7 Demand
For more information on these topics, see Varian Chapter 6: Demand.
The consumer’s demand function is the optimal bundle of goods as a function of prices and incomes faced by that consumer.
In this chapter, we’ll study how the demand for a good might change as prices and income change.
7.1 Income \(m\) Changes
What happens to the demand for good 1 when income \(m\) increases?
In the chapter on budget constraints, you proved that increasing the consumer’s income \(m\) will shift out the budget line. What will happen to the optimal choice \((x_1^*, x_2^*)\)?
We would normally think something like this would happen:
In the figure above, both \(x_1\) and \(x_2\) would be normal goods because as income \(m\) increases, the demand for both goods increase. But it’s easy to draw indifference curves where it’s not always true that demand increases as income increases. Goods are called inferior in that case:
Some examples of normal versus inferior goods: As your income increases, you might spend more on a personal vehicle, but less on the bus.
| Normal | Inferior |
|---|---|
| Personal car | Public transit |
| Organic vegetables | Canned vegetables |
| Own house | Rental apartment |
Summary: If good 1 is a normal good, then the demand for it increases when income increases. If good 1 is an inferior good, then the demand for it decreases when income increases.
7.2 Price \(p_1\) Changes
What happens to the demand for good 1 when \(p_1\) decreases?
Recall that if \(m\) and \(p_2\) stay constant and \(p_1\) increases, the y-intercept stays the same and the x-intercept moves out, flattening the slope:
Intuition tells us that if a good becomes cheaper, the ordinary case would be that we’d buy more of it, or at least buy the same amount of it (and not buy less of it). If this holds, then demand (with \(p_1\) on one axis and quantity demanded on the other axis) will slope down.
But it’s also possible to draw indifference curves such that if a good becomes cheaper, you’d buy less of it. This would result in upward-sloping demand curves.
Giffen goods are rare in practice, but for the sake of an example, consider a staple good like bread. Imagine a situation where the price of bread rises. Instead of buying less, people buy more bread because they effectively become poorer as a result of the price change and can no longer afford substitutes. Therefore, they rely even more on this staple good.
The opposite direction could hold as well: if the price of bread falls, people find that they can buy the bread that they need and also afford substitutes, so they end up buying less bread than before.
Summary: If good 1 is an ordinary good, then the demand for it increases when its price decreases. If good 1 is a giffen good, then the demand for it decreases when its price decreases.
7.3 Price \(p_2\) Changes
What happens to the demand for good 1 when \(p_2\) increases?
Perfect and Imperfect Complements
Perfect complements are goods that are consumed together, such as right and left shoes. You wouldn’t want one without the other. Imperfect complements, on the other hand, are goods that are often bought together but not necessarily always. For example, smartphones and phone cases are usually purchased as a pair, but not always. If the price of a particular smartphone were to rise significantly, the demand for its corresponding phone case might drop. Therefore, if \(x_1\) and \(x_2\) are complements (whether perfect or imperfect), an increase in \(p_2\) would typically result in a decrease in \(x_1\).
Perfect and Imperfect Substitutes
Perfect substitutes are goods that can be used interchangeably without any loss of utility, such as butter and margarine for baking cookies. Imperfect substitutes are goods that serve a similar function but differ in some aspects, like taking the bus versus taking a taxi. Both provide transportation, but they vary in terms of convenience, cost, and flexibility. If the price of bus fares were to rise significantly, people might opt to take taxis more often. Thus, if \(x_1\) and \(x_2\) are substitutes (whether perfect or imperfect), an increase in \(p_2\) would typically lead to an increase in \(x_2\).
Understanding the relationship between goods as complements or substitutes, whether perfect or imperfect, helps predict how changes in the price of one good can affect the demand for another.
7.4 Classwork 7 Math Skills Practice
First, here’s a chapter on Khan Academy on the concept of the slope-intercept form for an equation. Then, try these practice questions.
Practice Question: Rearrange the equation \(2 x_1 + 4 x_2 = 6\) into slope-intercept form with \(x_2\) as the dependent variable (on the y-axis).
Practice Question: Given the equation \(y = \frac{1}{2} x + 3.2\), what are the slope and y-intercept?
Practice Question: The budget line in slope-intercept form is \(x_2 = -\frac{p_1}{p_2} x_1 + \frac{m}{p_2}\). The slope of the budget line is:
7.5 Classwork 7
This chapter we learned that normal goods are ones where demand increases as income increases, and inferior goods are ones where demand decreases as income increases. An Engel Curve tells you what a consumer’s demand for good 1 is at any possible income (\(x_1\) on the x-axis and \(m\) on the y-axis). If the consumer has Cobb-Douglas utility \(u(x_1, x_2) = x_1^a x_2^b\),
- Prove that the Engel curve for \(x_1\) is a straight line with a y-intercept of 0 and a slope of \(\frac{p_1}{a}\). (You don’t have to prove again that \(x_1^* = \frac{ma}{p_1}\))
- Sketch the Engel curve. If \(p_1 = 1\) and \(a = 0.25\), how much of good 1 does the consumer choose when their income is 0? What about 1? What about 10?
- Does Cobb-Douglas utility indicate that goods are normal or inferior?
A demand curve tells you what a consumer’s demand for good 1 is at any possible \(p_1\) (it’s conventional to put \(x_1\) on the x-axis and \(p_1\) on the y-axis). Suppose the consumer has Cobb-Douglas utility with \(a = 0.5\) and an income of \(10\).
- Use R to draw the demand curve.
- If preferences are Cobb-Douglas, will goods be ordinary or giffen (is demand downward sloping or not)?
- Interpreting the demand curve: How low would you need to set the price for the consumer to buy at least one unit of good 1 (add a vertical line to your plot with the x-intercept at 1)? What about 2, 3, and 4 units? Viewed in this manner, the downward-sloping demand curve takes on a new meaning: when \(x_1\) is small, the consumer is willing to give up a lot of money (\(p_1\)) to obtain a little more of \(x_1\). However, as \(x_1\) increases, the consumer is willing to give up less money, on the margin, to acquire a bit more of good 1. Therefore, the marginal willingness to pay decreases as the consumption of good 1 increases.