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Chapter 3
Price Theory Applications
3.1 Elasticity: A Measure of Response
Imagine that you are the manager of the public transportation system for a large metropolitan area. Operating costs for the system have soared in the last few years, and you are under pressure to boost revenues. What do you do?
An obvious choice would be to raise fares. That will make your customers angry, but at least it will generate the extra revenue you need—or will it? The law of demand says that raising fares will reduce the number of passengers riding on your system. If the number of passengers falls only a little, then the higher fares that your remaining passengers are paying might produce the higher revenues you need. But what if the number of passengers falls by so much that your higher fares actually reduce your revenues? If that happens, you will have made your customers mad and your financial problem worse!
Maybe you should recommend lower fares. After all, the law of demand also says that lower fares will increase the number of passengers. Having more people use the public transportation system could more than offset a lower fare you collect from each person. But it might not. What will you do?
Your job and the fiscal health of the public transit system are riding on your making the correct decision. To do so, you need to know just how responsive the quantity demanded is to a price change. You need a measure of responsiveness.
Economists use a measure of responsiveness called elasticity. Elasticity is the ratio of the percentage change in a dependent variable to a percentage change in an independent variable. If the dependent variable is y , and the independent variable is x , then the elasticity of y with respect to a change in x is given by:
A variable such as y is said to be more elastic (responsive) if the percentage change in y is large relative to the percentage change in x. It is less elastic if the reverse is true.
As manager of the public transit system, for example, you will want to know how responsive the number of passengers on your system (the dependent variable) will be to a change in fares (the independent variable). The concept of elasticity will help you solve your public transit pricing problem and a great many other issues in economics. We will examine several elasticities in this chapter—all will tell us how responsive one variable is to a change in another.
3.2 The Price Elasticity of Demand
L EA RNING O BJECT IV ES
- Explain the concept of price elasticity of demand and its calculation.
- Explain what it means for demand to be price inelastic, unit price elastic, price elastic, perfectly price inelastic, and perfectly price elastic.
- Explain how and why the value of the price elasticity of demand changes along a linear demand curve.
- Understand the relationship between total revenue and price elasticity of demand.
- Discuss the determinants of price elasticity of demand.
We know from the law of demand how the quantity demanded will respond to a price change: it will change in the opposite direction. But how much will it change? It seems reasonable to expect, for example, that a 10% change in the price charged for a visit to the doctor would yield a different percentage change in quantity demanded than a 10% change in the price of a Ford Mustang. But how much is this difference?
To show how responsive quantity demanded is to a change in price, we apply the concept of elasticity. The price elasticity of demand for a good or service, e D, is the percentage change in quantity demanded of a particular good or service divided by the percentage change in the price of that good or service, all other things unchanged. Thus we can write
Because the price elasticity of demand shows the responsiveness of quantity demanded to a price change, assuming that other factors that influence demand are unchanged, it reflects movements along a demand curve. With a downward- sloping demand curve, price and quantity demanded move in opposite directions, so the price elasticity of demand is always negative. A positive percentage change in price implies a negative percentage change in quantity demanded, and vice versa. Sometimes you will see the absolute value of the price elasticity measure reported. In essence, the minus sign is ignored because it is expected that there will be a negative (inverse) relationship between quantity demanded and price. In this text, however, we will retain the minus sign in reporting price elasticity of demand and will say “the absolute value of the price elasticity of demand” when that is what we are describing.
Heads Up!
Be careful not to confuse elasticity with slope. The slope of a line is the change in the value of the variable on the vertical axis divided by the change in the value of the variable on the horizontal axis between two points. Elasticity is the ratio of the percentage changes. The slope of a demand curve, for example, is the ratio of the change in price to the change in quantity between two points on the curve. The price elasticity of demand is the ratio of the percentage change in quantity to the percentage change in price. As we will see, when computing elasticity
−0.10/0.75, or −13.33%. The price elasticity of demand between points A and B is thus 40%/(−13.33%) = −3.00.
This measure of elasticity, which is based on percentage changes relative to the average value of each variable between two points, is called arc elasticity. The arc elasticity method has the advantage that it yields the same elasticity whether we go from point A to point B or from point B to point A. It is the method we shall use to compute elasticity.
For the arc elasticity method, we calculate the price elasticity of demand using the average value of price, 𝑷̅, and the average value of quantity demanded, 𝑸̅. We shall use the Greek letter Δ to mean “change in,” so the change in quantity between two points is Δ Q and the change in price is Δ P. Now we can write the formula for the price elasticity of demand as
The price elasticity of demand between points A and B is thus:
With the arc elasticity formula, the elasticity is the same whether we move from point A to point B or from point B to point A. If we start at point B and move to point A, we have:
The arc elasticity method gives us an estimate of elasticity. It gives the value of elasticity at the midpoint over a range of change, such as the movement between points A and B. For a precise computation of elasticity, we would need to consider the response of a dependent variable to an extremely small change in an independent variable. The fact that arc elasticities are approximate suggests an important practical rule in calculating arc elasticities: we should consider only small changes in independent variables. We cannot apply the concept of arc elasticity to large changes.
Another argument for considering only small changes in computing price elasticities of demand will become evident in the next section. We will investigate what happens to price elasticities as we move from one point to another along a linear demand curve.
Heads Up!
Notice that in the arc elasticity formula, the method for computing a percentage change differs from the standard method with which you may be familiar. That method measures the percentage change in a variable relative to its original value. For example, using the standard method, when we go from point A to point B, we would compute the percentage change in quantity as 20,000/40,000 = 50%. The percentage change in price would be −$0.10/$0.80 = −12.5%. The price elasticity of demand would then be 50%/(−12.5%) = −4.00. Going from point B to point A, however, would yield a different elasticity. The percentage change in quantity would be −20,000/60,000, or −33.33%. The percentage change in price would be $0.10/$0.70 = 14.29%. The price elasticity of demand would thus be −33.33%/14.29% = −2.33. By using the average quantity and average price to calculate percentage changes, the arc elasticity approach avoids the necessity to specify the direction of the change and, thereby, gives us the same answer whether we go from A to B or from B to A.
Price Elasticities Along a Linear Demand Curve What happens to the price elasticity of demand when we travel along the demand curve? The answer depends on the nature of the demand curve itself. On a linear demand curve, such as the one in Figure 3.2 "Price Elasticities of Demand for a Linear Demand Curve", elasticity becomes smaller (in absolute value) as we travel downward and to the right.
Figure 3.2 Price Elasticities of Demand for a Linear Demand Curve
The price elasticity of demand varies between different pairs of points along a linear demand curve. The lower the price and the greater the quantity demanded, the lower the absolute value of the price elasticity of demand.
Figure 3.2 "Price Elasticities of Demand for a Linear Demand Curve" shows the same demand curve we saw in Figure 3.1 "Responsiveness and Demand". We have already calculated the price elasticity of demand between points A and B; it equals −3.00. Notice, however, that when we use the same method to compute the price elasticity of demand between other sets of points, our answer varies. For each of the pairs of points shown, the changes in price and quantity demanded are the same (a $0.10 decrease in price and 20,000 additional rides per day,
reduction in price increases the quantity demanded. The question is how much. Because total revenue is found by multiplying the price per unit times the quantity demanded, it is not clear whether a change in price will cause total revenue to rise or fall.
We have already made this point in the context of the transit authority. Consider the following three examples of price increases for gasoline, pizza, and diet cola.
Suppose that 1,000 gallons of gasoline per day are demanded at a price of $4. per gallon. Total revenue for gasoline thus equals $4,000 per day (=1,000 gallons per day times $4.00 per gallon). If an increase in the price of gasoline to $4. reduces the quantity demanded to 950 gallons per day, total revenue rises to $4,037.50 per day (=950 gallons per day times $4.25 per gallon). Even though people consume less gasoline at $4.25 than at $4.00, total revenue rises because the higher price more than makes up for the drop in consumption.
Next consider pizza. Suppose 1,000 pizzas per week are demanded at a price of $9 per pizza. Total revenue for pizza equals $9,000 per week (=1,000 pizzas per week times $9 per pizza). If an increase in the price of pizza to $10 per pizza reduces quantity demanded to 900 pizzas per week, total revenue will still be $9,000 per week (=900 pizzas per week times $10 per pizza). Again, when price goes up, consumers buy less, but this time there is no change in total revenue.
Now consider diet cola. Suppose 1,000 cans of diet cola per day are demanded at a price of $0.50 per can. Total revenue for diet cola equals $500 per day (=1, cans per day times $0.50 per can). If an increase in the price of diet cola to $0. per can reduces quantity demanded to 880 cans per month, total revenue for diet cola falls to $484 per day (=880 cans per day times $0.55 per can). As in the case of gasoline, people will buy less diet cola when the price rises from $0.50 to $0.55, but in this example total revenue drops.
In our first example, an increase in price increased total revenue. In the second, a price increase left total revenue unchanged. In the third example, the price rise reduced total revenue. Is there a way to predict how a price change will affect total revenue? There is; the effect depends on the price elasticity of demand.
Elastic, Unit Elastic, and Inelastic Demand
To determine how a price change will affect total revenue, economists place price elasticities of demand in three categories, based on their absolute value. If the absolute value of the price elasticity of demand is greater than 1, demand is termed price elastic. If it is equal to 1, demand is unit price elastic. And if it is less than 1, demand is price inelastic.
Relating Elasticity to Changes in Total Revenue
When the price of a good or service changes, the quantity demanded changes in the opposite direction. Total revenue will move in the direction of the variable that changes by the larger percentage. If the variables move by the same percentage, total revenue stays the same. If quantity demanded changes by a larger percentage than price (i.e., if demand is price elastic), total revenue will
change in the direction of the quantity change. If price changes by a larger percentage than quantity demanded (i.e., if demand is price inelastic), total revenue will move in the direction of the price change. If price and quantity demanded change by the same percentage (i.e., if demand is unit price elastic), then total revenue does not change. When demand is price inelastic, a given percentage change in price results in a smaller percentage change in quantity demanded. That implies that total revenue will move in the direction of the price change: a reduction in price will reduce total revenue, and an increase in price will increase it. When demand is price inelastic, a given percentage change in price results in a smaller percentage change in quantity demanded. That implies that total revenue will move in the direction of the price change: an increase in price will increase total revenue, and a reduction in price will reduce it. When demand is unit price elastic total revenue remains unchanged. Quantity demanded falls by the same percentage by which price increases.
Figure 3.3 Changes in Total Revenue and a Linear Demand Curve
Moving from point A to point B implies a reduction in price and an increase in the quantity demanded. Demand is elastic between these two points. Total revenue, shown by the areas of the rectangles drawn from points A and B to the origin, rises. When we move from point E to point F, which is in the inelastic region of the demand curve, total revenue falls.
A demand curve can also be used to show changes in total revenue. Figure 3. "Changes in Total Revenue and a Linear Demand Curve" shows the demand curve from Figure 3.1 "Responsiveness and Demand" and Figure 3.2 "Price Elasticities of Demand for a Linear Demand Curve". At point A, total revenue from public transit rides is given by the area of a rectangle drawn with point A in the upper right-hand corner and the origin in the lower left-hand corner. The height of the rectangle is price; its width is quantity. We have already seen that total revenue at point A is $32,000 ($0.80 × 40,000). When we reduce the price and move to point B, the rectangle showing total revenue becomes shorter and wider. Notice
any linear demand curve, demand will be price elastic in the upper half of the curve and price inelastic in its lower half. At the midpoint of a linear demand curve, demand is unit price elastic.
Constant Price Elasticity of Demand Curves
Figure 3.5 "Demand Curves with Constant Price Elasticities" shows four demand curves over which price elasticity of demand is the same at all points. The demand curve in Panel (a) is vertical. This means that price changes have no effect on quantity demanded. The numerator of the formula for the price elasticity of demand (percentage change in quantity demanded) is zero. The price elasticity of demand in this case is therefore zero, and the demand curve is said to be perfectly inelastic. This is a theoretically extreme case, and no good that has been studied empirically exactly fits it. A good that comes close, at least over a specific price range, is insulin. A diabetic will not consume more insulin as its price falls but, over some price range, will consume the amount needed to control the disease.
Figure 3.5 Demand Curves with Constant Price Elasticities
The demand curve in Panel (a) is perfectly inelastic. The demand curve in Panel (b) is perfectly elastic. Price elasticity of demand is −1.00 all along the demand curve in Panel (c), whereas it is −0.50 all along the demand curve in Panel (d).
As illustrated in Figure 3.5 "Demand Curves with Constant Price Elasticities", several other types of demand curves have the same elasticity at every point on them. The demand curve in Panel (b) is horizontal. This means that even the smallest price changes have enormous effects on quantity demanded. The denominator of the formulafor the price elasticity of demand (percentage change in price) approaches zero. The price elasticity of demand in this case is therefore infinite, and the demand curve is said to be perfectly elastic. Division by zero results in an undefined solution. Saying that the price elasticity of demand is infinite requires that we say the denominator “approaches” zero. This is the type of demand curve faced by producers of standardized products such as wheat. If the wheat of other farms is selling at $4 per bushel, a typical farm can sell as much wheat as it wants to at $4 but nothing at a higher price and would have no reason to offer its wheat at a lower price.
The nonlinear demand curves in Panels (c) and (d) have price elasticities of demand that are negative; but, unlike the linear demand curve discussed above, the value of the price elasticity is constant all along each demand curve. The demand curve in Panel (c) has price elasticity of demand equal to −1. throughout its range; in Panel (d) the price elasticity of demand is equal to −0. throughout its range. Empirical estimates of demand often show curves like those in Panels (c) and (d) that have the same elasticity at every point on the curve.
Heads Up!
Do not confuse price inelastic demand and perfectly inelastic demand. Perfectly inelastic demand means that the change in quantity is zero for any percentage change in price; the demand curve in this case is vertical. Price inelastic demand means only that the percentage change in quantity is less than the percentage change in price, not that the change in quantity is zero. With price inelastic (as opposed to perfectly inelastic) demand, the demand curve itself is still downward sloping.
Determinants of the Price Elasticity of Demand
The greater the absolute value of the price elasticity of demand, the greater the responsiveness of quantity demanded to a price change. What determines whether demand is more or less price elastic? The most important determinants of the price elasticity of demand for a good or service are the availability of substitutes, the importance of the item in household budgets, and time.
Availability of Substitutes
The price elasticity of demand for a good or service will be greater in absolute value if many close substitutes are available for it. If there are lots of substitutes for a particular good or service, then it is easy for consumers to switch to those substitutes when there is a price increase for that good or service. Suppose, for example, that the price of Ford automobiles goes up. There are many close substitutes for Fords—Chevrolets, Chryslers, Toyotas, and so on. The availability of close substitutes tends to make the demand for Fords more price elastic.
change. When demand is price elastic, total revenue moves in the direction of a quantity change.
The absolute value of the price elasticity of demand is greater when substitutes are available, when the good is important in household budgets, and when buyers have more time to adjust to changes in the price of the good.
3.4 Responsiveness of Demand to Other Factors
L EA RNING O BJECT IV ES
- Explain the concept of income elasticity of demand and its calculation.
- Classify goods as normal or inferior depending on their income elasticity of demand.
- Explain the concept of cross price elasticity of demand and its calculation.
- Classify goods as substitutes or complements depending on their cross price elasticity of demand.
Although the response of quantity demanded to changes in price is the most widely used measure of elasticity, economists are interested in the response to changes in the demand shifters as well. Two of the most important measures show how demand responds to changes in income and to changes in the prices of related goods and services.
Income Elasticity of Demand
We saw in the chapter that introduced the model of demand and supply that the demand for a good or service is affected by income. We measure the income elasticity of demand, eY , as the percentage change in quantity demanded at a specific price divided by the percentage change in income that produced the demand change, all other things unchanged:
The symbol Y is often used in economics to represent income. Because income elasticity of demand reports the responsiveness of quantity demanded to a change in income, all other things unchanged (including the price of the good), it reflects a shift in the demand curve at a given price. Remember that price elasticity of demand reflects movements along a demand curve in response to a change in price.
A positive income elasticity of demand means that income and demand move in the same direction—an increase in income increases demand, and a reduction in income reduces demand. As we learned, a good whose demand rises as income rises is called a normal good.
Studies show that most goods and services are normal, and thus their income elasticities are positive. Goods and services for which demand is likely to move in the same direction as income include housing, seafood, rock concerts, and medical services.
If a good or service is inferior, then an increase in income reduces demand for the good. That implies a negative income elasticity of demand. Goods and services for which the income elasticity of demand is likely to be negative include used clothing, beans, and urban public transit.
Figure 3.
When we compute the income elasticity of demand, we are looking at the change in the quantity demanded at a specific price. We are thus dealing with a change that shifts the demand curve. An increase in income shifts the demand for a normal good to the right; it shifts the demand for an inferior good to the left.
Cross Price Elasticity of Demand
The demand for a good or service is affected by the prices of related goods or services. A reduction in the price of salsa, for example, would increase the demand for chips, suggesting that salsa is a complement of chips. A reduction in the price of chips, however, would reduce the demand for peanuts, suggesting that chips are a substitute for peanuts.
The measure economists use to describe the responsiveness of demand for a good or service to a change in the price of another good or service is called the cross price elasticity of demand, eA, B. It equals the percentage change in the quantity demanded of one good or service at a specific price divided by the percentage change in the price of a related good or service. We are varying the price of a related good when we consider the cross price elasticity of demand, so the response of quantity demanded is shown as a shift in the demand curve.
The cross price elasticity of the demand for good A with respect to the price of good B is given by:
KE Y TA KEA WAYS
The income elasticity of demand reflects the responsiveness of demand to changes in income. It is the percentage change in quantity demanded at a specific price divided by the percentage change in income, ceteris paribus.
Income elasticity is positive for normal goods and negative for inferior goods.
The cross price elasticity of demand measures the way demand for one good or service responds to changes in the price of another. It is the percentage change in the quantity demanded of one good or service at a specific price divided by the percentage change in the price of another good or service, all other things unchanged.
Cross price elasticity is positive for substitutes, negative for complements, and zero for goods or services whose demands are unrelated.
T RY IT!
Suppose that when the price of bagels rises by 10%, the demand for cream cheese falls by 3% at the current price, and that when income rises by 10%, the demand for bagels increases by 1% at the current price. Calculate the cross price elasticity of demand for cream cheese with respect to the price of bagels and tell whether bagels and cream cheese are substitutes or complements. Calculate the income elasticity of demand and tell whether bagels are normal or inferior.
A NSWE R TO T RY IT! P RO BL E M
Using the formula for cross price elasticity of demand, we find that e AB= (−3%)/(10%) = −0.3. Since the eAB is negative, bagels and cream cheese are complements. Using the formula for income elasticity of demand, we find that eY = (+1%)/(10%) = +0.1. Since eY is positive, bagels are a normal good.
3.5 Price Elasticity of Supply
L EA RNING O BJECT IV ES
- Explain the concept of elasticity of supply and its calculation.
- Explain what it means for supply to be price inelastic, unit price elastic, price elastic, perfectly price inelastic, and perfectly price elastic.
- Explain why time is an important determinant of price elasticity of supply.
- Apply the concept of price elasticity of supply to the labor supply curve.
The elasticity measures encountered so far in this chapter all relate to the demand side of the market. It is also useful to know how responsive quantity supplied is to a change in price.
Suppose the demand for apartments rises. There will be a shortage of apartments at the old level of apartment rents and pressure on rents to rise. All other things unchanged, the more responsive the quantity of apartments supplied is to changes in monthly rents, the lower the increase in rent required to eliminate the shortage and to bring the market back to equilibrium. Conversely, if quantity supplied is less responsive to price changes, price will have to rise more to eliminate a shortage caused by an increase in demand. This is illustrated in Figure 3.10 "Increase in Apartment Rents Depends on How Responsive Supply Is". Suppose the rent for a typical apartment had been R 0 and the quantity Q 0 when the demand curve was D 1 and the supply curve was either S 1 (a supply curve in which quantity supplied is less responsive to price changes) or S 2 (a supply curve in which quantity supplied is more responsive to price changes). Note that with either supply curve, equilibrium price and quantity are initially the same. Now suppose that demand increases to D 2 , perhaps due to population growth. With supply curve S 1 , the price (rent in this case) will rise to R 1 and the quantity of apartments will rise to Q 1. If, however, the supply curve had been S 2 , the rent would only have to rise to R 2 to bring the market back to equilibrium. In addition, the new equilibrium number of apartments would be higher at Q 2. Supply curve S 2 shows greater responsiveness of quantity supplied to price change than does supply curve S 1.
Figure 3.10 Increase in Apartment Rents Depends on How Responsive Supply Is
The more responsive the supply of apartments is to changes in price (rent in this case), the less rents rise when the demand for apartments increases.
and rent out as additional units. In a short period of time, however, the supply response is likely to be fairly modest, implying that the price elasticity of supply is fairly low. A supply curve corresponding to a short period of time would look like S 1 in Figure 3.10 "Increase in Apartment Rents Depends on How Responsive Supply Is". It is during such periods that there may be calls for rent controls.
If the period of time under consideration is a few years rather than a few months, the supply curve is likely to be much more price elastic. Over time, buildings can be converted from other uses and new apartment complexes can be built. A supply curve corresponding to a longer period of time would look like S 2 in Figure 3.10 "Increase in Apartment Rents Depends on How Responsive Supply Is".
KE Y TA KEA WAYS
The price elasticity of supply measures the responsiveness of quantity supplied to changes in price. It is the percentage change in quantity supplied divided by the percentage change in price. It is usually positive.
Supply is price inelastic if the price elasticity of supply is less than 1; it is unit price elastic if the price elasticity of supply is equal to 1; and it is price elastic if the price elasticity of supply is greater than 1. A vertical supply curve is said to be perfectly inelastic. A horizontal supply curve is said to be perfectly elastic.
The price elasticity of supply is greater when the length of time under consideration is longer because over time producers have more options for adjusting to the change in price.
When applied to labor supply, the price elasticity of supply is usually positive but can be negative. If higher wages induce people to work more, the labor supply curve is upward sloping and the price elasticity of supply is positive. In some very high-paying professions, the labor supply curve may have a negative slope, which leads to a negative price elasticity of supply.
Summary
This chapter introduced a new tool: the concept of elasticity. Elasticity is a measure of the degree to which a dependent variable responds to a change in an independent variable. It is the percentage change in the dependent variable divided by the percentage change in the independent variable, all other things unchanged.
The most widely used elasticity measure is the price elasticity of demand, which reflects the responsiveness of quantity demanded to changes in price. Demand is said to be price elastic if the absolute value of the price elasticity of demand is greater than 1, unit price elastic if it is equal to 1, and price inelastic if it is less than 1. The price elasticity of demand is useful in forecasting the response of quantity demanded to price changes; it is also useful for predicting the impact a price change will have on total revenue. Total revenue moves in the direction of the quantity change if demand is price elastic, it moves in the direction of the price change if demand is price inelastic, and it does not change if demand is unit
price elastic. The most important determinants of the price elasticity of demand are the availability of substitutes, the importance of the item in household budgets, and time.
Two other elasticity measures commonly used in conjunction with demand are income elasticity and cross price elasticity. The signs of these elasticity measures play important roles. A positive income elasticity tells us that a good is normal; a negative income elasticity tells us the good is inferior. A positive cross price elasticity tells us that two goods are substitutes; a negative cross price elasticity tells us they are complements.
Elasticity of supply measures the responsiveness of quantity supplied to changes in price. The value of price elasticity of supply is generally positive. Supply is classified as being price elastic, unit price elastic, or price inelastic if price elasticity is greater than 1, equal to 1, or less than 1, respectively. The length of time over which supply is being considered is an important determinant of the price elasticity of supply.
3.6 Government Intervention in Market Prices:
Price Floors and Price Ceilings
L EA RNING O BJECT IV ES
- Use the model of demand and supply to explain what happens when the government imposes price floors or price ceilings.
- Discuss the reasons why governments sometimes choose to control prices and the consequences of price control policies.
So far in this chapter and in the previous chapter, we have learned that markets tend to move toward their equilibrium prices and quantities. Surpluses and shortages of goods are short-lived as prices adjust to equate quantity demanded with quantity supplied.
In some markets, however, governments have been called on by groups of citizens to intervene to keep prices of certain items higher or lower than what would result from the market finding its own equilibrium price. In this section we will examine agricultural markets and apartment rental markets—two markets that have often been subject to price controls. Through these examples, we will identify the effects of controlling prices. In each case, we will look at reasons why governments have chosen to control prices in these markets and the consequences of these policies.
Agricultural Price Floors
Governments often seek to assist farmers by setting price floors in agricultural markets. A minimum allowable price set above the equilibrium price is a price floor. With a price floor, the government forbids a price below the minimum. (Notice that, if the price floor were for whatever reason set below the equilibrium
