[PDF] [PDF] CONSTRAINED OPTIMIZATION: THEORY AND ECONOMIC

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CONSTRAINED OPTIMIZATION:

THEORY AND ECONOMIC EXAMPLES

Peter Kennedy

These notes provide a brief review of methods for constrained optimization. They cover equality-constrained problems only. Part 1 outlines the basic theory. Part 2 provides a number of economic examples to illustrate the methods. This material may be accessed by any person without charge at kennedy-economics.ca Posting it to any other website is a violation of copyright.

© Peter Kennedy 2019

Kennedy: Constrained Optimization © Peter Kennedy 2019 Posting this material to any site other than kennedy-economics.ca is a violation of copyright 2

PART 1: THEORY

1.1 THE CONSTRAINED OPTIMIZATION PROBLEM

We begin with a constrained optimization problem of the type x max ),...,( 1n xxf subject to bxxg n 1

The function ),...,(

1n xxf is called the objective function or maximand; the equation bxxg n 1 is called the constraint.

Remarks

1. We are restricting attention here to equality-constrained problems. An inequality- constrained problem would arise where the constraint is bxxg n 1 . The techniques we develop here can be extended easily to that case. 2. A minimization problem with objective function )(xf can be set up as a maximization problem with objective function )(xf.

An Example

Utility maximization subject to a budget constraint. (1.1) x max ),...,( 1n xxu subject to mxp n i ii 1

Suppose

2n and

ba xxxxu 2121
),( (Cobb-Douglas utility).

1.2 CHARACTERISTICS OF THE OPTIMUM

At the maximum of the objective function subject to the constraint, infinitesimal changes in the variables n xxx,...,, 21
which satisfy the constraint must have no effect on the value of the objective function. Otherwise, we could not be at a maximum. Thus, a necessary condition for the maximum is that

0df whenever 0dg. That is,

we require Kennedy: Constrained Optimization © Peter Kennedy 2019 Posting this material to any site other than kennedy-economics.ca is a violation of copyright 3 (1.2) 0... 2 21
1 w wwww n n dxxfdxxfdxxf for all 1 dx, 2 dx, ..., n dx satisfying (1.3) 0... 2 21
1 w wwww n n dxxgdxxgdxxg where i xf/ is the partial derivative of f with respect to i x, and i xg/ is the partial derivative of g with respect to i x. These conditions tells us that at the optimum there must be no way in which we can change the i x's such that we can change the value of the function (and in particular, increase the value of the function) and still satisfy the constraint.

1.3 THE UNCONSTRAINED OPTIMUM

Note that in the absence of the constraint we would be seeking conditions under which (1.2) holds for all 1 dx, 2 dx, ..., n dx, rather than only those changes in x that satisfy (1.3).

That is, we would require that the

i dx have zero coefficients in (1.2): (1.4) 0 i xf i

Note that this is a set of

n equations in n unknowns.

An Example

Profit maximization for a "competitive" firm with Cobb-Douglas technology, given by (1.5) ba xxxh 21

The profit maximization problem is

(1.6) max x ab px x w x w x

12 11 22

with first-order conditions (1.7) apx x w ab1121 and Kennedy: Constrained Optimization © Peter Kennedy 2019 Posting this material to any site other than kennedy-economics.ca is a violation of copyright 4 (1.8) bpx x w ab1212 We can solve these equations by first taking the ratio of (1.7) and (1.7) to obtain (1.9) ax bxw w 2 11 2

Now rearrange (1.9) to obtain:

(1.10) xbw x aw 211
2 Substitute (1.10) into (1.7) or (1.8) and solve for 1 x: (1.11) bab ba bwaw wbpwpx 11 1 2 11 2 1

Then substitute (1.11) into (1.10) to obtain xpw

2 ( , ). These are the factor demands or input demands. We can then construct the supply function by substituting these factor demands into the production function: (1.12) ypw x pw x pw ab 12

1.4 THE CONSTRAINED OPTIMUM: SOLUTION BY SUBSTITUTION

Rewrite (1.3) to isolate

1 dx: (1.13) 11 1 xgdxxg dx iii

Now substitute this expression for

1 dx into (1.2) to obtain (1.14) 0/)/( 111
1 i i ii ii dxxf xgdxxg xf

This can be written as

(1.15)

0/)/)(/(

1111
ii i iii dxxf xgdxxgxf Then by collecting terms under the summation operator, we have Kennedy: Constrained Optimization © Peter Kennedy 2019 Posting this material to any site other than kennedy-economics.ca is a violation of copyright 5 (1.16) 0/)/)(/( 111
i ii i dxxgxgxf xf The only solution to this equation is to set all of the coefficients on the i dx's equal to zero since the equation must hold for all possible values of the i dx's. That is, (1.17) 11 xgxgxf xf i i wwww ww 1i

This can in turn be written as

(1.18) 11 xgxf xgxf ii ww 1i

Note that (1.18) comprises

1n equations. Together with the constraint itself we

therefore have n equations which can be solved for the n unknowns (the i x's).

1.5 EXAMPLE: UTILITY MAXIMIZATION

Recall the utility maximization problem for

2n. For that example, equation (A1.18) -

which is a single equation in the

2n case - becomes

(1.19) 11 22
xgxf xgxf ww

This in turn can be rearranged as

(1.20) 21
21
xgxg xfxf ww

In the utility maximization problem we have

ii xuxf// andquotesdbs_dbs21.pdfusesText_27