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Unformatted text preview: Problem Set 4 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 8 The two welfare theorems of economics tell us that optimal and equilibrium resource alloca- tions coincide but only under certain conditions. Sometimes when market failures prevent the theorems from holding, however, government policy can help improve equilibrium out- comes, as the questions below reveal. 1. Optimal Allocations Consider an economy in which output is produced with capital k and labor (hours worked) h according to the Cobb-Douglas specification k h 1- , where 0 &lt; &lt; 1. In this static model, the capital stock k is taken as given, but hours worked h and consumption c are chosen by a benevolent social planner in order to maximize the utility ln( c )- h of a representative consumer, where ln denotes the natural logarithm. Hence, an optimal resource allocation solves the problem max h,c ln( c )- h subject to k h 1- c. Find solutions for the optimal choices for h and c in terms of the parameters k and . 2. Equilibrium Allocations Now consider the same economy, but where perfectly competitive markets for inputs and outputs replace the social planner in allocating resources. a. Now, the representative consumer (standing in for a large number of identical consumers) is endowed with k s units of capital, and chooses labor supply h s and consumption c to maximize utility subject to a budget constraint, that is, to solve max h s ,c ln( c )- h s subject to rk s + wh s c, where r is the rental rate for capital, w is the wage rate for labor, and output is the economys numeraire, so that the price of consumption equals one and all other prices are expressed in real terms. Find solutions for the optimal choices ofprices are expressed in real terms....
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This note was uploaded on 02/19/2010 for the course ECON 720 taught by Professor Ireland during the Fall '09 term at BC.
- Fall '09