Problem 3. Recall u is CES if for p < 1, a; > 0 and Σ; ai = 1, ² u(x) = (Σ α₁x² (1) Show that preferences are strictly convex and satisfy local nonsatiation. (2) Solve for its Hicksian demand directly.
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- There are 6 cups of espresso (x1) and 6 bars of chocolate (x2) in the house. Andrew and Eric have identical preferences, U₁ = alog(x) + (1 - a)log(x), where i = A, E and a € (0, 1). Suppose the Target allocation is ((2, 2), (4, 4)). (a) What is the relationship between UA and UE in the Target? (b) Find the ratio of equilibrium prices at the Target allocation. (c) Determine the lump-sum transfer necessary to achieve the Target if An- drew is initially endowed with (4, 1) and Eric with (2,5). Set x2 as the numeraire, i.e., assume pi 1. Under what condition would require Andrew to transfer to Eric or vice versa? Why? If a > 1/3, Andrew transfers to Eric and vice versa. = p and p2 =In the nonlinear function Y = ax ZC, the parameter c measures Multiple Choice O O O the percent change in X for a 1 percent change in Z. None of these options are correct. the elasticity of Y with respect to Z. ΔΥ/ΔΖ.Please answer all (a) - (e), whether they are True or False: (a) If a consumer spends her entire income, then she has a strictly monotone utility function. (b)The condition that ‘the marginal rates of substitution equal the ratio of prices’ is necessary but not sufficient for a given bundle to be a Walrasian demand. (c) If U, V: R2 → R are such that U is a strictly increasing transformation of V then U and V must represent the same preferences. (d) If the substitution effect is negative (in response to a price increase) then we know the Walrasian demand for the good in question (in response to the same price increase) will also be negative. (e) A consumer’s utility is continuous and strictly monotone and when prices are given by p and income is I her Walrasian demand yields a utility of 7. Then, any bundle that yields a utility of at least 8 must cost more than I.
- 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…Please answer all (a) to (e), whether they are True or False:(a) If a consumer spends her entire income, then she has a strictly monotone utility function.(b) The condition that ‘the marginal rates of substitution equal the ratio of prices’ is necessary but not sufficient for a given bundle to be a Walrasian demand.(c) If U, V: R2 → R are such that U is a strictly increasing transformation of V then U and V must represent the same preferences.(d) If the substitution effect is negative (in response to a price increase) then we know the Walrasian demand for the good in question (in response to the same price increase) will also be negative.(e) A consumer’s utility is continuous and strictly monotone and when prices are given by p and income is I her Walrasian demand yields a utility of 7. Then, any bundle that yields a utility of at least 8 must cost more than IM -)a+P (- a + B (II) Consider indirect utility function U* = (- P `P, (a) Find Marshallian demand functions for X1. (b) Find Minimum Expenditure function. (c) Find Hicksian demand functions for X2
- 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?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).2. The utility function is given by u(x,,x,) - In x, +– In x,. Check whether the following 4 ôg(p.1) _ ôf,(p.u)_ô9(p.1) др, Marshallian demand function, f(p,u) – the Hicksian demand function. Slutsky equation · P,(p,1) is satisfied. (p,1) - the