AEB 6106 Problem Set 2 Due Sept 12 At The Start Of Class

Question

Aeb 6106 Problem Set 2due Sept 12 At The Start Of Class 935aminst AEB 6106: Problem Set 2 Due Sept. 12 at the start of class (9:35AM) Instructions: 1. Please use your UF ID as the only identifying information on your problem set. Please do not include your name. 2. Please write your answers on separate paper, instead of squeezing answers into the margins of this paper. For the following utility functions in questions 1 - 4, maximize utility subject to a budget constraint to derive demand functions for goods x and y. 1) ð‘ˆ = ð‘¥ 4 5ð‘¦ ) ð‘ˆ = 3ð‘¥ + 10ð‘¦ 3) ð‘ˆ = min{5ð‘¥, 6ð‘¦} 4) ð‘ˆ = 4ð‘¥ 1 4 + 3ð‘¦ For all four utility functions in questions 5 - 8, maximize utility subject to a budget constraint when: ð‘ƒð‘¥ = 2, ð‘ƒð‘¦ = 1,ð¼ = ) ð‘ˆ = ð‘¥ 1 5ð‘¦ ) ð‘ˆ = 2ð‘¥ + 3ð‘¦ 7) ð‘ˆ = min{2ð‘¥, 7ð‘¦} 8) ð‘ˆ = 2ð‘¥ 1 2 + 2ð‘¦ For questions 9 through 12, derive the indirect utility functions for each utility function using your answers to questions 1 through 4. 9) ð‘ˆ = ð‘¥ 4 5ð‘¦ ) ð‘ˆ = 3ð‘¥ + 10ð‘¦ 11) ð‘ˆ = min{5ð‘¥, 6ð‘¦} 12) ð‘ˆ = 4ð‘¥ 1 4 + 3ð‘¦

Dr. Jack HW Helper · Accepted Answer

This problem set involves maximizing various utility functions subject to budget constraints to derive demand functions and then utilizing these results to compute the respective indirect utility functions. The exercise provides an essential understanding of consumer choice theory in microeconomics, particularly focusing on utility maximization and indirect utility derivation. Questions 1-4: Deriving Demand Functions The first four questions focus on maximizing specific utility functions given a fixed budget constraint, enabling us to solve for the demand functions for goods x and y. Question 1 Utility function: $ U = x^{4} y^{5} $. To maximize utility subject to a budget constraint $ p_x x + p_y y = M $, we set up the Lagrangian: $$ \mathcal{L} = x^{4} y^{5} + \lambda (M - p_x x - p_y y) $$ The first-order conditions are: $\frac{\partial \mathcal{L}}{\partial x} = 4 x^{3} y^{5} - \lambda p_x = 0 $ $\frac{\partial \mathcal{L}}{\partial y} = 5 x^{4} y^{4} - \lambda p_y = 0 $ Budget constraint: $ p_x x + p_y y = M $ Dividing the first FOC by the second: $$ \frac{4 x^{3} y^{5}}{5 x^{4} y^{4}} = \frac{\lambda p_x}{\lambda p_y} \Rightarrow \frac{4 y}{5 x} = \frac{p_x}{p_y} $$ which simplifies to: $$ y = \frac{5 p_x}{4 p_y} x $$ Substituting into the budget constraint: $$ p_x x + p_y \left( \frac{5 p_x}{4 p_y} x \right) = M \Rightarrow p_x x + \frac{5 p_x}{4} x = M $$ $$ p_x x \left( 1 + \frac{5}{4} \right) = M \Rightarrow p_x x \left( \frac{9}{4} \right) = M $$ $$ x^{*} = \frac{4 M}{9 p_x} $$ Similarly, $$ y^{} = \frac{5 p_x}{4 p_y} x^{} = \frac{5 p_x}{4 p_y} \times \frac{4 M}{9 p_x} = \frac{5 M}{9 p_y} $$ Question 2 Utility function: $ U = 3 x + 10 y $. Since the utility function is linear, the consumer will spend all income on the good with the higher marginal utility per dollar—here, comparing $ \frac{MU_x}{p_x} $ and $ \frac{MU_y}{p_y} $. Marginal utilities: $$ MU_x = 3, \quad MU_y = 10 $$ so the consumer prefers good y because: \[ \frac{10}{p_y} >

AEB 6106 Problem Set 2 Due Sept 12 At The Start Of Class

Aeb 6106 Problem Set 2due Sept 12 At The Start Of Class 935aminst

Paper For Above instruction

Question 1

Question 2

Question 3

Question 4

Questions 5-8: Maximizing Utility with Given Budget Constraints

Questions 9-12:

Conclusion

References