Suppose that a representative consumer has the following utility function: U(x, y) = x^{\alpha}...
Question:
Suppose that a representative consumer has the following utility function: {eq}U(x, y) = x^{\alpha} y^{t - \alpha}. {/eq} The price of good x is p_x and the price of good y is {eq}p_r {/eq} The consumer's income is equal to I.
(a) Write down the constrained optimization problem, and form the Lagrangian function.
(b) Provide an economic interpretation to the first-order conditions. Solve for optimal values of x, y and the Lagrange multiplier.
(c) Is your solution a maximum or a minimum? Verify using the second-order condition.
(d) How would you interpret the Lagrange multiplier?
LM
It is a tool to find out the maxima and minima. The condition includes that it should be subject to equality constraints. This concept was first brought and thus named after mathematician JL Lagrange.
Answer and Explanation: 1
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Ans. (a)
Maximize: {eq}U(x, y) = x^{\alpha} y^{t - \alpha} {/eq}
Subject to:
{eq}I = p_{x}x + p_{r}y {/eq}
Constrained Optimization Problem:
...
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Finding Minima & Maxima: Problems & Explanation
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Chapter 5 / Lesson 2
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One of the most important practical uses of higher mathematics is finding minima and maxima. This lesson will describe different ways to determine the maxima and minima of a function and give some real world examples.
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