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Optimization of consumer preferences – an example

100%

Fura B., Fura M.

Didactics of Mathematics

2015

issue 12(16)

61-68

In the article we discuss a standard example of an optimization problem. In our problem we are optimizing the objective function, i.e. a consumer utility function with two variables representing quantities of two commodities denoted by x1 and x2. We consider the standard optimization problem in which we maximize the defined utility function subject to a budget constraint. More precisely, the problem is to choose quantities of two commodities 1st and 2nd, in order to maximize u(x1, x2) function subject to the budget constraint. The aim of the article is to present how we can exemplify and solve this kind of problems in a classroom. In the paper we suggest four methods of finding the solution. The first one is based on the graphical interpretation of the problem. Based on this we can get the approximate solution of the defined optimization problem. Then, we present an algebraic approach to find the optimum solution of the given problem. In the first method we use the budget constraint to transform the utility function of two variables into the function of one variable. The second algebraic method of achieving the solution is based on the second Goosen’s law, which is known as the law of marginal utility theory. In the third algebraic method applied to find the maximum of the utility function, we use the Lagrange multipliers. The text emphasizes the educational aspect of the theory of consumer choice.

O jednolitym podejściu do rachunku wariacyjnego i sterowania optymalnego

51%

Grygierzec W.

Metody Ilościowe w Badaniach Ekonomicznych

2012

vol. 13

issue 1

118-126

Problemy rachunku wariacyjnego oraz sterowania optymalnego to z jednej strony dwie intensywnie rozwijane teorie matematyczne, z drugiej strony obie sprowadzają się do badania warunkowych zagadnień extremalnych. Zasada Lagrange'a pozwala zamienić poszukiwanie ekstremum warunkowego na poszukiwanie punktów stacjonarnych funkcji Lagrange'a. Idea ta może mieć zastosowania jeszcze w wielu zagadnieniach wychodzących poza pierwotne rozważanie jej twórcy.

Calculus of variations and optimal control theory are on one hand side intensively developing mathematical theories on the other at the center of both of them lies investigating of extremal problems. In connection with extremal problems there naturally arise questions important for mathematics and applications: 1) does there exist a solution of the problem? 2) is the solution unique? 3) how to really find the solution? For problems with constrains, a general principle was proposed by Lagrange. This idea can be generalized far beyond the limits of the problems that he considered. In the paper we present unified formulation of problems of calculus of variations and optimal control in connection with Lagrange principle.

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Optimization of consumer preferences – an example

O jednolitym podejściu do rachunku wariacyjnego i sterowania optymalnego