Kindly answer this question as soon as possible please. Let a consumer's preferences are reflected by a Cobb-Douglas utility function ie.,UCXX) = Xf X%3Da MBMX2 =P2The demand functions derived from this utility function are:P1Where, M is the income level, P1 and P2 are respective prices. a and B are respective shares ofX1 and X2, and a+B=1Calculate and prove the following:• For good X1, sum of all price and income elasticities is equal to zero.• Weighted sum of all income elasticities is equal to one.Weighted sum of all price elasticities with respective to the price of one good isnegative.

X1= αMP1 Question1: Price elasticity of good X1: = dX1dP1*P1X1 =-αMP12 * P1X1= -αMP1(αMP1) = -1…

Let a consumer's preferences are reflected by a Cobb-Douglas utility function i.e., UXLX) = Xf x5 BM X2 P2 αM The demand functions derived from this utility function are: P1 Where, M is the income level, Pi and P2 are respective prices. g and B are respective shares of X1 and X2, and at+B=1 Calculate and prove the following: For good Xı, sum of all price and income elasticities is equal to zero. Weighted sum of all income elasticities is equal to one. Weighted sum of all price elasticities with respective to the price of one good is negative.

Let a consumer's preferences are reflected by a Cobb-Douglas utility function i.e., UXLX) = Xf x5 BM X2 P2 αM The demand functions derived from this utility function are: P1 Where, M is the income level, Pi and P2 are respective prices. g and B are respective shares of X1 and X2, and at+B=1 Calculate and prove the following: For good Xı, sum of all price and income elasticities is equal to zero. Weighted sum of all income elasticities is equal to one. Weighted sum of all price elasticities with respective to the price of one good is negative.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.11P

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2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (x1, x2, 13) = A (x1 – T1)ª (x2 – ã2)° (x3 – ã3)" - where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + B+y= 1 without loss of generality? Do so for the rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.
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4. What kind of preferences are represented by a utility function of the form u(r1, r2) = VI1 + x2? WWhat about the utility function v(r1,r2) = 13.r, + 13.r2? %3D %3D
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5. What kind of preferences are represented by a utility function of the form u(x1, x2) = x1 + V*2? Is the utility function v(x1, x2) = xỉ + 2x1Vx2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = Vx1x2. What kind of pref- erences does it represent? Is the function v(1, x2) transformation of u(x1, x2)? Is the function w(x1, x2) = xỉx, a monotonic transformation of u(x1, x2)? = x²x2 a monotonic
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A consumer has a utility function U(X,Y ) = X1/2 Y1/2 . Prices of the goods are pX = £10 and pY = £5, respectively, and the consumer has income M = £200 to spend on X and Y. Currently, she buys 2 units of good X and spends the rest of her income on good Y.a) Determine this consumer’s current utility level. b) Computetheconsumer’smarginalutilitiesofgoodsXandY. c) Explain why the consumer’s current consumption of goods X and Y is not optimal. Should she substitute X for Y or vice versa? The consumer’s marginal rate of substitution is MRS = Y/X.d) Determine this consumer’s optimal consumption of both goods. e) By how much does the consumer’s utility increase when she consumes the optimal bundle?
. Answer all parts (a) (c) of this question. (a) Consider an agent whose preferences over any couple (x1, x2), where 2₁ ER+ and x2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R² → R+ which represent these preferences. A politician remarks "Our recent increases in the wage rates of teachers has been a total suc- cess! The shortage of teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely" (b) Explain and illustrate in a diagram what is meant by "income effects" and "substitution effects" of a wage rate change. (c) Explain and illustrate how you would model the labour supply decision of a potential teacher. Do you agree that the wage increase will increase the labour supply in this case? Carefully outline the assumptions underlying your argument.
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2. Jie has the utility function U, (a, b) = 2a + b and Daokui has the utility function U,(a, b) = v2ab, where a is the number of apples consumed and b is the number of bananas consumed. There are 10 units of apple and 10 units of banana in total. We draw an Edgeworth box with apple on the horizontal axis and banana on the vertical 4 axis and we measure Jie's consumptions from the lower left corner of the box. We assume that both apple and banana are infinitely divisible. Please answer the following questions. (a) Write down all pareto optimal allocations (you do not need to explain why). (b) If social welfare function is W min{U,(a, b), U(a, b)}, derive the social optimal allocation. Is this allocation pareto optimal? Suppose the endowment of Jie is 7 units of apple and 0 units of banana, and the endowment of Daokui is 3 units of apple and 10 units of banana. Please answer the following questions. (c) Assume that the price of apple is p, and the uniform price of banana is 1. Derive the…
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