Problem 2. Consider once again an economy in which there are two goods, 1 and 2, whose prices are p1 > 0 and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts x1 and x2, respectively. A consumer has preferences over R 2 + which are represented by the utility function u : R 2 + R, (x1, x2) 7 u(x1, x2) := x2 1 + x2 2 1/2 , where and are strictly positive real numbers. The consumer's income is I > 0. (Note: This is an example of CES utility function) (a) Formulate the consumer's utility maximization problem, find the firstorder conditions for utility maximization, and find the Marshallian demand functions x1(p1, p2, I) and x2(p1, p2, I) for goods 1 and 2, respectively. (Note: Use the Lagrangian method. Assume that the budget constraint holds with equality and that the solution is interior (i.e. x1 > 0 and x2 > 0), thus disregarding the non-negativity constraints on x1 and x2. Do not check second order conditions.) (b) Find the price elasticity of demand for good 1. Is demand for this good elastic or inelastic? (c) Find the derivative of the quantity of good 2 demanded with respect to the price of good 1. Use this to indicate whether goods 1 and 2 are substitutes or complements. Now consider a consumer whose preferences over R 2 + are represented by the CES utility function u : R 2 + R, (x1, x2) 7 u(x1, x2) := (x 1 + x2 ) 1/ , (1) where , and are real numbers such that 0 6= < 1, > 0 and > 0 The consumer's income is I > 0. (d) Why does the utility function v : R 2 + R, (x1, x2) 7 v(x1, x2) := x 1 + x 2 represent the same preferences over R 2 + as the function u in (1)? Now forget about the previous utility functions. For the next question, consider generic Marshallian demand functions for goods 1 and 2, x1(p1, p2, I) and x2(p1, p2, I), resulting from a consumer's utility maximization problem. Assume that Marshallian demands are differentiable functions of prices. (e) Write the budget constraint as p1x1(p1, p2, I) + p2x2(p1, p2, I) = I. Now take a derivative of this identity with respect to p1. (Note: Your derivative should have three terms, two of which are related to elasticity and one of which tells you whether these goods are substitutes or complements.) Use this derivative to argue that, for any demand function, if the demand for good 1 is inelastic, then the goods are complements. 3 (Note: This tight relationship does not generalize beyond the case of two goods. This is an unusual case in which a result depends on there being only two goods in our model.)