The expenditure function for a consumer with Cobb-Douglas preferences is e(p, u) upip a, where 0 < a < 1. Calculate the substitution matrix for the Cobb-Douglas 1-a demand system for this consumer. Verify that the diagonal terms are negative and the cross-price effects are symmetric. [Tip: for this question, remember that once you have derived the terms da¢ (p,u) and duk (p,u) i, j = 1, 2. for the symmetric matrix, you need to substitute for u with the indirect utility function, v(p, y)]

The expenditure function for a consumer with Cobb-Douglas preferences is e(p, u) upip a, where 0 < a < 1. Calculate the substitution matrix for the Cobb-Douglas 1-a demand system for this consumer. Verify that the diagonal terms are negative and the cross-price effects are symmetric. [Tip: for this question, remember that once you have derived the terms da¢ (p,u) and duk (p,u) i, j = 1, 2. for the symmetric matrix, you need to substitute for u with the indirect utility function, v(p, y)]

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Suppose that there are two goods, gl and g2, that Jane consumes. Demand functions for these two goods are given as follows: gl * = gl(pg1), pg2 ,Inc) g2 * = g2(pg1), pg2 ,Inc) where pg1, pg2 ,Inc refer prices of the goods and income, respectively. Explain and demonstrate (on the graph) the substitution and income effect for both an increase and decrease in price good g1. Use the notations given in the question.
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?
2. Suppose that a consumer's utility function is given by xy U (x, y) = x + y Denote the price of good x by pæ, the price of good y by Py, and consumer's income by w. Find this consumer's Marshallian demand functions and indirect utility function.
A student has a lunch utility function U(S, P) 8S + 3P, where S is number of salads per month and P is number of pizzas per month. The price of a salad is $5, the price of a pizza is 8, and his monthly income for lunch is $80. a) Derive the equation representing the consumer's demand for Salad. Represent the demand for Salad graphically. b) The price of Salad goes up to $25 and the price of Pizza does not change. Compute and represent graphically the new equilibrium of the student. What can you tell about the substitution effect on Salad and pizza
Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…
Karen has perfect complement preferences over toothbrushes B and toothpaste P, with the associated utility function U(B,P) = min (3B, 2P). Suppose the price of a toothbrush is $6, the price of toothpaste is $4 a tube, and she has $72 to buy her entire year's supply of toothbrushes and toothpaste. How many toothbrushes and tubes of toothpaste will she buy?
Consider the following utility function: U = 100x°.50,0.50 A consumer faces prices of P, = $2 and P, =$1 Assuming that graphically good X is on the horizontal axis and good Y is on the vertical axis, suppose the consumer chooses to consume 6 units of good X and 11 units of good Y. Then the marginal rate of substitution is equal to: MRS = (Enter your response rounded to two decimal places. Do not forget to include the negative sign.) Use absolute values. 15 stv МacВook Air 80 DI esc F1 F2 F3 F4 F5 F6 F7 F8 @ %23 2$ & 1 2 4 6 7 8 Q W T Y tab A S F H J K aps lock C V M control option command B H
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.
Consider the following utility function: U= 4X + Y. A consumer faces prices of P, = $2 and P, =1. Assuming that graphically good X is on the horizontal axis and good Y is on the vertical axis, suppose the consumer chooses to consume 10 units of good X and 19 units of good Y. Then the marginal rate of substitution (MRS) is equal to: MRS = (Enter your response rounded to two decimal places. Do not forget to include the negative sign.) étv Help Me Solve This eText P MacBook Air 80 DII esc F2 F3 F6 F8 F1 F4 F5 F7 @ 23 2$ & 1 2 4 6 7 8. Q W E R T Y U tab A S G caps lock C V shift fn control option command ーのエ B レ
Consider the following utility function: U = 100x.50,0.50 A consumer faces prices of P, = $2 and Py =$1. Assuming that graphically good X is on the horizontal axis and good Y is on the vertical axis, suppose the consumer chooses to consume 6 units of good X an units of good Y. Then the marginal rate of substitution is equal to: MRS = 1.83. (Enter your response rounded to two decimal places. Do not forget to include the negative sign.) Use absolute values. The consumer should consume to maximize utility. more Y and less X the same amount of X and Y more X and less Y 15 tv MacBook Air 80 esc DII F1 F2 F3 F4 F5 F6 F7 F8 @ #3 % 2 7 8. Q W E Y tab s lock A S F G H J Z C V N M Onteel 8 : < CO T D
4.13 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U(x, y) = (x° +y®)'/8 [in this function the elasticity of substitution o = 1/(1 – 6)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p,)¬/", where r = 8/(ò – 1) = 1 – 0. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by E = V(p', + p,)''". f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.
Repeat Question 1 with the utility function u(x₁, x₂) = x₁x2. (a) Determine the (Walrasian) demand functions for goods 1 and 2, x1(p1, p2, 1) and X2(P1, P2, I). Are goods 1 and 2 gross substitutes, gross complements, or neither. (b) Suppose that prices are p₁ = 1, p2 = 4 and income is I = 64. Compute the utility associated with the optimal choice of the consumer. (c) Then due to a tax of 3 dollars per unit, the price of good 1 increases to p₁ = 4. Compute the utility associated with the optimal choice of the consumer. (d) Determine the compensating variation in income CV. (e) Determine the equivalent variation in income EV.