# 4.1. Own-Price Elasticity of Demand

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Recall that when we introduced the concept of a demand function with Equation 1 earlier, we were simply theorizing that quantity demanded of some good, such as gasoline, is dependent on several other variables, one of which is the price of gasoline itself. We referred to the law of demand that simply states the inverse relationship between the quantity demanded and the price. Although that observation is useful, we might want to dig a little deeper and ask, Just how sensitive is quantity demanded to changes in the price of gasoline? Is it highly sensitive, so that a very small rise in price is associated with an enormous fall in quantity, or is the sensitivity only minimal? It might be helpful if we had a convenient measure of this sensitivity.

In Equation 3, we introduced a hypothetical household demand function for gasoline, assuming that the household’s income and the price of another good (automobiles) were held constant. It supposedly described the purchasing behavior of a household regarding its demand for gasoline. That function was given by the simple linear expression Qdx=11.2−0.4Px . If we were to ask how sensitive quantity is to changes in price in that expression, one plausible answer would be simply to recognize that, according to that demand function, whenever price changes by one unit, quantity changes by 0.4 units in the opposite direction. That is to say, if price were to rise by $1, quantity would fall by 0.4 gallons per week, so the coefficient on the price variable (−0.4) could be the measure of sensitivity we are seeking. There is a fundamental drawback, however, associated with that measure. Notice that the –0.4 is measured in gallons of gasoline per dollar of price. It is crucially dependent on the units in which we measured Q and P. If we had measured the price of gasoline in cents per gallon, instead of dollars per gallon, then the exact same household behavior would be described by the alternative equation Qdx=11.2−0.004Px . So, although we could choose the coefficient on price as our measure of sensitivity, we would always need to recall the units in which Q and Pwere measured when we wanted to describe the sensitivity of gasoline demand. That could be cumbersome. Because of this drawback, economists prefer to use a gauge of sensitivity that does not depend on units of measure. That metric is called elasticity, and it is defined as the ratio of percentage changes. It is a general measure of how sensitive one variable is to any other variable. For example, if some variable y depends on some other variable x in the following function: y = f(x), then the elasticity of y with respect to x is defined to be the percentage change in y divided by the percentage change in x, or %∆y/%∆x. In the case of own-price elasticity of demand, that measure is10 Equation (23) Edpx=%ΔQdx%ΔPx Notice that this measure is independent of the units in which quantity and price are measured. If, for example, when price rises by 10 percent, quantity demanded falls by 8 percent, then elasticity of demand is simply −0.8. It does not matter whether we are measuring quantity in gallons per week or liters per day, and it does not matter whether we measure price in dollars per gallon or euros per liter; 10 percent is 10 percent, and 8 percent is 8 percent. So the ratio of the first to the second is still –0.8. We can expand Equation 23 algebraically by noting that the percentage change in any variable x is simply the change in x (denoted “∆x”) divided by the level of x. So, we can rewrite Equation 23, using a couple of simple steps, as Equation (24) Edpx=%ΔQdx%ΔPx=ΔQdxQdxΔPxPx=(ΔQdxΔPx)(PxQdx) To get a better idea of price elasticity, it might be helpful to use our hypothetical market demand function: Qdx=11,200−400Px . For linear demand functions, the first term in the last line of Equation 24 is simply the slope coefficient on Px in the demand function, or −400. (Technically, this term is the first derivative of Qdx with respect to Px, dQdx/dPx , which is the slope coefficient for a linear demand function.) So, the elasticity of demand in this case is –400 multiplied by the ratio of price to quantity. Clearly in this case, we need to choose a price at which to calculate the elasticity coefficient. Let’s choose the original equilibrium price of$3. Now, we need to find the quantity associated with that particular price by inserting 3 into the demand function and finding Q = 10,000. The result of our calculation is that at a price of 3, the elasticity of our market demand function is −400 (3/10,000) = −0.12. How do we interpret that value? It means, simply, that when price equals 3, a 1 percent rise in price would result in a fall in quantity demanded of only 0.12 percent. (You should try calculating price elasticity when price is equal to, say, $4. Do you find that elasticity equals –0.167?) In our particular example, when price is$3 per gallon, demand is not very sensitive to changes in price, because a 1 percent rise in price would reduce quantity demanded by only 0.12 percent. Actually, that is not too different from empirical estimates of the actual demand elasticity for gasoline in the United States. When demand is not very sensitive to price, we say demand is inelastic. To be precise, when the magnitude(ignoring algebraic sign) of the own-price elasticity coefficient has a value less than one, demand is defined to be inelastic. When that magnitude is greater than one, demand is defined to be elastic. And when the elasticity coefficient is equal to negative one, demand is said to be unit elastic, or unitary elastic. Note that if the law of demand holds, own-price elasticity of demand will always be negative, because a rise in price will be associated with a fall in quantity demanded, but it can be either elastic or inelastic. In our hypothetical example, suppose the price of gasoline was very high, say \$15 per gallon. In this case, the elasticity coefficient would be −1.154. Therefore, because the magnitude of the elasticity coefficient is greater than one, we would say that demand is elastic at that price.11

By examining Equation 24, we should be able to see that for a linear demand curve the elasticity depends on where we calculate it. Note that the first term, ∆Q/∆P, will remain constant along the entire demand curve because it is simply the inverse of the slope of the demand curve. But the second term, P/Q, clearly changes depending on where we look. At very low prices, P/Q is very small, so demand is inelastic. But at very high prices, Q is low and P is high, so the ratio P/Q is very high, and demand is elastic. Exhibit 20 illustrates a characteristic of all negatively sloped linear demand curves. Above the midpoint of the curve, demand is elastic; below the midpoint, demand is inelastic; and at the midpoint, demand is unit elastic.

Exhibit 20. The Elasticity of a Linear Demand Curve Sometimes, we might not have the entire demand function or demand curve, but we might have just two observations on price and quantity. In this case, we do not know the slope of the demand curve at a given point because we really cannot say that it is even a linear function. For example, suppose we know that when price is 5, quantity demanded is 9,200, and when price is 6, quantity demanded is 8,800, but we do not know anything more about the demand function. Under these circumstances, economists use something called arc elasticity. Arc elasticity of demand is still defined as the percentage change in quantity demanded divided by the percentage change in price. However, because the choice of base for calculating percentage changes has an effect on the calculation, economists have chosen to use the average quantity and the averageprice as the base for calculating the percentage changes. (Suppose, for example, that you are making a wage of €10 when your boss says, “I’ll increase your wage by 10 percent.” You are then earning €11. But later that day, if your boss then reduces your wage by 10 percent, you are then earning €9.90. So, by receiving first a 10 percent raise and then a 10 percent cut in wage, you are worse off. The reason for this is that we typically use the original value as the base, or denominator, for calculating percentages.) In our example, then, the arc elasticity of demand would be:

E=ΔQQavgΔPPavg=−4009,00015.5=−0.244

There are two special cases in which linear demand curves have the same elasticity at all points: vertical demand curves and horizontal demand curves. Consider a vertical demand curve, as in Exhibit 21 Panel A, and a horizontal demand curve, as in Panel B. In the first case, the quantity demanded is the same, regardless of price. Certainly, there could be no demand curve that is perfectly vertical at all possible prices, but over some range of prices it is not unreasonable that the same quantity would be purchased at a slightly higher price or a slightly lower price. Perhaps an individual’s demand for, say, mustard might obey this description. Obviously, in that price range, quantity demanded is not at all sensitive to price and we would say that demand is perfectly inelastic in that range.

In the second case, the demand is horizontal at some price. Clearly, for an individual consumer, this situation could not occur because it implies that at even an infinitesimally higher price the consumer would buy nothing, whereas at that particular price, the consumer would buy an indeterminately large amount. This situation is not at all an unreasonable description of the demand curve facing a single seller in a perfectly competitive market, such as the wheat market. At the current market price of wheat, an individual farmer could sell all she has. If, however, she held out for a price above market price, it is reasonable that she would not be able to sell any at all because all other farmers’ wheat is a perfect substitute for hers, so no one would be willing to buy any of hers at a higher price. In this case, we would say that the demand curve facing a perfectly competitive seller is perfectly elastic.

Exhibit 21. The Extremes of Price Elasticity In finance, there exists the question of whether the demand for common stock is perfectly elastic. That is, are there perfect substitutes for a firm’s common shares? If so, then the demand curve for its shares should be perfectly horizontal. If not, then one would expect a negatively sloped demand for shares. If demand is horizontal, then an increase in demand (owing to some influence other than positive new information regarding the firm’s outlook) would not increase the share price. In contrast, a purely “mechanical” increase in demand would be expected to increase the price if the demand were negatively sloped. One study looked at evidence from 31 stocks whose weights on the Toronto Stock Exchange 300 Index were changed, owing purely to fully anticipated technical reasons that apparently had no relationship to new information about those firms.12 That is, the demand for those shares shifted rightward. The authors found that there was a statistically significant 2.3 percent excess return associated with those shares, a finding consistent with a negatively sloped demand curve for common stock.

In addition to the degree of substitutability, other characteristics tend to be generally predictive of a good’s elasticity of demand. These include the portion of the typical budget that is spent on the good, the amount of time that is allowed to respond to the change in price, the extent to which the good is seen as necessary or optional, and so on. In general, if consumers tend to spend a very small portion of their budget on a good, their demand tends to be less elastic than if they spend a very large part of their income. Most people spend only a little on, say, toothpaste each month, so it really doesn’t matter whether the price rises 10 percent or not. They would probably still buy about the same amount. If the price of housing were to rise significantly, however, most households would try to find a way to reduce the quantity they buy, at least in the long run.

This example leads to another characteristic regarding price elasticity. For most goods and services, the long-run demand is much more elastic than the short-run demand. The reason is that if price were to change for, say, gasoline, we probably would not be able to respond quickly with a significant reduction in the quantity we consume. In the short run, we tend to be locked into modes of transportation, housing and employment location, and so on. The longer the adjustment time, however, the greater the degree to which a household could adjust to the change in price. Hence, for most goods, long-run elasticity of demand is greater than short-run elasticity. Durable goods, however, tend to behave in the opposite way. If the price of washing machines were to fall, people might react quickly because they have an old machine that they know will need to be replaced fairly soon anyway. So when price falls, they might decide to go ahead and make the purchase. If the price of washing machines were to stay low forever, however, it is unlikely that a typical consumer would buy all that many more machines over a lifetime.

Certainly, whether the good or service is seen to be non-discretionary or discretionary would help determine its sensitivity to a price change. Faced with the same percentage increase in prices, consumers are much more likely to give up their Friday night restaurant meal than they are to cut back significantly on staples in their pantry. The more a good is seen as being necessary, the less elastic its demand is likely to be.

In summary, own-price elasticity of demand is likely to be greater (i.e., more sensitive) for items that have many close substitutes, occupy a large portion of the total budget, are seen to be optional instead of necessary, and have longer adjustment times. Obviously, not all of these characteristics operate in the same direction for all goods, so elasticity is likely to be a complex result of these and other characteristics. In the end, the actual elasticity of demand for a particular good turns out to be an empirical fact that can be learned only from careful observation and often, sophisticated statistical analysis.