# Nonlinear Budget Constraints

Initial Publication Date: March 26, 2020

## Summary

Teams graph both linear and nonlinear budget constraints and identify optimal choices for consumers with different preferences.

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## Context for Use

This activity may be appropriate for an ambitious introductory micro course, but it is likely better suited to an intermediate level microeconomic theory course. Students should have some exposure to standard (linear) budget constraints and indifference curve maps.

We recommend giving students less than 5 minutes for Part 1, about 10 minutes for Part 2, and another 5 minutes for Part 3. The surrounding discussion shouldn't take more than an additional 5-10 minutes.

## Overview

Teams are initially given three different pricing structures for socks. The first is pure linear pricing, and the other two give discounts for purchasing a large number of pairs. Teams draw the budget constraints for all three structures and identify optimal choices for consumers with different preferences.

## Expected Student Learning Outcomes

Students determine optimal choices in the context of nonlinear budget constraints.

## Information Given to Students

You are looking to purchase some socks and realize that the three local shops offer quite different pricing:

A. \$2 per pair
B. \$3 per pair and \$6 for a 4-pack of pairs
C. \$3 per pair for the first 3 and \$1 per pair after that

Part 1: Which shop offers the best deal?

A. Shop A
B. Shop B
C. Shop C
D. It depends.

Part 2: [Hand out three empty graphs]

Now let's put some economic structure on the problem. Consider socks and money as the two goods you care about. Assuming you have \$20, draw the possible consumption bundles associated with purchasing socks at each shop.

Part 3: [Hand out sheets with pre-printed indifference curves.]

Describe each of these consumers in words. Which deal would each customer prefer? Does this change your answer to Part 1? Which of the three consumers has attained the most happiness from their sock purchases?

Student Handout for Nonlinear Budget Constraints (Microsoft Word 2007 (.docx) 14kB Mar26 20)
Worksheets and Solutions for Nonlinear Budget Constraints (Acrobat (PDF) 8.4MB Mar26 20)

## Teaching Notes and Tips

This exercise is somewhat computational and is likely to be most useful early on in an intermediate microeconomics course.

Part 1: Some students will answer based on their own preferences, and some will recognize that the correct answer depends on the preferences of the individual making the choice.

Part 2: The first set of graphs represents someone for whom money and socks are perfect substitutes, the second represents someone who treats them as perfect complements, and the third represents someone with Cobb-Douglas preferences. Because socks can only be purchased in discrete pairs, don't let students draw their budgets sets as continuous curves. In fact, many products are only purchased in discrete quantities, although it is still useful to abstract away from this and consider consumer choices to be continuous. This is a more reasonable assumption when we consider the behavior of a whole group of consumers.

Part 3: In the discussion, students should be encouraged to think about what the indifference curves mean. That is, they should recognize the situations of perfect complements (right angles), perfect substitutes (straight lines), and somewhere in the middle (Cobb-Douglas curves). The last question in Part 3 is a bit of a trick as it is not meaningful to compare utility levels of different consumers.

At the end of the exercise, students should recognize that nonlinear budget constraints introduce complexity to the optimal choice problem—We cannot simply equate marginal utility with marginal cost here. The discrete nature of the choice also requires a different method than a continuous choice. Students should also know that they cannot answer the question of which consumer is best off because utility is ordinal. Perhaps the most important point is that optimal choices depend on both the budget set AND preferences.

## Assessment

One way to assess student learning here is to give another scenario with nonlinear pricing and have students graph the budget set. Students should also be able to identify the optimal choice of a consumer given their preferences (e.g., through indifference curves).