6 = 100 + 42c/6
52p3 = 600 + 42c
p3 = (300 + 21c)/26
which from (9) gives us:
pm = (60 + 2[(300 + 21c)/26] +3c)/6
= 60/6 + 2(300)/[(26)(6)] + 2(21)/[(26)(6)] + 3c/6
= 10 + 200/(52) + 14/52 + c/2
pm = (720 + 40c)/52 = p1 = p2
Find output levels from substituting these back into the demand functions:
q1 = (60 – 5p1 + 2p2 + 2p3)/7
= (60 – 3pm + 2p3)/7
= (60 – 3((720 + 40c)/52) + 2(300 + 21c)/26)/7
= [60 – 3(720)/52 + 4(300)/52 – 120c/52 + 42c/52]/7
= [(3120 – 960)/52 – 78c/52)]/7
= (2160 – 78c)/[14(52)]
q3 = (60 – 5p3 + 2p2 + 2p1)/7
= (60 – 5p3 + 4pm)/7
= (60 – 5(300 + 21c)/26 + 4[(720 + 40c)/52])/7
Substituting outputs and prices into the profit functions give profits:
πM = 3(60-c)2(144)/[14(26)2]
π3 = = (60-c)2(125)/[7(26)2]
Recall that pre-merger profit was:
π* = 5(60 – c)2/252
Clearly, post-merger profit is higher for both the merging and non-merging firms.
Consumers are worse off, because post-merger prices are higher and output is lower.
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.
The retailer solves:
MaxQ : (P – r – c2)Q