Behavior and Choice: EC 102

Since the decision to buy is an individual one, we began with constructing demand for one individual: Tara. However, it might be of interest to discuss demand for mountain bikes more generally. What if, for example, Aaron, Kate, and Kyle were also considering mountain bike purchases? Since each of them also has unique parameters determining their willingnesses to pay, their WTP may differ from Tara’s. Consider the range below:

Willingness to pay for a mountain bike, for Kyle, Tara, Kate, and Aaron.

It is possible, with each buyer’s WTP, to construct individual demand for each of them. Assuming these are the only four buyers in our market, it is also possible to construct the market demand, which gives the relationship between the price and the quantity of units demanded by all buyers in the market. For example, notice that at a price of $285, Kyle and Tara are willing to buy, while Kate and Aaron are not. Therefore, at a price of $285, the quantity demanded would be 2 bikes. The table above corresponds to the following demand schedule:

Market demand schedule for mountain bikes

Forming market demand is a matter of aggregating individual demand. In essence, we are looking at each price and summing up the number of units demanded by each individual consumer. At a price of $250, for example, Kyle, Tara, and Kate each demand 1 unit, while Aaron demands 0 units; this leads to a quantity demanded in the market of \(1 + 1 + 1 + 0 = 3\) bikes. The corresponding demand curve can be drawn to represent the same demand schedule. Once again, each horizontal segment indicates that an additional unit is demanded when that price is reached.

Let’s take this one step further: what if, instead of constructing the demand for mountain bikes for 4 people, we consider constructing demand for all of the residents of Tacoma? If we had access to the data for 200,000 residents, we could perform a very similar exercise. The WTP could range much more widely, perhaps with some people willing to pay very high prices and others (such as those who do not know how to ride a bike) willing to pay very little or nothing at all. Nevertheless, the data could easily be arranged to form the market demand schedule for mountain bikes in Tacoma, and visualized as a market demand curve for mountain bikes in Tacoma - such as in the graph below.

Let’s notice a few things. First, we show that there are very few bikes demanded at very high prices, while as prices drop, more and more consumers are willing to buy. Moreover, since there are 200,000 individuals to consider in the market demand curve, the horizontal axis stretches to higher values. Each horizontal segment still represents a price at which price equals the WTP for another buyer, but interestingly, the “steps” formed by those horizontal segments get extremely small! If we were to be very careful with the actual data, we would have to meticulously account for as many as 200,000 little horizontal segments.

This will lead us to use a simplification. An assumption. An approximation. The demand-in-Tacoma graph has steps so small, the demand curve essentially resembles a smooth line. This appearance of smoothness is the result of the very large number of potential consumers included in the market - recall that the market demand with only four consumers still looked very steplike. Therefore, in situations where there are many buyers, economists (and we) will treat the demand curve as if it were a smooth curve.

Mathematically, this smoothness would only literally be possible if there were an infinite number of consumers. In this way, when we draw the demand curve as a smooth curve, we are making an assumption. This assumption is that since the number of buyers is sufficiently large, and the steps therefore sufficiently small, the impact of those steps is very small and the demand can be reasonably approximated by a smooth curve. Like any assumption, this implies a tradeoff. Although we lose a bit of the realism of our demand curve as a representation of the real world (there are not infinite consumers in Tacoma), we gain a representation of market demand that is easier to work with graphically and mathematically.

This represents an important decision every economist must make when working with demand: stepwise demand or smooth demand? When does it matter? Consider a couple of scenarios. If we aim to analyze the market demand for mountain bikes in Tacoma, or the market demand for Toyotas in the United States, it is probably safe to assume that there are sufficiently many consumers to use a smooth demand curve. However, if the city of Tacoma looks to sell a large plot of land to a large business, and there are only three or four businesses who would be interested, we should be cautious in using a smooth demand curve.¹

This function illustrates that at a price of \(P = 20\text{,}\) the quantity demanded would be given by \(Q_D = 200 - (2)(20) = 160\text{.}\) At a price of \(P = 50\text{,}\) the quantity demanded would be given by \(Q_D = 200 - (2)(50) = 100\text{.}\) We are able to calculate the quantity demanded for any price. A demand function can be expressed with any rows we wish in a demand schedule, and can be graphed very precisely, two of many advantages of using a demand function in economic models.

The slope of the demand curve gives us useful information in many applications. But be careful! While the coefficient of the demand function \(Q_D = 200 - 2P\) is -2, the slope of the demand curve (more easily observed when graphed) is \(-\frac{1}{2}\text{.}\) A demand function is expressed as \(x = 200 - 2y\text{,}\) which means it is not initially in the \(y = mx + b\) form which allows us to interpret coefficient as slope. One useful construction is to solve the demand function for price (the \(y\) variable). This creates the inverse demand function - here \(P = 100 - \frac{1}{2}Q_D\) - and interpret the coefficient \(m = \frac{\Delta y}{\Delta x} = \frac{\Delta P}{\Delta Q_D} = -\frac{1}{2}\text{.}\)

The demand function and the inverse demand function represent the same relationship, but each plays its role in demand theory. The demand function \(Q_D = 200 - 2P\) is helpful because it is the typical way we intuitively think of demand. Price is the independent variable, so we can plug in a price to get the quantity demanded at that price. The inverse demand function \(P = 100 - \frac{1}{2}Q_D\) is more useful for graphing, though its intuition is a bit awkward. With \(Q_D\) as the independent variable, this tells us for a given quantity demanded, what is the highest possible price that will incentivize this \(Q_D\)? At \(Q_D = 100\text{,}\) for example, we can plug 100 into the inverse demand function to get a price of 50. This says that 50 is the highest price at which 100 units will be demanded: any price above 50 will yield fewer than 100 units.

At different prices and at different levels of income, we would anticipate different quantities demanded from consumers.²

To summarize, demand is a fundamental expression of a consumer (or consumers) willingness to pay for a good or service. Its simplest form is as the relationship between the price of the good and the number of units of the good the consumer(s) demand. Depending on context, demand can be expressed in a table, on a graph, with a mathematical function, or some combination of these.

Practice Problems: Subsection 1.7.2