Given the following indirect utility function: v(P,y) = y(p₁ + p₂) Derive the Marshallian demand function? Ensure to state the identity used to establish the demand function
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- Using the homogeneity of indirect utility, which of the following cannot be an indirect utility function? (a) V: = (b) V: = (c) V: = [2 PzPy I Pz+Py p²+p² (d) V = 1 PzDerive 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.6. Assume that utility is given by U(x,y) = x03J07 and Income I, price of good x = P and price of good y = Py 1 Show your work for each of the following parts (a) Use the uncompensated demand functions to compute the indirect utility function and the expenditure function (E) for this case. (b) Use the expenditure function calculated in part (a) together with Shephard's lemma to compute the compensated demand function for good x. (c) Use the results from part (b) together with the uncompensated demand function for good x to show that the Slutsky equation holds for this case.
- 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. 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…Which of the following is not a required characteristic for a function U(.) to be considered a valid utility function? (Select all that apply) U(.) must be monotonically increasing in all possible bundles U(.) must be able to assign a utility to every possible bundle of goods For any two bundles, A and B, either U(A) ≥ U(B) or U(B) ≥ U(A) must be true For any three bundles, A, B, and C, if U(A) ≥ U(B) and U(B) ≥ U(C), then U(A) ≥ U(C)
- Fi Consider the utility function: u(x1,x2) = Axf x}-« (-a where 0 0. (a) Compute the Marshallian demand functions. (b) Derive the indirect utility function. O Type here to :3 The following utility function is known as CES (constant elasticity of substitution) function: U (x, y) = (ax° + By')'/º, where a > 0, B > 0 %3D a) Is this function homothetic? b) How does the MRSY depend on the ratio x/y? Specifically, show that the MRSxy is strictly decreasing in the ratio x/y for all values d 1 and constant for 8 = 1. (Hint: take the derivative of the MRS with respect to the ratio x/y as z = and take the derivative with respect to z) x,y x/y c) Show that if x = y, the MRS of this function depends only on the ratio /B.The marginal utilities derived from the consumption of goods X and Y by a given consumer are given by MUx=40-6x and MUy=80-10y. If the prices of X and Y are Gh2 and Gh10 respectfully. determine the number of units of the commodities that the consumer must consume to optimize satisfaction granted that he/she earns an income of GH40 within the period.
- 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.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 %3DGiven a consumer has a money budget M = 90 and utility function ? ( ? , ?) = ? 1/4? 1 / 2 If she consumes two goods x and y with prices given by ? = 2 and ? = 4 state the ? ? budget constraint equation. Determine the utility maximizing quantities X and Y for the consumer