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Utility Maximization with a Kinked Budget Constraint

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This Demonstration shows the utility maximization problem and its solutions for a kinked budget line.

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Blue and red points in the upper-left plot represent the endowment and the optimal consumption plan, respectively.

Though the kinked budget is not commonly seen in textbooks, it is not unnatural.

Examples

1. When you sell used books at a store, their price is almost always lower than the price at which the store sells it again.

2. The interest rate on a deposit is lower than that for borrowing.

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Contributed by: Kazuki Kumashiro (December 2015)
Open content licensed under CC BY-NC-SA

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Details

The model is as follows.

Utility Function

Budget Constraint

if

if

The difference between the buying price and selling price causes a kinked budget constraint.

When the demand is determined on the kinked point, small changes in prices do not affect the demand. This is why the demand functions may not be smooth.

In this Demonstration, assume that is always greater than . This assures that the budget set is convex—that is, the solution is determined uniquely.

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