x 1 1 and p 1 1 Two points if you just talk about monotonicity (u (c) > 0).

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1 . (a) (8 points) What does it mean for observations x and p... x T and p T to be rationalized by a monotone utility function? Notice that this is a one good economy. For all t, p t x t function. p t x implies that u(x t ) u(x ), where u is some monotone In words, if a bundle is affordable, it gives you less utility than the one you chose. (You get full credit if you say this in words.) Two points if you just talk about monotonicity (u (c) > 0). (Note: you get full credit even if you don t write down that u is monotone.) (b) (8 points) What does it mean for these observations to satisfy the Generalized Weak Axiom of Revealed Preference? For all t, s, p t x t p t x s implies p sx t p sx s. In words, if x t is weakly revealed preferred to x s, then x s is not strictly revealed preferred to x s. Another way to put it: if x s was affordable at time t and the consumer chose x t, then x t must have been at least as expensive as x s at time s. Any of these answers (or something similar) give you full credit. (c) (5 points) Prove that if these observations are rationalized by a monotone utility function, then GWARP holds. Assume that these observations are rationalized by a monotone utility function u. Then w t = p t x t p t x t p t x u(x t ) u(x ) p t x t > p t x u(x t ) > u(x ) Assume by contradiction that GWARP doesnt hold. Then, there are times t, s such that p t x t p t x s and p sx t < p sx s. The first inequality implies that u(x t ) u(x s), and the second inequality implies the opposite: u(x t ) < u(x s). Since we have a contradiction, GWARP must hold.

2 2. Consider a consumer who has a utility function u(x, y) = x 2y (9 points) Draw the indifference curve through the bundle (x 0, y 0 ) = (0, 0) and indicate with an arrow the direction of preferences. (5 points for indifference curve, 4 points for direction of preferences) (8 points) Do these preferences satisfy convexity? (Note I m not asking about strict convexity.) You don t need to write down the mathematical definition of convexity, but justify your answer (you may use the graph to assist you in your justification/explanation if you wish) Yes they do. According to convexity, if you pick any two bundles on the same indifference curve, it must be the case that every point on the line connecting these bundles lies on an indifferent curve at least as high. This is true for these preferences because any two bundles you choose will have every bundle on the line lie on the same indifference curve. An example of two bundles on the indifference curve from part (a) and a line between them is in green above. Any explanation similar to this gets full credit. If you write about strict convexity correctly, you get 4 points. 2

3 (8 points) Do these preferences satisfy monotonicity? You don t need to write down the mathematical definition of monotonicity, but justify your answer. (8 points) No they do not. MU y < 0: more of good y makes the consumer worse off. Any explanation similar to this gets full credit. 3. Given a consumer with utility function u(c, l) = log c + l in a 2 good economy, where c is food, and l is leisure. (a) (6 points) This person has L = 2 hours in their work day. Given a price p for food and a wage rate w, write down the consumer s maximization problem. max {log c + l} c 0,l 0 s.t. pc + wl = 2w (b) (9 points) What are the demand functions for c and l for this consumer? Show your work. You may use the back of the page for the calculations. If you realized that the solution is interior based on what you learned in homework 3 (M > p y, which means 2w > w, or 2 > in this case), you had to state this explicitly. I.e., write the solution is interior because... If you have a good explanation and derived the solution correctly, you have full credit. If you assumed the solution is interior and used the Interior Lagrangian Method, but did not say why the solution is interior, you had 2 points taken off. Another way to get full credit is to use the General Lagrangian Method. The first order conditions are: Case c λ p, and if c > 0, then c = λ p () λ w, and if l > 0, then = λ w (2) pc + wl 2w, and if λ > 0, then pc + wl = 2w (3) c = 0, l = 0. You get an immediate contradiction since 0 is undefined. Case c > 0, l = 0 3

4 From condition (), we get c = λ p. Therefore, λ > 0 and from condition (3) pc + wl = 2w. So, c = 2w. At this p point, we need to check that the second condition is satisfied. λ w if only if w if and only if. This is impossible (a contradiction). 2w 2 Case 3 c = 0, l > 0. You get an immediate contradiction since 0 is undefined. Case 4 c > 0, l > 0. From the first order conditions, you get c = λ p () = λ w (2) pc + lw = 2w (3) Divide (2) by () to get c = w 2w w. Plug into (3) to solve for l = =. p w 4

5 (c) (6 points) For the next two questions, put consumption on the horizontal axis, and leisure on the vertical axis. Draw the indifference curves of this consumer. (Your drawing does not have to be very precise, but I want you to get the general shape right). To draw indifference curves, fix utility at k and solve for the quantity you re going to put on the vertical axis (leisure). l = k logc The thing to notice is that preferences are quasilinear, and hence indifference curves intersect the horizontal (consumption) axis. If you drew indifference curves that intersect the horizontal axis but are tangent to the vertical axis, you got full credit. 5

6 (d) (3 points) Now, assume the wage rate is 4 and the price of consumption is 6. Draw the budget constraint of the consumer below. (e) (2 points for bundle, point for indifference curve) Draw and label the optimal bundle (c, l ) in the picture above. Draw the indifference curve passing through this bundle. From the demand function (c, l ) = (2/3, ). I label this on the picture with a black dot on the old (green budget constraint). I draw the indifference curve as a black line. (f) Now, assume the wage rate w increases to 5. (2 points) What is the total change in l? The answer to this question is a number. The new consumption bundle is (5/6, ). Therefore, the total change in l is 0. (4 points) What is the income effect for l? The answer to this question is a number. 6

7 By Slutsky (you don t have to mention Slutsky), total change in demand = income effect + substitution effect. I solve for the substitution effect below using the same method as on the homework. We can get the income effect later as total change - substitution effect. (g) (4 points)what is the substitution effect for l? The answer to this question is a number. m = w =. Therefore, to calculate the substitution effect you need to solve the problem with compensated income: max {logc + l} c 0,l 0 6c + 5l 48 + = 59 If you wrote down this problem, but didn t solve it, you get 3 points. Since 59 > 5, you can remember that this problem has an interior solution from homework 3. If you assume the solution is interior without explaining why, you get point taken off. Another way to get full credit is General Lagrangian Method. The first order conditions are c λ 6, and if c > 0, then c = λ 6 () λ 4, and if l > 0, then = λ 5 (2) 6c + 5l 59, and if λ > 0, then 6c + 5l = 59 (3) As before, in both cases when c = 0, we get a contradiction If l = 0, c > 0, then = c λ 6 by (). Therefore, λ > 0. Therefore, c = 59/6 by (3). We now need to check that (2) is satisfied. λ = 6 =. Now, λ if and only if 59, which is a contradiction. Therefore, the only possible solution is l > 0, c > 0. Divide (2) by () to get c = 5 6. From (3), l = = Therefore, the substitution effect is 0.8- = -.2 and the income effect is.2 (2 points) After you calculate them, draw the income and substitution effects in the graph on this page, like we did in class. You don t need to draw the indifference curves. The new budget constraint is in blue. The compensated budget constraint is in red. 7

8 4. (a) (5 points) What does it mean for preferences to be represented by a utility function u? x y u(x) u(y). In words, x weakly preferred to y if and only if utility of x is greater than utility of y. (this is in review sheet no. ) You get full credit for either words or mathematical formulation. (b) (5 points) What assumption do have to satisfy in order to be represented by a utility function? You don t need to write down the mathematical expressions; just write down the names of the assumptions. Rational and continuous (this is in review sheet no. ) 5. What is the definition of a game, as discussed in class? (5 points) Players, a set of strategies for every player, and a utility function for every player. (this is in review sheet no. 2) 8

9 6. EXTRA CREDIT. Here is John Nash s PNAS paper. Summarize the paper in one or at most two sentences at the bottom of the page. Nash defines mixed strategy Nash Equilibrium, and proves that it exists in any game. 9