8.6 Demand Curves

Having derived the demand functions, we can visualize them in different ways. The most important visualization of a demand function is a demand curve, which is fundamentally a way to understand how the quantity of one good — without loss of generality, good 1 — varies as its own price changes.

For historical reasons, economists plot demand curves with the quantity of the good on the horizontal axis, and its price on the vertical axis. This is a little counterintuitive, because most students are used to seeing the exogenous (independent) variable on the horizontal axis; so remember to follow the right convention in this course!

While we can draw the demand curve for any utility function, let’s look at the three canonical functions we’ve been looking at in this chapter: Cobb-Douglas, perfect complements, and perfect substitutes.

Demand curve for Cobb-Douglas

Let’s start by plotting our demand curve for a Cobb-Douglas utility function. In the top graph, we have the budget line/indifference curve diagram demonstrating the constrained optimization problem for a consumer with Cobb-Douglas preferences. As the price of good 1 varies, the budget line pivots around the vertical axis: a lower price of good 1 means a larger budget set (and a larger horizontal intercept, $m/p_1$), while a higher price of good 1 means a smaller budget set. As you can see, as the price of good 1 increases, the quantity demanded of good 1 — that is, the value of $x_1^\star$, or the amount of good 1 in the optimal bundle — decreases.

The demand curve, which is shown in the lower graph, plots the relationship between the price of good 1 and the quantity demanded directly. The horizontal axis is the same as in the top graph: that is, it’s the quantity of good 1 in the optimal bundle. The vertical axis here shows the price. Try changing the price of good 1 to see how each diagram changes:

Note that we label the curve in the bottom diagram $d_1(p_1 | p_2,m)$. We can read this as: “the quantity demanded of good 1 at price $p_1$, holding $p_2$ and $m$ constant.” This is the familiar ceteris paribus assumption from Econ 1.

When plotting a demand curve, the easiest way is often to choose a few prices and plot the quantity demanded at those prices. If you check the “Show $p_1 = 2, 4, 6, 8$” box in the diagram above, it will add the four budget lines corresponding to those prices in the top diagram, and grid lines for those prices in the bottom graph. You can see that the horizontal coordinates of the optimal points subject to each of those budget lines correspond to the horizontal coordinates of the demand curve. We’ll do this exercise in class for a number of functions.

Let’s now take a preliminary look at what some of those demand functions look like, and how they reflect the behavior implied by the utility functions.

Demand curves for perfect complements

The demand curve for perfect complements is constructed in the same way: for each budget line, find the corresponding quantity demanded, and plot quantity as a function of price:

The interesting part about this demand curve is that it has a horizontal intercept. That is, even at a price of zero, you won’t buy an infinite amount of the good. This is because you always want to consume these two goods in a specific ratio, so even if one of the goods is free, you’ll stop consuming it when you achieve that desired ratio.

This can help explain, for example, why restaurants put salt and pepper out for free on a table. Salt, in fact, used to be one of the most valuable commodities in the world, before refrigeration, because it acts as a preservative. Nowadays, you can get it for free! Why? Because even though it’s free, you’re not going to just pour it over your meal in a gleeful frenzy at all this free salt; rather, you’ll just use a sprinkle or two to adjust the taste of your meal. Because you’re only interested in salt as a complement to your other food, you don’t use an infinite amount even if it’s free.

Demand curves for perfect substitutes

The behavior for goods that are perfect substitutes was different than these other kinds of goods, because it’s characterized by a discontinuity: below a certain price of good 1, you’ll spend all you money on good 1; but above that price, you’ll spend none. At the exact cutoff price, you’re indifferent between all the bundles on the budget line. This results in a demand curve that “jumps” at a single price from $x_1 = 0$ to $x_1 = m/p_1$:

To see how this works, check the box marked “Show $m/p_1$.” A dotted curve will appear showing this relationship. All along this curve, the consumer is spending all her money on good 1. Of course, along the vertical axis, she’s spending none of her money on good 1. The horizontal portion of her demand curve occurs at the price of good 1 such that $MRS = p_1/p_2$; that is, when $p_1 = MRS \times p_2$.

Try changing the price of good 2, and see how this affects the demand curve. Changing $p_2$ means this cutoff price shifts: the higher the price of good 2, the higher the price of good 1 at which you’re indifferent between buying these two goods.