This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Constrained Optimization: Examples Consumers problem : Suppose that a consumer has a utility function U ( x,y ) = x . 5 y . 5 , the price of x is $2, the price of y is $3 and the consumer has $100 in income. How much of the two goods should the consumer purchase to maximize her utility? Firms problem : Suppose that a firms production function is given by q = K . 5 L . 5 , the price of capital is $2 and the price of labour is $5. What is the least cost way for the firm to produce 100 units of output? 1 Relevant Mathematics Both of the above problems have a common mathematical structure: max x 1 ,...,x n f ( x 1 ,x 2 ,...,x n ) subject to g ( x 1 ,x 2 ,...,x n ) = 0 It is possible that instead of maximizing f ( x 1 ,...,x n ) we could be minimizing f ( x 1 ,...,x n ). The above problems can be translated into the above mathematical framework as max ( x,y ) x . 5 y . 5 subject to 2 x + 3 y 100 = 0 and min ( K,L ) 2 K + 5 L subject to K . 5 L . 5 100 = 0 2 One approach When the constraint(s) are equalities, we can convert the problem from a constrained optimization to an unconstrained optimization problem by substituting for some of the variables. In the consumers problem, we have 2 x +3 y = 100, so x = 50 (3 / 2) y . We can use this relation to substitute for x in the utility function which gives U ( x,y ) = x . 5 y . 5 = (50 (3 / 2) y ) . 5 y . 5 This is now a function of just y and we can now maximize this func tion with respect to y . It is important to observe that this is an unconstrained optimization problem since we have incorporated the constraint by substituting for x . 3 The first order conditions for the maximization of (50 (3 / 2) y ) . 5 y . 5 gives us 1 2 3 2 50 3 2 y . 5 y . 5 + 1 2 50 3 2 y . 5 y . 5 = 0 Solving this gives y = 50 / 3, and since x = 50 (3 / 2) y , we have x = 25. For the firms minimization problem, we can proceed similarly: the constraint gives us K 1 / 2 L 1 / 2 = 100 or K = 10000 /L 2 . Therefore, the objective function becomes 20000 /L 2 + 5 L . This can be minimized easily with respect to L , and then the corresponding K found easily. 4 The Lagrangean approach Two reasons for an alternative approach: 1. In some cases, we cannot use substitution easily: for instance, suppose the con straint is x 4 + 5 x 3 y + y 2 x + x 6 + 5 = 0. Here, it is not possible to solve this equation to get x as a function of y or vice versa. 2. In many cases, the economic constraints are written in the form g ( x 1 ,...,x n ) 0 or g ( x 1 ,...,x n ) 0. While the Lagrangean tech nique can be modified to take care of such cases, the substitution technique cannot be modified, or can be modified only with some difficulty....
View Full
Document
 Fall '10
 riley
 Utility

Click to edit the document details