10fB_ps03solutions

# 10fB_ps03solutions - Microeconomics ECON 100A Problem Set 3...

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Microeconomics ECON 100A Problem Set 3 Solutions Due October 13, 2010 (Solutions Posted October 15, 2010) 1. Toni’s utility is 3 2 3 ) , ( z x z x U = a. Does Toni consider X and Z goods? (“Goods” are items we like more of, ie. more is better. “Bads” are items that we like to have less of, ie. less is better. For example, pizza is a good. Garbage is a bad.) Prove your answer. b. What is Toni’s marginal rate of substitution between goods x and z (her MRS)? c. Are Toni’s indifference curves convex? Prove you answer. a. X and Z are goods (assuming x and z>0) because MU is positive for x and z. 3 6 ) , ( xz x z x U = >0 and 2 2 9 ) , ( z x z z x U = >0 b. The MRS is x z z x xz z z x U x z x U MRS 3 2 9 6 ) , ( ) , ( 2 2 3 - = - = - = , Yes, the indifference curves are convex since the absolute value of the MRS falls as z decreases and x increases. 2. Consider the constrained optimization problem. [This question is especially important because it enables you to solve for consumer demand functions from scratch.] Y z p x xz z x U z z x = + = x 3 , p subject to 3 ) , ( max a. List the choice variables. The choice variables are x and z b. List the parameters. The parameters are P x , P z , Y. c. Are they goods or bads? Prove this. Yes, they are goods. One way to prove this is to show that U x and U z are positive for all positive x, z (i.e. utility is increasing in x and z).

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9 ) , ( and 0 3 ) , ( 2 3 > = > = xz z z x U z x z x U d. Are they convex? Yes, they are convex. You can use any of the four methods shown in class. One way is to show that MRS is decreasing as x increases and z decreases along an indifference curve. Here, the MRS= x z z x U z x U z x 3 ) , ( ) , ( - = - which falls as z decreases and x increases. e. Draw an indifference curve diagram and budget line and show the optimal bundle of goods for the consumer. See class notes. Graph with X on horizontal axis and Z on vertical axis. Budget line has X intercept of I/P x and Z intercept of I/P z . Then draw indifference curves (level sets of utility function) with goal of reaching indifference curve tangent to budget constraint. f. Set up the Lagrangian. The Langragian is )) , ( ( ) , ( z x h z x f L λ + = where f(.) is the objective function (what we want to maximize) and h(.) is our constraint (something our solution must satisfy). In our case: ) ( 3 3 z P x P Y xz L z x - - + = g. Write down the first order conditions. F.O.C (first-order conditions) are that the first-partial derivatives of L with respect to each choice variable and λ equal 0 for a maximum, i.e.: 0 0 9 0 3 2 3 = - - = = - = = - = z P x P Y L P xz z L P z x L z x z x 2 3 9 3
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10fB_ps03solutions - Microeconomics ECON 100A Problem Set 3...

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