PS5_solutions,_100A - Problem set 5 suggested solutions...

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Unformatted text preview: Problem set 5 - suggested solutions Question 1 (exercise 21.12 except e&g, add i&j) • (a) Draw a graph with the aggregate demand curve D0 for the “greenies.” Assume that green cars are competitively supplied at a market price p∗ — and draw in a perfectly elastic supply curve for green cars at that price. This is illustrated in panel (a) of Graph 21.12. • (b) There are two types of externalities in this problem. The first arises from the positive impact that green cars have on the environment. Suppose that the social marginal benefit associated with this externality is some amount k per green car and illustrate in your graph the efficient number of cars x1 that this implies for “greenies”. Then illustrate the Pigouvian subsidy s that would eliminate the market inefficiency. This is also illustrated in panel (a) of Graph 21.12. At price p∗, greenies buy x0 cars — but the marginal social benefit curve lies k above D0 — which implies the efficient quantity of cars that greenies should buy is x1 . A subsidy of size s = k will result in exactly that — because the entire subsidy will go toward reducing the price of cars (given the elastic supply curve in the competitive market.) • (c) The second externality emerges in this case from the formation of social norms — a form of network externality. Suppose that the more green cars 1 the “meanies” see on the road, the more of them become convinced that it is “the right thing to do” to buy green cars even if they are somewhat less convenient right now. Suppose that the “meanies’s” linear demand D1 for green cars when x1 green cars are on the road has vertical intercept below (p∗ −k). In a separate graph, illustrate D1 — and then illustrate a demand curve D2 that corresponds to the demand for green cars by “meanies” when x2 (> x1 ) green cars are on the road. Might D2 have an intercept above p∗? This is done in panel (b) of Graph 21.12. As the number of green cars increases, the demand for green cars by “meanies” increases — implying D2 lies above D1 and can certainly have intercept higher than p∗. • (d) Does the subsidy in (b) have any impact on the behavior of the “meanies”? In the absence of the network externality, is this efficient? No, the subsidy in (b) has no effect on the meanies because it does not reduce price sufficiently for any meanie to buy a green car given the number x1 of green cars bought by the greenies. In the absence of the network externality, it would indeed be efficient for only the greenies to buy green cars — because the meanies simply do not value them sufficiently to buy them even when they get the subsidy that takes the positive externality from green cars into account. • (f ) Explain how the imposition of a larger initial subsidy has changed the “social norm” — which can then replace the subsidy as the primary force that leads people to drive green cars. The social norm here is like peer pressure — the more green cars, the greater the peer pressure felt by those who don’t drive green cars. As the social norm changes, green cars are valued by meanies because others are driving them. Whether meanies simply want to look good in front of others — or whether we interpret the increased demand as an in- crease in the perception that driving green cars is “the right thing to do”, the subsidy might eventually be replaced by the new social norm. • (h) How could sin taxes like this be justified as means of maintaining social taboos and norms through network externalities? People who advocate such sin taxes might also be thinking of the impact that indi- vidual behavior has on social norms. The more people smoke, the more socially acceptable it is to smoke — and thus the greater will be the demand for cigarettes. The more available pornography is, the more socially acceptable it might be for others to consume pornog- raphy. To the extent to which consumption of pornography causes changes in behavior — such as extramarital affairs or casual relationships — that some might wish would not occur, we have a network externality that re-inforces other negative externalities. If these network externalities are strong, and if you believe the social norms that might form from less use of cigarettes or pornography to be important, you might advocate sin taxes primarily on those grounds. 2 • (i) Consider an alternative policy to a subsidy on green cars to be a pollution tax on polluting cars. Illustrate how the markets for polluting & green cars change with this new policy. (Assume all cars in each category to produce the same amount of pollution.) Graph: Since we have negative externalities, our long run supply curve, incorporating social pollution costs, or SMC, is upward sloping/exponential with pollution costs. Private marginal cost curve (PMC) is horizontal, not incorporating social pollution costs. Demand is downward sloping. Intersection of PMC and Demand curve represents an inefficient level of output supplied on the market at x∗ and p∗ . Applying pollution tax will cause a decrease in output supplied down to an optimal/desired level of output xopt and popt . The vertical distance between SMC and PMC at xopt represents the size of the tax required so that inefficiencies are eliminated. What could happen as a side effect of taxing the cars that pollute is that due to the network externalities, there would be less and less polluting cars on the street, which could make people demand more green cars. Thus, on the green car market, we may observe an upward shift in demand. • (j) Finally consider a cap & trade policy for pollution, now with older cars producing more pollution than younger cars. Green cars consume no pollution. Explain what happens the “demographics” of cars on the road. How will that change if greenies decide to purchase some of the pollution rights? Graph: With cap and trade policy, new market of vouchers is created. Due to this policy, we can then expect that there will be less older cars (that produce more pollution) on the roads. If greenies decide to purchase some of the pollution rights, demand on the market for vouchers will shift upwards, which will unambigiously increase the vouchers’ rental price. That means that even less people will decide to purchase an old car, so we will be seeing more younger than older cars on the streets. (Accepted alternative: Supply of vouchers for nongreenies/polluting cars decreases, which given their demand, increases rental price of vouchers. Result follows as above.) Question 2 1 1 2 The N consumers in the economy have the utilty function xn Y 2 where xn is the amount individual n spends on private consumption and Y is the public good national defense. Each consumer has an income I. The price of x is 1 and the cost of y is y 2 . Of course, y1 + y2 + y3 + . . . + yn = Y • (a) What is the socially optimal level of provision of this public good?1 1 Your maximization problem setup looks something like this (check p.1060 - ... for details) 3 Since our utility function is Cobb-Douglas, you know that the form of deI mand function of u(x, y ) = xα y 1−α for y looks like y = (1 − α) P . Here, α = 1 . 2 1 I Thus, consumer i ￿ s demand is P = 2Y i . Since we have N consumers with identical preferences and equal incomes, we obtain the market demand for Y ￿N i by vertically summing individual demands for the public good Y ( i=1 M BY ). 1 NI Market demand for public good Y is thenP = 2 Y . ￿N i Optimal level of public good is provided when it holds i=1 M BY = M CY . 2 Since M C = Y , socially optiml level of provision of the public good Y equals ￿ to 1 NI 2Y =Y2 ￿Y = 3 NI 2 . Suppose, as in the book, MC=1. Then, Y = NI 2. • (b) If it not provided by the government, what is the individual contribution of each person toward purchasing the public good? Optimization problem2 : 1 maxzi ,xi xi2 Y 1 2 st. I = x i + zi Y = zi + (N − 1)z, , where zi represents individual’s contribution towards the public good Y, where z is the amount everyone but i contributes to the public good. ￿ ￿ 1 1 maxzi ln(I − pi zi ) + ln(zi + (N − 1)z ) . 2 2 Take FOC wtr. to zi and equate it to zero. 1 (−1) 1 1 I (N − 1) + = 0 ⇒ zi ( z ) = − z, 2 (I − pi zi ) 2 (zi + (N − 1)z ) 2 2 which represents the i ￿ s best response function of contribution to public good Y given actions from all other than i . In Nash equilibrium, each individual best responds to each other. Also, we are in a symmetric case, where, repeating the same approach for “all other but i ” consumers, will give us a symmetric best response function. To solve for how much each individual contributes to a public good in an equilibrium , you can maxx1 ,x2 ( 1 lnx1 + 1 lnY ) st. uN −1 (xN −1 , Y ) = u and Y = N I − x1 − ¯ 2 2 ￿ maxx1, x2 ( 1 lnx1 + 1 ln(N I − x1 − N −1 xi )) st. uN −1 (xN −1 , Y ) = u ¯ i=1 2 2 ￿N ￿N − 1 1 1 ⇒ L = ( 2 lnx1 + 2 ln(N I − x1 − i=1 xi )) + λi (¯ − ui (xi, Y )) u i=2 2 Assume p = p = 1. x i 4 N −1 ￿ i=1 xi (i) plug best reponse function z (zi ) into (zi (z ) and obtain z eq , or (ii), since you know that each individual contributes the same amount towards the public good Y, equate zi (z ) = z and solve for z eq . Due to simplicity, I will show (b). I (N − 1) I − z = z ⇒ z eq = 2 2 N +1 • (c) What is the total amount of the public good provided? Is this society more efficient when it is large or small? Total amount of the public good provided, assuming there are N individuals, is: NI . N +1 The society is more efficient when it is small. Why? Look at the best response functions - when N increases, my individual contribution towards the public good will decrease. For instance, if it is just two people in the world, each of them contributes half of the whole contribution towards the public good. If one extra person enters, it seems to you as if another person is actually giving twice as much. That means that you can decrease your contribution to the public good more and more as N increases - you have more and more people to free ride on, as N increases. Why is this inefficient? Each individual contributes up to the level where his private marginal benefits equal to marginal costs. However, ￿N in the case of free-riding, our total marginal benefits (M B1 + i=2 M Bi ) of the public good’s consumtion are larger than MC. You can then see that the amount contributed by an individual is less than efficient, because an individual only takes into account M B1 . Thus, as N increases, we are producing inefficiently lower quantity of the public good, because we can free-ride more and more. Y = Nz = • (d) If the government supported individual contributions with a tax refund of .5 for every unit of Y purchased by the individual, how would this change the individual contribution? Repeat the same optimization problem as in b), only add taxation and subsidy into the story: ￿ ￿ 1 1 maxzi ln((I + 0.5zi ) − pi zi ) + ln(zi + (N − 1)z ) . 2 2 You obtain the following best response function: (N − 1) z. 2 In equilibrium, each individual will contribute: zi ( z ) = I − zi (z ) = z ⇒ z eq = Y. 2I . (N + 1) Individual will increase his individual contribution towards the public good 5 • (e) Is it possible to arrive at the efficient level of production with a subsidy? If so, what should the subsidy be? When N = 2, there are two of us, and each one therefore takes into consideration half the overall benefit from his private contribution to the public good. When N = 3, the amount of the overall benefit each of us takes into account falls to 1/3 — and when N = 4, it falls to 1/4. For a population of N , each person therefore only takes into account 1/N of the overall benefit from his private contribution — − leaving (NN 1) that he does not take into account. As N get large, the fraction of the overall benefit of a private contribution that each person takes into account approaches zero. If I don’t take into account half the benefit that I create by giving to the public good, then a subsidy of s = 0.5 will cause me to internalize the externality by having me pay for only half of the actual contributions I am making. When I don’t take into account 2/3 of the benefits I create (as when N = 3), the subsidy must rise to s = 2/3 — and when N = 4, it must rise to s = 3/4 because I now only take into account 1/4th of the benefits I create. For − a population of N , the subsidy must therefore rise to s = (NN 1) because that is the fraction of the benefit I create with my contribution that I do not take into account. This implies that the optimal subsidy will approach s = 1 as N gets large. In short - It is possible to arrive at the efficient level of production wth a − subsidy equal to s = (NN 1) , since individuals internalize the externality they are making with their own contribution towards the public good. When N is getting large, optimal subsidy approaches 1.3 Question 3 Now there are only 2 consumers in the society, with demand curves y = −3P +10 and y = −2P + 10. • (a) Suppose y is a public good like national defense. Draw the market demand curve. Sum both individual demand curves vertically. 4 • (b) The government has decided to protect only those who have passports. What will be the level of national defense provided and what will be the price of the passport that each citizen purchases? Use the Lindahl Pricing concept. We find SMB (vertical summation): 10 y y 1 5y − ) + (5 − ) = 8 − 3 3 2 3 6 In order to find socially optimal level of public good, we equate SMB=MC P =( 3 Note that this case (when subsidy is approaching 1) is basically equivalent to the case of government funding all the private contributions/providing the public good. 4 Graphs not drawn to scale. 6 1 5y 50 8− = 2y ⇒ y = 3 6 17 Now the government uses price discrimination and has each individual pay a different amount such that each individual is paying the price that he or she would have paid if he or she were demanding this amount. Thus, we use individual’s demand curve to see what each individual would pay. This is then called the Lindhal equilibrium. • Type 1 consumer pays: P= 50 2 10 − 13 = 2 3 3 39 • Type 2 consumer pays: P =5− 50 13 2 =3 1 13 • (c) Now suppose y is a private good like electricity. Draw the market demand curve. Sum both individual −2P + 10 y = −5P + 20 0 demand curves horizontally. , if 0 < y ≤ 3 1 3 , if 3 1 < y ≤ 20 3 , otherwise 5 • (d) What is the efficient level of production if the cost of electricity is y 2 ? Optimality condition: M B = M C . We have TC = y 2 ⇒ M C = 2y , private good. • Case 1: 0 < y ≤ 3 1 3 y = −2P = 10 ⇒ P = 5 − y 2 y = 2y ⇒ y = 2 2 Since y = 2 falls in the Case 1 interval, this is a possible optimal solution. 5− • Case 2: 3 1 < y ≤ 20 3 y = −5P + 20 ⇒ P = 4 − y 5 y 20 = 2y ⇒ y = 5 11 does not fall in the Case 2 interval, this is not an optimal 4− Sine y = 20 11 solution. We conclude that y = 2 is an efficient level of production if the total cost of electricity is y 2 . 5 Graphs not drawn to scale. 7 • (e) If a monopolist is providing the good, what is the profit maximizing price and quantity? How much profits are made? Optimality condition: MR=MC. We have TC = y 2 ⇒ M C = 2y , private good. • Case 1: 0 < y ≤ 3 1 3 5 − y = 2y ⇒ y = Since y = solution. 5 3 5 3 falls in the Case 1 interval, this is a possible optimal monopolist’s • Case 2: 3 1 < y ≤ 20 3 4− 2y 20 = 2y ⇒ y = 5 12 Sine y = 20 does not fall in the Case 2 interval, this is not a possible optimal 12 monopolist’s solution. We conclude that y = 5 is an optimal level of monopolist’s production if the 3 total cost of electricity is y 2 . • (f) If the monopolist is able to discriminate, what is the price each consumer must pay? In our case, there are only 2 types of consumers, therefore the monopolist can charge a different price to each of them (assuming he can recognize each type of the consumer) and uses third degree price discrimination. Prices charged will be: • Consumer of type 1: M R1 = M C 10 2y 5 10 ( 5 ) 11 = 2y ⇒ y = and P = − 4 =2 3 3 4 3 3 12 Profit: π = yP − y 2 = 2 1 12 • Consumer of type 2: M R2 = M C 5 − y = 2y ⇒ y = (5) 5 1 and P = 5 − 3 = 4 3 2 6 Profit: π = yP − y 2 = 4 Together, he earns: 6 1 . 4 Profit without discrimination: 4 1 6 8 1 6 • (g) Does discriminating monopolist have more profits? Why? (answer with logic, not math) Yes, the monopolist has higher profit, because he can, in our case, where he knows willingness to pay for each type of the consumer, extract consumer surplus from each of them by charging them different price. Also, while discriminating, he can reach customers with a lower demand that were previously exluded from the market due to a very high price and a limited output. • (h) Is the discriminating monopolist more efficient than the non-discriminating monopolist? Justify by comparing monopolist outputs to the efficient output. Doing welfare analysis, one obtains the following results (see graphs for help): • Discriminating monopolist: – ydisc, type1 = y d1 = 5 , pd1 = 2 11 4 12 – ydisc, type2 = y d2 = 5 , pd2 = 4 1 3 6 ∗ efficient outcomes and prices on each market: · y ef,1 = 10 , pef,1 = 2 6 7 7 · y ef,2 = 2, pef,1 = 4 ∗ price at intersection M Ri = M C at y d,i 2( 5 ) · pM R1 = 10 − 34 = 2 1 3 2 · pM R 2 = 5 − 5 = 3 1 3 3 25 – DW L1 = (2 11 − 2 1 ) ∗ ( 10 − 5 ) ∗ 1 = 672 12 2 7 4 2 5 – DW L2 = (4 1 − 3 1 ) ∗ (2 − 5 ) ∗ 1 = 36 6 3 3 2 – Total DW L1+2 from both markets equals: 355 2016 • Non-discriminating monopolist – ynondisc = y nd = 5 , pnd = 4 1 3 6 ∗ efficient outcomes and prices on one market: · y ef = 2, pef = 4 ∗ price at intersection M R = M C at y nd · pM R = 5 − 5 = 3 1 3 3 – DW L = (4 1 − 3 1 ) ∗ (2 − 5 ) ∗ 6 3 3 1 2 = 5 36 DW L1+2 > DW L We are dealing with the third degree price discrimination. We know that, compared to an efficient outcome, some DWL will arise, however it is ambigious to say, whether it is more efficient from the society’s point of view to allow the monopolist to discriminate or not. As you can see from calculations above, third degree price discrimination by the monopolist causes more DWL compared to the case when monopolist is treating the market as one and is not discriminating at all. 9 ...
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This note was uploaded on 09/11/2011 for the course ECON 100A taught by Professor Woroch during the Spring '08 term at Berkeley.

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