1. This question will let you examine/explore a more interesting utility func- tion than the simple example discussed in class as there will be both cross-price elas- ticity and an inferior good. Suppose you are told a consumer has the following utility function: U (9219₂) = 9z+√9z+ 9₂ You should assume income is Y, the price of good r is P2, and the price of good z is P₂. This question will ask about several concepts discussed in lecture. (a) What is the Marshallian demand for goods x and z? I.e. find (qq) for both interior solutions and corner solutions. Note: the outcome is "ugly" for the interior section and both corner should include constraints, i.e. limits using Y relative to f(P, P₂). Hint: if solving using MRS = MRT, can use this information again in part (d). (b) Suppose you are told P₂ = $8 and P₂ = $1. Create a graph showing the Income Consumption Curve (ICC) for Y = {$12.25, $35, $49, $70, $98). Be sure to clearly label the graph. Based on the ICC, which of the goods is the inferior good? Note: this should help with the corner solutions found in part (a), i.e. you should be able to check the "criteria," i.e. Y vs f(P, P₂). (c) Graph the Engel curve for good z from Y = $0 to Y = $100 given P₁ = $8 and P₂ = $1. Hint: there should be 3 parts as the demand function has 3 parts with the changes to the Engel curve consistent with the outcome in part (b). (d) Find the Hicksian Demand for the interior solutions as a function of Pr, P₂, and Ū, i.e. find (qq). What is the Expenditure function E(P, P₂, Ū)? Hint: you can use MRS = MRT you found in part (a) and solve for q₂ = f(qz, PT, P₂) as this will make finding a solution much easier. (e) Using from part (a) and q from part (d) with P₂ = $8, P₂ = $1, and Y = $70 find what the Substitution Effect and Income Effect from a +$0.05 change in the price of good z. Does your answer make sense? Explain why or why not. Note: values for q with P₂ = {$0.95, $1.05} are not "nice" values and you should include 4 decimal places when giving the answer. (f) Suppose the consumer's income remains Y = $70, but the price of good x increases to P₁ = $18 and the price of good z increases to P₂ = $2. What is the Compensating Variation? What is the Equivalent Variation? Note: given you found the Expenditure function in part (d), this should be relatively straight forward.

1. This question will let you examine/explore a more interesting utility func- tion than the simple example discussed in class as there will be both cross-price elas- ticity and an inferior good. Suppose you are told a consumer has the following utility function: U (9219₂) = 9z+√9z+ 9₂ You should assume income is Y, the price of good r is P2, and the price of good z is P₂. This question will ask about several concepts discussed in lecture. (a) What is the Marshallian demand for goods x and z? I.e. find (qq) for both interior solutions and corner solutions. Note: the outcome is "ugly" for the interior section and both corner should include constraints, i.e. limits using Y relative to f(P, P₂). Hint: if solving using MRS = MRT, can use this information again in part (d). (b) Suppose you are told P₂ = $8 and P₂ = $1. Create a graph showing the Income Consumption Curve (ICC) for Y = {$12.25, $35, $49, $70, $98). Be sure to clearly label the graph. Based on the ICC, which of the goods is the inferior good? Note: this should help with the corner solutions found in part (a), i.e. you should be able to check the "criteria," i.e. Y vs f(P, P₂). (c) Graph the Engel curve for good z from Y = $0 to Y = $100 given P₁ = $8 and P₂ = $1. Hint: there should be 3 parts as the demand function has 3 parts with the changes to the Engel curve consistent with the outcome in part (b). (d) Find the Hicksian Demand for the interior solutions as a function of Pr, P₂, and Ū, i.e. find (qq). What is the Expenditure function E(P, P₂, Ū)? Hint: you can use MRS = MRT you found in part (a) and solve for q₂ = f(qz, PT, P₂) as this will make finding a solution much easier. (e) Using from part (a) and q from part (d) with P₂ = $8, P₂ = $1, and Y = $70 find what the Substitution Effect and Income Effect from a +$0.05 change in the price of good z. Does your answer make sense? Explain why or why not. Note: values for q with P₂ = {$0.95, $1.05} are not "nice" values and you should include 4 decimal places when giving the answer. (f) Suppose the consumer's income remains Y = $70, but the price of good x increases to P₁ = $18 and the price of good z increases to P₂ = $2. What is the Compensating Variation? What is the Equivalent Variation? Note: given you found the Expenditure function in part (d), this should be relatively straight forward.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.7P

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