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1. Suppose that a consumer with utility U(x1, 12) is given an initial endowment 71, 12 so that his total budget set is pill , p232 (a) Derive a formula for the slutsky equation with endowments (b) Suppose that pi, p2 are such that consumption is exactly equal to the initial en- dowment. Show that of = #52 in this special case. 2. You are given the following partial information about a consumer's purchases. he consumes only two goods.: Year 1 Year 2 Quantity Price Quantity Price Good 1 100 100 Good 1 120 100 Good 2 100 100 Good 2 80 Over what range of quantitites of good 2 consumed in year 2 would you conclude: (a) That his behaviour is inconsistent (i.e. it contradicts the weak axiom of revealed preference) (b) That the consumer's consumption bundle in year 1 is revealed preferred to that in year 2? (c) That the conumer's consumption bundle in year 2 is revealed preferred to that in year 1? (d) That there is insufficient information to justify (a),(b),(c) (e) "That good 1 is an inferior good (at some price) for this consumer? Assume that the weak axiom is satisfied (f) *That good 2 is an inferior good (at some price) for this consumer? Assume that the weak axiom is satisfied. 3. Recall that for demand functions to come from a utility maximization function: . They must be homogeneous of degree 0. That is r(tp1, tp2, tm) = I(p1, p2, m) . They must satisfy the budget constraint p . r = m . The slutsky matrix must be symmetric. (#2 + 2x = un + er,) 8p2 . The slutsky matrix is equal to the matrix of hicksian demand functions: up1 Up1 (a) Suppose that we have 2 goods and consider the demand functions x(p,m) defined by: (p, m) = P2 m" Pi + P2 PI 12(p, m) = Bpi m P1 + P2 P2 Determine o and B (b) "Suppose that we have 2 goods and the hicksian demand function for good one hi(p, u) : hi(p, u) = 24 PI 1. Given that 201 = ", show that for pz > 0, h2(p, u) = In ( ) u + g(pz, u) where g is an arbitrary function. 2. Suppose that when pi = pz, hi(p, u) = u, ha(p, u) =0. Determine h2(p, u). 4. Consider the utility function In(x) + y. (a) Optional: Solve for r(p,m), y(p,m), "(p.m), e(p,u), a" (p,m),y" (p,m). Follow the solutions from PS2 if necessary. (b) Suppose that there are 5 people in the economy each with endowments m', i = 1,2, 3, 4, 5. 1. Suppose that m' > p, Vi. Construct the aggregate demand function for a and y. What properties do the individual demand functions have that simplifty this problem?"Deriving the Envelope Thoerem: Consider the more general problem M(o, p) = max, f(r, 0, B) subject to g(r, a, 8) = 0. Show that: dM(a,B) of(x*, a, b) + jog(I', a, b) do da da 2. For each of the following, derive x(p,m), h(p,u), v(p.m), h(p,u) using the standard budget constraint pill + P212 = m: (a) u(x1, 12) = max(1 1, 12) (b) u(:1, 12) = min(1 1, 12) (c) u(x1, 12) = 201 + 12 (d) u(r, 12) = 1 1,43 (e) u(x1, 12) = = Inn + ; Inx2 (f) What happens if we replace u by e" in part (e)? Compare this to part (d). Can you work out an easy way to derive the hicksian demand functions of a function when you make monotonic transformations of the original function? 3. *A besotted mathematician named Donic consumes either gin or tonic. His preferences are rare 'cause he thinks in the square in a way that is almost sardonic. Specifically, Donic prefers larger drinks to smaller drinks but requires that the square of the amount of lime in a drink equal the sum of the squares of the amounts of gin and tonic. Find a utility function that represents Donic's preferences. Find Donic's Marshalian demand functions for lime, tonic, and gin. 4. (From Midterm 2005) Consider a consumer with a utility function u(21, 12) = e(film(z2))1/ (a) What properties about utility functions will make this problem easier to solve? (b) Which of the non negativity input demand constraints will bind for small m? (c) Derive for the marshallian (uncompensated) demand functions and the indirect utility function. (d) Derive the expenditure function in terms of original utils u. 5. Consider the indirect utility function given by: v(P1, P2, m) = m P1 + p2 (a) What are the demand functions (b) What is the expenditure function? (c) What is the direct utility function? 6. 'Consider the utility function: u(x1, 12) = min(2x1 + 12, X1 + 212) (a) Draw the indiference curve for u(x1, 12) = 20. Shade the area where u(x], T2) 2 20 (b) For what values of 2 will the unique optimum be x1 = 0 (c) For what values of , will the unique optimum be x2 = 0 (d) If neither x1 and x2 is equal to zero, and the optimum is unique, what must be the value of #? 7. Assume that there is a consumer with weakly monotonic, convex preferences and who is a utility maximizer. For each of the following pairs of bundles, specify if bundle 1 is ~, 5, or uncomparable to bundle 2.14.04 - Problem Set 1 Due Sept 22nd in recitation 1) Start with an arbitrary utility function u(x1.x2) that is differentiable. Let v(1) be a monotonic transformation of u. a) Solve: max w (x1-*2) ST : PIXI + pzX2 = m b) Solve: max vu (x1.x2)) ST : PIXI + pzx2 = m c) Discuss the relationship between these problems. What characteristics of the utility function is generating this result? 2) Consider the following problem: maxx"y Subject to: x + py = 10,x 2 0,y 20 a) Show formally that the utility function ."y is at least weakly monotonic and strongly convex for a > 0. You may use ideas from problem 1 to simplify the problem. b) Find V(a.p).x(a.p).y(a.p) 3) Solve the following: max Inx + y ST : 2r + y = 10.x 2 0,y 20 4) One way to rule out the potential that the non negativity constraints aren't binding is to look at the marginal rate of substitution (MRS) when one of the factors gets arbitrarily close to zero. Suppose that we have a function f(x1,*2). The MRS12(x1,x2) is the amount of x] required to keep the function f the same when x2 changes by a small amount. MRS12(x1.x2) is read "the marginal rate of substitution of good 1 for good 2 at (11,x2)" Formally: MRS 12 (x1,12) = - dx1 dx2 (n.) a) Consider the function f(x, y) = xy. Starting from a point where x.y > 0. what happens to the MRS., as y grows smaller and approaches zero (ie lim, MRS.,(x,y)) ? What happens to limoMRS,? b) Consider the function x + y. What is lim,-DMRS.,? What is lim,-MRS y.? c) Consider Inx + y. What is lim, .MRS.,? What is lim,-.MRS.?TRADE AND MIGRATION This question explores migration in a two-good Heckscher-Ohlin model. Supposed we have a country (Home) that is endowed with capital K and labor L. It can produce two goods, clothing (C) and food (F) using its K and L endowment. It takes 4 units of labor and 2 unit of capital to make food; and it takes 2 units of labor and 5 units of capital to make clothing. Capital and labor are complements in production of both goods. Home has 160 units of labor and 20 units of capital, while Foreign has 120 units of labor and 120 units of capital. In summary, we have the following information: Technology: Factor Endowments: alf = 4 akr = 2 Home: L = 160; K = 20 aLc = 2 ake = 5 Foreign: L* = 120; K* = 120 10. Given the information above, show which country is capital abundant and which country is labor abundant. Show which good is capital intensive and which good is labor intensive. Which product will Home export? Which product will Foreign export? 1 1. Now suppose that Foreign is closed to trade but experiences an increase in the labor force through immigration from the Home country. This results in "biased" growth. Biased towards which sector? What happens to the relative price of clothing in the Foreign country? Does this change conform to the Rybczynski's Theorem? Explain.TRADE AND MIGRATION This question explores migration in a two-good Heckscher-Ohlin model. Supposed we have a country (Home) that is endowed with capital K and labor L. It can produce two goods, clothing (C) and food (F) using its K and L endowment. It takes 4 units of labor and 2 unit of capital to make food; and it takes 2 units of labor and 5 units of capital to make clothing. Capital and labor are complements in production of both goods. Home has 160 units of labor and 20 units of capital, while Foreign has 120 units of labor and 120 units of capital. In summary, we have the following information: Technology: Factor Endowments: alf = 4 akr = 2 Home: L = 160; K = 20 aLc = 2 ake = 5 Foreign: L* = 120; K* = 120 10. Given the information above, show which country is capital abundant and which country is labor abundant. Show which good is capital intensive and which good is labor intensive. Which product will Home export? Which product will Foreign export? 1 1. Now suppose that Foreign is closed to trade but experiences an increase in the labor force through immigration from the Home country. This results in "biased" growth. Biased towards which sector? What happens to the relative price of clothing in the Foreign country? Does this change conform to the Rybczynski's Theorem? Explain