325_PracticePS1_Soln

325_PracticePS1_Soln - Department of Economics University...

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Department of Economics University of Maryland Economics 325 Intermediate Macroeconomic Analysis Practice Problem Set 1 Suggested Solutions Professor Sanjay Chugh Spring 2011 1. Partial Derivatives. For each of the following multi-variable functions, compute the partial derivatives with respect to both x and y . Solution: In order to compute the partial derivative with respect to x, momentarily pretend that y is a constant (for example, imagine momentarily that y = 5) and proceed to differentiate using the usual rules of calculus. Likewise, in order to compute the partial derivative with respect to y , momentarily pretend that x is a constant (for example, imagine momentarily that x = 5) and proceed to differentiate using the usual rules of calculus. Applying this algorithm to each of the given functions: a. (, ) f xy x y We have ( , ) x y and y x . b. ( ,) 2 3 x y ± We have ( , ) 2 x fxy and (, ) 3 y . c. 24 xy We have 4 (, ) 2 x x y and 23 (, ) 4 y . d. (, ) l n 2 l n x y ± We have ( , ) 1/ x x and (, ) 2 / y y . e. 2 x y ± Recall from principles of basic mathematics that we can write this function as 1/2 2 x y ± . Hence, the partial derivatives are 1 / x x x ² and 1 / y y y ² .

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2 f. (, ) x fxy y Recall from principles of basic mathematics that we can write this function as 1 f xy x y ± . Hence, the partial derivatives are 1 1 / x y y ± and 22 / y x y x y ± ± ± . g. y x Recall from principles of basic mathematics that we can write this function as 1 y x ± . Hence, the partial derivatives are / x y x y x ± ± ± and 1 1 / y x x ± 2. Properties of Indifference Maps. For the general model of utility functions and indifference maps developed in class, explain why no two indifference curves can ever cross each other. Your answer must explain the economic logic here, and may also include appropriate equations and/or graphs. Solution: The proof proceeds by contradiction. Consider the following indifference curves that cross each other. c 2 c 1 A C B
3 The consumption bundle A lies on both indifference curves. Because bundle A and bundle B lie on the same indifference curve, they yield the same level of utility. Likewise, because bundle A and bundle C lie on the same indifference curve, they must yield the same level of utility. This then implies that bundle B and bundle C yield the same level of utility (transitive property of preferences). But if this were true, then bundle B and bundle C should lie on the same indifference curve, which they do not by assumption. Thus, we have reached a logical contradiction – indifference curves cannot cross each other. 3. A Canonical Utility Function. Consider the utility function 1 1 () , 1 c uc V ± ± ± where c denotes consumption of some arbitrary good and (the Greek letter “sigma”) is known as the “curvature parameter” because its value governs how curved the utility function is. In the following, restrict your attention to the region 0 c ! (because “negative consumption” is an ill-defined concept). The parameter is treated as a constant.

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This document was uploaded on 11/01/2011 for the course ECON 325 at Maryland.

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325_PracticePS1_Soln - Department of Economics University...

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