---
title: "Income and Substitution Effects"
format:
  html:
    self-contained: TRUE
---

*For more information on these topics, see Varian Chapter 8: Slutsky Equation.*

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In this assignment I'll help you explore the sometimes surprising effects of price changes. For example, when people are offered higher wages, we often assume they'll work more: like Uber drivers taking on extra shifts during surge pricing. But what if your wage jumped from ten dollars an hour to a *million dollars an hour*? Would you work more, or would you decide you can afford to sometimes take some leisure time? In this assignment, we'll unpack these ideas and see how these effects play out.

## Review: Budget Lines

<iframe width="756" height="525" src="https://www.youtube.com/embed/WhUfvMxNqAY" title="Practice with Budget Constraints" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

### 1) Let $p_1 = 2$, $p_2 = 2$, and $m = 10$. Calculate the x and y intercepts of the budget constraint.

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### 2) Consider question 1 but suppose $p_2$ increases to 5. What are the new x and y intercepts? Does the budget line shift but maintain the same slope, or does it pivot?

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## Income and Substitution Effects

When the price of one good changes, it affects the budget line in two ways:

1. **The slope changes**: The budget line pivots around the consumer's original choice $(x_1^*, x_2^*)$, altering the rate at which they can trade one good for another.

2. **The position shifts**: The budget line moves in a way that $(x_1^*, x_2^*)$ may no longer be affordable, reflecting a change in their purchasing power.

The change in the slope of the budget line means the relative prices of the two goods have changed. When one good becomes relatively more expensive compared to the other, the resulting change in demand is called the **substitution effect**.

At the same time, the shift in the budget line reflects a change in the consumer's overall purchasing power. For example, if good 1 becomes cheaper, your income can now go further. This increase in purchasing power leads to a change in demand, known as the **income effect**. Together, these effects help explain how price changes influence your consumption choices.

### 3) Imagine candy bars originally cost 50 cents each. If the price goes up by 10 cents, does the consumer's purchasing power increase or decrease?

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<iframe width="855" height="481" src="https://www.youtube.com/embed/qaSi5Fm-yeE" title="Example: Income and Substitution Effects" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>

### 4) Income vs Substitution. Consider the demand function $x_1 = 20 + \frac{m}{6 p_1}$. Let $x_2$ be the composite good.

   a) Find the consumer's demand for goods 1 and 2 if $p_1 = 1$ and $m = 36$.
   
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   b) Now find the consumer's demand for goods 1 and 2 if $p_1 = 2$ and $m$ is still $36$.
   
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   c) Comparing parts a and b, what is the total change to the demand for good 1 $\Delta x_1$?
   
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   d) Now we'll decompose the total change $\Delta x_1$ into two parts: the substitution effect versus the income effect. Start with the substitution effect: the change in demand for $x_1$ due only to the change in relative prices, and not the change in purchasing power. The first step is to find $m'$: under the new prices ($p_1 = 2$), what would the consumer's income need to be to afford their initial choice from part a?
   
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   e) Find the consumer's demand $x_1^S$ for good 1 under the new prices ($p_1 = 2$) and under $m'$. I'll call this $x_1^S$ to indicate this is the demand due to substitution (purchasing power held constant). Use R as a calculator.
   
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   f) The difference between $x_1$ in part A and $x_1$ in part E is the substitution effect. The difference between $x_1$ in part E and $x_1$ in part b is the income effect. Report these values; which was larger in magnitude?
   
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