文档介绍：ECN 770, Fall 2004
Professor Lin Zhou
Lecture Note 11: Comparative Statics of Optimal Solutions
1. Introduction
So far we have studied the following static optimization model and have presented conditions under which an optimal solution always exists:
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While the existence of optimal solution is one of the important subjects concerning optimization, it is not the only subject we are interested in. In this lecture, we discuss how optimal solutions change when some of the parameters of an optimization problem change. This subject is called the study of parative statics of optimal solutions.
Eaxmple 1 The basic profit maximization problem petitive firm faces is:
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In this problem, x is the choice variable (possibly a vector), which represents various inputs and is the production function. The other two variables: w is the input price and p is the output price. They are not chosen by the firm. Rather, the firm takes them as given in making the profit maximization decision. Different values of w and p lead to different objective functions, and consequently, different optimal solutions. We are interested in the functional relationship between the values of w, p and the profit maximizing choices (the supply function). Is it continuous in w and p?
Eaxmple 2 The basic utility maximization problem a consumer faces is
In this problem, x – the consumption bundle – is the choice variable, and p – the price –
and I – the e – are given exogenously. Different values of p and I lead to different budget sets, and consequently, different optimal consumption bundles. Again, we are interested in the functional relationship between p and I and the optimal consumption bundles (the demand function). Is the optimal solution continuous in p and I?
As we can see from these two examples of optimization problems, various parameters either affect the objective function, or the constraints, or both. A general problem of this type can be formulated as:
(CSy)
in which x i