4. Consider the following two-good utility function: u(x1, x2) = VT1 + V#2. Answer the questions set out below. ITip: Go back to the definition of monotonic transformations on page 17 (theorem 1.2) to confirm that the transformation is effected by a strictly increasing function, in other words if v(r, u) = f(u(x, y)), f is a strictly increasing function on the set of values taken on by u. 1 (a) What information is contained in the marginal rate of substitution of a utility function (apart from the fact that it represents the slope of the indifference curves)? Derive the marginal rate of substitution for the utility function u(x1, x2) above. (b) Explain why Hicksian demand functions are often referred to as "compensated demand functions". (c) If the corresponding budget constraint is given by y = P1x1 + P2x2, show that the con- %3D 2 sumer's Hicksian demand functions are x (p, u) = („)*u² and a (p, u) = („)* u² - Pi+p2 respectively. (d) What is the degree of homogeneity in prices of the Hicksian demand functions in (c) above? Provide an economic interpretation for this.

4. Consider the following two-good utility function: u(x1, x2) = VT1 + V#2. Answer the questions set out below. ITip: Go back to the definition of monotonic transformations on page 17 (theorem 1.2) to confirm that the transformation is effected by a strictly increasing function, in other words if v(r, u) = f(u(x, y)), f is a strictly increasing function on the set of values taken on by u. 1 (a) What information is contained in the marginal rate of substitution of a utility function (apart from the fact that it represents the slope of the indifference curves)? Derive the marginal rate of substitution for the utility function u(x1, x2) above. (b) Explain why Hicksian demand functions are often referred to as "compensated demand functions". (c) If the corresponding budget constraint is given by y = P1x1 + P2x2, show that the con- %3D 2 sumer's Hicksian demand functions are x (p, u) = („)u² and a (p, u) = („) u² - Pi+p2 respectively. (d) What is the degree of homogeneity in prices of the Hicksian demand functions in (c) above? Provide an economic interpretation for this.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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1. Consider a consumer who consumes two goods. The utility function is given by u(x1, x₂) /2x1+1+√√√x₂, - where x > 0 denotes the amount of good i = the price of good 1 is 3, the price of good 2 is p2 > 0, and the income is 12. 1,2 consumed. Unless stated otherwise, (1) Calculate the marginal rate of substitution (measuring the value of good 1) at the consumption plan (x₁, x₂) = (3, 4). (2) Calculate the consumption plan that is optimal for the consumer. (3) Let v(p₂) denote the utility level at the optimal consumption plan. Calculate its derivative with respect to p2, i.e., v'(P2), at p2 = 3. (4) Suppose that p2 = 1 and the utility level is 10. Calculate the compensated demand for each good.
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Elements of Mathematical Economics
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