=Problem 3. Consider the Cobb-Douglas utility function u(x, y) = x²y². Let the budgetconstraint be Pxx+PyY I where pa, py are the prices and I denotes the income.(a) Write the Lagrangian for this utility maximization problem and solve the first-order con-ditions to find the demand functions for both good x and good y. [Hint: Your results should onlydepend on the parameters pr, Py, I.](b) In the optimal consumption bundle, how much money is spent on x? How much on y?Express your answer in proportion of the total income I.(c) Check the second order conditions.(d) Assume the price p, increases. How does the optimal consumption bundle change?(e) Assume the income doubles. What is the new optimal consumption bundle? How doesyour answer in (b) change?

***Since the student has posted a question with multiple subparts, the expert is required to solve…

Problem 3. Consider the Cobb-Douglas utility function u(x, y) = x²y². Let the budget constraint be pax +Pyy = I where pe, py are the prices and I denotes the income. (a) Write the Lagrangian for this utility maximization problem and solve the first-order con- ditions to find the demand functions for both good x and good y. [Hint: Your results should only depend on the parameters pa, py, I.] (b) In the optimal consumption bundle, how much money is spent on x? How much on y? Express your answer in proportion of the total income I. (c) Check the second order conditions.

Problem 3. Consider the Cobb-Douglas utility function u(x, y) = x²y². Let the budget constraint be pax +Pyy = I where pe, py are the prices and I denotes the income. (a) Write the Lagrangian for this utility maximization problem and solve the first-order con- ditions to find the demand functions for both good x and good y. [Hint: Your results should only depend on the parameters pa, py, I.] (b) In the optimal consumption bundle, how much money is spent on x? How much on y? Express your answer in proportion of the total income I. (c) Check the second order conditions.

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

Question

=
Problem 3. Consider the Cobb-Douglas utility function u(x, y) = x²y². Let the budget
constraint be Pxx+PyY I where pa, py are the prices and I denotes the income.
(a) Write the Lagrangian for this utility maximization problem and solve the first-order con-
ditions to find the demand functions for both good x and good y. [Hint: Your results should only
depend on the parameters pr, Py, I.]
(b) In the optimal consumption bundle, how much money is spent on x? How much on y?
Express your answer in proportion of the total income I.
(c) Check the second order conditions.
(d) Assume the price p, increases. How does the optimal consumption bundle change?
(e) Assume the income doubles. What is the new optimal consumption bundle? How does
your answer in (b) change?

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Step 1: Define consumer theory

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Step 2: State the Lagrangian and derive the first order conditions and the demands for x and y

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Step 3: Find out the proportion of income spent on both goods

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