PS4-Solution. Mehrdad Esfahani. Fall Arizona State University. Question 1 Question 2 Question 3 Question 4 Question 5

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1 PS4-Solution Mehrdad Esfahani Arizona State University Fall 2016 Mehrdad Esfahani PS4-Solution 1 / 13

2 Part d Part e Question 1 Choose some 1 k l and fix the level of consumption of the goods index by i k is fixed and choose two bundles with different levels of all other goods: x, y x y u(x) u(y) k l u i (x i ) + u i ( x i ) i=1 k u i (x i ) i=k+1 i=1 i=1 k u i (y i ) k l u i (y i ) + u i ( x i ) i=1 i=k+1 Mehrdad Esfahani PS4-Solution 2 / 13

3 Part d Part e Question 1 Let w 1 > w 0 and for t = 0, 1 let x t solves the utility maximization problem. Rtp: x 1 x 0. If x 0 i = 0, then obviously x 1 i x0 i. So let s assume x0 i > 0. Since w 1 > w 0, x 1 l x 0 l for some good l. Let λ i be the marginal utility of wealth level i. we have: u i (x1 i ) p i λ 1 = u l (x1 l ) p l < u l (x0 l ) p l λ 0 = u i (x0 i ) p i Since u i ( ) is strictly decreasing, x1 i > x0 i. Mehrdad Esfahani PS4-Solution 3 / 13

4 Part d Part e Question1 For compensated demand: h(p, v(p, w)) = d(p, w). If w increases, v(p, w) increases because of LNS and hence compensated demand increases. Mehrdad Esfahani PS4-Solution 4 / 13

5 Part d Part e Question 1 If good l is not the one with non-concave function, then the analysis is exactly the same as the previous part. If one the other hand, good l is the one that increases, then depending on the sign of the derivative of the marginal utility, other goods are either all inferior locally (assuming consumption of each is positive of course). So we must have some inferior good in this case. An example of such function is: u(x 1, x 2 ) = e x 1 + e x 2 Mehrdad Esfahani PS4-Solution 5 / 13

6 Part d Part e Question 1 Example Complementary goods: computer, mouse, keyboard Habit formation: the utility of dinner depends on what was eaten for lunch and breakfast. Mehrdad Esfahani PS4-Solution 6 / 13

7 Part a By Walras law, αp x(αp, w) αw, so that p x(αp, w) w. By the weak axiom, if x(p, w) x(αp, αw), then αp x(p, w) > αw, so that p x(p, w) > w, which is a violation of Walras law. Therefore, x(p, w) = x(αp, αw). Mehrdad Esfahani PS4-Solution 7 / 13

8 Part a For this part, we need two expressions that I leave their proof as an exercise: S(p, w) p = 0 S T (p, w) p = 0 For the two good case, we have [ ] s11 s 12 s 21 s 22 [ s11 s 21 s 12 s 22 ] [ p1 p 2 [ p1 p 2 ] = ] = [ ] 0 0 [ ] 0 0 Therefore, p 1 s 11 + p 2 s 12 = p 1 s 11 + p 2 s 21 = 0 s 12 = s 21. Mehrdad Esfahani PS4-Solution 8 / 13

9 Part a Using Roy s identity: d i l (p, wi ) = a i (p) p l + b(p) w i p l b(p) For this being homothetic, we need the demand to be HD1 in w. Therefore a i (p) is a constant. For it to be quasilinear in good l, b(p) should be a function of p l. 1 1 Any change in prices rather than p l should not change the demand and the wealth effect disappears. Mehrdad Esfahani PS4-Solution 9 / 13

10 Part a Using the fact that v(p, e(p, u)) = u, we can find the expenditure function: e(p, u) = u a(p) b(p) We do not know anything about the curvature of consumer s problem. So no inference about the expenditure function can be done. Mehrdad Esfahani PS4-Solution 10 / 13

11 Part a I propose the following redistribution: w i = ai (p) a i (q) b(q) + b(p) b(q) wi Summing up over all consumer to check that if this redistribution is feasible: I i=1 w i w i = 1 (V (p, w) V (q, w)) 0 b(q) From the redistribution equation, we have: a i (q) + b(q) w i a i (p) + b(p)w i This means everyone is weakly prefer this redistribution. Mehrdad Esfahani PS4-Solution 11 / 13

12 The indifference between the prices means p 1 p 2 = q 1 q 2. Suppose we have a wealth redistribution to make consumer 1 indifferent, then w 1 = q 1. Doing the same for consumer 2, w 2 = q 2. For 2p 1 2p 2 feasibility of this redistribution, we need w 1 + w 2 1 (q 1 p 1 ) So, there is no redistribution. 2 Keeping in mind that p 1p 2 = q 1q 2 Mehrdad Esfahani PS4-Solution 12 / 13

13 Since 0 Y, then π(p) p 0 = 0. Consider two possibilities: 1 y Y : p y 0 π(p) = 0. 2 y 0 Y such that p y > 0 βy Y, β 1 βy 0 Y π(p) + Mehrdad Esfahani PS4-Solution 13 / 13

Microeconomics, Block I Part 1

Microeconomics, Block I Part 1 Piero Gottardi EUI Sept. 26, 2016 Piero Gottardi (EUI) Microeconomics, Block I Part 1 Sept. 26, 2016 1 / 53 Choice Theory Set of alternatives: X, with generic elements x,