2. The utility function is given by u(x₁,x2)==lnx +±ln x₂. Check whether the following Slutsky equation 1 1 4 (p.1) дрг f(pu)_ĉo (p.1) ap2 ·Q2(p,1) is satisfied. (p,I) - the al Marshallian demand function, ƒ(p,u) – the Hicksian demand function.
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- 1. For each of the following utility function, compute the Hicksian demand function h(p, v), the expenditure function e(p, v) and the Slutsky (Substitution) matrix S(p, w). In your com- putation procedure, feel free to use your (or my) answers in Assignment 5. 3 a) u(*1, 12) = VI, + ¤2 b) u(x1, 12) = log(1) +3 log r2 c) u(r1, 12) = min(r1, 12) d) u(x1,r2) = max(r1, r2) %3D %3D %3D %3D1. Consider the utility function given by u (x1, x2) = x1x3, and budget constraint given by P1x1 + P2x2 = w. (a) Solve the EMP to find the Hicksian demand function, h (p, u). (b) Find the expenditure function e (p, u). (c) Recover h (p, u) from e (p, u). (d) Noting that the indirect utility function corresponding to the UMP for this form is 4w3 v (P1, P2, w): 27p1p3' demonstrate that v (.) and e (.) are inverses of each other. (e) Using only the indirect utility function above, recover the optimal consumption bundle of the UMP (i.e., Walrasian demand function). (f) Using your answer to (e), identify the substitution effect on the quantity demanded of good 1 of a change in the price of good 1. 1; and consider the Hicksian demand curve for good 1 corresponding to , and Walrasian demand curve for good 1 corresponding to w = 1. At what price and quantity for good 1 do they intersect? Which is steeper at this point of intersection? (g) Assume P2 What does that signify? (h) Using your…3- Assuming that the equation F(U, x, X2 = f(x1, x2,, ... Xn): ******* **** ,xn) = 0 implicitly defines a utilityfunction U 073 a) Find the expressions for 60, 6U and 6x4 " 3 6x2 6xn 6x2 6xn b) Interpret their respective economic meanings. c) Now. assume the utility function is U(x, y) = y√x. Does the consumer believe that more is better for each good? Do the consumer's preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing?
- Derive Ryan's demand function for q₁, given his utility function is where o = = (9₁) P + (9₂)P, 1 1-p The demand curve for q₁ as a function of P₁, P2, and Y is (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ 9₁ = character.) U= Let the price of q₁ be p₁, let the price of q2 be p2, and let income be Y.Let a consumer's indirect utility function for two goods be described as follows: 20m о(p, т) — Pi 5p1 P2 where p1, P2 > 0 and m > 0. (a) Construct a direct utility function u that generates v as its indirect utility function. (You may use that v is quasi-convex in p.) (b) Determine the consumer's Marshallian demand x(p, m) for good 1. (c) Determine the consumer's expenditure function e(p, u). (d)) Determine the consumer's Hickisan demand x#(p, u) for good 1.A utility function is called separable if it can be written as U(x, y) = U (x) + U,(y), where U > 0, U" < 0, and U1, U, need not be the same function. a. What does separability assume about the cross-partial derivative U„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.
- Consider a consumer with indirect utility function v(p, w) = w - B1p1 - B2p2 √P1P2 where 3₁ and 32 are nonnegative constants. (a) Find the consumer's expenditure function. (b) Find the Hicksian and Marshallian demands for Good 1. (c) Find expressions for the substitution and income effects on Good 1 associated with a marginal increase in the price of Good 2. (d) Suppose that the price of good 1 changes from pi to kp₁ where 0 < k < 1. Find the compensating variation of this price change.Suppose you have the following indirect utility function: V(Pa, Py, I) = In PxPy What are marshallian demands for x and y? I (a) (9x9y) = (22) (b) (9,9y) = (In, In 2) (c) (9, 9y) = (exp(2p/py), exp(2ppy)) I (d) (9x, gy) = (2pr+py' px+2py) What is the expenditure function for the associated expenditure minimization problem? (a) E(pa, Py, U) = (P + Py) ln(U) (b) E(pa, Py, U) = √exp(U)Papy (c) E(pa, Py, U)= (p²+p²) In(U) (d) E(pa, Py, U) = exp(U)²papy What are the individual's Hicksian demands for goods x and y? (a) (h₂, hy) = ((BU)¹/², (PU) ¹/²) (b) (ha, hy) = (RU, DU) (c) (ha, hy) = ((2 exp(U))¹/², (exp(U))¹/²) -1/2 (d) (hx, hy) = ((P₂PzU)−¹/², (P₂PzU)-¹/2) Are x and y complements or substitutes?2. Consider the following utility function, u (x1, x2) = x{x", where y e (0, 1) %3D (a) Derive the Hicksian demand functions and the expenditure function. (b) perive the indirect utility functions.
- 3. Consider the following utility function, u (21, x2) = min V#1, Varz), where a > 0 Derive the Marshallian demand functions. (Explain your derivation in details.) Does the Marshallian demand increase with price? Are the two consumption goods normal goods? Show two different ways to derive the Hicksian demand functions. (b) Does the Hicksian demand increase with price?1. Prove that MRS = - holds when the utility function is U = U₂ = U (91, 92).Answer the following questions using the following information. Columns 1 and 2 in the table below show the marginal utility that Cody gets by purchasing products A and B. Column 3 shows the marginal utility Cody gets from saving Assume that the price of A is $13 the price of B is $10, and Cody has an income of $129. a) Find the following series of MU/$ for each column. Note: Keep as much precision as possible during your calculations. Your final answer should be accurate to at least two decimal places Column 1 Column 2 Column 3 Units of A MU MU/S Units of B MU MU/$ Number of $ saved MU MU/S 68 6.8 1 80 6.15 1. 13 0. 2 71 5.46 2 63 6.3 2 10 0. 3 66 5.08 3 56 5.6 3 7 4 61 4.69 4. 46 46 0. 54 4.15 40 4 45 3.46 31 3.1 6. 1 39 3 22 2.2 0. 29 2.23 8 17 1.7 8 b) What quantities of A and B will Cody purchase in maximizing his utility? Quantity of A: 0 Quantity of B: 0 c) How many dollars will Cody save? Dollars Saved = $0 SAVE AND CLOSE