3.1. The Demand Function and the Demand Curve

We first analyze demand. The quantity consumers are willing to buy clearly depends on a number of different factors called variables. Perhaps the most important of those variables is the item’s own price. In general, economists believe that as the price of a good rises, buyers will choose to buy less of it, and as its price falls, they buy more. This is such a ubiquitous observation that it has come to be called the law of demand, although we shall see that it need not hold in all circumstances.

Although a good’s own price is important in determining consumers’ willingness to purchase it, other variables also have influence on that decision, such as consumers’ incomes, their tastes and preferences, the prices of other goods that serve as substitutes or complements, and so on. Economists attempt to capture all of these influences in a relationship called the demand function. (In general, a function is a relationship that assigns a unique value to a dependent variable for any given set of values of a group of independent variables.) We represent such a demand function in Equation 1:

Equation (1) 

Qdx=f(Px,I,Py,...)

where Qdx represents the quantity demanded of some good X (such as per household demand for gasoline in gallons per week), P_x is the price per unit of good X (such as $ per gallon), I is consumers’ income (as in $1,000s per household annually), and P_y is the price of another good, Y. (There can be many other goods, not just one, and they can be complements or substitutes.) Equation 1 may be read, “Quantity demanded of good X depends on (is a function of) the price of good X, consumers’ income, the price of good Y, and so on.”

Often, economists use simple linear equations to approximate real-world demand and supply functions in relevant ranges. A hypothetical example of a specific demand function could be the following linear equation for a small town’s per-household gasoline consumption per week, where P_y might be the average price of an automobile in $1,000s:

Equation (2) 

Qdx=8.4−0.4Px+0.06I−0.01Py

The signs of the coefficients on gasoline price (negative) and consumer’s income (positive) are intuitive, reflecting, respectively, an inverse and a positive relationship between those variables and quantity of gasoline consumed. The negative sign on average automobile price may indicate that if automobiles go up in price, fewer will be purchased and driven; hence less gasoline will be consumed. As will be discussed later, such a relationship would indicate that gasoline and automobiles have a negative cross-price elasticity of demand and are thus complements.

To continue our example, suppose that the price of gasoline (P_x) is $3 per gallon, per household income (I) is $50,000, and the price of the average automobile (P_y) is $20,000. Then this function would predict that the per-household weekly demand for gasoline would be 10 gallons: 8.4 − 0.4(3) + 0.06(50) − 0.01(20) = 8.4 − 1.2 + 3 − 0.2 = 10, recalling that income and automobile prices are measured in thousands. Note that the sign on the own-price variable is negative, thus, as the price of gasoline rises, per household weekly consumption would decrease by 0.4 gallons for every dollar increase in gas price. Own-price is used by economists to underscore that the reference is to the price of a good itself and not the price of some other good.

In our example, there are three independent variables in the demand function, and one dependent variable. If any one of the independent variables changes, so does the value of quantity demanded. It is often desirable to concentrate on the relationship between the dependent variable and just one of the independent variables at a time, which allows us to represent the relationship between those two variables in a two-dimensional graph (at specific levels of the variables held constant). To accomplish this goal, we can simply hold the other two independent variables constant at their respective levels and rewrite the equation. In economic writing, this “holding constant” of the values of all variables except those being discussed is traditionally referred to by the Latin phrase ceteris paribus (literally, “all other things being equal” in the sense of “unchanged”). In this reading, we will use the phrase “holding all other things constant” as a readily understood equivalent for ceteris paribus.

Suppose, for example, that we want to concentrate on the relationship between the quantity demanded of the good and its own-price, P_x. Then we would hold constant the values of income and the price of good Y. In our example, those values are 50 and 20, respectively. So, by inserting the respective values, we would rewrite Equation 2 as

Equation (3) 

Qdx=8.4−0.4Px+0.06(50)−0.01(20)=11.2−0.4Px

Notice that income and the price of automobiles are not ignored; they are simply held constant, and they are “collected” in the new constant term, 11.2. Notice also that we can rearrange Equation 3, solving for P_x in terms of Q_x. This operation is called “inverting the demand function,” and gives us Equation 4. (You should be able to perform this algebraic exercise to verify the result.)

Equation (4) 

P_x = 28 – 2.5Q_x  

Equation 4, which gives the per-gallon price of gasoline as a function of gasoline consumed per week, is referred to as the inverse demand function. We need to restrict Q_x in Equation 4 to be less than or equal to 11.2 so price is not negative. Henceforward we assume that the reader can work out similar needed qualifications to the valid application of equations. The graph of the inverse demand function is called the demand curve, and is shown in Exhibit 1.1

Exhibit 1. Household Demand Curve for Gasoline

This demand curve is drawn with price on the vertical axis and quantity on the horizontal axis. Depending on how we interpret it, the demand curve shows either the highest quantity a household would buy at a given price or the highest price it would be willing to pay for a given quantity. In our example, at a price of $3 per gallon households would each be willing to buy 10 gallons per week. Alternatively, the highest price they would be willing to pay for 10 gallons per week is $3 per gallon. Both interpretations are valid, and we will be thinking in terms of both as we proceed. If the price were to rise by $1, households would reduce the quantity they each bought by 0.4 units to 9.6 gallons. We say that the slope of the demand curve is 1/−0.4, or –2.5. Slope is always measured as “rise over run,” or the change in the vertical variable divided by the change in the horizontal variable. In this case, the slope of the demand curve is ΔP/ΔQ, where “Δ” stands for “the change in.” The change in price was $1, and it is associated with a change in quantity of negative 0.4