CONSTRAINED OPTIMIZATION: THEORY AND ECONOMIC
)( = (Cobb-Douglas utility). 1.2 CHARACTERISTICS OF THE OPTIMUM. At the maximum of the objective function subject to the constraint
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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 2PART 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 1The 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 1Suppose
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,...,, 21which 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... 2 211 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 iNote 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 21The profit maximization problem is
(1.6) max x ab px x w x w x12 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 2Now rearrange (1.9) to obtain:
(1.10) xbw x aw 2112 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 121.4 THE CONSTRAINED OPTIMUM: SOLUTION BY SUBSTITUTION
Rewrite (1.3) to isolate
1 dx: (1.13) 11 1 xgdxxg dx iiiNow substitute this expression for
1 dx into (1.2) to obtain (1.14) 0/)/( 1111 i i ii ii dxxf xgdxxg xf
This can be written as
(1.15)0/)/)(/(
1111ii 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 1iNote 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 the2n case - becomes
(1.19) 11 22xgxf xgxf ww
This in turn can be rearranged as
(1.20) 2121
xgxg xfxf ww
In the utility maximization problem we have
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