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Unformatted text preview: ECON7020: MACROECONOMIC THEORY I Martin Boileau II. TWO-PERIOD ECONOMIES: A REVIEW 1. Introduction These notes brie°y review two-period economies. This includes the consumer's problem, the producer's problem, and general equilibrium. In what follows, I borrow freely from Chiang (1984), Farmer (1993), Feeney (1999), Obstfeld and Rogo® (1996), and Smith (1997). 2. Constrained Optimization Throughout, we make extensive use of constrained optimization. That is, we aim to solve problems of the following form: max f ( x 1 ;:::;x n ) subject to g ( x 1 ;:::;x n ) · y: This problem is solved using the Lagrangian: L = f ( x 1 ;:::;x n ) + ¸ [ y ¡ g ( x 1 ;:::;x n )] ; where ¸ is a Lagrange multiplier. Assuming an interior solution, the necessary ¯rst-order necessary conditions for a maximum are @L @x j = @f @x j ¡ ¸ @g @x j = 0 for j = 1 ;:::;n: The second-order su±cient conditions for a maximum are that the bordered hessian, the matrix of second derivatives of the function f ( ¢ ) bordered by the ¯rst derivatives of the function g ( ¢ ), is negative de¯nite (i.e. the bordered principal minors alternate in sign). To simplify our life, most of our applications will deal with a simple case for which there exists an interior absolute maximum. That is, we will assume that the objective function f ( ¢ ) is explicitely quasiconcave and that the constraint set g ( ¢ ) is convex. ² A function f ( ¢ ) is explicitely quasiconcave if for any pair of distinct points u and v in the domain of f , and for 0 < μ < 1, f ( v ) > f ( u ) ) f [ μu + (1 ¡ μ ) v ] > f ( u ). 1 y A twice continuously di®erentiable function f ( ¢ ) is quasiconcave if its Hessian bordered by its ¯rst derivatives is negative de¯nite. ² A function g ( ¢ ) is convex only if for any pair of distinct points u and v in the domain of g , and for 0 < μ < 1, μg ( u ) + (1 ¡ μ ) g ( v ) ¸ g [ μu + (1 ¡ μ ) v ]. y A twice continuously di®erentiable function g ( ¢ ) is convex if its hessian is every- where positive semide¯nite. 3. Consumption 3.1. The Standard Consumer's Problem The standard consumer's problem is max U ( c 1 ;c 2 ) subject to p 1 c 1 + p 2 c 2 = W; where c 1 an c 2 denote consumption of goods 1 and 2, p 1 and p 2 are the prices of these goods, and W is the consumer's income. The above optimization problem is composed of an objective function U ( ¢ ) and a constraint. The objective function is the utility function. It summarizes the consumer's preferences for goods c 1 and c 2 . Your basic graduate microeconomics course should layout the conditions for this function to exist and be quasiconcave. In general, it requires that preferences adhere to the assumptions of completeness, re°exivity, transitivity, continuity, strong monotonicity, non-satiation, and convexity. The constraint is simply a convex budget set. Finally, we assume throughout that the consumer is a price taker: the consumer takes prices as given....
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This note was uploaded on 10/16/2011 for the course ECONOMICS 640 taught by Professor Kepang during the Fall '11 term at Wilfred Laurier University .
- Fall '11