EconS Industrial Organization Assignment 6 Homework Solutions

Transcription

1 EconS 45 - Industrial Organization Assignment 6 Homework Solutions Assignment 6-1 Return to our vertical integration example we looked at in class today. Suppose now that the downstream rm requires two units of the upstream rm s output in order to produce one unit of their own output. 1. Calculate the equilibrium quantity, price and pro ts of both the upstream and downstream rms. Starting with the downstream rm, we need to take into consideration that they need two units of the input. Setting up their pro t imization problem, (a bq D )q D p U q D q D = a bqd p U = 0 and solving this for q D gives us our downstream quantity as a function of the upstream price, q D = a pu b From here, we need to use our market clearing condition. Since the downstream rm needs two units of the upstream rm s product, we know that their equilibrium quantity is half the upstream rm s, i.e., q D = qu. Substituting this into our downstream demand function gives us the upstream rm s demand, q U = a pu b and solving this expression for p U gives us the upstream rm s inverse demand function, p U = a b qu Now we can set up the upstream rm s pro t imization function, a b q U qu q U cq U = a 1 bq U c = 0

2 and solving for q U, we obtain the upstream quantitiy, q U = a c b Plugging it back into the upstream inverse demand function gives the upstream price, p U = a b a c = a + c b 4 and lastly, upstream pro ts U = (p U c)q U = (a c) 8b Moving back to the downstream market, we can plug in the upstream price to get our downstream quantity, q D = a a+c 4 = a c b 4b which is exactly half of the upstream rm s quantity (a good thing to check). Lastly, the downstream price is a c p D a + c = a b = 4b 4 and the downstream pro t level is D = (p D p U )q D = (a c) 16b. Suppose the two rms vertically integrated. How much do the pro ts of the vertically integrated rm increase? The vertically integrated simply uses the downstream inverse demand and the upstream production costs (it still takes double the upstream production cost to make the downstream product), (a bq)q cq = a bq c = and solving this expression for q gives us our equilibrium quantity, q = a c b Plugging this back into the inverse demand function gives us the equilibrium price, a c p = a b = a + c b

3 and lastly, equilibrium pro ts, = (p c)q = (a c) 4b Taking the di erence of the vertically integrated rm s pro ts and the sum of the upstream and downstream pro ts, (a c) (a c) (a c) (a c) + = 4b 8b 16b 16b gives us the increase the vertically integrated rm experiences since it removes double marginalization. Assignment 6- 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 1. If no vertical integration is possible, what are the equilibrium pro ts of both the upstream and downstream rms? Starting with downstream rm 1, their pro t imization problem is q D 1 (a bq D 1 bq D )q D 1 p U q D D D 1 = a bq D 1 bq D p U = 0 and solving this for q1 D gives us downstream rm 1 s best response to any quantity produced by rm, q1 D = a pu q D b We can perform the same steps for downstream rm, but since both downstream rms are identical, we can invoke symmetry and set q1 D = q D, updating our above best response function, and solving for q1 D rm s price, q D 1 = a pu b gives both downstream rms quantity as a function of the upstream q D 1 = q D = a pu

4 For the market clearing condition, we know that the upstream rm has to supply both downstream rms, and thus q U = q1 D + q D. Substituting, q U = a pu + a pu = (a pu ) and solving this expression for p U gives us the upstream rm s inverse demand function, q U p U = a qu Setting up the upstream rm s pro t imization problem, a q U U = a qu and solving for q U gives us the upstream rm s quantitiy, q U = a Lastly, plugging this back into the upstream inverse demand function gives the upstream price, p U a = a = a which gives us the upstream pro t level, U = p U q U = a 6b Now we return to the downstream market and nd our downstream quantities, and equilibrium downstream price q1 D = q D = a a p D = a which gives us downstream pro ts a b 6b = a 6b a b = 6b a D 1 = D = (p D p U )q D = a 6b 4

5 . If the upstream rm can vertically integrate with one downstream rm, what are the equilibrium pro ts of both rms? (a bit tricky) Starting with the unintegrated rm, their pro t imization problem is (a bq I bq D )q D p U q D q D = a bqi bq D p U = 0 and solving this expression for q D, we have the unintegrated rm s best response to the integrated rm s quantity and the upstream price, q D = a pu b For the integrated rm, they have pro t from two sources: selling in the downstream market and supplying the unintegrated downstream rm. Their pro t imization problem is q I ;p U (a bq I bq D )q I + p U q U For now, I ll just take one rst-order condition here (we ll come back), q I = a bqi bq D = 0 and solving this for q I, we have the integrated rm s best response function q I = a b q D From this, we can now express the quantities of the integrated and unintegrated rms as a function of the upstream rm s price. q D = a pu 1 a q D b b q D = a pu 4 4b q D = a pu q I = a 1 a b p U = a + pu Now, the amount the upstream rm supplies to the downstream rm has to equal how much they demand, i.e., q U = q D, q U = a pu 5

6 Let s return to our vertically integrated rm s pro t imization problem, (a bq I bq D )q I + p U q U q I ;p U Pretty much everything here is a function of p U. Intuitively, the integrated rm knows that if it raises the price it charges to the unintegrated rm, it lowers that rm s ability to compete as much in the downstream market. Substituting everything we have thusfar, q I ;p U q I ;p U with rst-order condition a + p U a p U a + p U a p a b b + p U U a + p U a + p U a p + p U U and solving this expression for p I a + pu 9b p U = a + a 4pU = 0 Substituting these values back into the downstream components, q I = a + a = a b q D = a a = 0 Intuitively, the vertically integrated rm sets their price so high, that the unintegrated rm does not enter the market at all. Thus, the downstream rm makes no pro ts, while the upstream rm enjoys monopoly pro ts, U = a 4b Assignment 6- Consider the situation in our second example, where a rm charges a squeeze price to its competitor to prevent them from vertically integrating. If you were a regulator, how could you detect such behavior in this market? What could you do to either punish or prevent this behavior? There are many answers to this problem, but basically, if a rm is o ering a product for sale in a competitive market, but isn t selling anything, wouldn t that seem strange to a regulator? Why even have that product for sale? A regulator could ne a rm for doing this or simply impose a price ceiling to prevent it. They could also subsidize a merger from the two remaining rms. 6

7 Assignment 6-4 Consider our example from today, but instead of one downstream rm, let there be two and they compete in quantities. Inverse market demand becomes p = a q D 1 q D 1. Calculate the equilibrium downstream market price (not as a function of the upstream price). In the downstream market, rm 1 s pro t imization problem is q D 1 (a q D 1 q D )q D 1 (c D + p U )q D D = a qd 1 q D c D p U = 0 and solving this expression for q D 1 gives downstream rm 1 s best response to rm s quantity and the upstream price, q D 1 = a cd p U From here, we know that the downstream rms are identical, so let s invoke symmetry, q D 1 = q D, q D q1 D = a cd p U q1 D = q D = a cd p U Our market clearing condition tells us that the upstream rm must supply exactly what the downstream rms demand, thus q U = q1 D + q D. Substituting, q U = (a cd p U ) and solving for p U, we have the upstream inverse demand function p U = a c D qu which allows us to setup the upstream rm s pro t imization problem, a c D q U qu q U c U q U = a cd q U c U = 0 7

8 and nally, solving this expression for q U gives us the upstream rm s quantity, q U = a cd c U Plugging this back into the upstream inverse demand function gives us the upstream price, p U = a c D a c D c U = a cd + c U From here, we can return to the downstream market to get the downstream quantity, a c D a q D 1 = q D = and nally, the downstream price, a c p D D c U = a b 6 c D +c U = a cd c U 6 a c D c U b = a + cd + c U 6. Compare the result in part 1 with the monoply price from our example. Do the downstream rms charge a higher price than the monopoly price? You can assume a > c D + c U. Comparing, a + c D + c U a + c D + c U = a cd c U 4 1 which is positive, given our assumption. Thus, the price charged in a downstream duopoly is less than what we see in a downstream monopoly. This should make sense, as competitive pressure drives prices down.. What could the upstream implement as a RPM agreement? How would this change the pro ts of all three rms? The upstream rm could implement a simple franchising agreement where rms charge the monopoly price in the downstream market, and pay a lump sum of their pro ts to the upstream rm. This will work out identically to the example in class where the downstream rms have zero pro ts and the upstream rm makes monopoly pro ts. Assignment 6-5 No Assignment. No Class. 8

9 Practice Problem Suppose we had an upstream and downstream monopolist. The upstream monopolist pays a constant marginal cost of c U and the downstream rm must simply purchase from the upstream rm at a price of p U. The downstream rm can also o er retail services at a cost of s. Market inverse demand in the downstream market is as follows, p = a q D s 1. If the upstream and downstream rms vertically integrate, what is the equilibrium level of retail services? In this case, the single rm s pro t imization problem is a q c U q s with rst-order = a q s q s = q s s = 0 c U = 0 which is two-equations and two unknowns. Rearranging the second equation, s = q s = q (this is pretty awful). Substituting this into the rst equation, and lastly, s = q q 1 1 = a c U 4 q 1 = a c U q = (a cu ) 16 (a c U ) 16 1 = (a cu ) 8. If no vertical integration occurs, what is the equilibrium level of retail services? Starting with the downstream rm, their pro t imization problem is q D a q D p U q D s q D ;s s 9

10 with = a q D p U = 0 = (qd ) s s = 0 We can follow the same steps as in part 1 to obtain the downstream quantity and price as functions of the upstream price (Just replace c U with p U ) q D = (a pu ) 16 s = (a pu ) 8 Once again, we can use the market clearing condition q D = q U to generate an upstream demand function q U = (a pu ) 16 To keep things (relatively) simple, I ll setup the upstream rm s pro t imization problem di erently, (a p (p U c U U ) ) p U = (a pu ) U 16 (a p U ) (p U c U ) = 0 16 Rearranging terms, and solving for p U, (p U c U ) = a p U p U = a + cu 4 From here, we can nd our level of retail services, s = (a a+c U 4 8 ) = 9 16 (a c U ) 8. Compare parts 1 and. What happens to retail services? It s fairly easy to see that retail services fall. Remember that the downstream rm s incentives are not the same as the upstream s, and they ll cut back on retail services to increase their pro ts. Interestingly, they almost cut retail services in half. 10