Section 2.6 Deriving demand
Let’s explore the important relationship between the consumer’s utility maximization problem and demand. They should seem related: the optimal bundle gives the number of units of good \(x\) the consumer wants to purchase, and demand is about \(\ldots\) the quantity of units a consumer demands. Are optimal bundles and demand the same thing?
To test the hypothesis, we return to the utility maximization problem. The consumer, per usual, will choose an optimal bundle of good \(x\) and good \(y\text{.}\) Only this time, instead of solving for a numerical solution, we will leave the prices of the two goods as parameters. Suppose the consumer has \(MU_x = \frac{1}{x}\) and \(MU_y = \frac{1}{y}\text{,}\) and income \(I = 20\text{.}\) Denote each good’s price with \(P_x\) and \(P_y\text{,}\) respectively. We know the optimal bundle \((x^*, y^*)\) should satisfy the condition 1, the budget constraint:
\begin{equation*}
P_x x + P_y y = 20
\end{equation*}
and condition 2, the marginal utility per dollar condition:
\begin{equation*}
\frac{1}{P_x x} = \frac{1}{P_y y}
\end{equation*}
We know to solve two equations-two unknowns to find the optimal bundle, so let’s solve for \(x\text{.}\) Condition 2 can be expressed as
\begin{equation*}
P_x x = P_y y
\end{equation*}
which can be substituted into the budget constraint 1 :
\begin{equation*}
P_x x + P_y y = 20
\end{equation*}
\begin{equation*}
P_x x + \textcolor{red}{P_x x} = 20
\end{equation*}
Now we can solve for \(x\text{:}\)
\begin{equation*}
2P_x x = 20
\end{equation*}
\begin{equation*}
x^* = \frac{20}{2P_x}
\end{equation*}
\begin{equation*}
x^* = \frac{10}{P_x}
\end{equation*}
The optimal choice of good \(x\) for the consumer is not numerical, since we do not know the price of good \(x\text{.}\) Rather, we have the optimal choice of good \(x\) for any possible price the consumer might face! If \(P_x = 1\text{,}\) for example, then the consumer will optimally want to buy \(\frac{10}{1} = 10\) units. If \(P_x = 2\text{,}\) the consumer will optimally want to buy \(\frac{10}{2} = 5\) units. And if the price is very high, like \(P_x = 10\text{,}\) the consumer will only want to buy one unit.
This solution tells us, given a collection of prices, how many units the consumer wants to purchase. This relationship should sound familiar: it is just demand! In fact, the solution to the consumer’s problem \(x^* = \frac{10}{P_x}\) is exactly the consumer’s demand function for \(x\text{!}\) If we plot the points given above, and draw the function directly with \(P_x\) on the y-axis, and units of \(x\) on the x-axis, it is plain to see:
This conclusion draws an important connection for us. We began the course with a look at demand, a discussion of which factors influence demand, and how to model consumer characteristics in tables, on graphs, and in functions. With utility maximization, we have shown that the demand function is derived directly from the consumer’s optimal choice problem. If we want to know how many units of a good a consumer will demand, we can look to her optimal choice problem, since this is exactly the objective of utility maximization.
Importantly, if we want to study how changes in parameters will shift demand, this comes directly from utility maximization as well. For example, in the problem above, we could ask how demand changes when income increases from 20 to 40. We have seen that this will shift the consumer’s budget constraint outward, generating more affordable bundles. But how does the increase in income impact demand? Using the same mathematical approach, we would show that the new optimal choice for the consumer is
\begin{equation*}
x^{**} = \frac{20}{P_x}
\end{equation*}
At this new solution, if \(P_x = 1\text{,}\) the consumer will optimally want to buy \(\frac{20}{1} = 20\) units. If \(P_x = 2\text{,}\) the consumer will optimally want to buy \(\frac{20}{2} = 10\) units. At \(P_x = 10\text{,}\) the consumer will only want to buy two units. After the income increase, notice that at every price, the consumer wants to purchase more units than she did prior to the increase. This sounds like - and exactly is - a demand increase!
The corresponding graph shows this more precisely. On the left, we see that at each of the prices of 1, 2, and 10, the quantity demanded has increased, which is indicative of an increase in demand. On the right, the two demand functions are directly plotted. Before the income increase, demand is \(x^* = \frac{10}{P_x}\text{;}\) after, demand is \(x^{**} = \frac{20}{P_x}\text{.}\) Side by side, it is clear that the income increase leads to an increase in demand, since the new demand curve is shifted to the right. In fact, since we observe a positive relationship between income and demand, we can conclude that in this example, good \(x\) is a normal good!
