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Existence of representative consumer Suppose households 1 and 2 have one-period utility functions u(c1) and w(c2), respectively, where u and w are both increasing, strictly concave, twice-differentiable functions of a scalar consumption rate. Consider the Pareto problem:
Subject to the constraint c1 + c2 = c. Show that the solution of this problem has the form of a concave utility function vθ(c), which depends on the Pareto weight θ. Show that vθ(c) = θu (c1) = (1 − θ)w (c2). The function vθ(c) is the utility function of the representative consumer. Such a representative consumer always lurks within a complete
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- A consumer is faced with the followlling Utility Function, U( x 1 x2) = ( xp +xp ) 1/ρ, where 0<ρ<1. The consumer also faces the prices and and has income level m. 1. Set up the Lagrangian 0ptimisation function for the consumer and Compute the optimal consumption bundle for the consumer. 2. The solution in (a) represents the Marshallian demand function for and . Using the solution in (a) compute the indirect utility function. 3. Derive the corresponding expenditure function for the consumer and the Hicksian demand function.Consider a three-commodity consumer setting with the expenditure function:e(p, u) = up1α p2βp3γ 1. Find the indirect utility function2. Find the Walrasian demand function3. Verify Roy's identity4. Recover consumer's direct utility functionIf total utility increases as wealth increases, the first derivative of the utility function is negative. TRUE OR FALSE
- Consider a three-commodity consumer setting with the expenditure function: e(p, u) = up1α p2βp3γ Find the indirect utility function Find the Walrasian demand function Verify Roy's identity Recover consumer's direct utility functionQ11. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F. Suppose as Case A, Total income is $120 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively. a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e (F*,C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space.A consumer is faced with the following utility function, U(x1 x2)=(xp1 1+xp2)1/p, where 0<p<1. The consumer also faces the prices p1 and p2 and has income level m. Set up a Lagrangian optimisation function for the consumer and compute the optimal consumption bundle for the consumer
- Consider the following function describing the utility of a consumer: U(x1, x2, x3) = a1*ln(x1) + a2*ln(x2) + a3*ln(x3), where ln = natural logarithm and a1, a2, a3 constants a. Pose the primal problem (using Langrange's method), obtaining the Marshallian demands for each good and the individual's indirect utility function. b. From the results obtained from question a., find the minimum expenditure function and the Hicksian demands.Emma has a utility function U(x1, x2, x3) = log x1 + 0.8 log x2 + 0.72 log x3 over her incomes x1, x2, x3 in the next three years. This is an example of (A) expected value; (B) quasi-hyperbolic utility function; (C) standard discounted utility; (D) none of the above. Emma’s preferences can exhibit which of the following behavioral patterns? (A) preference for flflexibility; (B) context effffects; (C) time inconsistency; (D) intransitivity.Suppose the consumer solves the following UMP: max (x1)^2 + (x2)^2 , s.t. p1x1 + p2x2 ≤ w where p1,p2 > 0. a) Plot the indifference curves b) Find the Marshallian demand functions. Show graphically the utility maximizing choices.
- Consider an individual whose preferences are represented by the utility function U(x1,x2) = min {3x1+x2 , x1+3x2}. For this individual, Plot the indifference curve giving a utility level of 6. Calculate her MRS at consumption bundle (x1,x2), does MRS depend on the value of (x1,x2)? Calculate her optimal consumption bundle when she has an income of 24, and P1=1, P2=2. Calculate her optimal consumption bundle when she has an income of 24, and P1=1, P2=4. Calculate the income and substitution effects resulting from the change in price of Good 2 (on the level of Good 2 consumption!) Calculate her demand function X1* (P1, P2,m) Calculate her own price, cross price, and income elasticities at X1*(1, 2, 24) and at X1*(1, 4, 24). Based on these, can you say the goods 1 and 2 are (gross, or Marshallian) complements or substitutes?Q3. Consider a utility function with one consumption good q1 and one type of leisure q2 c) Show mathematically and diagrammatically the decomposition of totaleffect.The one-period model with quasi-linear utility predicts that a decrease in marginal income tax rates could increase tax collection if:Group of answer choices Substitution effects dominate income effects so that the percent change in taxes is greater than the percent change in GDP Substitution effects dominate income effects so that the percent change in taxes is less than the percent change in GDP Income effect dominate substitution effects so that the percent change in taxes is less than the percent change in GDP Income effects dominate substitution effects so that the percent change in taxes is greater than the percent change in GDP