Econ 210, Final, Fall 2015.

Transcription

1 Econ 210, Final, Fall Prof. Guse, W & L University Instructions. You have 3 hours to complete the exam. You will answer questions worth a total of 90 points. Please write all of your responses on the exam itself in the space provided. If you need additional space, there is a blank sheet included at the end. You may refer only to your own handwritten, cheat sheet. Calculators and all other references materials are not allowed. If a question asks for a numeric quantity you may leave your answer in expression form for full credit. (For example, would be perfectly acceptable in place of 2.) Be sure to label any diagrams you draw, to show your work and to explain your reasoning. Finally, take note that questions are printed on BOTH sides of each page. You may keep your cheat sheet. Thank you and good luck! 1. (9 Points) Consider the insurance problem we discussed in class. maxπ b v(ω b +(1 γ)k)+(1 π b )v(ω g γk) k where k is the amount of insurance, γ is the price of the insurance per dollar of bad-state payout, ω b andω b aretheendowmentsinthegoodandbadstate, π b isprobabilitythatthebad state will occur and v is the Bernoulli utility in the consumer s Von-Neumann Morgenstern expected utility function. In the context of this problem... (a) (3 points) Define fair (a.k.a. actuarially fair) insurance ASNWER Fair insurance sells at a price that gives the seller zero expected profit. In this case, we re talking about state contingent bonds that sell for the price of the probability of the bad state. In other words, insurance is fair if γ = π b. (b) (3 points) Define what it would mean to be fully insured. (i.e. how much insurance is that?) ANSWER An agent is fully insured when their asset portfolio is such that they would experience the same level of consumption in all states of nature. In this case, it means choosing k equal to k full so that ω b +(1 γ)k full = ω g γk full 1

2 or you could simplifiy this to k full = ω g ω b Either answer is acceptable. (c) (3 points) State (but don t prove) the full insurance principle. ANSWER The FIP states that if fair insurance is offered, then the optimal choice for any risk-averse agent is to fully insure. So if γ = π b and v < 0 (risk averse), then k full as defined above is the optimal choice. 2. (3 Points) The payoffs to the row and column players in the game Battle of the Sexes are shown below. For example, when both players choose O, the row player receives 1 and the column player receives 2. Identify all of the pure strategy Nash equilibria in this game, if any exist. O F O 1,2 0,0 F 0,0 2,1 ANSWER (O,O) and (F,F) are the two PSNE in this game. (O,O) is a PSNE because when the row player chooses O, the best response for the column player is to choose O because 2 > 0. Also when the column player choose O, it a best response for the row player to choose O because 1 > 0. Hence in the strategy profile (O,O), each player is playing a best response given what the other player is choosing. Similar reasoning for the profile (F,F). 3. (3 Points) In general, when there is perfect certainty about demand for pollution, regulating that pollutant with either a tax or a system of tradable permits have roughly equivalent efficiency properties. However, as shown by Martin Weitzman, if there is uncertainty about demand, then a tax will be more efficient than tradable permits when (a) the marginal damage curve is relatively flat. (b) the marginal damage curve is relatively steep. (c) the total damage to human health caused by the pollution is large. (d) the total damage to human health caused by the pollution is small. (e) monitoring costs are large. (f) monitoring costs are small. ASNWER (a) the flatter the MD curve, the better a price-approach (tax) works. It has little to do with the scale of the damage, so (c) and (d) are out. Also, both price (tax) and quantity 2

3 (tradable permits) approaches require accurate pollution measurement and monitoring to work, so there s no advantage of one approach over the other on that dimension which rules out (d) and (e). 4. (3 Points) Circle all that apply. For a price change, the Slutsky-compensated budget line always (a) goes through the endowment and is parallel to the horizontal axis (b) goes through the endowment and is parallel to the old budget line (c) goes through the endowment and is parallel to the new budget line (d) goes through the original choice and is parallel to the horizontal axis (e) goes through the original choice and is parallel to the old budget line (f) goes through the original choice and is parallel to the new budget line. (g) intersects the wage-compensated supply curve ANSWER (f) goes through the original choice and is parallel to the new budget line. 5. (6 Points) In the Spence model the we covered in class, e*(h) and e*(l) represent the efficient levels of education for the high and low types while w*(h)=y(h, e*(h)) and w*(l)=y(l, e*(l)) represent the competitive wages paid to each type in the case of perfect information ( equal to the respective types productive outputs evaluated at their efficient education levels). Meanwhile c(l,e) and c(h,e) are functions that describe the cost experienced by each type of worker of obtaining education level e. Using these definitions, write down an inequality that defines the envy case. ANSWER As discussed in class, the shape of the cost functions and production functions combine to generate envy, if the when market is paying w*(h) for e*(h) and w*(l) for e*(l), the low-type would be better off to mimic the high-type. That is w (H) c(l,e (H)) > w (L) c(l,e (L)) 6. (6 Points) V is Alice s maximum willingness to pay for the object being sold in a second price auction. Let Z be the highest bid among all bidders who are not Alice. Consider a strategy for Alice of submitting a bid B < V. Explain why this isn t a good idea compared to bidding V. (Hint. You should layout and briefly analyze 3 cases). ANSWER If Alice bids below her maximum willingness to pay there are three cases to consider. (a) Z < B < V. In this case, there is no damage. Should would have won with her bid B or by bidding her true value V and would have paid the same (Z) for the object either way. 3

4 (b) B < Z < V. In this case, the low-ball bid B causes a loss. If she had bid her true value V, she would have won the aucition and gained surplus V Z > 0, but instead she loses the auction and gains 0. (c) B < V < Z. In this case, there is no damage; bidding B or V would have caused her to lose the auction get 0 surplus. So we see that, bidding B < V, never does better than bidding V and sometime does worse. 7. (9 Points) The following bids represent net willingness to pay for a public good in a Groves- Clarke mechanism where the question is should the project be built or not. Bidder Net WTP Bids A -20 B -10 C 15 D 35 (a) (3 Points) Will the project go through? ANSWER. Yes, because the sum of the bid is positive. v i = = 20 i (b) (3 Points) Which bidder or bidders, if anyone, owe a Clarke Tax? ANSWER Only bidder D owes a Clarke tax. In general, only pivotal bidders owe a Clarke tax. A pivotal bidder is one whose bid, if taken out of the sum, would have changed the sign of the sum. In this case, we rule out any bidder whose bid was negative. (Since taking out their bid would only have made the result a higher positive number.) How about C? No, taking out C s bid of $15 would still leave a positive sum of $5. How about D? Yes, without D s bid of 35, the sum of bid would have been -$15. v i = = 15 i D (c) (3 Points) For each bidder that owes a non-zero Clarke tax, how much do they owe? ANSWER The Clarke tax owed by D is exactly the (absolute value of) of the sum of bid excluding her own - so $15 as calculated above. 8. (15 Points) First-price sealed bid auction. Assume there are 2 bidders. Each bidder i has a true value (maximum willingness to pay), v i distributed uniformly on [0,1]. Each bidder knows her own value but does not know the values of the other bidders in the game. Show 4

5 that there is an equilibrium in linear strategies. In other words, if player 2 employs a strategy with the form b 2 (v 2 ) = a 2 v 2, derive player 1 s best response and show that it also has the linear form.answer Player 1 s problem is to maximize expected consumer surplus Can we quantify Pr(win b 1 )? Yes. max b 1 Pr(win b 1 )(v 1 b 1 ) Pr(win b 1 ) = Pr(b 1 > b2) = Pr(b 1 > a 2 v 2 ) = Pr(b 2 < b 1 a 2 ) = b 1 a 2 where the last line follows due to the assumption that v 2 is uniformaly distributed on [0,1]. Plug this back in to the maximization problem to get The FOC is b 1 max (v 1 b 1 ) b 1 a 2 b 1 a 2 + (v 1 b 1 ) a 2 = 0 2b 1 = v 1 b 1 = v 1 2 Therefore 1 s best response is a linear strategy (bid is a linear function of value). 9. (15 Points) Madison Cheese makes cheddar out of labor (l) and capital (k). The following production function represents their technology ( f(,x k ) = 3log min{, x ) k 3 }+1 Let w l and w k be the constant prices of labor and capital face by this firm. Derive the factor demands for labor and capital. ANSWER The easiest way to do this is to realize that labor 5

6 and capital are perfect complements in this production function; each unit of labor goes with exactly 3 units of capital and any combination of labor and capital not in that ratio could not possibly be the profit maximizing choice. Therefore we know that whatever the optimal factor choices are, we will have x k = 3 making this subsitution we can write the profit maximization problem as This gives a FOC: max = p out 3log( +1) w l w k (3 ) 3p out +1 w l 3w k = 0 3p out = ( +1)(w l +3w k ) (p out,w l,w k ) = 3p out w l +3w k 1 x k (p out,w l,w k ) = 9p out w l +3w k 3 So demand for both factors is positively related to the price of the output (p out ) and inversely related to the sum of the factor prices which is consistent with the factors being perfect complements in production. An especially astute interpretation would note here that there are price combinations that would lead to a negative demand for the input. What does this mean? Taken literally it suggests that if input prices are high enough relative to the output price, then the firm would want to buy cheese and run it backwards through its production process to generate labor and capital. However, this ignores, of course, that we typically do not think of production functions as two-way reversible processes. In most cases the 2nd law of thermodynamics would prohibit implementing such a strategy. Therefore a more reasonable interpretation is that those price combinations would result in corner solutions (zero demand for both inputs). 10. (21 Points) Consider a system of income taxation and support in which every adult receives gauranteed income of G each week and pays proportion t (0,1) of all of their earnings in taxes. Suppose that every adult i has preferences for consumption (c) and leisure (l) described by the following utility function 6

7 u i (,x c ) = x α i l x 1 α i c Note that each individual i has its own α, though you should assume that every α i (0,1). Also each individual has access to a job that pays wage w i > 0 per hour. Also assume that a worker can choose to work any amount of hours in a week between 0 and 100. (Equivalently, they can choose the quantity of leisure,, to be any number between 0 and 100). Hence, to be clear, the total consumption of individual i under this system would be x c = G+w i (1 t)(100 ) (a) (20 Points) Using the methods we learned in class, derive an expression involving α i, t and w i that represents the threshhold level of G above which worker i would prefer to stay home and not work at all. Diagrams may be helpful. ANSWER. Let s set up the maximization problem... maxx α i x 1 α i c c, s.t.x c = G+w i (1 t)(100 ) 100 There are two approach one could take here. (1) Use MRS = MRT along with the budget line equation to solve for demands of leisure and consumption or (2) Use the budget line equation to subsitute out one of the choice variables in the utility function and then differentiate w.r.t. the remaining choice variable. I m going to follow the second approach, but either approach should result in the same answers. Using the budget line equation, let s remove x c from the utility function. max x α i l [G+w i (1 t)(100 )] 1 α i It strikes me as easier to take a monotonic transformation here - in particular the log. (The argument of a function that maximizes it, is the same that maximizes its log) max α i log( )+(1 α i )log[g+w i (1 t)(100 )] 7

8 The first order condition ( i.e. the first derivative w.r.t set equal to zero) is α i + (1 α i)( w i (1 t)) G+w i (1 t)(100 ) = 0 α i = (1 α i)(w i (1 t)) G+w i (1 t)(100 ) α i [G+w i (1 t)(100 )] = (1 α i )(w i (1 t)) α i [G+w i (1 t)100] = [(1 α i )(w i (1 t))+αw i (1 t)] α i [G+w i (1 t)100] = w i (1 t) = α i[g+w i (1 t)100] w i (1 t) The last line gives us the demand for leisure (l) - with the important caveat that leisure demand cannot exceed 100 hours. Note that we could have guessed this from the start from what we know about Cobb-Douglas preferences - specifically that they generate constant income share demands. G + w i (1 t)100 is the value of the consumer s endowment (analagous to the m from the pizza-beer model. w i (1 t) is the consumer s after-tax wage (which is the price of leisure), and α i is the income share of leisure. To answer the question, we need to know when this expression would equal or exceed leading the consumer to a corner solution. (100 hours of leisure demand implies 0 hours of work.) = 100 α i[g+w i (1 t)100] w i (1 t) 100 α i [G+w i (1 t)100] 100w i (1 t) α i G 100w i (1 t)(1 α i ) G 100w i(1 t)(1 α i ) α i (b) (1 Point - Almost purely for the fun of it bonus question) Suppose that the joint density of α and w in the population is given by f(α,w). Write down an expression for the labor force participation rate involving f, G and t. Interpret your expression. ANSWER Note that by the definition of a density function the following expression evaluates to 1 8

9 1 α=0 w=0 f(α,w)dwdα = 1 and that is what the participation rate would be if the grant didn t induce anyone to stop working completely. However as we determined in the previous parts there are part of α w space where people would choose to work zero for some given value of G. To see this more explicitly just re-arrange the threshhold equation derived above by putting w i on the LHS. w i Gα i 100(1 t)(1 α i ) To get the participation rate, we just need to skip over those parts in the integral above as follows. 1 α=0 Gα w= i 100(1 t)(1 α i ) f(α,w)dwdα 9