twoperiodmodels

# twoperiodmodels - ECON7020: MACROECONOMIC THEORY I Martin...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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 .

### Page1 / 23

twoperiodmodels - ECON7020: MACROECONOMIC THEORY I Martin...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online