1.21 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
By way of example, let’s plot the 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:
[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/demand/demand_cobb_douglas ]
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.