14.1 Introduction

As we saw on Friday, demand functions describe a consumer’s optimal bundle as a function of prices and income. For example, we can write the demand for good 1 as \(x_1^*(p_1,p_2,m)\) These demand functions represent three relationships:

• The relationship between the quantity demanded of a good and its own price (how $x_1^*$ varies with $p_1$)
• The relationship between the quantity demanded of a good and the prices of other goods (how $x_1^*$ varies with $p_2$)
• The relationship between the quantity demanded of a good and the consumer’s income (how $x_1^*$ varies with $m$)

When we investigate any of these relationships, the natural thing to do is to hold the other variables constant. Therefore, when we plot the demand curve for good 1, we plot $x_1$ as a function of $p_1$, holding $p_2$ and $m$ constant (the ceteris paribus assumption); a change in $p_1$ is therefore reflected in the diagram as a movement along the demand curve for good 1. If $p_2$ or $m$ affect the relationship between $x_1$ and $p_1$ – that is, if the prices of other goods or a consumer’s income affect how that consumer responds to the price of good 1 – then changes in $p_2$ or $m$ will cause a shift of the demand curve for good 1.

The key economic concepts we’ll be analyzing will have to do with the direction of the responses to changes, and should be familiar from Econ 1. Specifically:

• If the consumer’s income increases:
  • If the demand for good 1 increases, we say that good 1 is a normal good
  • If the demand for good 1 decreases, we say that good 1 is an inferior good
• If the price of good 2 increases:
  • If the demand for good 1 increases, we say the two goods are substitutes
  • If the demand for good 1 decreases, we say the two goods are complements
  • If the demand for good 1 is unchanged, we say the two goods are independent goods

For the most part, we will be concerned with normal goods in this class, as the mathematics of inferior goods are pretty complicated. So for now, let’s focus on analyzing the differences between complements, substitutes, and independent goods. We’ll first see the effect they have on demand curves, and then we’ll decompose the effect of a price change into two separate effects – the income effect and the substitution effect – to see how the curvature of the indifference curves affect consumer behavior.