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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:
maxf(x1; :::; xn) subject to g(x1; :::; xn) · y:
This problem is solved using the Lagrangian:
L = f(x1;:: :;xn) + ¸[y ¡ g(x1;: ::; xn)];
where ¸ is a Lagrange multiplier.
Assuming an interior solution, the necessary ¯rst-order necessary conditions for a
maximum are @L
@xj =
@f
@xj ¡ ¸
@g
@xj = 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
...

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