3.3 Price Elasticity

[3.2 Demand] [3.4 Supply] [3.5 Production] [3.6 Costs]

Why is it that members of the Organisation of Petroleum Exporting Countries (OPEC) can increase their total revenue by selling less oil? The answer is, by selling less oil, they can drive up its price on the world market. The world's oil consumers may decrease their consumption of oil in response to the price increase, but the decrease in consumption is fairly small. In fact, it’s small enough that the increase in price more than offsets the decrease in quantity sold. The total revenue that oil producers receive rises as a result, as does the motivation of producers to restrict the output of oil.

The OPEC example illustrates a fundamental concept in economics known as elasticity. Elasticity measures the relative response in one variable to a change in another variable. For example, the price elasticity of demand for oil measures the relative response in the quantity demanded of oil to a change in the price of oil.

Similarly, the price elasticity of supply of tomatoes measures the relative response in the quantity supplied of tomatoes to a change in the price of tomatoes. The cross price elasticity of demand of coffee with respect to tea measures the relative response in the quantity demanded of coffee to a change in the price of tea. The income elasticity of demand of housing measures the relative response in the quantity demanded of housing to a change in consumer income.

Click here to take a look at the impact of elasticity on America Online.

Now view the following animation to find out more about price elasticity.

You can then calculate the price elasticity at any point along a linear demand curve by applying the point-slope formula.

slope = the value -b from the demand equation

Sometimes you have a set of price/quantity combinations for which you want to calculate elasticity. For instance, suppose that you are given the following demand information for sales of a music CD and would like to calculate the price elasticity of demand at $12.

Price	Quantity (Sales per Month)
$11	1,100
$12	1,000
$13	925

You decide to do this by examining the effects of a price change from $12 to $13. It will help to solve this if you rewrite the formula for elasticity in the following way:

Q₀ and P₀ represent the starting price and quantity (in this example, $12), while Q₁ and P₁ represent the ending price and quantity (in this example, $13). To find the elasticity, simply insert the demand information given.

You may be wondering if the calculation of elasticity at $12 would change if it were calculated instead by examining a price increase from $11 to $12. The answer is yes. To see this, recalculate elasticity using the formula above.

Click here to see two examples of how to calculate elasticity using demand information.

Price elasticity of demand using the point-slope formula

Another method of calculating elasticity is using the point-slope formula. Suppose a coffee shop faces the following linear daily demand curve for cups of coffee:

What is the price elasticity of demand at a price of $4 per cup?

At a price of $4, consumers would want to buy 20 cups of coffee per day.

Using P = $4, Q_d = 20, and slope = -0.1, the price elasticity of demand is

Note that it is also common to see price elasticity reported as an absolute value. Here, at a price of $4, E_d = 2.

Elasticity along a linear demand curve

The coffee shop example allows for the demonstration of another important point. The value of elasticity of demand typically varies along a demand curve. In fact, it will always vary along a linear demand curve.

For example, calculate the price elasticity of demand for coffee at the coffee shop at a price of $2 per cup. When P = $2,.03 the demand function yields quantity demanded of 40 cups.

The slope of the demand curve is still -0.1, so price elasticity of demand is

Notice that demand is more inelastic (that is, E_d is smaller in absolute value) at the lower price of $2 than at $4. It may seem odd that the elasticity of demand differs at prices of $2 and $4, when the slope of this demand curve is the same at all points. This fact can be explained by considering the percentage changes in price and quantity in each case, rather than the absolute changes.

Because the slope of the demand curve in the coffee shop example is -0.1, a one-unit increase in quantity demanded will occur for each $0.10 decrease in price. Notice that if the quantity demanded is 20, an increase in quantity of one unit is a 5 percent change in quantity demanded.

At a price equal to $4 (which corresponds to a quantity demanded of 20), a $0.10 decrease in price is a -2.5 percent change in price.

Therefore, the price elasticity of demand, defined as the percentage change in quantity demanded divided by the percentage change in price is -2.

However, when quantity demanded is 40, a one-unit change is only a 2.5 percent change, which is smaller than the 5 percent change calculated above.

Because quantity is already large, a one-unit increase is not as significant. On the other hand, a price change of $0.10, when the price is only $2, is a 5 percent change.

Because the price is starting out low, this $0.10 change is more significant in percentage terms. The price elasticity of demand when P = $2 and Q_d = 40 is -0.5, as you saw before.

Even though the absolute changes in price and quantity are the same in these two examples, the percentage changes are quite different.

Understanding percentage changes and elasticity will help you answer questions such as: Is a $0.10 price increase significant? The answer is, it depends. If the product sells for thousands of dollars (a car, for instance), then an extra dime would go unnoticed. If the product is a first-class postage stamp that sells for around $0.35, an increase of a dime seems extreme. Similarly, an increase in quantity demanded of 100 sandwiches would be insignificant to a fast-food chain that sells millions of sandwiches, but the same increase would be meaningful to a local sandwich shop that may sell only 500 sandwiches a week.

To review why elasticity differs along a linear demand curve, take a look at the following animation.

For perfectly competitive firms that often have no control over the price they can charge (that is, they must simply accept the existing market price), marginal revenue is constant and equal to the market price. The additional revenue a firm earns by increasing the quantity it produces by one unit (marginal revenue) is simply the market price.

Firms in competitive markets, then, must decide only how many units to produce and sell. (See Topic 4.4 for more information.) Note, however, that many producers acting together can influence the market price even in competitive markets.

The OPEC illustration described earlier provides one such example. Understanding the relation between the price elasticity of demand and total producer revenue helps OPEC determine the quantity of oil that each of its member countries should produce in order to maximise total OPEC revenue.

A firm that has some degree of market power (that is, a firm that can choose the price of its product) must consider both the price to charge and the number of units to sell. (You can also see Topic 4.2 for more information.) This is where price elasticity of demand becomes important, as it determines the optimal markup over cost a firm can charge to maximise its profits.

The key is to understand what happens to a firm's total revenue if price is changed by a small amount. The example below should help clarify this concept.

Example: Price Elasticity of Demand and Total Revenue

Suppose the coffee shop in the example above lowers the price of a cup of coffee by a small amount (from, say, $4 per cup to $3.50 per cup). What would you expect to happen to the firm's total revenue from its coffee sales?

The linear demand curve used in the coffee shop example is

Consumers would purchase 20 cups of coffee at $4 per cup.

Therefore, total revenue would be

If the shop decreased its price to $3.50 per cup, quantity demanded would increase to 25 cups per day.

Therefore, total revenue would be

In this case, the firm's total revenue from coffee sales increased when it decreased the price of its coffee. There are two opposing effects on total revenue when there is a decrease in price. The first effect comes from the firm charging a lower price on each unit it sells. This first effect decreases total revenue. However, we know from the law of demand that as price falls, the quantity of a good that consumers will be willing to purchase rises. This second effect of a price decrease tends to increase total revenue because quantity sold increases.

The net effect of a price decrease on total revenue depends on what effect dominates, which is exactly what the price elasticity of demand tells us. The fact that total revenue increased when price decreased from $4 to $3.50 is consistent with the fact that demand was found to be elastic at a price of $4. For an elastic demand, the percentage change in quantity demanded is larger than the percentage change in price.

To summarise:

· If demand is elastic at the current price, then total revenue will increase (implying positive marginal revenue of output) when price is decreased. The additional revenue generated by increasing quantity will be positive. This means that the percent change in price will be smaller than the percent change in quantity demanded, causing total revenue to increase.

· If demand is unit elastic, then total revenue will not change. The additional revenue generated by increasing quantity is zero. Thus, price and quantity demanded change by the same percent, leaving total revenue unchanged.

If demand for the product is inelastic at the current price, then total revenue will decrease (implying negative marginal revenue of output) when the price is decreased. The additional revenue generated by increasing quantity will be negative. This means that the percent change in price will be larger than the percent change in quantity demanded, causing total revenue to decrease.

The implications of this analysis for a firm's price-setting decision are important. In general, a firm with some market power will not maximise profits by setting a price at which demand is inelastic. When demand is inelastic, a firm could increase revenue and decrease costs (because it sells fewer units) by raising price.

This is because the percentage change in quantity sold increases but the percentage change in price decreases, as one moves upwards along the demand curve towards higher prices.

The logical conclusion of the analysis thus far is that, for a linear demand, the price that will maximise a firm's total revenue will be the price at which demand is unit elastic. The following example illustrates this result.

Example: Revenue-maximising prices and price elasticity

For consistency, return to the coffee shop example, where daily demand for cups of coffee is given by

What price per cup should the coffee shop charge to maximise total revenue?

Because a firm's total revenue is maximised where demand is unit elastic, you need to find the unit elastic price and quantity combination for this linear demand curve. To find the price-quantity pair, begin with the point-slope formula for price elasticity of demand.

Set the elasticity equal to -1 and substitute the appropriate slope and expression for price (taken from the demand function).

Then solve for Q_d.

The result indicates that demand is unit elastic at a quantity of 30 cups. To find the corresponding price, substitute the value for Q_d into the demand function.

The demand for coffee at the shop is unit elastic at a price of $3, which corresponds to a quantity sold of 30 cups per day. Therefore, a price of $3 maximises total daily revenue. Total daily revenue would be $90.

It is also true that the firm's marginal revenue is zero at Q = 30 units, which you can demonstrate using calculus.

A firm's total revenue is price times quantity. From the demand function, you know the relation between price and the number of cups that will be sold.

Therefore,

The firm's marginal revenue is the derivative of total revenue with respect to quantity.

Notice that if you set marginal revenue equal to zero and solve for Q, you find that Q = 30.

Marginal revenue is positive for all units up to 30 and is negative for all units after 30. The graph below illustrates this result. Notice that at 30 units of output, where demand is unit elastic, P = $3 and total revenue is maximised at $90.

Does this mean that firms should set price where demand is unit elastic? Again, the answer is generally no. Firms are interested in maximising profits, not total revenue. The only time the two will coincide is when it costs the firm nothing to produce additional units of its product (that is, when the firm's marginal cost of output is zero).

Test your knowledge now by evaluating a proposal by a wildlife-conservation organisation to increase the price of its T-shirts in order to increase total revenue.

Exercise: Wildlife Conservation Organisation

For additional information on profit-maximising with positive marginal costs, see Topic 4.3.

What determines the price elasticity of demand for a good? In short, goods that are necessities, have few good substitutes, and/or constitute a small portion of a consumer's budget tend to have more inelastic demand than goods that are luxuries, have many good substitutes, and/or constitute a large portion of a consumer's budget. Also, the longer the time period consumers have to react to price changes, the more elastic demand will be. For example, if the price of petrol doubled overnight, few households and businesses would respond with any significant decrease in petrol consumption in the very short term. Over time, however, consumers would find alternative ways to decrease petrol consumption (such as moving closer to work, carpooling, taking public transportation, or purchasing more fuel-efficient cars). This implies more elastic demand and a greater response in quantity demanded to a price change over time.

The following is a link to an advanced topic regarding demand curves where elasticity remains the same over the entire curve.

Cross-Price Elasticity and Income Elasticity of Demand

A cross-price elasticity of demand measures the relative change in the quantity demanded of one good in response to a change in the price of a related good. Formally,

You can apply the cross-price elasticity of demand to market demand for a good or to individual demand for a good. For example, if the price of tea increases by 10 percent, causing the total quantity demanded of coffee to increase by 20 percent, then the cross-price elasticity of demand for coffee with respect to tea would be 2 (that is, 20/10 = 2).

Notice that tea and coffee are substitute goods for most people. Substitute goods will have a positive cross-price elasticity, because an increase in the price of one causes an increase in the quantity demanded of the other, as individuals substitute away from the good whose price has increased.

On the other hand, two complement goods, such as bread and butter, should have negative cross-price elasticities. That is, an increase in the price of bread will lead not only to a decrease in the quantity demanded of bread but also to a decrease in the quantity demanded of its complement, butter. Two goods that are unrelated should have cross-price elasticities equal to zero.

Leisure airline fares are sometimes accompanied by the reservation of a rental car. Recently, because of increases in new-car prices, car rental companies have increased their prices by 5 percent.

For some people planning vacations, this increase in rental car prices might persuade them to rethink their plans. Instead of flying to their destination they might decide to drive there instead, eliminating the need to rent a car.

Assume that there is a 3 percent decrease in leisure fare purchases attributable to the increase in rental car rates. With this information, calculate the cross-price elasticity of the price change in rental cars and demand for leisure fares.

Because this cross-price elasticity is negative, you can infer that the goods or services being measured are complements. If the cross-price elasticity was positive, they would be substitutes.

The income elasticity of demand measures the relative response in the quantity demanded of a good to a change in consumer income. Formally,

You can apply income elasticity of demand to market demand or individual demand for a good.

For example, if your income increases by 50 percent and you go to see 25 percent more movies, then the income elasticity of your demand for movies is 0.5 (that is, 25/50 = 0.5).

For some goods, you might have a negative income elasticity of demand. For example, if your income increases by 50 percent and your consumption of fast-food hamburgers decreases by 10 percent, then your income elasticity of demand for fast-food hamburgers is -0.2 (that is, -10/50 = -0.2).

Goods that have positive income elasticities of demand are referred to in economics as normal goods. Goods that have negative income elasticities of demand are referred to as inferior goods.

Click on each of the following links for some more cases of applying elasticity

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Philip Morris

Airline Travel

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