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answer4 - Solutions to Problem Set 4 EC720.01 Math for...

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Solutions to 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 α h 1 - α , where 0 < α < 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 α h 1 - α c. Define the Lagrangian for this problem as L ( h, c, λ ) = ln( c ) - h + λ ( k α h 1 - α - c ) and note that the first-order conditions L 1 ( h * , c * , λ * ) = - 1 + λ * (1 - α ) k α ( h * ) - α = 0 and L 2 ( h * , c * , λ * ) = 1 /c * - λ * = 0 can be combined with the binding constraint L 3 ( h * , c * , λ * ) = k α ( h * ) 1 - α - c * - 0 to form a system of three equations in the three unknowns h * , c * and λ * . Combine the two first-order conditions to obtain c * = (1 - α ) k α ( h * ) - α and then combine this last expression with the binding constraint to obtain (1 - α ) k α ( h * ) - α = k α ( h * ) 1 - α which can be used to find h * = 1 - α. 1
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Finally, substitute this last expression back into the binding constraint to find c * = k α (1 - α ) α . 2. Equilibrium Allocations Now consider the same economy, but where perfectly competitive markets for inputs and outputs replace the social planner in allocating resources.
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