EconS 425 - 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, and pro…ts of both the upstream and down- stream …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 maximization problem,

max (a bqD)qD 2pU qD qD with …rst-order condition,

@D = a 2bqD 2pU = 0 @qD

and solving this for qD gives us our downstream quantity as a function of the upstream price, a 2pU qD = 2b From here, we need to use our clearing condition. Since the downstream …rm needs two units of the upstream …rm’sproduct, we know that their equilibrium quantity D qU is half the upstream …rm’s, i.e., q = 2 . Substituting this into our downstream demand function gives us the upstream …rm’sdemand,

qU a 2pU = 2 2b and solving this expression for pU gives us the upstream …rm’sinverse demand function, a b pU = qU 2 2 Now we can set up the upstream …rm’spro…t maximization function,

a b max qU qU cqU qU 2 2   with …rst-order condition, @U a = bqU c = 0 @qU 2

1 and solving for qU , we obtain the upstream quantitiy,

U a 2c q  = 2b Plugging it back into the upstream inverse demand function gives the upstream price,

U a b a 2c a + 2c p  = = 2 2 2b 4   and lastly, upstream pro…ts

2 U U U (a 2c)   = (p  c)q  = 8b Moving back to the downstream market, we can plug in the upstream price to get our downstream quantity, a+2c D a 2 4 a 2c q  = = 2b 4b  which is exactly half of the upstream …rm’squantity (a good thing to check). Lastly, the downstream price is

D a 2c 3a + 2c p  = a b = 4b 4   and the downstream pro…t level is

2 D D U D (a 2c)   = (p  2p )q  = 16b

2. 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 up- stream production costs (it still takes double the upstream production cost to make the downstream product), max (a bq)q 2cq q with …rst-order condition, @ = a 2bq 2c = 0 @q and solving this expression for q gives us our equilibrium quantity, a 2c q = 2b Plugging this back into the inverse demand function gives us the equilibrium price, a 2c a + 2c p = a b = 2b 2   2 and lastly, equilibrium pro…ts, (a 2c)2  = (p 2c)q = 4b Taking the di¤erence of the vertically integrated …rm’s pro…ts and the sum of the upstream and downstream pro…ts, (a 2c)2 (a 2c)2 (a 2c)2 (a 2c)2 + = 4b 8b 16b 16b   gives us the increase the vertically integrated …rm experiences since it removes double marginalization.

Assignment 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

p = a bq1 bq2 and for simplicity, cD = cU = 0.

1. If no vertical integration is possible, what are the equilibrium pro…ts of both the up- stream and downstream …rms?

Starting with downstream …rm 1, their pro…t maximization problem is max (a bqD bqD)qD pU qD D 1 2 1 1 q1 with …rst-order condition,

D @1 D D U D = a 2bq1 bq2 p = 0 @q1 D and solving this for q1 gives us downstream …rm 1’s best response to any quantity produced by …rm 2, a pU qD qD = 2 1 2b 2 We can perform the same steps for downstream …rm 2, but since both downstream D D …rms are identical, we can invoke symmetry and set q1 = q2 , updating our above best response function, 3qD a pU 1 = 2 2b D and solving for q1 gives both downstream …rms’quantity as a function of the upstream …rm’sprice, a pU qD = qD = 1 2 3b

3 For the market clearing condition, we know that the upstream …rm has to supply both U D D downstream …rms, and thus q = q1 + q2 . Substituting, a pU a pU 2(a pU ) qU = + = 3b 3b 3b and solving this expression for pU gives us the upstream …rm’sinverse demand function, 3b pU = a qU 2 Setting up the upstream …rm’spro…t maximization problem,

3b max a qU qU qU 2   with …rst-order condition, @U = a 3bqU @qU and solving for qU gives us the upstream …rm’squantitiy,

U a q  = 3b Lastly, plugging this back into the upstream inverse demand function gives the up- stream price, U 3b a a p  = a = 2 3b 2 which gives us the upstream pro…t level,  

2 U U U a   = p q  = 6b Now we return to the downstream market and …nd our downstream quantities,

a D D a 2 a q  = q  = = 1 2 3b 6b and equilibrium downstream price

D a a 2a p  = a b b = 6b 6b 3     which gives us downstream pro…ts

2 D D D U D a   =   = (p  p )q  = 1 2 36b

4 2. 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 maximization problem is

max (a bqI bqD)qD pU qD qD with …rst-order condition, @D = a bqI 2bqD pU = 0 @qD and solving this expression for qD, we have the unintegrated …rm’s best response to the integrated …rm’squantity and the upstream price, a pU qI qD = 2b 2 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 maximization problem is max (a bqI bqD)qI + pU qU qI ;pU For now, I’ll just take one …rst-order condition here (we’ll come back),

@I = a 2bqI bqD = 0 @qI and solving this for qI , we have the integrated …rm’sbest response function a qD qI = 2b 2 From this, we can now express the quantities of the integrated and unintegrated …rms as a function of the upstream …rm’sprice. a pU 1 a qD qD = 2b 2 2b 2   3qD a 2pU = 4 4b a 2pU qD = 3b a 1 a 2pU a + pU qI = = 2b 2 3b 3b   Now, the amount the upstream …rm supplies to the downstream …rm has to equal how much they demand, i.e., qU = qD, a 2pU qU = 3b

5 Let’sreturn to our vertically integrated …rm’spro…t maximization problem, max (a bqI bqD)qI + pU qU qI ;pU Pretty much everything here is a function of pU . 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, a + pU a 2pU a + pU a 2pU max a b b + pU qI ;pU 3b 3b 3b 3b          a + pU a + pU a 2pU max + pU qI ;pU 3 3b 3b       with …rst-order condition @I 2a + 2pU a 4pU = + = 0 @pU 9b 3b and solving this expression for pU ,

U a p  = 2 Substituting these values back into the downstream components, a I a + 2 a q  = = 3b 2b a D a 2 2 q  = = 0 3b  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 pro…ts, 2 U a   = 4b

Assignment 6-3

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 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 qD qD 1 2 1. Calculate the equilibrium downstream market price (not as a function of the upstream price).

In the downstream market, …rm 1’spro…t maximization problem is

max (a qD qD)qD (cD + pU )qD D 1 2 1 1 q1 with …rst-order condition, @D = a 2qD qD cD pU = 0 @qD 1 2

D and solving this expression for q1 gives downstream …rm 1’sbest response to …rm 2’s quantity and the upstream price, a cD pU qD qD = 2 1 2 2 From here, we know that the downstream …rms are identical, so let’sinvoke symmetry, D D q1 = q2 , 3qD a cD pU 1 = 2 2 a cD pU qD = qD = 1 2 3 Our market clearing condition tells us that the upstream …rm must supply exactly U D D what the downstream …rms demand, thus q = q1 + q2 . Substituting, 2(a cD pU ) qU = 3 and solving for pU , we have the upstream inverse demand function 3 pU = a cD qU 2 which allows us to setup the upstream …rm’spro…t maximization problem, 3 max a cD qU qU cU qU qU 2   with …rst-order condition, @U = a cD 3qU cU = 0 @qU

7 and …nally, solving this expression for qU gives us the upstream …rm’squantity,

D U U a c c q  = 3 Plugging this back into the upstream inverse demand function gives us the upstream price, D U D U U D 3 a c c a c + c p  = a c = 2 3 2   From here, we can return to the downstream market to get the downstream quantity,

D a cD+cU a c D U D D 2 a c c q  = q  = = 1 2 3  6 and …nally, the downstream price,

D U D U D U D a c c a c c 2a + c + c p  = a b b = 6 6 3    

2. 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 > cD + cU .

Comparing, 3a + cD + cU 2a + cD + cU a cD cU = 4 3 12 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 down.

3. 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 Practice Problem

Suppose we had an upstream and downstream monopolist. The upstream monopolist pays a constant of cU and the downstream …rm must simply purchase from the upstream …rm at a price of pU . The downstream …rm can also o¤er retail services at a cost of s2. Market inverse demand in the downstream market is as follows, qD p = a s 1. If the upstream and downstream …rms vertically integrate, what is the equilibrium level of retail services?

In this case, the single …rm’spro…t maximization problem is q max a q cU q s2 q;s s   with …rst-order conditions @ 2q = a cU = 0 @q s @ q2 = 2s = 0 @s s2 which is two-equations and two unknowns. Rearranging the second equation, 2s3 = q2 2 q 3 s = 1 2 3 (this is pretty awful). Substituting this into the …rst equation,

2q U 2 = a c q 3 1 2 3 4 1 U 2 3 q 3 = a c (a cU )3 q = 16 and lastly, 2 (a cU )3 3 U 2 16 (a c ) s = 1 =  2 3  8

2. If no vertical integration occurs, what is the equilibrium level of retail services?

Starting with the downstream …rm, their pro…t maximization problem is qD max a qD pU qD s2 qD;s s   9 with …rst-order conditions @ 2qD = a pU = 0 @qD s @ (qD)2 = 2s = 0 @s s2 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 cU with pU )

(a pU )3 qD = 16 (a pU )2 s = 8 Once again, we can use the market clearing condition qD = qU to generate an upstream demand function (a pU )3 qU = 16 To keep things (relatively) simple, I’ll setup the upstream …rm’s pro…t maximization problem di¤erently, (a pU )3 max (pU cU ) pU 16   with …rst-order condition,

@U (a pU )3 3(a pU )2 = (pU cU ) = 0 @pU 16 16 Rearranging terms, 3(pU cU ) = a pU and solving for pU , a + 3cU pU = 4 From here, we can …nd our level of retail services,

a+3cU 2 U 2 (a 4 ) 9 (a c ) s = =  8  16 8  

3. Compare parts 1 and 2. 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.

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