Econ Economics of Public Sector HW #1 Suggested Solutions.

Transcription

1 Econ Economics of Public Sector HW #1 Suggested Solutions. by Anna Rubinchik-Pessach October 7, 1 Problem 1: Chapter 5, q.4 Cash-for-Clunkers Program The goal of the EPA policy is to limit emission of hazardous elements (hydrocarbons and carbon monoxide) in the air. The SOURCE of the emission is not important: whether it is emitted as a side-effect of some production processoritisemittedbyoldcars,itisequallyhazardous. As these elements can be released in the air as a part of a technological processes (say, coal-based power plants), the EPA standard implicitly limits the output of the firms that use these technologies. Assume that the production technology gives rise to the following marginal cost schedule. Given the externality, the optimal output level from the firm s perspective will be higher than the socially optimal level (the intersection of the total marginal cost, the solid line and the downward sloping demand). 1

2 y The downward sloping line is the demand for firm s output. The dashed line is the marginal cost to society due to the emission of CO and HC,the dotted upward sloping line is the private (firms) marginal cost of producing, the solid line is the sum of the two marginal costs, the private cost of producing and the cost of polluting. x Once the limit on emissions is in place, the quantity produced by the firm is limited (they would want to produce more, but that will violate the emission standards) unless it invests in a newer technology that reduces emission of the hydrocarbons and carbon monoxide per unit of output. It is quite possible that buying the clunkers is cheaper than investing in the new technology. Cash-for-clunkers program offers a way for the polluting firms to satisfy the standard without changing their technology. The purchase of the clunkers will be viewed by the firm as an implicit tax that the government imposes on the polluters. The tax is paid for each unit that is being produced after the emissions have reached the maximum level allowed by the EPA: If the firm decides to produce these additional units of output, it will have to clean the pollution so created by eliminating clunkers from the roads.

3 q Similar to the previous graph for quantities q<1.8, at which the firm produces allowed levels of pollution, the two marginal costs are different. If the firm decides to produce more that 1.8 units, it buys the clunkers so as to neutralize the pollution created by the additional units. Therefore the private MC jumps up at this quantity. On the other hand, the MC of pollution is zero for q>1.8. Thus, it creates no externality for the output in excess of 1.8 and so the private MC (dotted line) and social MC coincide. Problem Consider the following. Two suspects are arrested and charged with a crime. They are held in separate cells and are explained the consequences of their actions. If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess, then both will be sentenced for 6 months. Finally, if one confesses but the other denies, then the confessor will be released immediately, but the other will be sentenced to 9 months Describe this situation as a normal form game. a. What are the strategies (actions) available to each player? answer: confess, do not confess b. Describe outcomes under all possible scenarios (actions taken by players) answer: see the problem description c. Assume players dislike imprisonment. The longer is the sentence, the smaller is the utility of the player. Characterize the utility that a player associates with each of the possible outcomes. answer: 3

4 confess do not confess confess -6,-6,-9 do not confess -9, -1,-1 d. What will the suspects do during the trial? (Find Nash equilibrium of the game.) answer: Both confess is a unique Nash equilibrium of the game. Best response of either player is to confess no matter what the opponent is doing: if the partner (column player) chooses to confess, then it is better for the row player to confess (in case of confession player one gets 6 month in jail (utility of -6) as opposed to the case when he does not confess (utility of -9)).If the partner (column player) decides not to confess, then the choice for player one is between 1 month in jail (in case he does not confess) and being free (in case he confesses). Exactly the same reasoning will be used by column player. Thus, both will confess. e. Is it an efficient outcome? answer: If the society consists of the two prisoners, then, no. They could have denied the crime and get smaller punishment. f. Repeat the analysis (a-e) for the same story with the only difference being that the last sentence reads: Finally, if one confesses but the other denies, then the confessor will be released immediately, but the other will be sentenced to 5 months. answer: thegamepayoffs change. The new payoffs are as follows: confess do not confess confess -6,-6,-5 do not confess -5, -1,-1 There will be two (pure strategy) Nash equilibria. 1. Row player is confessing and column player is denying. Row player is denying and column player is confessing First (row) player s best action now depends on what his partner is doing. If player (column player) is confessing, then it is better for the first player not to confess (-5>-6). On the other hand, if his partner is denying the crime, then the row player s best response is to confess (>-1). Identical reasoning will be employed by the second (column) player. Thus, there are two symmetric (pure strategy) equilibria, in which one of the players is confessing and the other one is denying. 4

5 3 Problem 3: Chapter 5, q.7 In the absence of intervention, the amount produced (provided perfect competition) have to satisfy the equality between the marginal benefit (demand) and marginal cost (the supply) 1 X = 5 (1) X = 5 () The efficient (socially optimal) level of producing X has to satisfy the equality between the marginal benefit and the total marginal cost, 1 X = 7 (3) X = 3 (4) In the following graph the quantity of good X is measured on the horizontal axes and the cost/benefit (in dollars) is measured on the vertical axes X Thegaintothesocietyinvolvedinmovingfromproducing5 units to producing 3 units is the excess of marginal costs over the marginal benefits that will be eliminated by the decreased production. From the graph it is evident that each unit produced over and above 3 units has a constant marginal cost equal to 7 and a decreasing marginal benefit below7. Therefore the gain is 1 (7 5) (5 3) =. A tax of $ per unit will lead the producer to choose the socially optimal level of production. The revenue raised will be $ 3=$6. 5

6 4 Problem 4 (from Robert Frank) A small village has six people. Each can either fish in a nearby lagoon or work in a factory. Wages in the factory are $4/day. Fish sell in competitive markets for $1 a piece. If L persons fish the lagoon, the total number of fish caught is given by F =8L L. Each fishermangetsanequalshareofthe total catch. People prefer to fish unless they expect to make more money working in the factory. a. If people decide individually whether to fish or work in the factory, how many will fish? What will be the total earnings for the village? answer Each fishermen gets $1 F (L) (5) L from fishing given there are L fishermen in the lagoon. Alternatively, a person can get $4 for working in the factory. A villager will fish as long as $1 F (L) L $4 (6) 8 L 4 (7) Thus, it will be worthwhile to go fishingaslongasthetotalnumberof fishermen is below. Once there are two fishermeninthelagoon,thethird one will prefer to go work in the factory. Total earnings are: the two fishermen bring 8 =8dollars and four workers bring 4 4=16dollars. Total earnings are = 4 dollars per day b. What is the socially optimal number of fishermen? What will be the total earnings of the village if the optimal number of the fishermen are operating? answer: the socially optimal quantity of fishermen is determined by the marginal catch of the fishermen. The social planner will send another villager to fish as long as the value of his (marginal) catch is above 4: $1 F (L) $4 (8) 8 4L 4 (9) 6

7 Therefore it is socially optimal to have one villager fishing and all the rest working in the factory. (Note that the first fishermen brings the value of 6 dollars, while the second fishermen s catch is worth only $, which is less than $4 that he can alternatively earn in the factory.) Total earnings are: 6+5 4=6> 4. c. Why is there a difference between the equilibrium and socially optimal numbers of fishermen? answer: there is an externality that is generated by a fishermen in the lagoon. Once an additional fishermen goes out fishing, not only his catch is smaller than the average catch before him joining, but also the catch of the other fishermen decreases. Clearly, when the fishermen make their decisions independently they overlook the negative externality they impose on the others. 7