Chapter 4
Elasticity 4.0 Preliminaries Practitioners in almost all fields of study including all scientific disciplines are interested in the following question: How does a change in one variable affect another variable? Economics is no exception. We want to know for example, how changes in the price of a good affect its quantity demanded or how changes in wages affect the level of employment in an industry. In previous chapters we discussed several such questions. For example, in Chapter 2 we discussed the law of demand which states that price and quantity demanded are inversely related, holding other variables constant. The type of analysis that results in statements like the law of demand is often called qualitative analysis. All it can tell us is the direction in which a variable moves: if the price of a good or service increases (and other variables don’t change) the quantity demanded will decrease, and vice versa. But in many circumstances we want to know by how much a variable will change: If the price increases by some amount, by how much will the quantity demanded decrease in response? EXAMPLE 4.1:
Craig Alberts, C.E.O. of Prometheus, Inc., a mediumsize manufacturer and distributor of novelty electrical products, is considering a 15% increase in the price of one of their bestselling items, a batteryoperated storm lantern. He knows that quantity demanded will fall, but by how much? By 15%, less than 15% or more than 15%?
EXAMPLE 4.2:
Jim Hightower, a tax analyst for the state of Caltex was given the assignment of estimating the effect on tax revenues of a one percentagepoint increase in the state sales tax on cigarettes. He realizes that an increase in the sales tax is equivalent to a price increase and that a higher price will usually lead to a lower quantity demanded. So to calculate the effect of the tax on revenues he has to estimate by how much quantity demanded will decrease.
The type of analysis involved in answering questions like these requires quantitative analysis and in economics such questions are usually discussed under the heading of elasticity. There are many elasticity concepts both in microeconomics and macroeconomics; we shall discuss several of them in this chapter. 1
Probably the best known and most widely used of these concepts is price elasticity of demand.
4.1 Price Elasticity of Demand In broad terms, price elasticity of demand deals with the sensitivity of buyers to price changes. But expressed this way the concept is too vague to be useful. To make it more precise we define the coefficient of price elasticity of demand.
DEF 4.1:
The coefficient of price elasticity of demand of good X is defined as the percentage change in the quantity demanded of good X divided by the percentage change in its price.
Using the symbol ep for this coefficient, we write:
(4.1)
Alternatively, using Q for quantity demanded, P for price, ∆Q and ∆P for changes in these variables, and x for the multiplication sign, we can write: ,
In EQ (4.2) the numerator shows the percentage change in the quantity demanded of good X and the denominator shows the percentage change in its price. After canceling the 100s, the formula can be rewritten more conveniently as follows:
(4.3)
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COMMENT 4.1: It clearly does not make sense to ask what the effect of a change in price on the quantity demanded of some good or service is, if other variables which also affect demand change at the same time! So just like in our definition of demand and in our discussion of the law of demand in Chapter 2, we assume in DEF 4.1 that these other variables remain unchanged. In other words, price elasticity of demand can only be calculated along a given demand curve. (This comment applies to all the other elasticity concepts we discuss in this chapter.) Case 4.1:
The price of good X increases by 1% and in response the quantity demanded decreases by 1.7%. Using EQ (4.1) we get:
When economists calculate numbers like the coefficient of price elasticity of demand, the first question they probably ask is: What is the expected algebraic sign (positive or negative) of the resulting coefficient? The expected sign of the coefficient of price elasticity of demand is negative. COMMENT 4.2: The law of demand states that for any good or service, price and quantity demanded are inversely related, holding other variables constant. The expected negative sign of the coefficient of price elasticity of demand merely reflects this relationship: the numerator and denominator of ep should move in opposite directions, i.e., they should have opposite signs. Hence the resulting coefficient is expected to be negative. It is sometimes inconvenient to deal with negative numbers. We often handle this by ignoring negative signs. In Case 4.1 we could simply say that the coefficient of price elasticity of demand is 1.7. More formally, we take the absolute value of the coefficient and write ep = 1.7. We now introduce the following definition:
DEF 4.2:
If ep > 1, the demand is elastic. In words, if in some interval of a demand schedule (or demand curve) the coefficient of price elasticity of demand is greater than one, we say that the demand in the interval is elastic.
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Analogous definitions follow for Case 4.2 (ep < 1), Case 4.3 (ep = 1), Case 4.4 (ep = 0) and Case 4.5 (ep = ∞). These definitions are summarized in Table 4.1. Table 4.1
Coefficient of Price Elasticity of Demand Demand Elasticity ep > 1 Elastic ep < 1 Inelastic ep = 1 Unit (or unitary) elastic ep = 0 Completely inelastic ep = ∞ Perfectly elastic QUESTION 4.1:
How do we interpret these concepts?
ANSWER 4.1:
When buyers are relatively sensitive to price changes we say that the demand is elastic; when they are relatively insensitive to price changes we say the demand is inelastic and unit elastic demand constitutes the boundary between these two conditions. When buyers are totally insensitive to price changes we say that the demand is completely inelastic (ep = 0) and when demand is perfectly elastic buyers are “infinitely” sensitive to price changes. (The last statement is difficult to interpret at this stage but its meaning will become clear in Chapter 8. See also EXAMPLE 4.5 below.)
QUESTION 4.2:
Why do we define ep as the percentage change in quantity demanded divided by the percentage change in price instead of simply the change in quantity demanded divided by the change in price? That is, we could hypothetically define the “coefficient of price elasticity of demand” as shown in Equation 4.3’ but in fact we don’t!
ANSWER 4.2:
Defining ep as we do in DEF 4.1 and EQ (4.1), that is, in terms of relative instead of absolute changes in the variables, means that our measure is “unitfree”: To say that ep = 2 has the same meaning whether we are talking about chewing gum, cars or concert tickets. In other words, we get the same numerical result (and the same
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meaning) whether we measure quantities in tons, kilos or ounces and prices in dollars, cents, euros or yen. EXAMPLE 4.3
The Tempos Co. designs and markets electronic watches which currently are priced at $80. Unit sales are 600,000 per year. They are in the process of rethinking their pricing policy and Linda Goodheart, the firm’s marketing V.P. has asked her staff to come up with an estimate of the price elasticity of demand for this product. Two teams went to work on this task and they came up with different results: Team A estimated that ep ≈ 1.32 and Team B’s estimate was that ep ≈ 0.68. These possibilities are shown in FIG 4.1: The current price ($80) and quantity demanded (600,000) are shown by point a. Based on these estimates, two possible demand curve can be drawn (both passing through point a). If Team A is right the resulting demand curve is shown by the straight line labeled D2. For example, a 25% increase in the price (from $80 to $100) leads to a 33% drop in the quantity demanded (from 600,000 to 400,000) shown by the movement from point a to point c along D2. Using EQ (4.1), we can calculate the coefficient of price elasticity of demand:
If Team B is right the resulting demand curve is shown by the straight line labeled D1. A 25% increase in the price leads to a 17% drop in the quantity demanded (from 600,000 to 500,000) shown by the movement from point a to point b along D1 and we can again calculate the coefficient of price elasticity of demand:
It is easy to see that in this case the relatively “flat” demand curve (D2) represents an elastic demand (in the neighborhood of the intersection point a) and the relatively “steep” demand curve (D1) represents an inelastic demand. This in fact is generally the case: If two demand curves intersect at some point, then in the “neighborhood” of the intersection point the “flatter” demand curve represents a more elastic demand than the “steeper” demand curve. There are two obvious cases in which the shape or “look” of a demand curve immediately provides us with information about the nature of price elasticity of demand: First, when demand is completely inelastic (ep = 0), buyers are totally 5
insensitive to price changes (they buy the same amount of a good or service no matter what the price) so the demand curve is a vertical straight line. Second, when demand is FIG 4.1
D1
b c
a
D2
perfectly elastic (ep = ∞), buyers are “infinitely” sensitive to price changes and the demand curve is a horizontal straight line. EXAMPLE 4.4:
The BeltonDixon Co., a major New Jerseybased drug manufacturer, developed and is about to start selling the wonder drug Prosaic. At a meeting at which the marketing campaign for the drug was discussed, Jim Madison, product manager for Prosaic stated that according to the company’s market research there exists a “potential market” for the drug of 200 million doses per year. When asked if this “potential market” is in any way affected by the price the company plans to charge he replied in the negative. Madison must therefore believe (rightly or wrongly) that the “demand curve” for Prosaic looks like FIG 4.2 below, i.e., the demand is completely inelastic (ep = 0).
EXAMPLE 4.5:
The Beaver Co. is an Illinoisbased, multimillion dollar grower of agricultural crops such as wheat. Although a large operation by most standards, it is “small” in relation to the wheat market as a whole and is therefore unable to influence the price of wheat (except perhaps through political action). We can therefore assume Beaver is a “price6
taker,” (REM Chapter 2), i.e., they must accept the “going” market price; But at this price they are able to sell practically any quantity they wish to sell. To illustrate, assume the going price of wheat is $40 per metric ton. Then Beaver’s “demand curve” would look like FIG 4.3 on the next page: It says in effect that at a price above $40 they would not be able to sell any amount but at any price below $40 they could sell an “infinite” amount. Hence the demand is perfectly elastic (ep = ∞) and the demand curve is a horizontal straight line. Figure 4.2
4.2 Arc Price Elasticity of Demand We want to calculate the coefficient of price elasticity of demand for some interval of a demand schedule such as the one given in Table 4.2 below. To do this we use EQ (4.3). But since we want to calculate ep for the interval from P = $4 to P = $5, we must decide which pricequantity combination to use as our “base,” i.e., where to start our calculations. We compromise by using both, i.e., we calculate ep “going up” and “going down” the demand schedule.
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Figure 4.3
Table 4.2
Price Quantity 1 $5 1 2 $4 2 PROBLEM 4.1:
Calculate the coefficient of price elasticity of demand for the demand schedule given in Table 4.2 “going up,” i.e., from P = $4 to P = $5 and “going down,” i.e., from P = $5 to P = $4.
SOLUTION 4.1:
Using EQ (3), it is easy to show that ep = 2 “going up” and ep = 5 “going down.” (a) “Going up”:   (b) “Going down”:
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So in both cases ep > 1 and we say that demand in the interval is elastic. But we are unhappy about the fact that we get such different numerical results depending on which way we proceed in our calculations. We would like to have a single number for the coefficient of price elasticity of demand in any interval. (The simplest explanation for this problem is that the formula which we introduced in EQ (4.1) applies strictly speaking only when the relative changes in prices and quantities are “very small” – mathematicians say “infinitesimally small.”) When the relative changes in prices and quantities are not very small (as in the example in Table 4.2) we use a different approach. We rename the formula given in EQ (4.1), EQ (4.2) and EQ (4.3) the coefficient of point price elasticity of demand and construct a new formula called the coefficient of arc price elasticity of demand. (This is often called the “midpoint formula.”) We use the symbol Ep for this coefficient. Instead of choosing one or the other pricequantity combination as the base for our calculation, we use the averages (i.e., the “midpoints”) of the two prices and quantities. Arbitrarily naming one of the pricequantity combinations “1” and the other “2” (See Table 4.2) we can write the formula as follows:
But ∆Q is simply the “new” minus the “old “quantity, i.e., Q2 – Q1 (and similarly for ∆P) and the “2s” and “100s” cancel out. We can therefore rewrite EQ (4.4) as follows:
This equation can then be rewritten more conveniently as follows:
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PROBLEM 4.2:
Calculate Ep, the absolute value of the coefficient of arc price elasticity of demand for the demand schedule given in Table 4.2 “going up” i.e., from P = $4 to P = $5 and “going down,” from P = $5 to P = $4.
SOLUTION 4.2:
It is easy to show that both calculations lead to the result that E p = 3. (a) “Going up”: 



(b) “Going down”:
This in fact is the purpose of this exercise: to show that the result is the same no matter in which direction the calculation is made. So when we use the arc elasticity formula we come up with a single answer to the question, what is the coefficient of price elasticity of demand in an interval of a demand schedule or demand curve.
4.3 Price Elasticity of Demand and Total Revenue There is an important relationship between price elasticity of demand and the effect of price changes on the total revenue of producers or sellers.
DEF 4.3:
Total Revenue (TR), also called sales revenue or simply sales by accountants and business people, is defined, for the case of a singleproduct firm (or industry), as price times quantity. That is, the price of the product is multiplied by the quantity produced and sold. In symbols: TR = P x Q
EXAMPLE 4.6:
The Kettle Corp., an Ohiobased manufacturer of hand saws, produced 120,000 saws in March of 201X and sold them at a wholesale price of $6.70 each. Their total revenue in March therefore was: TR = $6.70 x 120,000 = $804,000
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FIG 4.4
Consider the linear demand curve shown in FIG 4.4 above. When the price of good X is equal to the distance OP1 the quantity demanded (which can be read off the demand curve at point A) is equal to the distance OQ1. Total revenue at this pricequantity combination is represented geometrically by the area of the rectangle OP1AQ1, i.e., price (OP1) times quantity (OQ1). Next, when the price is lower (shown by the distance OP2) quantity demanded is higher (shown by distance OQ2). (This of course is just an instance of the law of demand.) At the price OP2 total revenue is equal to the area of the rectangle OP 2BQ2. It is clear from FIG 4.4 that the second rectangle is larger than the first; hence at the lower price total revenue is higher. QUESTION 4.3:
How can we tell that total revenue is higher at the lower price?
ANSWER 4.3:
The two (“total revenue”) rectangles have rectangle OP2CQ1 in common. The second rectangle is “smaller” than the first by the area P2P1AC (shown in red). It is “larger” than the first by the area Q1CBQ2, (shown in green). It is obvious to the naked eye that the “gain” (green rectangle) is larger than the “loss,” (red rectangle), hence we conclude that the area of the second rectangle is larger than that of the first and total revenue is higher at the lower price. 11
DEF 4.4:
If in an interval of a demand schedule (or demand curve) a lower price is associated with higher total revenue (or a higher price is associated with lower total revenue) we say that the demand in the interval is elastic.
QUESTION 4.4:
What is the economic interpretation of these rectangles?
ANSWER 4.4:
The area of the (red) rectangle P2P1AC represents the loss in revenue resulting from the lower price and the area of (green) rectangle Q1CBQ2 represents the gain in revenue resulting from the larger quantity demanded.
COMMENT 4.3: It is easiest to think of this as simply a new definition of elastic demand, but you should be aware that this definition is logically identical to that given in DEF 4.2, but we won’t go into the justification for this statement. QUESTION 4.5:
Can you figure out what happens to total revenue when the price drops from P3 to P4 (or when it rises from P4 to P3)?
ANSWER 4.5:
You should use the same reasoning as in ANSWER 4.3. When the price drops from P3 to P4 the decline in total revenue due to the lower price is shown by the area of the (red) rectangle P4P3EG. The increase in total revenue resulting from the larger quantity demanded is shown by the area of the (green) rectangle Q3GFQ4. The second rectangle looks smaller than the first, hence we conclude that a lower price is associated with lower total revenue (and a higher price is associated with higher total revenue).
QUESTION 4.5 and ANSWER 4.5 lead us to the following definition: DEF 4.5:
If in an interval of a demand schedule (or demand curve) a lower price is associated with lower total revenue (or a higher price is associated with higher total revenue) we say that the demand in the interval is inelastic.
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EXAMPLE 4.7:
Consider the demand schedule shown in the table below: Price Quantity $7 2 6 3 5 4 4 5 3 6 2 7
TR $14 18 20 20 18 14
We want to determine the coefficient of price elasticity of demand in the (shaded) interval between P = $4 and P = $5. Using the arc elasticity formula, we find that Ep = 1. (The calculation of Ep is left as an exercise for the reader.) But notice the effect of a price change on total revenue in this interval: An increase in the price from P = $4 to P = $5 leaves total revenue unchanged at TR = $20 and a decrease in the price from P = $5 to P = $4 also leaves total revenue unchanged. This leads to the following definition:
DEF 4.6:
If in an interval of a demand schedule (or demand curve) a change in price (i.e., an increase or a decrease) results in no change in total revenue, we say that the demand in the interval is unit elastic.
DEF 4.4, DEF 4.5 and DEF 4.6 and an additional case are summarized in the table below. Table 4.3
Price P↑ P↓ P↑ P↓ P ↑or ↓ P ↑or↓ HINT 4.1:
Total Revenue TR ↓ TR ↑ TR ↑ TR ↓ No change TR↑ or ↓by the same %
Elasticity of Demand Elastic Elastic Inelastic Inelastic Unit elastic Completely inelastic
One can think of the relationship between price elasticity of demand, price changes and their effects on total revenue in the following way: if price and total revenue move in opposite directions, demand is
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elastic and if price and total revenue move in the same direction demand is inelastic.
HINT 4.2:
There is an opportunity for error in two of the concepts we have discussed in this chapter. REM: (i) If there is a change in price but no resulting change in quantity demanded, we are dealing with a completely inelastic demand (ep=0) but (ii) If there is a change in price but no resulting change in total revenue, we are dealing with a unitelastic demand (ep = 1). It is easy to confuse the two situations.
PROBLEM 4.3:
Lay Kenny, CFO of HTGT, Inc., believes that increasing the price of the XL screw from $40 per gross to $48 per gross will raise revenues derived from this product from $8 million to $9.6 million per year. What is the implied coefficient of price elasticity of demand for the XL screw?
SOLUTION 4.3:
A price change from $40 to $48 represents a 20% increase, that is: ($48  $40)/$40 = 0.2 or 20%. Similarly, a change from $8 million to $9.6 million represents a 20% increase in total revenue. A close look at FIG 4.2 and Table 4.3 indicates that when price and total revenue move in the same direction and in the same proportions, i.e., they rise (or fall) by the same percentage, then the demand is completely inelastic and ep = 0.
COMMENT 4.4: The discussion so far and a careful examination of FIG 4.4 suggest the following: a (downsloping) linear demand curve has a segment where demand is elastic (near the “top”) and a segment where demand is inelastic (near the “bottom”). Hence there must be a boundary point where demand is unit elastic. (In Chapter 8 we shall find out how to locate this point.) PROBLEM 4.4:
The Metropolitan Transportation Authority (MTA) proposes raising the subway fare in New York City from $2.50 to $3.00. They expect revenues to increase from $2.40 billion to 2.76 billion. Do they think the demand for subway rides is elastic, inelastic or what?
SOLUTION 4.4:
Since the MTA believes that a fare increase will lead to an increase in total revenue, (and the percentage increases will not be the same!) they must believe that the demand for subway rides is inelastic. 14
QUESTION 4.6:
Does the following calculation offer an alternative solution to Problem 4.4?
E’p < 1, so the answer is inelastic, as before. ANSWER 4.6:
No! The solution is incorrect even though the answer seems to be the same as before. Notice that in all the price elasticity formulas we have discussed it is quantities (measured in physical units) which appear in the numerator, not revenues (measured in dollars or other currency.)
4.4 Differences in Price Elasticity of Demand If we want to find the coefficient of price elasticity of demand for a particular good or service, we must do the hard work of collecting data on prices, quantities and the “other variables” that affect demand and use fairly complicated statistical techniques to estimate the appropriate coefficient. (In this chapter we assume that most of this work has been done for you!) EXAMPLE 4.8:
Economists Hsiangtai Cheng and Oral Capps Jr. wanted to calculate the coefficients of price elasticity of demand for seafood. They obtained information on household consumption and prices for individual species as well as for two broad categories of seafood (“finfish” and shellfish.) They also obtained information on a large number of other variables which they thought might influence the demand for seafood. (Economists call these “independent variables”.) Their source was the Seafood Consumption Survey conducted for the National Marine Fisheries Administration of the U.S. Department of Commerce. The table below lists some of these independent variables: Selected Independent Variables Occupation of household manager Number of children in household Religion of household Season Price of poultry Price of red meat Household income 15
APPLICATION 4.1 PROBLEM:
The Rockford Co., a Vermontbased manufacturer of hiking boots has “market power,” that is, within limits they are able to determine the price at which they sell their product. They find themselves in the inelastic range of their demand curve. What should they do?
SOLUTION:
Consider the following simple accounting equation: Profit = TR – TC Here TC stands for total cost, i.e., the total cost of production for a given level of output. (Total cost will be discussed in greater detail in Chapter 6.) If Rockford is in the inelastic range of their demand curve, raising the price leads to higher total revenue. A higher price leads to a smaller quantity demanded, hence to a lower output level and lower total costs. Higher total revenue and lower total cost lead to higher profit; so Rockport should raise the price. How far should they go in raising the price? Certainly to the point where demand is unit elastic but we do not yet have enough information to advise them what to do beyond that point.
Using statistical tools which are part of the branch of economics called econometrics Cheng and Capps came up with estimates for the coefficients of price elasticity of demand which are shown in Table 4.4. As can be seen, the demand for all the individual species (except oysters) and both broad categories (shellfish and finfish) was inelastic. Over the years economists and statisticians have tried their hands at estimating the coefficients of price elasticity of demand for many
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different goods and services. Some additional examples are shown in Table 4.5 below. Table 4.4 Category Price Elasticity of Demand Shellfish Crabs – 0.77 Oysters –1.13 Shrimp – 0.70 Shellfish (total) – 0.89 Finfish Cod – 0.54 Flounder/Sole – 0.45 Haddock – 0.56 Perch – 0.70 Snapper – 0.97 Finfish (total) – 0.67 Source: Hsiangtai Cheng and Oral Capp, Jr., “Demand Analysis of Fresh and Frozen Finfish and Shellfish in the United States,” American Journal of Agricultural Economics, August 1988
Table 4.5 Price Elasticity of Demand Price Elasticity of Demand (ShortRun) (LongRun) Air Travel –0.10 –2.40 Automobiles –1.88 –2.21 China, Glassware –1.55 –2.55 Commuter Rail –0.62 –1.59 Electricity (Household) –0.10 –1.90 Gasoline –0.20 –0.50 – 1.50 Medical Care, Hospitalization –0.31 –0.92 Physicians’ Services –0.10 –0.60 Tires –0.86 –1.19 Wine –0.70 –0.84 – 1.20 Good or Service
Economic analysis as well as a careful look at data such as those shown in Table 4.5 suggest a number of possible explanations for differences in price elasticity of demand across goods and services. The following items constitute an incomplete list.
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REM 4.1:
Almost every explanation in economics has to be supplemented with the statement “all other things equal” or ceteris paribus. So we may expect that the demand for some good may be inelastic for one of the reasons discussed below but elastic for another reason.
Availability of Substitutes Generally, the existence of close substitutes is one of the major factors explaining differences in price elasticity of demand between one product and another. If there is some good with many close substitutes, an increase in its price will cause buyers to substitute away from it and its quantity demanded will drop sharply (and vice versa). We describe the demand for this good as elastic. It is for this reason that we expect the demand for unique lifesaving drugs to be highly inelastic and the demand for Chevrolets to be more elastic than the demand for cars in general (since there are many substitutes for Chevrolets).
Proportion of Expenditures If spending on a good makes up a small proportion of household budgets (in the case of consumer goods) a change in its price will have a small impact on people’s purchasing decisions and hence quantity demanded will change relatively little and the demand will be inelastic but if the item involved plays a large role in household budgets a price change will have a larger impact and the demand is likely to be elastic. This is one of the reasons why we expect the demand for newspapers to be inelastic but the demand for housing to be elastic.
“Durability” of a Good or Service If a good is relatively durable, potential buyers may purchase it now or postpone purchasing it until prices are more favorable from their point of view. So in this case buyers are sensitive to price changes and we would expect the demand to be elastic. But in the case of relatively nondurable goods such a “postponement” may not be possible; Buyers are relatively insensitive to price changes and we would expect the demand to be inelastic.
Luxury versus Necessity The very word necessity suggests a good or service which is urgently required. Hence we would expect buyers of such goods to be insensitive to price changes and demand to be inelastic. But luxuries by their very nature are not “urgently required” and buyers therefore would be expected to be sensitive to prices. The resulting demand will be elastic. But note the famous saying: “One person’s luxury is another person’s necessity.” So in practice it is frequently hard to draw a sharp line between these two categories. 18
Time Frame Notice that in all the examples in Table 4.5 the calculated coefficients of price elasticity of demand are higher in the long run than in the short run. This is generally (but not always!) the case. In other words, demand is usually more elastic when viewed from a longrun than a shortrun perspective. Why is this so? The answer is that in the long run people have more time to adjust to price changes. EXAMPLE 4.9:
In the last quarter of the 20th century there were a number of episodes of sharp increases in the price of gasoline, usually brought about by successful efforts by the O.PE.C. cartel to raise the price of oil. At first, the resulting drop in oil consumption was small. But with the passage of time, people were able to make adjustments: They bought smaller and more fuelefficient cars and generally changed their driving habits so that several years after each price increase the outcome was a considerable (relative) drop in gasoline consumption. In other words, the demand for gasoline is more elastic in the longrun than in the shortrun.
The results of this discussion are briefly summarized in Table 4.6 Table 4.6 The demand for a good is likely to be: elastic if it: inelastic if it: has many close substitutes. has few close substitutes. represents a large proportion of represents a small proportion of expenditures. expenditures. is durable. is nondurable. is a luxury. is a necessity. is viewed from a longrun is viewed from a shortrun perspective. perspective.
4.5 Price Elasticity of Supply Just as we are interested in the sensitivity of buyers to price changes, we are also interested in the sensitivity of producers or sellers to price changes. This leads to a consideration of the concept price elasticity of supply and the coefficient of price elasticity of supply. COMMENT 4.5: In Chapter 2 we pointed out that supplyanddemand analysis strictly speaking applies to competitive markets, i.e., markets with many 19
sellers who individually are unable to influence the market price. In the same way the concept of price elasticity of supply applies to competitive markets and not to markets where sellers have market power. In these cases it makes no sense to ask how sellers will react to a price change since, within limits, they are able to set their own price! (Think of monopoly as an extreme example!)
DEF 4.6:
The coefficient of price elasticity of supply of good X is defined as the percentage change in the quantity supplied of good X divided by the percentage change in its price.
Using the symbol es for this coefficient, we write:
Using Qs for quantity supplied, and P and the ∆ symbol as before, we can write:
Simplifying as in Equation (4.3) we write:
COMMENT 4.6: Note that EQ (4.9), has the same “look,” or “structure” as EQ (4.3), and thus represents the coefficient of point price elasticity of supply.
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QUESTION 4.7:
What is the expected algebraic sign of the coefficient of price elasticity of supply?
ANSWER 4.7:
The law of supply predicts that the numerator and denominator in EQ (4.7) will move in the same direction, hence they will have the same sign. So the expected algebraic sign of es is positive.
QUESTION 4.8:
If in an interval of a supply schedule (or supply curve) e s > 1, i.e., the coefficient of price elasticity of supply is greater than one, what do you think we will call this situation?
ANSWER 4.8:
Clearly we will call the supply in the interval elastic. This and related definitions are summarized in Table 4.7.
NOTE 4.1:
The issue discussed in PROBLEM 4.1 and in Section 4.3 applies to all the elasticity concepts in this chapter, including of course price elasticity of supply. That is, the point elasticity formula is defined strictly speaking only for “very small” changes in the variables. So when these changes are not very small we have to construct an appropriate arc elasticity formula. But all the arc elasticity formulas again have the same “structure,” or “look.” Construction of arc elasticity formulas (including the coefficient of arc price elasticity of supply) is therefore be left as an exercise for the reader. In addition, the concepts “elastic,” “inelastic,” etc., are defined in similar ways for all the elasticity concepts. So constructing the resulting definitions for these terms will also be left as an exercise for the reader. Table 4.7 Coefficient of Price Elasticity of Supply Supply Elasticity eS > 1 Elastic eS < 1 Inelastic eS = 1 unit (or unitary) elastic eS = 0 completely inelastic eS = ∞ perfectly elastic
4.6 Differences in Price Elasticity of Supply Why are there differences in price elasticity of supply? Why is the supply of one good or service elastic while that of another good or service is inelastic? There are several technical reasons for such differences which we will discuss in later chapters but the 21
major explanation of interest to us is time: that is, the time available to producers to respond to price changes. The idea involved can be best explained with the help of an example. EXAMPLE 4.10
Consider the town of Durham, a fishing harbor in New England. Its fishing fleet consists of about two dozen boats, which during the fishing season leave the harbor early in the morning and return in the late afternoon with their catch, which they sell to fish merchants at the docks. The prices which they receive vary, depending on supply and demand conditions. Imagine four situations: (1) The price of fish increases (or decreases) significantly over the course of a few days. Call this period the very short run (VSR). (This is sometimes called the “production period.”) Boat owners and crews have little opportunity to respond and they will supply the same quantity of fish, regardless of the price change. The supply of fish is completely inelastic in this case. (See FIG 4.5) (2) The price of fish increases significantly over the course of several months. Now boat owners can increase the size of their crews to some degree, boats can stay in the fishing grounds longer, and in general the existing fleet can be employed more intensively. (The opposite response would occur if prices decreased significantly.) Call this period the short run (SR). In the short run the supply will turn out to be inelastic (but not completely inelastic). (3) The price of fish increases significantly over the course of several years. Call this period the long run (LR). Now entrepreneurs will invest in more (or larger, or more advanced) fishing boats, more individuals will enter the fishing industry, etc. The response will be much greater and we can expect the supply of fish to be elastic. (Again, the opposite response would occur if prices decreased significantly.) (4) In some industries the required resources are nonspecialized and abundant; such industries are able to supply any quantity at “normal” prices if enough time is available to adjust to changes in demand. The time required would then be called the very long run (VLR) and the supply would be perfectly elastic.
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FIG. 4.5
4.7 Some More Elasticity Concepts Whenever we ask, how does a change in one variable (call it “X”) affect another variable (call it “Y”), where both X and Y are measured in relative (or percentage) terms, we have an elasticity concept. But there are a handful of such concepts in addition to the ones we discussed so far, which are well known and widely used.
Income Elasticity of Demand We ask the following question: How does a change in income (for example, the average income of the buyers of a particular good or service) affect its quantity demanded? This naturally leads to the definition of the coefficient of income elasticity of demand.
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APPLICATION 4.2
Price ($/Q)
Price ($/Q)
A
B
S2 S1 e
23
S1
j 37
f
33 30
S2
30 27
D1
g
e m D2
0
70
100
Quantity
0
90 100
Quantity
QUESTION:
Who pays sales taxes?
ANSWER:
On the surface the answer seems obvious. Buyers pay sales taxes. But closer examination shows that things are not so simple: the answer to the question depends on the relative elasticities of supply and demand of the taxed good(s).
COMMENT:
Many states and localities impose taxes on purchases of goods (and sometimes services) but some goods are often excluded from this form of taxation (usually food and medicine). What is typically called a sales tax is sometimes also called a value tax.
DEFINITION: As the name implies, a value tax is calculated as some percentage of the price (or “value”) of a good or service. EXAMPLE:
The sales tax rate in New Jersey currently is 7%. So if the price tag on a taxable item in a store is $100, the buyer pays $107. Examining the effects of taxes of this type is easiest if one studies a somewhat different kind of tax called a quantity tax.
DEFINITION: A quantity tax is a tax imposed per unit (not on the value) of a good or service.
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EXAMPLE:
The Federal tax on gasoline is approximately 19 cents per gallon, regardless of the price. Consider the supplyanddemand diagram above. In both Panel A and Panel B the initial position is shown by point e with an equilibrium price of $30 and an equilibrium quantity of 100.
REM:
The major determinant of supply is the cost of production. So for example, in the graph above, costs are such that producers/sellers are willing and able to supply 70 units at a price of $23 and 100 units at a price of $30 (Panel A) and 90 units at a price of $27 (Panel B).Assume now that a $10 quantity tax is imposed. Whatever the quantity sold, sellers must now pay $10 per unit to some level of government. They are therefore willing to produce and sell any given quantity only if they receive an additional $10 per unit .This is shown in both panels as an upward (or leftward) shift of the supply curve (from S1 to S2). The vertical distance between the two lines shows the amount of the tax (in this case $10). Consider Panel A. The demand curve (D1) is relatively flat so it represents a relatively elastic demand. With the “old” demand curve and the new supply curve the new equilibrium point is f and the new equilibrium price is $33. This is the amount buyers pay; but sellers only receive $23 (as shown by point g on the old supply curve). Why? Because out of the $33 collected they must pay $10 to some government (shown by the vertical distance fg). So buyers pay $3 more than before but sellers receive $7 less than before. This example then illustrates the point that when demand is elastic, sellers pay a larger portion (and buyers a smaller portion) of a quantity tax. Now consider Panel B. With the “old” demand curve and the new supply curve the new equilibrium point is j and the new equilibrium price is $37. Again, this is the amount buyers pay but sellers only receive $27 (as shown by point m on the old supply curve). So buyers pay $7 more than before and sellers receive $3 less than before. (The amount of the tax is shown by the distance mj.) This example illustrates the point that when demand is inelastic, buyers pay a larger portion (and sellers a smaller portion) of a quantity tax.
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QUESTION:
These conclusions follow from a careful examination of the supplyanddemand diagrams in Panels A and B. But can we explain these facts in terms of economic behavior and not just graphs?
ANSWER:
Think of it this way: An elastic demand means that buyers can relatively easily “substitute away” from (or do without) the particular good or service. This makes it more difficult for sellers to impose higher prices on buyers and forces them to “absorb” a larger proportion of a quantity tax. By contrast, an inelastic demand means that it is more difficult for buyers to substitute away from the good or service and sellers are able to “pass on” a larger proportion of the tax onto buyers.
NOTE:
This discussion was framed entirely in terms of differences in price elasticity of demand. It should be clear that price elasticity of supply also has an effect on the outcome. It is left as an exercise for the reader to think through the question how differences in price elasticity of supply determine whether buyers or sellers pay a larger proportion of a quantity tax.
DEF 4.7
The coefficient of income elasticity of demand of good X is defined as the percentage change in the quantity demanded of good X divided by the percentage change in the incomes of buyers of good X.
Using the symbol eI for this coefficient we have:
NOTE 4.2:
It is left as an exercise for the reader to construct the point elasticity and arc elasticity formulas for the coefficient of income elasticity of demand using EQ (4.3) and (4.6) for the coefficient of price elasticity of demand as a model. 26
QUESTION 4.8:
What is the expected algebraic sign of eI?
ANSWER 4.8:
Generally (but not always) one would expect eI to be positive: As income increases the demand for many goods and service increases, so with a given supply, the (equilibrium) quantity demanded increases. (The opposite is also true: As income decreases the demand for many goods and service decreases.) So with the numerator and denominator in EQ (4.10) moving in the same direction, they have the same signs, so eI would be positive. But sometimes the opposite holds: As income increases (or decreases) the demand for some goods decreases (or increases) and ei is negative.
REM 4.2:
If the demand for a good or service moves in the same direction as income (or wealth) we call the good a normal good. If they move in opposite directions we call the good an inferior good.
EXAMPLE 4.11: Jim Morrison, marketing V.P. of Lewis Ballon, Inc., a diversified designer and manufacturer of luxury goods, wanted a “quick and dirty” estimate of next year’s unit sales of one of their signature handbags. In the most recent period approximately 72,000 of these handbags were sold per year. Sherry Miller, a staff economist, gave him two pieces of information: the coefficient of income elasticity of demand for this product was around 1.7 and the consensus estimate of next year’s average income was 3 percent above this year’s. His quick calculation looked like this:
So based on the expected increase in average income alone he estimated that sales of the item would rise by 5.1% to 75,672. (Of course Morrison realized that a number of other factors might affect unit sales; this was just the first step in his estimating procedure.)
Cross Elasticity of Demand We would like to answer the following question: How does a change in the price of one good (or service) affect the quantity demanded of another good (or service)? The question of course only makes sense if there is some sort of relationship between the two goods.
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REM 4.3:
From the point of view of buyers, two goods are either substitutes (they can be used in place of each other) or complements (they are used together).
The question we ask above naturally leads to the definition of the coefficient of cross (or crossprice) elasticity of demand.
DEF 4.8:
NOTE 4.3:
The coefficient of cross elasticity of demand between good 1 and good 2 is defined as the percentage change in the quantity demanded of good 1 divided by the percentage change in the price of good 2.
“Good 1” and “good 2” are arbitrary names we give to the two (related) goods.
Using the symbol ec we can write:
QUESTION 4.9:
If good 1 and good 2 are substitutes, what is the expected sign of the coefficient of cross elasticity of demand between them?
ANSWER 4.9:
If good 1 and good 2 are substitutes an increase in the price of good 2 will lead to a decline in its quantity demanded (because of the law of demand!) Hence buyers will switch to its substitute, good 1. The demand for good 1 will increase and, given the supply, the (equilibrium) quantity demanded will also increase. So the numerator and denominator in EQ (4.11) will move in the same direction, hence ec is expected to be positive. (A similar analysis of the case where the price of good 2 decreases would lead to the identical conclusion.)
QUESTION 4.10: If good 1 and good 2 are complements, what is the expected sign of the coefficient of cross elasticity of demand between them? ANSWER 4.10:
The answer to this question is left as an exercise for the reader.
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APPLICATION 4.3
Price ($/Q)
40
a
b
c
20
d D2
D1
0
80
130
180
Quantity (Q)
Chris Matthews is the owner of Oxalite, an importer and distributor of small lighting fixtures, mainly in the Midwest. Due to a rise in the costs of imports he felt it necessary to double the average price of his products over a relatively short period of time, from an average of $20 to an average of $40. Based on his intuitive knowledge of price elasticity of demand, he expected sales to drop from an average of 130 units to an average of 80 units per week. He was pleasantly surprised that his sales remained approximately the same! At first he assumed that he had simply overestimated the price elasticity of demand for his products and that the demand was in fact highly inelastic, but he soon realized that this was unlikely. After some further investigation he noted that average income in his region had risen from $16,000 to $20,000 and that this probably explained why his sales remained constant despite a doubling of his prices. He did some quick calculations, the results of which are summarized in the diagram shown above. NOTE:
D1 represents the demand when the average income in Oxalite’s market area is $16,000 and D2 represents the demand when the average income is $20,000.
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PROBLEM A1:
Calculate the coefficient of arc price elasticity of demand when average income in Oxalite’s market region is (a) $16,000 and (b) $20,000.
SOLUTION A1:
The solution to this problem is left as an exercise for the reader.
PROBLEM A2:
To avoid incorrect sales forecasts in the future, Matthews asked his assistant, JudyAnn Donohue, to calculate the coefficient of income elasticity of demand for Oxalite’s products. How should she go about this assignment?
SOLUTION A2:
Clearly she should use the coefficient of arc income elasticity of demand in this case. (Why?) The formula for calculating this coefficient is shown below; As before, the subscript “2” represents the “new” value of income and “1” represents the “old” value.
REM A1:
When the arc elasticity formula is used the choice of “old” and “new” values is arbitrary. There are two possible ways Ea can be calculated based on the data in the diagram: between points a and b and between points c and d. Why? Because a and b lie on the same straight line parallel to the horizontal axis. Along this line the price remains constant (P = $40). The same is true for points c and d: Along the horizontal straight line between them the price remains constant at P = $20.
REM A2:
In calculating all the elasticity coefficients discussed in this chapter we have to assume that only two variables change at one time. So in calculating the coefficient of income elasticity of demand we assume that other variables, such as price, remain constant. Otherwise we would simply obtain nonsensical results.
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The results of Donohue’s calculations are as follows:
CONCLUSIONS:
(1)
Clearly the coefficient of income elasticity of demand is not constant. It depends on the level of other variables, in this case the price.
(2)
Since EI is positive in both calculations we conclude that the goods in question are normal goods.
(3)
Since EI is greater than one in both calculations we conclude that the demand for the goods in question is incomeelastic.
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APPLICATION 4.4 The Cellophane Case More than half a century ago the government brought an antitrust case against the Du Pont Company, charging it among other things with monopolizing and attempting to monopolize the market for cellophane, a transparent, flexible packaging material. Du Pont produced approximately 75% of the cellophane sold in the United States at the time, so it was a “near monopoly” in this market. But the company and its lawyers argued that it was incorrect to focus on the product cellophane; instead the focus should have been on a broader market, one for “flexible wrapping materials,” which would include besides cellophane various transparent films, wrapping papers and other products. In this larger market Du Pont would have had a much smaller share of the market and could not be accuse of “monopolizing and attempting to monopolize.” How can one decide which market definition is more appropriate? Clearly, if two goods are close substitutes they can be thought of as part of the same market. One way to decide whether two goods are close substitutes is to ask experts: Do the product s perform similar functions? Do they have similar technical characteristics? Do users actually think of them as close substitutes? But another way is to ask: What is the coefficient of cross elasticity of demand between the two goods? In deciding the case the Supreme Court used both approaches. As Justice Reed wrote in his decision: “If a slight decrease in the price of cellophane causes a considerable number of customers of other flexible wrapping materials to switch to cellophane, it would be an indication that a high crosselasticity of demand exists between them; that the products compete in the same market.” Source: U.S. v. Du Pont (The Cellophane Case); 351 U.S. 377 (1956)
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Advertising Elasticity of Demand In the business world the following question might be asked: How does advertising affect the quantity demanded of a good or service? If the focus is on changes in advertising expenditures and on changes in the quantity demanded (and those changes are expressed in percentage terms) the result is another elasticity concept: advertising elasticity of demand. This leads to the definition of the coefficient of advertising elasticity of demand.
DEF 4.9
The coefficient of advertising elasticity of demand is defined as the percentage change in the quantity demanded of good X divided by the percentage change in advertising expenditures on that good.
Using eA for this coefficient, we write:
QUESTION 4.11: You are an entrepreneur and you produce and sell gizmos. Assuming you hire a competent advertising agency, what is the expected sign of the coefficient of advertising elasticity of demand for gizmos? ANSWER 4.11:
The answer to this question is left as an exercise for the reader.
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PROBLEMS: (1)
When the price of “Ekstra” brand pens was $159 (per gross), the quantity sold was 3,460 (gross) while when the price was $287 the quantity sold was 3,100. Calculate the coefficient of price elasticity of demand in the interval “going up” and “going down” the demand schedule using the point price elasticity formula. Then recalculate using the arc elasticity formula. Compare the results. Is the demand for these pens elastic, inelastic or what?
(2)
Assume the O.P.E.C. cartel succeeds in raising the world price of oil from $47 to $55 per barrel and as a result both member and nonmember countries’ revenues increase from $357 to $406 billion. Is the demand for oil elastic, inelastic or what? Explain.
(3)
An economic forecaster for American Airlines calculates that per capita (i.e., per person) income in the United States in 201X will be 5% above the previous year’s level. He predicts that for this reason alone passenger traffic will increase from 23.6 billion revenue passenger miles (RPMs) to 25.5 billion RPMs. Calculate the implied coefficient of income elasticity of demand for air travel. According to this estimate is the demand for air travel incomeelastic, incomeinelastic or what? Is air travel a normal or an inferior good? Explain. (Note: Airlines measure their “output” in units called revenue passenger miles or RPMs)
(4)
If the supply of good X is completely inelastic and the demand for it increases what will be the effect on the equilibrium price and quantity of good X? Draw a graph and explain.
(5)
Consider the figure below. D1 shows the demand for gizmos when the XYX Corp. spends $6 million on advertising while D2 shows the demand when they spend $15 million on advertising. (a)
Calculate the coefficient of price elasticity of demand when the company spends $6 million on advertising.
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(b)
Calculate the coefficient of price elasticity of demand when the company spends $15 million on advertising. NOTE: There are several possible calculations of these coefficients, but the sets of prices and quantities you use should be adjacent, i.e., as close as possible given the information in the graph.
(c)
Calculate the coefficient of advertising elasticity of demand (Ea) when the price is $20. Does the sign of EA surprise you? Explain.
(d)
Calculate the coefficient of advertising elasticity of demand when the price is $40.
80 70
Price ($/Q)
60
50 40
D2
30
D1
20 10 0 0
500
1000
1500
2000
2500
3000
3500
4000
Quantity (Q)
(6)
Use the concept of price elasticity of demand and your common sense knowledge of agricultural markets (especially the market for food staples) to explain why a good harvest may be bad new for farmers.
(7)
Car prices are expected to rise by approximately 3% and as a result car sales are expected to drop by 2%. On the basis of these data calculate the coefficient of price elasticity of demand using both the point and arc elasticity formulas. Compare the results
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