solve_ge_example2011plus

solve_ge_example2011plus - Marc-Andr Letendre e ECONOMICS...

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Marc-Andr´ e Letendre January 24, 2011 ECONOMICS 2HH3 Solving for a Competitive equilibrium: An example Contents 1 Introduction 1 2 Consumer Optimization 2 3 Firm Optimization 4 4 Government 5 5 Solving for the Competitive Equilibrium Outcome 6 6 A Numerical Example 7 6.1 Solving the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.2 An Increase in Government Expenditures . . . . . . . . . . . . . . . . . . . . 8 6.3 An Increase in Total Factor Productivity (TFP) . . . . . . . . . . . . . . . . 12 1 Introduction This handout explains how to proceed to solve for the general equilibrium of the static model described in chapters 4 and 5 of Williamson. In general equilibrium all markets clear ( i.e. demand equals supply on all markets). Since the consumers and ±rms are behaving competitively in the model we are solving, we are solving for the model’s competitive equilibrium. 1
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Solving for the equilibrium of the model involves solving for the endogenous variables (vari- ables determined inside the model) as functions of exogenous variables (variables determined outside the model) and parameters. Moreover, the solutions we get for the endogenous vari- ables must satisfy the four conditions of a competitive equilibrium listed on pages 128 and 129 of the book (3rd edition). To simplify, these conditions state that all markets must clear, consumers and Frms optimize and the government satisFes its budget constraint. In order to solve for the general equilibrium of the model, we need demand and supply func- tions which are obtained from the representative consumer and representative Frm decision making processes. Thus, that’s where we start. 2 Consumer Optimization Note: Also read textbook pages 589-590 (up to equation (A.4)). Suppose the utility function of the representative consumer is U ( C, ` )= C 1 - a 1 - a + θ ` 1 - a 1 - a , θ > 0 , 0 <a< 1 . (1) The consumer chooses C and ` to maximize utility subject to a time constraint N s + ` = h (2) and a budget constraint given by C = wN s + π - T (3) which can be combined to eliminate N s C = wh - w ` + π - T. (4) Equation (4) is the consumer’s budget constraint we use in our analysis. Maximizing utility function (1) taking into account of constraint (4) can be done using the Lagrange-multiplier method which involves setting up a Lagrangian function. Without going 2
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into technical details, 1 we set up the Lagrangian function L ( C, ` , λ )as L ( C, ` , λ )= C 1 - a 1 - a + θ ` 1 - a 1 - a + λ [ wh - w ` + π - T - C ] (5) where λ is a Lagrange multiplier and the arguments of L are the consumer’s choice variables and the Lagrange multiplier. The term in square brackets in equation (5) corresponds to the right-hand side of 0 = - w ` + π - T - C which is obtained by subtracting C o f both sides of (4). To solve the maximization problem we set the partial derivatives L ( - ) / C , L ( - ) / ∂` ,and L ( - ) / ∂λ equal to zero and solve for C , ` and λ (although we will not pay attention to the solution for λ here).
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This note was uploaded on 01/30/2012 for the course ECON 2hh3 taught by Professor Leatandre during the Spring '11 term at McMaster University.

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solve_ge_example2011plus - Marc-Andr Letendre e ECONOMICS...

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