lagrange-econ - ECONOMIC APPLICATIONS OF LAGRANGE...

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ECONOMIC APPLICATIONS OF LAGRANGE MULTIPLIERS MATH 232 Maximization of a function with a constraint is common in economic situations. The first section considers the problem in consumer theory of maximization of the utility function with a fixed amount of wealth to spend on the commodities. We consider three levels of generality in this treatment. The second section presents an interpretation of a Lagrange multiplier in terms of the rate of change of the value of extrema with respect to the change of the constraint constant. The first sub- section gives a general presentation, the second subsection illustrates this formula for a particular situation, and then the third subsection derives the formula in general. In the third section, we calculate a rate of change of the minimal cost of the output with respect to the change of price of one of the inputs. 1. Maximize Utility with Wealth Constraint 1.1. Three commodities. Assume there are three commodities with amounts x 1 , x 2 , and x 3 , and prices p 1 , p 2 , and p 3 . Assume the total value is fixed, p 1 x 1 + p 2 x 2 + p 3 x 3 = w 0 , where w 0 > 0 is a fixed positive constant. Assume the utility is given by U = x 1 x 2 x 3 . The maximum of U on the commodity bundles given by the wealth constraint satisfy the equations x 2 x 3 = λp 1 x 1 x 3 = λp 2 x 1 x 2 = λp 3 w 0 = p 1 x 1 + p 2 x 2 + p 3 x 3 . If we multiply the first equation by x 1 , the second equation by x 2 , and the third equation by x 3 , then they are all equal: x 1 x 2 x 3 = λp 1 x 1 = λp 2 x 2 = λp 3 x 3 . One solution is λ = 0 , but this forces one of the variables to equal zero and so the utility is zero. If λ = 0 , then p 1 x 1 = p 2 x 2 = p 3 x 3 w 0 = 3 p 1 x 1 w 0 3 = p 1 x 1 = p 2 x 2 = p 3 x 3 . Thus, one third of the wealth is spent on each commodity. This gives the maximization of U . 1.2. Maximize of a Weighted Utility. We make the same assumptions on the commodities as the last example, but assume the utility is given by U = x a 1 1 x a 2 2 x a 3 3 with a 1 > 0 , a 2 > 0 , and a 3 > 0 . This utility function gives a weight to the preference of the commodities as the solution to the 1
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2 MATH 232 maximization problem shows. The maximum of U on the commodity bundles given by the wealth constraint satisfy the equations a 1 x a 1 - 1 1 x a 2 2 x a 3 3 = λp 1 a 2 x a 1 1 x a 2 - 1 2 x a 3 3 = λp 2 a 3 x a 1 1 x a 2 2 x a 3 - 1 3 = λp 3 w 0 = p 1 x 1 + p 2 x 2 + p 3 x 3 .
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