3. (23 points) A consumer consumes two goods: 1 and 2. His utility function is U (11,02) = v(11) + where v(x1) is a posi

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3 23 Points A Consumer Consumes Two Goods 1 And 2 His Utility Function Is U 11 02 V 11 Where V X1 Is A Posi 1 (95.98 KiB) Viewed 17 times

3. (23 points) A consumer consumes two goods: 1 and 2. His utility function is U (11,02) = v(11) + where v(x1) is a positively sloped function and 21,19 > 0. As- sume v'(11) > 0 and v"(x) < 0 for all 11 > 0. The consumer has a budget constraint Plutp:12 = m where pi and P2 are the prices of goods 1 and 2 and m is his budget, show that he will consume such that v' (31) a) (4 points) Given the utility function and the budget constraint, find an implicit expres- sion for the optimal consumption of good 1 by substituting 3, from the budget constraint in the utility function and maximizing (note, you cannot find an explicit solution for G1). b) (2 points) Verify the second order conditions for your optimization to make sure you found a maximum c) (3 points) We know that optimal consumption is found when the marginal rate of substitution, MC, is equal to the relative prices, Use this knowledge to confirm your result found in a). d) (3 points) Consider a function of total expenditure of the consumer: (21,0) = P + Find the gradient of U(* 1,02) and the gradient of E($1,12). e) (3 points) Use your answer in d) to build a Jacobian matrix for the functions U (81,12). E(+1,12). Confirm that the consumption is at the optimum when the gradients are collinear, i.e. when the determinant of the Jacobian is zero. 1) (4 points) A reminder of microeconomics, the price clasticity of demand is er where x is the consumption of the good and p is its price. Also remember that an ordinary good has negative price elasticity. Use total derivatives, your utility maximization result (in a), c) or e), and the properties of v to show that good 1 is an ordinary good. g) (4 points) Use total derivatives and the budget constraint to show that the price elasticity of good 2 is pi d. dp # -1