Suppose an individual has preferences over goods x and y, and their expenditure minimization problem has the following expenditure function: E(px, Py, U) = (px + 3p,)U. What is the person's Hicksian demand? O(h, hy) = (2p U, pU) O(h, hy) = (U, 3U) O(h, hy) = (Up,',Up,) O(hx, hy) = (3U, U) What is the individual's indirect utility? OV = 3p! p OV = P.+3p, OV = P Py OV = P.P
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- 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?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…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.
- Morgan has the following utility function: u(x, y) = 5 ln(x) + 3y. Her income is given by I = 15 and the prices originally are pr = 2 and py = 3. = (a) What are Morgan's Marshallian demands? (b) How much of each good is Morgan currently consuming? (c) What is the utility level that Morgan can achieve? (d) Assume the price of x increases to p = 4, find Morgan's new levels of consumption. X X (e) Find the total, substitution and income effects for good x caused by the price change. Consider this price change a "large" price change (Apz = Pz - Px=4-2=2).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 %3DNaomi has the following utility function, U(x, y) = x + 3y. Suppose that py > 4pr. What are the hicksian demands of x and y? (a) (hr, hy) = (U, 0) (b) (h, hy) = (0, U/3) (c) (ha, hy) = (U, U/3) (d) (ha, hy) = (In (U), In (U/3))
- Suppose Marcel's preferences over consumption bundles (X, Y) can be represented by the utility function U(X, Y) = X³Y. Which of the following expressions gives Marcel's Marshallian demand? (a) (X*,Y*) = ( 31 3px+py' (b) (X*,Y*) = (px (PX+py)) (d) (X*,Y*) : = 3px' +py) 3px+py (c) (X*,Y*) = (-1,0) Ipx px (px+py)' py (py+PX) 1 I 3 I 4px' 4 py (e) (X*,Y*) = = ( px + 4py ² : 31 px +4py' 4px+pyDerive the Hicksian demand and the expenditure function for u(x,y)=(.4X^.5+.6Y^.5)^23. 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?
- 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.A consumer chooses non-negative consumption levels of two goods x1 and x2 and has a utility function given as: U = = 0xfx% where 0 < a <1 and 0 is a constant. Assume that the prices of goods x1 and x2 are respectively Piand p2 Set up the expenditure minimization problem of the consumer and derive the Compensated (Hicksian) demand function for goods 1 and 2 and investigate whether the two goods are complements or substitutes. i.(b) U(x, y) = min [ax, y]