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Midterm 1 Answer Key - Spring 2010

Midterm 1 Answer Key - Spring 2010 - Economics 1480 spring...

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Economics 1480 – spring 2010 Midterm #1 – ANSWER KEY Directions: Please place your final answer in the space provided below. Partial credit will be given in problems C and D for work that is both carefully done and legible. Good luck! Problem A (20 points – 2.5 points each): Consider an economy with two individuals (1 and 2), each of whom consumes good Z and good Y. The economy is endowed with 6 units of Z and 6 units of Y. Each has a utility function given by: U = Z × Y Consider the following allocations Individual 1 Individual 2 Y Z Y Z Allocation A 6 6 0 0 Allocation B 6 0 0 6 Allocation C 2 4 4 2 Allocation D 3 3 3 3 U 1 = 6·6 = 36 U 2 = 0·0 = 0 U 1 = 6·0 = 0 U 2 = 0·6 = 0 U 1 = 2·4 = 8 U 2 = 4·2 = 8 U 1 = 3·3 = 9 U 2 = 3·3 = 9 Indicate below whether the statements are true or false by placing a check in the appropriate box. (explanations on following page) Statement True False Allocation A is Pareto Efficient TRUE Allocation B is Pareto Efficient FALSE Allocation C is Pareto Efficient FALSE Allocation D is Pareto Efficient TRUE Allocation C is a Pareto improvement over (or relative to) B TRUE Allocation D is a Pareto improvement over (or relative to) C TRUE Allocation D is a Pareto improvement over (or relative to) A FALSE Allocation C is a Pareto improvement over (or relative to) A FALSE

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Allocation A is Pareto efficient (PE). Any Pareto improvement over allocation A would require U 1 36, which is only possible if agent 1 has everything (Y 1 = 6, Z 1 = 6). Allocation D represents a Pareto improvement over situation B because each does better off in D relative to B (i.e., 9 > 0). Hence, B is not PE, because a PE allocation by definition cannot be Pareto-improved upon. Similarly, D is a Pareto improvement over C (so C is not PE), and C is a Pareto improvement over B. Note that at any interior point (meaning each of the agents consumes a positive amount of each good), Pareto efficiency requires equality of two agents’ marginal rates of substitution. This yields a simpler but equivalent condition as follows: MRS 1 = MRS 2 ( U 1 / Y 1 ) / ( U 1 / Z 1 ) = ( U 2 / Y 2 ) / ( U 2 / Z 2 ) Z 1 /Y 1 = Z 2 /Y 2 Then it is clear that C is not PE (because 2/4 4/2). However, allocation D is PE (because 3/3 = 3/3). (This condition will not work to test allocations A and B for Pareto efficiency, because they are corner points (e.g., Y 1 = 0), not interior points.)
Problem B (20 points – 2.5 points each): Consider an economy with two individuals (1 and 2), each of whom consumes a continuous public good and a private good. Public goods are financed by individual contributions, which are denoted by g 1 and g 2 . Indifference curves for individual 1 are labeled U 1 and U 1 , while the indifference curves for individual 2 are labeled U 2 and U 2 . Best response functions are given by g 1 (g 2 ) and g 2 (g 1 ). Indicate below whether the statements are true or false by placing a check in the appropriate box.

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