EconS 425 - Vertical Integration Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 26
Introduction Let s continue our discussion of vertical mergers. Last time, we saw that when one upstream and one downstream rm acted as monopolists, vertical integration increases pro ts and consumer surplus. These would be quite desirable for society. Today, we ll expand to several rms and look at the e ects of imperfect competition. Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 26
Suppose we had a case where there was one upstream rm and two downstream rms. To make it simple, we ll assume that the downstream rms serve di erent markets. For example, think about a single American aluminum company that is the only aluminum provider to both the US and the UK. The rms in the US and UK likely don t compete with one another, but the Aluminum rm has a monopoly on the supply to both of them. To make things more interesting, we ll assume that the two downstream rms face di erent market demands. Under these conditions, the upstream rm will want to price discriminate between the two rms. Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 26
Can the upstream rm implement price discrimination though? Remember that they need market power, di erentiated consumers, and the ability to prevent arbitrage. While the upstream rm de nitely has both market power and di erentiated consumers, they have no way of preventing arbitrage. What prevents the more elastic downstream rm from selling excess upstream product to the other rm? The upstream rm could impose a "no resale" clause in any contractual agreement, but almost all of those get struck down as anticompetitive. Let s look at what happens when no price discrimination is possible. Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 26
Suppose that the two downstream rms face the following inverse market demand curves p D i = a i b i q D i i = 1, 2 and must each purchase one unit of the upstream rm s output as an input for their product at a price of p U. The downstream rms also face a constant marginal cost of production of c D while the upstream rm faces a constant marginal cost of production of c U. Firm i s pro t maximization problem is max q i (a i b i q D i )q D i (c D + p U )q D i and taking a rst-order condition with respect to q D i gives us π D i qi D = a i 2b i q D i c D p U = 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 26
a i 2b i q D i c D p U = 0 Solving this expression for qi D gives us rm i = 1, 2 s output level as a function of the price charged by the upstream rm, q D i = a i c D p U 2b i Notice that for both downstream rms, nothing that the other rm does a ects the quantity that they product. Each rm is a local monopoly, so there is no strategic interaction between the rms. We ll explore that possibility later. Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 26
q D i = a i c D p U 2b i Remember before that the upstream rm s equilibrium output level must be exactly the input level demanded by the downstream rms. This is a market clearing condition from general equilibrium. Since one unit of the upstream rm s output is required to make one unit of each downstream rm s output, the total amount of inputs required are q U = Q D = q D 1 + q D 2 = a 1 c D p U 2b 1 + a 2 c D p U = a 1b 2 + a 2 b 1 (c D + p U )(b 1 + b 2 ) 2b 1 b 2 2b 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 26
q U = a 1b 2 + a 2 b 1 (c D + p U )(b 1 + b 2 ) 2b 1 b 2 Doing some algebra, I can solve this expression for p U, which gives us the inverse demand function for the upstream rm s output, q U = a 1b 2 + a 2 b 1 c D (b 1 + b 2 ) 2b 1 b 2 b 1 + b 2 p U 2b 1 b 2 b 1 + b 2 p U = a 1b 2 + a 2 b 1 c D (b 1 + b 2 ) 2b 1 b 2 2b 1 b 2 q U p U = a 1b 2 + a 2 b 1 c D (b 1 + b 2 ) 2b 1 b 2 q U b 1 + b 2 b 1 + b 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 26
From here, we can set up the upstream rm s pro t maximization problem max q U a1 b 2 + a 2 b 1 c D (b 1 + b 2 ) 2b 1 b 2 q U q U b 1 + b 2 b 1 + b 2 with rst-order condition c U q U π U q U = a 1b 2 + a 2 b 1 c D (b 1 + b 2 ) 4b 1 b 2 q U c U = 0 b 1 + b 2 b 1 + b 2 Solving this expression for q U gives us our output level for the upstream rm, q U = a 1b 2 + a 2 b 1 (c U + c D )(b 1 + b 2 ) 4b 1 b 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 26
We can then plug this value back into our derived inverse demand function to obtain the upstream rm s price p U = a 1b 2 + a 2 b 1 c D (b 1 + b 2 ) b 1 + b 2 2b 1 b 2 b 1 + b 2 q U = a 1b 2 + a 2 b 1 (c D c U )(b 1 + b 2 ) 2(b 1 + b 2 ) And lastly, we have our upstream rm s pro t level, π U = (p U c U )q U = a 1b 2 + a 2 b 1 (c U + c D )(b 1 + b 2 ) 2 8b 1 b 2 (b 1 + b 2 ) Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 26
Now we can return to our downstream rms and calculate their equilibrium values. To save some space, I ll just present the results. Firm i s output as a function of the other rm s (j s) parameters is, q D i = a i (2b i + b j ) a j b i (b i + b j )(c U + c D ) 4b i (b i + b j ) with accompanying price and pro ts of p D i = a i b i q D i = a i (2b i + 3b j ) + a j b i + (b i + b j )(c U + c D ) 4(b i + b j ) π D i = (p D i c U p U )q D i = a i (2b i + b j ) a j b i (b i + b j )(c U + c D ) 2 16b i (b i + b j ) 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 26
Let s compare some values to see how the di erent downstream markets compare. If we look at the di erence between prices, p D 1 p D 2 = a 1 a 2 2 if a 1 > a 2, then the rst downstream rm has consumers that are more willing to pay for the good than downstream rm 2. They will pay a higher price. The upstream rm would love to extract some of these pro ts from downstream rm 1, but they can t since they can t price discriminate. Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 26
What about pro ts? They re a lot harder to compare. We can look at some simple e ects by making a couple of assumptions. Let b 1 = b 2 = b, such that both downstream market inverse demand curves have the same slope. If we take the di erence of our pro ts, π D 1 π D 2 = (a 1 a 2 )(a 1 + a 2 2(c U + c D )) 8b In this case, if a 1 > a 2, then downstream rm 1 has higher pro ts than downstream rm 2. Again, the consumers in market 1 have a higher willingness to pay, so this should make sense. Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 26
Now, let a 1 = a 2 = a, such that consumers in both markets have the same willingness to pay. If we di erence our pro ts, π D 1 π D 2 = (b 2 b 1 )(a c U c D ) 2 18b 1 b 2 Which is positive when b 1 < b 2. Thus, the rm with the atter (more elastic) market inverse demand function will receive higher pro ts. We can combine those two results; a market that has a higher willingness to pay and more elastic consumers will yield higher pro ts. Without price discrimination, the downstream rms that receives higher pro ts is able to retain more of those pro ts for themselves. Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 26
How can an upstream rm get around this? What if they vertically integrated with one of the downstream rms? It s possible that they could integrate with both of the downstream rms, but that s not interesting. We ll assume that costs make it feasible to only integrate with one rm. We don t know which rm is the better choice for integration, so we ll say that the upstream rm integrates with downstream rm i. Firm j remains as the unintegrated downstream rm. Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 26
Starting with integrated rm i, their pro t maximization problem is with rst-order condition, max q i (a b i q i )q i (c U + c D )q i π i q i = a 2b i q i (c U + c D ) = 0 Solving this expression for q i gives us rm i s equilibrium output level, q i = a cu c D 2b i Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 26
q i = a cu c D 2b i From here, we can calculate the integrated rm i s market price and pro t level p i = a b i q i = a + cu + c D π i = (p i c U c D )q i = (a cu c D ) 2 which as you can notice, is quite similar to a monopolist s pro t level (and what we saw last time with an integrated rm) if we allow b = b i and c = c U + c D. 2 4b i Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 26
For the unintegrated downstream rm j, they must still purchase their input from integrated rm i. Now, however, the upstream component of our integrated rm can e ectively charge di erent prices to each downstream rm. It charges 0 to rm i since they are vertically integrated, and p U to rm j. The equilibrium values for rm j are identical to what we saw last time with one upstream and one downstream rm. We must simply let b = b j in that model and c = c U + c D. The equilibrium results are Upstream Downstream q a c U c D a c U c D 4b j 4b j p a+c U +c D 3a+c U +c D 2 4 π (a c U c D ) 2 (a c U c D ) 2 8b j 16b j Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 26
Let s compare these results. For the integrated market, if we di erence the integrated price and the preintegrated output levels, q i qi D = a i b j + a j b i (b i + b j )(c U + c D ) > 0 4b i (b i + b j ) we nd that as long as a i > c D + c D, the output from the integrated rm is higher than the output of the preintegrated rm. These consumers will be made better o by this arrangement. Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 26
Now let s compare prices. Di erencing the price paid by the unintegrated rm and their preintegration price, which is ambiguous. p U j p U = b j (a j a i ) 2(b i + b j ) If a j > a i, meaning that the upstream rm integrates with the downstream rm whose consumers have a lower willingness to pay, the price paid by the upstream rm increases. Their consumers are made worse o. Otherwise, both rms receive a lower price for the output of the upstream rm. Intuitively, when we can t price discriminate, markets that would normally receive a low price must pay a higher price due to the higher willingness to pay consumers. Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 26
The question remains which downstream rm the upstream rm should integrate with. After vertically integrating with rm i, the integrated rm s total pro t is Πi = πi + π U i = 1 2(ai c U c D ) 2 + (a j c U c D ) 2 8 b i b j and to make the notation a bit easier, let each rm s pro t marging be expressed as ã i a i c U c D. Substituting this into our pro t function,! Π i = 1 8 2ã 2 i b i + ã2 j b j Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 26
The upstream rm will want to integrate with downstream rm 1 instead of downstream rm 2 if 1 2ã 2 1 + ã2 2 8 b 1 b 2 Π1 > Π2 > 1 8 ã 2 1 b 1 > ã2 2 b 2 2ã 2 2 b 2 + ã2 1 b 1 Interpreting this, if the downstream rms have inverse demand functions with the same slope (b 1 = b 2 ), then the upstream wants to integrate with the rm with the higher willingness to pay in their market. If the downstream rms have markets with the same willingness to pay (a 1 = a 2 ), the upstream rm wants to integrate with the rm that has the atter inverse demand curve. Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 26
ã 2 1 b 1 > ã2 2 b 2 What s interesting about this is that there remains a range of values for a 1, a 2, b 1, and b 2 where vertical integration helps all consumers. If we let a 1 > a 2, but b 1 < b 2 ã1 ã 2 2, the upstream rm will integrate with rm 1, but both markets receive a lower price than before the integration occured. This would be an e cient arrangement for all parties involved. Unfortunately, this range is limited, and it s much more likely that prices will rise for some consumers. Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 26
Summary By vertically integrating with a downstream rm, an upstream monopolist can e ectively implement price discrimination. Depending on the market structure, it s possible that pro ts will rise and prices will fall for all those involved, which would be e cient. Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 26
Next Time Vertical Mergers in an Imperfect Market Context What happens when rms start limiting who they will supply or buy from? Reading: Chapter 12.3 Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 26
Homework 6-2 Return to our vertical integration example we looked at in class today. Suppose that instead of each downstream rm acting as a monopolist, they instead compete in quantities against one another. Market price is given as and for simplicity, c D = c U = 0. p = a bq 1 bq 2 1. If no vertical integration is possible, what are the equilibrium pro ts of both the upstream and downstream rms? 2. If the upstream rm can vertically integrate with one downstream rm, what are the equilibrium pro ts of both rms? (a bit tricky) Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 26