Midterm #2 EconS 527 [November 7 th, 2016]

Transcription

1 Midterm # EconS 57 [November 7 th, 16] Question #1 [ points]. Consider an individual with a separable utility function over goods u(x) = α i ln x i i=1 where i=1 α i = 1 and α i > for every good i. Assume that the consumer faces a strictly positive price vector p >> and his wealth is given by w >. (a) Find the Walrasian demands, and the shadow price of wealth. [1 points] (b) et us next find the shadow price of wealth using an alternative approach. First, find the indirect utility function, v(p,w), resulting from the previous UMP. Then, measure how it is affected by a marginal increase in wealth, i.e., find the derivative v(p,w). Does your result coincide with what you found in part (a)? [15 points] w a) The consumer solves a UMP given by max u(x) s. t px w Using the shortcut MRS i,j=p i/p j, we obtain interior solutions α i x i = p i p j or α i p i p j = x i, which together with the budget constraint yields a Walrasian demand of x i (p, w) = α iw p i (1) For every good i. In addition, we can obtain the angrage multiplier, λ, from the first order-condition Which combined with (1) yields u x i = λp i or α i x i = λp i α i α i w p i = λp i And solving for λ we obtain λ(p, w) = 1 w Hence, the marginal value of relaxing the constraint (i.e., the shadow price of wealth) is 1 w. b) The indirect utility function is 1

2 Hence, the marginal utility of wealth is v(p, w) w v(p, w) = α i ln α iw i=1 p i 1 α i = α i α i w = 1 p i w α 1 i = w i=1 p i=1 i Interestengly, this result is generalizable to settings in which, given the separatable nature of the utility function, the consumer focuses on a subset of goods {1,,, 1} where 1<, { 1,, }, etc. and solves a separated UMP for each of these subsets of goods, i.e., one UMP for goods {1,,, 1}, another UMP for goods { 1+1,, }, etc. The consumer s solution to these separated UMPs must coincide with that in part (a), where the consumer simultaneously considers all goods. Question # [1 Points]. Explain verbally and graphically the Hicksian wealth compensation for a reduction in the price of x 1.

3 The Hicksian wealth compensation is a compensation that is used to adjust a consumer s wealth so that he/she can maintain his/her original utility level when the price(s) of one or more goods change. When the price of x_1 reduces, the Hicksian wealth compensation results in that the consumer will be take a part of his/her wealth away to maintain the original utility level. From the diagram we can find that the Hicksian wealth compensation will be more significant than Slutsky wealth compensation and the consumer will consume a slightly more of goods 1. Question # [ Points]. Consider an individual with the following utility function, where income. x denotes x if u( x) = 5 x if 5 x 5 x > (a) Depict the utility function with u(x) on the vertical axis and income, x, on the horizontal axis. Show that this individual is (weakly) risk averse. (b) Suppose that there are three states of the world, each equally likely. There are two assets, x and y. The asset x is the random variable with payoffs (1,5,9) and the asset y is the random variable with payoffs (,,1). (Note that assets specify a payoff triple, to indicate the payoff arising in each of the three equally likely states of the world.) Calculate the expected utility of asset x and of asset y. Which asset, hence, would be preferred by this individual, if both of them were offered at the same price? (c) Calculate the expected value of each asset (you previously found the expected utility). Calculate the variance of both assets. Which asset would be chosen by this individual if he were variance averse? a. This individual s utility can be expressed as the minimum of x and 5/ + x. In particular x is the minimum of these linear functions for all x satisfying x 5/ + x, or x 5/ ; and 5/ + x is the minimum of both linear functions for x > 5/. Hence, the function can be represented u(x) = min{x; 5/ + x}, as the figure below depicts. Specially, min{x; 5/ + x} considers x for the interval x 5/, and 5/ + x for value of x beyond that cutoff. As the figure illustrates, the function u(x) = min{x; 5/ + x}, depicted the lower envelope of the lines 5/ + x and 5/, and it is a (weakly) concave function.

4 b. et us first find the expected utility of asset x, EU(x) = 1 min { 1, 5 + 1} + 1 min { 5, 5 + 5} + 1 min {9 1, 5 + 9} where the first term represents that the first outcome occurs, yielding a payoff of $9, and the second (third) term reflect the second (third) outcome with payoff $5 ($9; respectively). Simplifying this expression, we obtain And similarly for the expected utility of asset y, = = 1 EU(y) = 1 min {, 5 + } + 1 min {, 5 + } + 1 min {9 1, 5 + 1} = = Therefore, EU(x) < EU(y), making asset y to be preferred by the individual, if both assets were o ered at the same price. c. The expected value of each asset is E(x) = = 5 E(y) = = 5 Hence, both assets have the same expected value. et us now find their variance. For convenience, we use the formula Var(x) = E(x ) - E(x). We already know E(x) and E(y), let us then and E(x ) for asset x and E(y ) for asset y, E(x ) = = 17 E(y ) = = 114 Therefore, we can use this information, together with the expected values of the assets, E(x) and E(y), found above, and compute the variance of every asset, Var(x) = E(x ) E(x) = 17 5 = Var(y) = E(y ) E(y) = = Hence, V ar(y) > V ar(x). Therefore, if this individual were variance averse, he would select asset x since, given the same mean for both assets, asset x has the lowest variance. 4

5 Question #4 [15 Points]. Define and discuss the Independence Axiom (IA). Provide an example in which the IA is violated. Definition: A preference relation satisfies I.A. if, for any three lotteries,, and, and α (,1) we have if and only if α +(1- α α +(1- α). Example: Extreme preference for certainty. Question #5 [ Points]. Consider a bargaining game between Player 1 (proposer) and Player (responder). Player 1 makes a take-it-or-leave-it offer to Player, specifying an amount s={, v/, v} out of an initial surplus v, i.e., no share of the pie, half of the pie, or all of the pie, respectively. If Player accepts such a distribution, Player receives the offer s, while Player 1 keeps the remaining surplus v - s. If Player rejects, both players get a payoff of zero. (a) Draw the extensive form game representing the bargaining game and find the Subgame Perfect Nash Equilibrium. (b) Describe the strategy space for every player (c) Provide the normal form representation of this bargaining game. (d) Does any player have strictly dominated pure strategies? Find the Nash equilibria. a. Extensive form game Player 1 s=v s=v/ s= Player Player Player A R A R A R v v/ v/ v SPNE: {s=, AAA;s=,AAR} b. Strategy set for player 1 5

6 S 1 = {, v, v} Strategy set for player S = {AAA, AAR, ARR, RRR, RRA, RAA, ARA, RAR} c. Normal form representation: Using the three strategies for Player 1 and the eight available strategies for Player, the 8 matrix of figure. represents the normal form representation of this game Player AAA AAR ARR RRR RRA RAA ARA RAR s = v,v,v,v,,,,v, Player 1 s = v/ v/,v/ v/,v/,,, v/,v/, v/,v/ s = v,,,, v, v, v,, Figure.. Normal-form representation of the Bargaining game d. No player has any strictly dominated pure strategy: Player 1. For player 1, we find that s = v yields a weakly (not strictly) higher payoff than s = v, that is u 1 (s = v, s ) u 1 (s = v, s ) for all strategies of player, s S (i.e., some columns in the above matrix), which is satisfied with strict equality for some strategies of player, such as ARR, RRR or RRA. Similarly, s = yields a weakly (but not strictly) higher payoff than s = v. That is, u 1(s = v, s ) u 1 (s = v, s ) for all s S, with strict equality for some s S, such as ARR and RRR. Similarly, s = v yields a weakly higher payoff than s = for some strategies of player, such as RRR, but a strictly larger payoff for other strategies, such as RAR. Hence, there is no weakly dominated strategy for Player 1. Player. Similarly, for Player, u (s, s 1 ) u (s, s 1 ) for any two strategies of Player s s and for all s 1 S 1 with strict equality for some s 1 S 1. Question #6 [15 Points]. et us consider the following lobbying game in figure.7 where two firms simultaneously and independently decide whether to lobby Congress in favor a particular bill. When both firms (none of them) lobby, congress decisions are unaffected, implying that each firm earns a profit of 1 if none of them lobbies (-5 if both choose to lobby, respectively). If, instead, only one firm lobbies its payoff increases to 15 (since it is the only beneficiary of the policy), while that of the firm that did not lobby collapses to zero. Firm 1 obby Not obby Firm 1 obby -5, -5 15, Not obby, 15 1, 1 Figure 1. obbying game (Normal-form) 6

7 (a) Find the mixed strategy Nash equilibrium of the lobbying game. Assume that p is the probability that Firm 1 chooses obby and q is the probability that Firm chooses obby. (b) Graphically represent each player s best response function. Firm 1. et q be the probability that Firm chooses obby, and (1 q) the probability that she chooses Not obby. The expected profit of Firm 1 playing obby (fixing our attention on the top row of the matrix) is EU 1 (obby) = 5q + 15(1 q) = 15 q When, instead, firm 1 chooses not lobby, its expected profit is: EU 1 (Not obby ) = q + 1(1 q) = 1 1q If Firm 1 is mixing between obbying and Not lobbying, it must be indifferent between obbying and Not lobbying. Otherwise, it would select the pure strategy that yields the highest expected payoff. Hence, we must have that EU 1 (obby) = EU 1 (Not obby) rearranging and solving for probability q yields 15 q = 1 1q q = 1/ Therefore, for Firm 1 to be indifferent between obbying and Not lobbying, it must be that Firm chooses to obby with a probability of q = 1/. Firm. A similar argument applies to Firm. If Firm chooses to lobby (fixing the right-hand column), it obtains an expected profit of EU lobby 5p 15 1 p When firm, instead selects to not lobby (in the right-hand column), it obtains an expected profit of EU (not lobby)=p+1(1-p) Hence, firm, must be indifferent between obbying and Not lobbying, that is, solving for probability p yields EU (obby) = EU (Not obby) 5p + 15(1 p) = p + 1(1 p) p = 1/ Hence, Firm is indifferent between obbying and Not lobbying as long as Firm 1 selects to obby with a probability p = 1/. Then, in the lobbying game the mixed strategy Nash equilibrium (msne) prescribes that: 7

8 msne = {( 1 obby, 1 No obby), ( 4 obby, 1 No obby)} 4 That is, p = 1/ and q = 1/. b) Figure 1 below depicts every player s best response function, and the crossing points of both best response functions identify the two psne of the game (p, q) = (,1), illustrating (Not lobby, obby), (p, q) = (1,) which corresponds to (obby, Not lobby), and the msne of the game (p, q) = ( 1, 1 ) found above. q (obby, Not obby) 1 msne 1/ (Not obby, obby) 1/ 1 p Figure 1. Best Response Functions, psne and msne 8