Answer Key: Problem Set 1

Transcription

1 Answer Key: Problem Set 1 Econ Fall Question 1 a The profit function (revenue minus total cost) is π(q) = P (q)q cq The first order condition with respect to (henceforth wrt) q is P (q )q + P (q ) = c Note that P (q) = β in this example Using this, the FOC leads to βq + (α βq ) = c, or q = α c β Plug this into the demand function to obtain P = P (q ) = α + c b Recall the demand elasticity is ε := dq P dp q Therefore, by dividing the FOC by P we have c The maximized profit is q P dp (q ) } dq {{ } 1/ε +1 = c P, or P = c 1 + 1/ε π(q ) = P q cq = (P c)q = α c α c β (α c) = 4β d We can compute the consumer s surplus by integrating the excees of marginal WTP (the inverse demand 1

2 function P (q)) over the equilibrium price P up to the equilibrium quantity q : q q (P (q) P )dq = 0 0 (α βq P )dq = (α P )q β (q ) = β(q ) = (α c), 8β where the penultimate equality uses P = α βq e The efficient quantity q e satisfies P (q e ) = c (ie, the marginal WTP equals the social marginal cost c), or q e = (α c)/β The deadweight loss is the integral of the marginal WTP (net of the marginal cost) that would have been additionally realized in the efficient equilibrium: q e q (P (q) c)dq = q e (α βq c)dq = (α c)(q e q ) β (qe ) (q ) q = (α c)(q e q ) β (qe + q )(q e q ) = α c 4 α c β (α c) = 8β = [ α c β(qe + q ) ] (q e q ) Remark In fact, we could as well have answered these using a graph more easily; see Figure 1 P α CS π DW L P c MC P (q) O q MR q e q Figure 1: Welfare implication of monopoly f Since the demand (of an individual) is q(p ) = 10 P, the market demand is Q(P ) = 100q(P ) = 100(10 P ), and the inverse market demand is P (Q) = 10 Q/100 Therefore we can plug α = 10,

3 β = 1/100 into the results above: π = DW L = (α c) 4β (α c) 8β = 100 = 100 (10 c) 4 (10 c) 8 Remark Note that we would get the same answer when we applied α = 10 and β = 1 (the individual demand) and then scaled up the resulting profit and deadweight loss by 100 This holds more generally; when the composition of consumers remains the same, scaling the market size does not change the equilibrium price, while the equilibrium quantity and the total surplus are scaled accordingly (as long as the consumers are price-takers and the marginal cost is constant) Intuitively, facing 100 identical consumers in a market is equivalent for the monopolist to dealing with 100 duplicate markets each of which has one price-taking consumer with the individual demand function, provided that the consumer behavior and the marginal cost are scale-invariant g Consider a two-part tariff T = a + bq Since there is only one type of consumers, the monopolist sets b = c to induce the efficient quantity q e and then set a = q e 0 (P (q) c)dq = qe (α c) = (10 c) to extract all the surplus from each consumer The deadweight loss equals zero, and the profit per consumer is a + bq e cq e = a = (10 c) / Hence the profit is 100 (10 c) / = 50(10 c) Question Assuming that the firm sells the good to both consumers, it chooses a and b in the two-part tariff T (q) = a+bq to maximize the profit (total revenue from two consumers minus total cost) π = (a + bq 1 + a + bq ) c(q 1 + q ) = a + (b c)(q 1 + q ), 3

4 where q i is the demand of consumer i = 1, The firm knows q i depends on the tariff it offers 1 Then what level of q i does consumer i chooses given a and b? Given a and b, consumer i s optimization problem max[θ i v(q) a bq] q yields the FOC θ i v (qi ) = b Since in this question v (q) = 1 q, the demand satisfies θ i (1 qi ) = b, or q i (b) = 1 b/θ i By the way we know that at the optimal (a, b), the low type (consumer 1) has zero surplus: θ 1 v(q 1) (a + bq 1) = 0 Plugging the last display into this we have θ 1 1 (b/θ 1) = a + b(1 b/θ 1 ), or a = (θ 1 b) Now the firm s problem is in terms of b; the consumer s reaction q i a also depends on b: depends on b and the optimal cover max[a(b) + (b c)(q1(b) + q(b))] b Take the derivative wrt b to obtain the FOC: a (b) + (q 1 + q ) + (b c)((q 1) (b) + (q ) (b)) = 0 (b/θ 1 ) + ( b/θ 1 b/θ ) + (b c)( 1/θ 1 1/θ ) = 0 b = (θ 1 + θ )c θ 1 Accordingly the optimal a is a = a(b ) = [θ 1 (θ 1 + θ )c] 8θ 1 and the profit is π = a + (b c)(q 1 (b ) + q (b )), provided that the firm intends to sell the good to both consumers Alternatively, the firm may sell the good only to the high type (consumer ) In this case the firm sets 1 As such, we cannot simply take the partial derivative of π wrt a and b to derive the optimal tariff here; we need to take into account the effect a and b have on q 1 and q 4

5 b = c and ( ) â = θ v q( b) cq( b) to extract all the surplus from consumer The profit, then, is π = â + bq( b) }{{} cq }{{ ( b) } csr expenditure total cost = θ 1 (c/θ ) If π π (depending on the given parameters θ 1, θ, and c), then the firm sets two-part tariff T = a +b q Otherwise, the firm sets T = â + bq Question 3 a We may normalize the total number of consumers to be, so that we have one consumer in each group Since they have different demand functions, the market demand is a little tricky Adding the two individual demand curves horizontally (see Figure ), we obtain the market demand function 10 p 18/5 p 5 5 Q/ 0 Q 8 Q =, or p =, (10 p) + (18 5p) 0 p 18/5 4 Q/7 8 Q 8 and hence 5 Q 0 < Q < 8 MR(Q) = 4 Q/7 8 < Q < 8 By MR(Q ) = c, the monopoly quantity Q is 7 and the price P is 3 (although the quantity is not meaningful in itself because it is subject to the size of the market; if there are 0 patrons in total, for example, then the equilibrium price and quantity would be 3 and 70, respectively) We can solve it in a more intuitive way The manager has two options: (i) attract young patrons only (ii) attract old patrons only (iii) attract both types of patrons Option (i) is infeasible; to attract young patrons, the price should not exceed 18/5 Since 18/5 is less than the highest (marginal) WTP 5 of an old patron, any such price will also attract old patrons In (ii), the inverse demand faced by the manager is p = 5 q/ Applying Question 1a, the optimal price is (5 + )/ = 35 This attracts the young patrons 5

6 p 5 P = 3 c = MC D MR O 8 Q = 7 8 Q Figure : Market demand as the horizontal sum as well, contradicting that the manager wants to attract the old patrons only In (iii), the inverse demand faced by the manager is p = 4 q/7 Applying Question 1a, the optimal price is (4 + )/ = 3 As this price attracts both types, it is consistent with the assumption of (iii), and hence the answer Remark If in case (ii) the optimal price were, say, 45, then we would have two candidates for the optimal price, namely 45 and 3; price at 45 would attract old patrons only (high price and small quantity), keeping young patrons out, whereas price at 3 would attract both types (low price and large quantity) The optimal price, then, depends on the profits from these two scenarios b The manager now runs two monopolistic markets (3rd PD) In one market, there are only the patrons under 5 with the inverse demand function p = (18 q)/5 Again using the results from Question 1a, the price for young patrons is (18/5 + )/ = 8 Analogously we have the price (5 + )/ = 35 for patrons over 5 c We approach this part as in Question 1g The manager sets b = c = for both groups For patrons under 5, the demand would then be q e = 18 5b = 8 according to the demand function Using the results from Question 1g, the manager sets a = q e (α c)/ = 8 (18/5 )/ = 64 to exploit the total surplus created (be sure to use α = 18/5 from the inverse demand function p = (18 q)/5 when applying Question 1g); the two-part tariff for patrons under 5 is therefore T = 64 + q, and the profit is 64 per young patron Similarly, the manager employs the two-part tariff T = 9 + q for old patrons, and the profit is 9 per old patron This is consistent with Figure In the region to the left of Q = 8, only the old patrons are willing to enter the nightclub But MR never reaches the MC, implying that it is not optimal to operate in this region 6

7 d As in part c (since the drink price is the same as in c), the manager charges 64 as the cover charge If the cover charge were lower than 64, then the consumer surplus would be strictly positive for both groups Thus the manager could slightly increase the cover fee to increase the profit while still attracting both groups If the cover were higher than 64, then the students would not come to the nightclub The profit is 64 per young patron, and also 64 per young patron; since the drink price equals the marginal cost, the profit comes not from selling additional drinks but only from the cover fee e The after-midnight market is just a market comprising of the patrons under 5 only Using the result in part b, the price after midnight is 8 Using the result from Question 1c, the profit is (α c) 4β = (18/5 ) 4 1/5 = 3 per young patron For before-midnight market, let s normalize the total number of patrons to be 1; we have /7 young patrons and 5/7 old patrons As before, it is not feasible to attract only young patrons Suppose the manager wants to attract only old patrons, thus facing the demand function q = (5/7) (10 p), or the inverse demand function p = 5 (7/10)q Applying Question 1a, the optimal price is then (5+)/ = 35 Since the highest marginal WTP of a young patron is 36 > 35, this price will attract young patrons as well, leading to a contradiction Now suppose the manager wants to attract both types of patrons, thus facing the demand function q = (5/7) (10 p) + (/7) (18 5p), or the inverse demand function p = 43/10 7q/0 Applying Question 1a, the optimal price is 315, which attracts both types of patrons In sum, the manager sets price at 315 in the before-midnight market The profit is then (α c) 4β = (43/10 ) 4 7/0 378 by Question 1c Question 4 a Let type 1 refer to those who have the demand curve p = 5 q 1 /, and type those who have p = 10 q ; we have λ of type 1 consumers and 1 λ of type consumers Consider a two-part tariff T = a + bq As can be seen in Figure 3, as long as the monopolist wants to induce both types of consumers to buy, the optimal a is the surplus type 1 consumers would enjoy under the 7

8 p 10 D 5 b a O 10 b D 1 10 q Figure 3: Two-part tariff with two types price b (before introducing a) Therefore, a(b) = (5 b) (the red triangle) Check that type consumers still have positive surplus (the blue area) even after paying a, and hence such a is consistent with the scenario where both types buy the good Now the problem boils down to choosing the optimal b: max b λ Rewrite the objective function as [a(b) + (b c)q 1(b)] }{{} profit per a unit of type 1 consumer +(1 λ) [a(b) + (b c)q (b)] }{{} profit per a unit of type consumer λ[a(b) + (b c)q 1 (b)] + (1 λ)[a(b) + (b c)q (b)] = a(b) + (b c)[λq 1 (b) + (1 λ)q (b)] = (5 b) + (b )[λ(10 b) + (1 λ)(10 b)] = (5 b) + (b )[10 (1 + λ)b] where the second equality uses a(b) we derived above, the demand functions, and c = The FOC wrt b yields (5 b ) + (10 (1 + λ))b (b )(1 + λ) = 0, or b = 1 + λ λ Note that b = 1 + 1/λ > since λ < 1 Also this result makes sense only if b 5, or λ 1/4, for otherwise type 1 consumer will not buy any amount 8

9 Now the optimal two-part tariff (attracting both types) is T = ( λ ) λ λ λ q, and the optimized profit is a + λ((b c)q 1 (b )) + (1 λ)((b c)q (b )), or π a = ( λ ) ( ) ( 1 + λ + λ λ ) ( ) ( 1 + λ + (1 λ) λ ) λ λ λ λ λ b Based on the comparison of demand functions q 1 = 10 p < 10 p = q, we see that type consumers are high-demand consumers If only high-demand consumers buy the good, the monopolist sets b = c = and a to be the surplus a type consumer enjoys under the unit price b (before introducing a), ie, a = 3 (the total area of red and blue regions when b = ) Therefore the two-part tariff is T = 3 + q, and the profit is π b = 3(1 λ) c If λ = 1/, then π a = 95 < 16 = π b Therefore it is optimal to employ the tariff as in part b in this case If λ = 3/4, then π a 9083 > 8 = π b, making the tariff as in part a be more profitable The result makes sense; as low-demand consumers become dominant, the monopolist has incentive to invite them to buy the product, forgoing some of the surplus extracted from the high-demand consumers Question 5 a Figure 4 shows the fully nonlinear tariff as proposed in the question Consider the consumer 1 s indifference curve that achieves zero utility (so that the profit from consumer 1 can be maximized while she is voluntarily participate) The maximization of the profit π 1 from consumer 1 occurs when a iso-profit line is tangent to the indifference curve, namely (q 1, T 1 ) in the figure Given this, to guarantee incentive compatibility while maximizing the profit from consumer, we consider the indifference curve that passes through (q 1, T 1 ) Similarly the maximization of π occurs when an iso-profit line is tangent to the indifference curve, say at (q, T ) b An idea is to move (q 1, T 1 ) slightly along to (q 1, T 1) consumer 1 s indifference curve See Figure 5 Since the iso-profit curve was tangent to the indifference curve at (q 1, T 1 ), π 1 rarely changes However, since consumer s indifference curve was not tangent to consumer 1 s indifference curve, consumer s new pretending-to-be-consumer-1 option (q 1, T 1) becomes less tempting than the previous one Since this outside option became worse for consumer, the monopolist can charge higher T and increase π, while inducing consumer to choose (q, T ) 9

10 T T = π + cq T π T 1 T = π 1 + cq u 1 (q, T ) 0 π 1 u (q, T ) u (q 1, T 1 ) O q 1 q q Figure 4: An example of fully nonlinear tariff (Question 5a) T T π T 1 T 1 π 1 π 1 O q 1 q 1 q q Figure 5: Another example of fully nonlinear tariff (Question 5b) Mathematically, let s lower (q 1, T 1 ) slightly to (q 1, T 1) along the indifference curve Then π 1 is almost unchanged since the iso-profit line was tangent to the indifference curve Now hold q fixed but raise T to make consumer be indifferent between (q, T ) and (q 1, T 1) (incentive compatibility) Note that voluntary participation (VP) holds for consumer 1 before and after the reform: T 1 = θ 1 v(q 1 ) T 1 = θ 1 v(q 1) 10

11 Also the incentive compatibility (IC) holds for consumer before and after the reform: θ v(q ) T = θ v(q 1 ) T 1 = T = θ (v(q ) v(q 1 )) + T 1 θ v(q ) T = θ v(q 1) T 1 = T = θ (v(q ) v(q 1)) + T 1 Therefore the profits from consumer before and after the reform are π = T cq = θ (v(q ) v(q 1 )) + T 1 cq π = T cq = θ (v(q ) v(q 1)) + T 1 cq, so that π π = θ (v(q 1 ) v(q 1)) + T 1 T 1 = θ (v(q 1 ) v(q 1)) + θ 1 (v(q 1) v(q 1 )) = (θ θ 1 )(v(q 1 ) v(q 1)) > 0, finishing the proof A more rigorous proof would require a formal application of calculus One such an approach was introduced in the lecture on September 10 Here we consider another approach, which follows the above argument more closely Note that given the choice of q 1, we can choose T 1, q, T optimally (optimal conditional on the choice of q 1 ); (q 1, T 1 ) should be on the indifference curve of consumer 1 that achieves zero utility (VP1), the indifference curve that passes through (q 1, T 1 ) (IC) should have slope c at q and T In other words, given q 1, the other three variables T 1, q, T are determined by θ 1 v(q 1 ) T 1 = 0 (VP1) θ v(q ) T = θ v(q 1 ) T 1 (IC) θ v (q ) = c (efficient q ) The third equation shows that q does not depend on q 1, and hence dq /dq 1 = 0 The first two equations 11

12 imply T 1 = θ 1 v(q 1 ) and T = θ v(q ) (θ θ 1 )v(q 1 ) Using these, π 1 = T 1 cq 1 = θ 1 v(q 1 ) cq 1 π = T cq = θ v(q ) + (θ 1 θ )v(q 1 ) cq Therefore (recall dq /dq 1 = 0) dπ 1 dq 1 (q 1 ) = θ 1 v (q 1 ) c dπ dq 1 (q 1 ) = (θ 1 θ )v (q 1 ) < 0 Let q e 1 be the level of q 1 in part a Note that q e 1 and the corresponding T 1 (q e 1), q (q e 1), and T (q e 1) represent the very tariff in part a We had θ 1 v (q e 1) = c This implies dπ 1 dq 1 (q e 1) = θ 1 v (q e 1) c = 0, which amounts to our intuition that slightly changing q 1 (and T 1 accordingly) would not change the profit from consumer 1 Now we have dπ dq 1 (q e 1) = dπ 1 dq 1 (q e 1) + dπ dq 1 (q e 1) = (θ 1 θ )v (q e 1) < 0 which means that the choice q e 1 in part a was not optimal, and the monopolist can increase the profit by decreasing q 1 (and adjust T 1 and T accordingly): downward distortion for low-type consumers Question 6 a The monopolist would charge each consumer her willingness to pay The profit would be = 45 b First consider the sports channel The monopolist may sell it either (i) to both consumers or (ii) to the consumer with the higher WTP (consumer 1 in this case) In (i), the monopolist sets the price at 8 to collect 16 in total In (ii), it sets the price at 15 to collect 15 Therefore the monopolist sets the price at 8 to obtain 16 from sports In a similar way, we see that the monopolist optimally sets the price of cooking channel at 10 to sell it to both consumers (because 10 > 1), collecting 0 The profit, therefore, is = 36 1

13 c The WTP of consumer 1 and consumer for the bundle are = 5 and = 0, respectively The monopolist accordingly wants to attract both consumers by setting the price at 0 (because 0 > 5), collecting 40 in total 13