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I : Solution problem set

Oligopoly

  

Question 1

(a) An equilibrium in an starts with the same idea as in perfect or :

The clears (QD = QS). But, it adds the requirement that each …rm is doing the best it can (that is, choosing the quantity/ that maximizes its pro…t), conditional on the actions taken by its competitors (that is, the quantities/ chosen by all other …rms). As such, an oligopoly equilibrium has to be stable not only in equating total quantity supplied and demanded, but also in remaining stable among the individual producers in the market.

(b) Since products are (perfect) substitutes in a Cournot model, a …rm’s optimal rises as the other …rm’s output falls, which explains why reaction curves are downward-sloping. In a Bertrand model, however, a …rm’soptimal price decreases when its competitor’sprice decreases, which explains why reaction curves are upward-sloping.

(c) While are illegal in most developed countries, the lure of collusive pro…ts entices many …rms to enter into agreements. At the same time, cartel members have an incentive to cheat by producing more than the agreed-upon amount (or by lowering the price) and increasing their own pro…ts at the expense of other cartel members. In the absence of legal means to enforce the cartel agreement, cartels tacitly rather than explicitly coordinate their activities to avoid running afoul of competition . Cooperating to put informal operation rules in place to regulate activities was a means used by the Dutch cartels to build trust among members, reduce instability, and prolong the duration and pro…tability of the cartel.

(d) By di¤erentiating its product, a …rm makes the residual it faces less elastic every- where. For example, no consumer will buy from that …rm if its rival charges less and the are homogeneous. In contrast, some consumers who prefer this …rm’sproduct to that of its rival will still buy from this …rm even if its rival charges less. Hence, the lower the of demand at the equilibrium, the higher the price that a …rm sets.

(e) If the …rms produce identical goods, the equilibrium price is lower if they set price rather than quantity. If the goods are heterogeneous, we cannot answer this question de…nitively.

Question 2

(a) If the …rms collude and act like a monopolist, they will set MR = MC 32 4Q = 16 () () Q = 4. P  = 32 2 4 = $24 per ton. Assuming the …rms split output equally, the pro…t for m m 

1 1 1 each …rm is: SC = SBB = (P  AT C) Q = (24 16) 4 = $16. 2  m  m 2   (b) If Squeaky Clean produces one more ton, total quantity rises to 5. The new price is: P =

32 2 5 = $22, and Squeaky Clean’spro…t is: SC = (P AT C) qSC = (22 16) 3 = $18,    which is larger than the $16 under . Yes, Squeaky Clean has an incentive to cheat and produce one more ton of chlorine.

(c) If Squeaky Clean cheats, the price falls to $22. This reduces Biobase’spro…ts to:

BB = (P AT C) qBB = (22 16) 2 = $12.   (d) If each …rm produces one more ton of chlorine, the new price is: P = 32 2 6 = $20 and pro…ts 1 1  for each …rm are: SC = SBB = (P AT C) Q = (20 16) 6 = $12. Hence, both 2   2   …rms are worse o¤. When producing 4 tons of chlorine, the new price is: P = 32 2 7 = $18,  and Squeaky Clean’s pro…t is: SC = (P AT C) qSC = (18 16) 4 = $8, which is lower   than $12, so Squeaky Clean has no incentive to cheat.

Question 3

(a) If each …rm’s output is 12 and there are 4 …rms, then Q = 12 4 = 48. Hence, P = 100 Q =  100 48 = 52.

(b) member = (P AT C) q = (52 20) 12 = 12 32 = 384.   

(c) Qnew = 3 12+22 = 58, Pnew = 100 58 = 42. Firm 1’spro…t is: 1;new = (42 20) 22 = 484,   while the other …rms earn (42 20) 12 = 264.  (d) Firm 1’stotal revenue increase from 52 12 = 624 to 42 22 = 924, hence by 300. The quantity   e¤ect is (58 48) (42 0) = 420, which is a gain. The price e¤ect is (48 0) (52 42) = 480,   which is a loss. So the e¤ect on the cartel is negative ( 60). However, the private incentives of Firm 1 show why this cartel is unstable. Firm 1’sgain is the entire quantity e¤ect of 10 42 = 420, th  but its loss is just 1 of the price e¤ect = 1 48 10 = 120, so its net gain (in terms of increased 4 4  total revenue) is 420 120 = 300. (e) Each of the other …rms’pro…ts fall from 384 to 264, so pro…ts fall by 384 264 = 120.

Question 4

(a) In an oligopoly, when the products are identical, in this case identical quality of babysitting, the expected outcome is that price will equal . In this case, marginal cost is $5, and therefore the predicted price is $5 per hour.

(b) If the products of the two …rms, in this case two babysitters, are di¤erentiated from each other, then we expect that the price will be higher than $5 as each of the …rms (babysitters) will charge a price above marginal cost.

2 (c) The optimal price for the cartel is found the same way as the optimal price for a monopolist (MR = MC). One method of solving it is to create the following table: Price ($ per hour) Quantity (hours) TR MR per additional hour of babysitting 9 0 0 80 8 10 80 10 = 8 126 80 7 18 126 18 10 = 5:75 144 126 6 24 144 2418 = 3 150144 5 30 150 3024 = 1 136 150 4 34 136 3430 = 3:5 The optimal price is $7 per hour.

(d) An example of issues that could make one of them more likely to cheat would be how easy it is to detect cheating, how much they the pinky swear, and how they could be punished for breaking the promise.

Question 5

(a) Begin by substituting Q = qO + qG into the market inverse demand curve: P = 100 2Q = 100 2qO 2qG. From this inverse demand curve, derive each …rm’s MR-curve:

MRO = 100 4qO 2qG MRG = 100 2qO 4qG

Each …rm will set its equal to its marginal cost to maximize pro…t. Since marginal revenue is a function of the other …rm’sproduction choice, this represents the reaction curve.

MRO = 100 4qO 2qG = 12 qO = 22 0:5qG () MRG = 100 2qO 4qG = 20 qG = 20 0:5qO ()

(b) To solve for the equilibrium, substitute one …rm’sreaction curve into the other’s:

qO = 22 0:5qG qO = 22 0:5 (20 0:5qO) ()  q = 16 and q = 20 0:5 16 = 12 () O G 

Therefore, OilPro produces 16; 000 oil changes per year, while GreaseTech produces 12; 000. See graph below.

3 (c) The market price is found by substituting the market quantity into the market inverse demand

curve: P = 100 2q 2q = 44. The market price will be $44 per oil change. O G

(d) O = TRO TCO = (44 16; 000) (12 16; 000) = $512; 000. G = TRG TCG = (44    12; 000) (20 12; 000) = $288; 000. Hence, the …rm with the lower marginal cost provides more  oil changes and makes more pro…t.

(e) Since OilPro is going to move …rst and knows that GreaseTech’soutput is a function of its output, we need to substitute GreaseTech’sreaction curve into the market demand curve to solve for the inverse demand curve for OilPro:

P = 100 2qO 2qG P = 100 2qO 2 (20 0:5qO) ()  P = 60 qO ()

Hence, the MR-curve for OilPro is: MRO = 60 2qO. Setting MRO = MCO gives OilPro’s pro…t-maximizing output:

MRO = 60 2qO = MCO = 12 q = 24 () O

Substitute q = 24 into GreaseTech’sreaction curve to …nd q : q = 20 0:5 24 = 8. OilPro O G G  will produce 24; 000 oil changes, while GreaseTech will only produce 8; 000. Use the inverse

4 demand curve to …nd the market price: P = 100 2q 2q = $36. O;S = TRO;S TCO;S = O G (36 24; 000) (12 24; 000) = $576; 000. G = TRG;S TCG;S = (36 8; 000) (20 8; 000) =     $128; 000.

(f) If GreaseTech is the …rst mover, use OilPro’sreaction curve to …nd the inverse demand curve for OilPro and follow the same procedure as in (e).

P = 100 2qO 2qG P = 100 2 (22 0:5qG) 2qG ()  P = 56 qG () MRG = 56 2qG = MCG = 20 q = 18 () G q = 22 0:5 q = 13 O  G

Hence, GreaseTech will produce 18; 000 oil changes, while OilPro will only produce 13; 000.

P = 100 2q 2q = $38. G = TRG;S TCG;S = (38 18; 000) (20 18; 000) = $324; 000. O G   O;S = TRO;S TCO;S = (38 13; 000) (12 13; 000) = $338; 000.  

Question 6

Since the MR-curve is twice as steeply sloped as is the demand curve, the MR-curves corresponding to the inverse demand functions are:

MRA = 197 30:2qA 0:3qI MRI = 490 20qI 6qA Equating the MR-functions to the MC gives the best-response functions. For ADM, this gives:

MRA = 197 30:2qA 0:3qI = MC = 40. Solve for qA to obtain AMD’s best-response func- 157 0:3qI 450 6qA tion: qA = 30:2 . Similarly, Intel’s best-response function is: qI = 20 . By solving simul- taneously the system of best-response functions, we …nd the Nash-Cournot equilibrium quantities: 450 6q 157 0:3q A I 157 0:3( 20 ) 450 6( 30:2 ) qA = q 5 million CPU’s, and qI = q 21 million CPU’s. 30:2 , A  20 , I  Substituting these values into the respective inverse demand functions, we obtain the corresponding prices: P  = 197 15:1q 0:3q = $115:2 and P  = 490 10q 6q = $250 per CPU. A A I I I A

Question 7

(a) Your rival should undercut your price by the smallest possible increment, thereby stealing the entire market. If one eurocent is the smallest increment, then all 100 customers will pay e9:99, leaving your rival with a 100% market share and pro…ts of 100 (9:99 8:00) = e199. Meanwhile,  your pro…ts are zero.

5 (b) Your options include shutting down, innovation (reducing your costs to be more competitive), collusion (if you can’tbeat your rival, join him), and di¤erentiating your product. Use advertising to convince buyers that your product is special, and thus deserving a premium price.

Question 8

We are looking for an equilibrium in a di¤erentiated-products Bertrand market in which each …rm sets its price to maximize its pro…t, taking the prices of its competitors as given, that is, a Nash equilibrium. Burton and K2 set their price so that marginal revenue is equal to marginal cost (assumed to be zero). Hence:

MRB = 900 4PB + PK = MCB = 0 PB = 225 + 0:25PK () MRK = 900 4PK + PB = MCK = 0 PK = 225 + 0:25PB () These are the reaction curves for Burton and K2: As the competitor’sprice rises, the own price rises. To …nd the equilibrium prices, plug one company’sreaction curve into the other’s:

PB = 225 + 0:25 (225 + 0:25PB)  P  = $300 () B P  = 225 + 0:25 300 = $300 K  At equilibrium, both …rms charge the same price, $300. This isn’t too surprising. After all, the two …rms face similar-looking demand curves and have the same (zero) marginal costs. Graphically:

6 To know the quantity each …rm sells, plug each …rm’sprice into its demand equation. Burton’squantity demanded is qB = 900 2 300 + 300 = 600 snowboards. K2 also sells qK = 900 2 300 + 300 = 600   snowboards. Again, the fact that both …rms sell the same quantity is not surprising because they have similar demand curves and charge the same price. Total output is therefore 1; 200 snowboards.

Question 9

(a) HHI = 512 + 142 + 102 + 72 + 52 + 22 = 2:975.

2 2 2 2 2 (b) HHI = 61 + 14 + 7 + 5 + 2 = 3:995: Then HHI = HHINew HHIOld = 3; 995 2; 975 = 1:020:

$17:5 billion $2:5 billion (c) If $17:5 billion is 7% of the market, then each 1% of the market is worth 7% = 1% , and since iTunes and iBooks account for 14% + 10% = 24% of the market, they are collectively worth 24% $2:5 billion = $60 billion.  1%

Question 10

10.1.(a) 10.2.(d) 10.3.(c) 10.4.(a) 10.5.(b) 10.6.(b) 10.7.(c) 10.8.(c) 10.9.(c) 10.10.(b)

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