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shortage constraints.

shortage constraints.

Microeconomic Theory
Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter3: Preferences And Utility
Section: Chapter Questions
Problem 3.9P
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Question 1 Consider a consumer living during a time of civil war. In order to maximise her utility, the consumer can purchase goods by paying cash or by using coupons that the government hands out (also referred to as “rationing"). Suppose you wish to model the consumer's behaviour as a constrained maximisation problem. To simplify matters, you assume that there are only two goods in the economy, x and y, and that the consumer's utility function is given by U(, y) = xy?. You further assume the price of good r and y in the cash market is 20. However, in the coupon market you assume the price of good r is 40, while the price of good y is 20. You also make the assumption that the consumer has a cash budget of 2000 and a total allotment of coupons of 2400. Last, you assume that the consumer cannot consume negative quantities of goods, so that x >0 and y >0 (the non-negativity constraints). 1.1. First set up the consumer's constraint functions as inequality constraints (in other words, allowing the consumer to not consume the entire budget or allotment of coupons). 1.2. Now set up the Lagrangian function for the constrained maximisation prob- lem and derive the Kuhn-Tucker conditions without solving for a and y. 1.3. Discuss why the non-negativity constraints are non-binding. 1 1.4. Assume that the ration constraint is no-binding but that the budget con- straint is binding. Show why these assumptions do not provide a feasible solution within the current framework. 1.5. Now assume that the ration constraint is binding and the budget constraint is non-binding and solve for r*, y* and X. 1.6. Set up the bordered Hessian matrix for the problem and confirm that the solution in 1.5 is indeed a maximum. 1.7. If the price of good x in the coupon market changed from 40 to 30, estimate the new optimal level of utility without recalculating the optimal values of x and y. 1.8. Now suppose that you would like to introduce two additional constraints into your model, more specifically, you introduce the constraints that the consumer can only purchase a maximum of 50 units of good x and a maxi- mum of 100 units of good y ("the shortage constraints"). • Write down the two new shortage constraints. • Suppose the consumer was at the point on her utility function calculated in 1.5 above prior to the introduction of the shortage constraints. Would the introduction of these shortage.constraints make any difference to 2 of 3 the amount of goods which the consumer is consuming at the current maximum? • Are the shortage constraints therefore binding or non-binding?
Transcribed Image Text:Question 1 Consider a consumer living during a time of civil war. In order to maximise her utility, the consumer can purchase goods by paying cash or by using coupons that the government hands out (also referred to as “rationing"). Suppose you wish to model the consumer's behaviour as a constrained maximisation problem. To simplify matters, you assume that there are only two goods in the economy, x and y, and that the consumer's utility function is given by U(, y) = xy?. You further assume the price of good r and y in the cash market is 20. However, in the coupon market you assume the price of good r is 40, while the price of good y is 20. You also make the assumption that the consumer has a cash budget of 2000 and a total allotment of coupons of 2400. Last, you assume that the consumer cannot consume negative quantities of goods, so that x >0 and y >0 (the non-negativity constraints). 1.1. First set up the consumer's constraint functions as inequality constraints (in other words, allowing the consumer to not consume the entire budget or allotment of coupons). 1.2. Now set up the Lagrangian function for the constrained maximisation prob- lem and derive the Kuhn-Tucker conditions without solving for a and y. 1.3. Discuss why the non-negativity constraints are non-binding. 1 1.4. Assume that the ration constraint is no-binding but that the budget con- straint is binding. Show why these assumptions do not provide a feasible solution within the current framework. 1.5. Now assume that the ration constraint is binding and the budget constraint is non-binding and solve for r*, y* and X. 1.6. Set up the bordered Hessian matrix for the problem and confirm that the solution in 1.5 is indeed a maximum. 1.7. If the price of good x in the coupon market changed from 40 to 30, estimate the new optimal level of utility without recalculating the optimal values of x and y. 1.8. Now suppose that you would like to introduce two additional constraints into your model, more specifically, you introduce the constraints that the consumer can only purchase a maximum of 50 units of good x and a maxi- mum of 100 units of good y ("the shortage constraints"). • Write down the two new shortage constraints. • Suppose the consumer was at the point on her utility function calculated in 1.5 above prior to the introduction of the shortage constraints. Would the introduction of these shortage.constraints make any difference to 2 of 3 the amount of goods which the consumer is consuming at the current maximum? • Are the shortage constraints therefore binding or non-binding?
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