Chapter 4

Using the Elasticity Shortcut

In This Chapter

Grasping the basics of elasticity

Understanding and calculating quantity demanded changes

Identifying the effect of price changes on revenue

Examining how income, other goods’ prices, and advertising influence demand

This chapter brings you up to speed on the concept of elasticity and how it works. It explains how elasticity determines a business’s revenue side and tells you what price to charge, how much advertising to do, and how changes in other prices or income affect your sales. If you remember only one concept in managerial economics, elasticity is it. The fact that a single concept provides all this information makes it magical. Calculating an elasticity value is like pulling a rabbit out of a hat; one number tells nearly everything to the amazement of those watching (your coworkers). And when you combine elasticity and revenue information with production costs, you can determine how the firm will maximize its profit.

Using Elasticity Is The Key to Flexibility

The law of demand states that increasing a good’s price reduces the good’s quantity demanded (the amount of the good that customers purchase given its price). This relationship is important, but somewhat obvious. Similarly, demand reacts to changes in incomes, the price of related goods, and advertising efforts. Elasticity measures the responsiveness of one economic variable to another and is the concept you use to determine these relationships. For example, when the price of movie tickets goes up, you and other customers will buy fewer tickets. So, for the theater manager, the critical question is how much will the number of tickets sold decrease — will it be a little or a lot? Similarly, the theater manager needs to know how movie ticket demand reacts to changes in incomes, the price of popcorn at the concession stand, and advertising for the blockbuster movie.

Think of elasticity as a measure of flexibility. Elasticity tells you how flexible customers are to change. Generally, if customers are very flexible to a given change, they’re considered elastic. If customers aren’t very flexible to a change, they’re inelastic.

As a general rule, you hope customers are inelastic. That way, when you increase price, they will still buy a lot of your product. But remember, general rules are just that — general. And there is a reason for the old saying, “The exception proves the rule.”

Customers respond to many things, so focus on the things that are most important to them. The most important elasticity concepts describe how customers respond to changes in

The good’s price

Income

The prices of other goods

Advertising

Managers typically control two of these factors: the good’s price and advertising. Sometimes managers at least partially control a third factor, the prices of other goods. For example, a movie theater manager controls the ticket’s price and the prices of concessions (although the manager doesn’t control the prices other theaters charge). Finally, managers can’t control the general level of customers’ income, but demand is often affected by whether the general income level is increasing, a period of prosperity, or decreasing, a recession. A recession decreases movie ticket sales.

Knowing the Price Elasticity of Demand: The Fundamental Trade-Off

The price elasticity of demand measures the most important elasticity relationship — how much quantity demanded changes given a price change. In other words, the price elasticity of demand allows you to project how a price change impacts revenue. For example, if the price of movie tickets increases from $8 to $10, does quantity demanded decrease from 5,000 tickets per week to 4,500 or from 5,000 to 3,000? It really matters. In the first case, the movie theater’s revenue increases from $40,000 ($8 × 5,000) to $45,000 ($10 × 4,500). In the second case, the theater’s revenue decreases from $40,000 to $30,000 ($10 × 3,000).

Before changing price, you need to know if the result will be similar to the first or second situation, and the price elasticity of demand tells you which it will be.

Determining the price elasticity of demand: Formulas are your friend

Mastering managerial economics involves calculating values, with the ultimate goal of determining how to maximize profit. The usefulness of the price elasticity of demand depends upon calculating a specific value that measures how responsive quantity demanded is to a price change.

The formula used to calculate the price elasticity of demand is

The symbol η represents the price elasticity of demand. The symbol Q0 represents the initial quantity demanded that exists when the price equals P0. The symbol Q1 represents the new quantity demanded that exists when the price changes to P1.

In this formula, the price elasticity of demand will always be a negative number because of the inverse relationship betweYou get the advertising elasticity of demand equaen price and quantity demanded. As price went up, quantity demanded went down, or vice versa. When price goes down, quantity demanded goes up. Price and quantity demanded always move in opposite directions, hence the price elasticity of demand is always negative.

Suppose that you own a company that supplies vending machines. Currently, your vending machines sell soft drinks at $1.50 per bottle. At that price, customers purchase 2,000 bottles per week. In order to increase sales, you decide to decrease the price to $1, and sales increase to 4,000 bottles.

To calculate the price elasticity of demand with this information, here’s what you do:

1. Plug in the values for each symbol.

Because $1.50 and 2,000 are the initial price and quantity, put $1.50 into P0 and 2,000 into Q0. And because $1.00 and 4,000 are the new price and quantity, put $1.00 into P1 and 4,000 into Q1.

2. Work out the expression on the top of the formula.

Start by dividing the expression on top of the equation. (Q1 – Q0) equals 2,000, and (Q1 + Q0) equals 6,000. Dividing 2,000 by 6,000 equals ¹⁄₃.

3. Work out the expression in the bottom of the equation.

(P1 – P0) equals –$0.50, and (P1 + P0) equals $2.50. Dividing –$0.50 by $2.50 equals –¹⁄₅.

4. Do the final division of the remaining values on the top and bottom of the equation.

Divide the top result, ¹⁄₃, by the bottom result, –¹⁄₅, to get the price elasticity of demand of –⁵⁄₃ (or –1.67).

So the price elasticity of demand for soft drinks equals

The price elasticity of demand is simply a number; it is not a monetary value. What the number tells you is a 1 percent decrease in price causes a 1.67 percent increase in quantity demanded. In other words, quantity demanded’s percentage increase is greater than the percentage decrease in price. Thus, when you decrease the price of soft drinks, you will sell a lot more soft drinks, and your revenue will go up (from $3,000 to $4,000).

Whenever the absolute value of demand is greater than one, price decreases will increase revenue.

Recognizing degrees of flexibility with inelastic or elastic

Economists use the words inelastic and elastic to describe how responsive quantity demanded is to a price change. When demand is inelastic, quantity demanded changes very little when price changes. When the price elasticity of demand is between 0 and –1, demand is inelastic. In an extreme case, if demand is perfectly inelastic (the price elasticity of demand equals 0), quantity demanded doesn’t change at all when price changes. This situation results in a vertical demand curve. Life-saving medicines may have a perfectly inelastic demand over large price ranges. In other words, even if the price of the medicine doubled, you’d still buy the same amount in order to save your life.

If quantity demanded changes a lot when price changes, demand is considered elastic, and the price elasticity of demand will be a negative number larger than -1. In an extreme situation, demand is perfectly elastic when the slightest change in price causes an incredibly large change in quantity demanded. Farmers often face a perfectly elastic demand. A farmer who tries to sell wheat at a price one cent higher than the market price won’t sell any wheat. Buyers will go to any one of a million other wheat farmers and buy the wheat for a penny less. (And you know what Ben Franklin said: “A penny saved is a penny earned.”) When that farmer lowers price by one cent to the market price, the farmer can sell all the wheat that is grown; a one cent decrease in price leads to an incredibly large change in quantity demanded.

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As a manager, you hope the demand for your good is inelastic. You know that if you raise price, customers will buy less. But you hope they only buy a little less.

Influencing the price elasticity of demand

Four primary factors influence customers’ reactions to price changes:

Substitutability: The first factor affecting customer response to price increases is the number and closeness of substitute goods — what economists call substitutability. If you know of a small number of not-so-good substitutes available for your good, then the demand for your good is inelastic. Customers will find it difficult to switch to another good if you increase price. However, if a large number of close substitutes exist with a high degree of substitutability, when you increase price, your customers find it very easy to switch. As a result, the quantity demanded for your good goes down a lot with an elastic demand.

Luxury or necessity: Whether your good is a luxury or necessity also influences how customers respond to price changes. If your good is a necessity, when you increase price, customers still have to have the good; their quantity demanded can’t be very responsive to the price change, and demand is inelastic. If the good is a luxury, customers can do without, and quantity demanded will be very responsive to the price change. Demand for luxuries is elastic. (For more on this topic, see the section “Identifying necessities and luxuries,” later in this chapter.)

Proportion of income spent: Customers also respond to how much income they spend on a good. When you buy a new car, you tend to shop around. A 10 percent difference in price means a lot. For a $30,000 car, it means $3,000. In this case, your demand is elastic; you’re very responsive to price differences. But a 10 percent difference in the price of a pizza might mean spending $11 rather than $10. This percentage difference is the same as for a car, but you’re not likely to drive all over town trying to save a dollar. In this case, because pizza takes less of your income, you’re not as responsive to the 10 percent price difference as with the car.

Time: The longer the period of time since the price change, the easier it is to adjust your spending. When the price of gasoline increases, you can’t do very much right away. The day after the price increases, you still have to get to work from where you live. Your demand is inelastic. But over a longer period of time, you may be able to join a car pool or move closer to work or find a new job. Thus, over a longer period of time, adjusting becomes easier, and your demand becomes more elastic.

These four factors determine whether customers respond to higher prices by purchasing a lot less, in other words, demand is elastic, or a little less, demand is inelastic.

Think about a local store trying to decide whether or not to raise the prices on the high-definition televisions it sells. How will customers respond? If a large number of other stores exist in the area, a high degree of substitutability occurs. Most customers are likely to think buying a new television is a luxury. Televisions are a big-ticket item; they tend to take a large proportion of your income. Finally, televisions are used over a very long period of time. You can wait for a sale before buying one. Each of these factors indicates that the demand for high-definition televisions will be elastic. Customers will buy a lot fewer televisions at the higher price.

Identifying the bottom line, almost: The price elasticity of demand and revenue

Total revenue equals the good’s price multiplied by the quantity sold. Because the price elasticity of demand shows the relationship between price and quantity sold, the elasticity number captures all the information you need to anticipate changes in total revenue.

If demand is inelastic (the price elasticity of demand is between 0 and –1), the quantity sold does not change very much when price changes. As a result, a higher price causes a very small decrease in the quantity sold and total revenue increases. (The higher price you receive for the goods you sell more than offsets the slightly smaller number you sell.) On the other hand, charging a lower price does not cause much of an increase in quantity demanded; total revenue decreases.

As these situations illustrate, when demand is inelastic, price and total revenue change in the same direction; they both increase or decrease together.

For an elastic demand (the price elasticity of demand is bigger than –1), the opposite situation occurs; price and total revenue move in opposite directions. If the good’s price increases, quantity demanded decreases a lot and total revenue decreases. (The higher price you receive isn’t enough to offset the very large decrease in the amount of goods you sell.) If you decrease the good’s price, a large increase occurs in quantity demanded, and total revenue increases.

Thus, when demand is elastic, price and total revenue change in opposite directions.

Say that your vending machine company initially charges a $1.50 per bottle and sells 2,000 bottles per week. Decreasing the price to $1 increases sales to 4,000 bottles. The price elasticity of demand equals –1.67; demand is elastic. The price decrease increases total revenue from $3,000 to $4,000 because the $0.50 decrease in price is more than offset by selling 2,000 more bottles. Thus, price and total revenue move in opposite directions given the elastic demand.

Another useful concept is marginal revenue, the change in total revenue that occurs when one additional unit of a good is sold. The formula describing the relationship between marginal revenue, MR, and the price elasticity of demand, η, is

Use this formula with the point price elasticity of demand. (For information on how to calculate the point price elasticity of demand, see the section “Calculating Elasticity with Calculus (If You Must),” later in this chapter.)

If your good is currently selling at price P, and you know the point price elasticity of demand η, you can quickly determine how much your revenue changes if you lower price to sell one additional unit of the good.

Assume that your company charges a $1.25 per bottle of soft drink and the point price elasticity of demand is –⁵⁄₃. To determine marginal revenue (the amount of revenue you add by selling one more bottle):

1. Insert $1.25 for P and -⁵⁄₃ for η.

2. Calculate the value in the parentheses.

equals

or ²⁄₅.

3. Multiply $1.25 by ²⁄₅.

The marginal revenue equals $0.50.

So the marginal revenue received when an additional bottle is sold is

Measuring the Income Elasticity of Demand

Elasticity measures consumer or customer flexibility — how responsive they are to changes in various factors. More precisely, elasticity measures the change in the quantity purchased given changes in other things. The most important of these relationships is the one between the good’s price and the quantity demanded. But the beauty of elasticity is you can determine the relationship between demand or quantity purchased, and other factors in exactly the same way!

The income elasticity of demand measures the responsiveness of a good’s demand to changes in income, just like the name suggests. So, one of the useful hints to note with elasticity is the term tells you what relationship you’re examining. And although businesses can’t control the general income level, it can have a strong effect on demand. For example, restaurants typically experience a decrease in demand during a recession. In recessions, incomes drop, and people eat out less. The restaurant owner can’t change the income level or end the recession, but the owner must make adjustments based on how the recession affects the number of customers eating at the restaurant.

Determining the income elasticity of demand: Yet another formula friend

Calculating the income elasticity of demand is essentially the same as calculating the price elasticity of demand, except you’re now determining how much the quantity purchase changes in response to a change in income.

The formula used to calculate the income elasticity of demand is

The symbol ηI represents the income elasticity of demand; η is the general symbol used for elasticity, and the subscript I represents income. In the formula, the symbol Q0 represents the initial demand or quantity purchased that exists when income equals I0. The symbol Q1 represents the new demand that exists when income changes to I1.

In this formula, the income elasticity of demand can be a positive or negative number, and it makes a real difference which it is. If the income elasticity of demand is negative, then the commodity is an inferior good. An inferior good is one whose demand decreases as incomes increase or demand increases as incomes decrease. (As an example, rice and potatoes are inferior goods.) In other words, an inverse relationship exists between demand and income, and the income elasticity of demand is negative. This relationship is unusual.

The opposite situation is a normal good — normal because you get the expected or normal relationship. For a normal good, as income increases, the good’s demand increases. That’s what you expect, and most goods are normal. As your income increases, your demand for movie tickets, restaurant meals, cars, and maybe even asparagus increases. And the opposite will happen if your income decreases. Therefore, normal goods have a direct relationship between income and demand, and the income elasticity of demand is positive.

Finally, the larger the number (either positive or negative) for the income elasticity of demand, the more responsive demand is to a change in income. A large number for the income elasticity of demand means a large change in demand occurs when income changes.

Say that you own a company that supplies vending machines. Currently, your vending machines sell soft drinks at $1.50 per bottle, and at that price, customers purchase 2,000 bottles per week. For your community, the weekly income is $600. Then a major employer in the community closes, and a number of workers lose their job. The average weekly income in the community falls to $400 and you note that your vending machine sales decrease to 500. You didn’t change the price of soft drinks, but your sales decreased dramatically due to the change in income. The income elasticity of demand will tell you how responsive soft drink sales are to the change in income.

The method for calculating the income elasticity of demand is similar to the method used to calculate any elasticity. Here’s what you do:

1. Because $600 and 2,000 are the initial income and quantity, put $600 into I0 and 2,000 into Q0.

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2. Because $400 and 500 are the new income and quantity, put $400 into I1 and 500 into Q1.

3. Start by dividing the expression on top of the equation.

(Q1 – Q0) equals –1,500, and (Q1 + Q0) equals 2,500. Dividing –1,500 by 2,500 equals –³⁄₅.

4. Divide the expression in the bottom of the equation.

(I1 – I0) equals –$200, and (I1 + I0) equals $1,000. Dividing –$200 by $1,000 equals –¹⁄₅.

5. Divide the top result, –³⁄₅, by the bottom result, –¹⁄₅.

You get the income elasticity of demand 3.

So the income elasticity of demand for soft drinks equals

The income elasticity value tells you a 1 percent decrease in income causes a 3 percent decrease in demand. Thus, because you sell fewer soft drinks when incomes decrease, soft drinks are a normal good.

Identifying necessities and luxuries

Inferior goods have an inverse relationship between income and demand, and normal goods have a direct relationship between income and demand (see preceding section). But you can subdivide normal goods into two groups:

Necessities are goods you have to have. So, even if your income decreases, you will still purchase nearly the same amount, perhaps just a few less. As a result, the income elasticity of demand for necessities will be between 0 and 1.

Luxuries are things you like, but you can do without. Thus, if your income decreases, you’ll purchase a lot fewer luxuries. For example, you probably won’t eat out so often. In the case of luxuries, the income elasticity of demand is greater than 1. The larger the value is, the greater the luxury.

Finally, a negative income elasticity of demand means the commodity is an inferior good.

Looking at the Cross-Price Elasticity of Demand

The cross-price elasticity of demand measures the responsiveness of a good’s demand — say, the infamous brand x — to changes in the price of a second good, brand y. This relationship is crucial because the amount of your good customers purchase is influenced by the prices rival firms charge for similar or substitute goods. Also, the price you charge for one good — hamburgers, for example — influences the amount you sell of a second good, french fries.

Determining the cross-price elasticity of demand: Never enough friends or formulas

Calculating the cross-price elasticity of demand requires determining how good x’s demand changes in response to a different price for good y.

The cross-price elasticity of demand’s formula is

Note how similar this formula is to other elasticity formulas. In this case, the symbol ηx,y represents cross-price elasticity of demand. The x represents the good whose quantity is changing, and the y represents the good whose price is changing. So, in the formula, the symbol Qx0 represents the initial demand or quantity purchased for good x when the price of good y is Py0. The symbol Qx1 represents good x’s new demand when good y’s price changes to Py1.

As with all elasticity values, the larger the number (either positive or negative), the more flexible or responsive quantity is. For the cross-price elasticity of demand, a larger number indicates good x’s demand will change a lot when good y’s price changes.

Your vending machine company currently sells soft drinks at $1.50 per bottle, and at that price, customers purchase 2,000 bottles per week. At the same time, a local convenience store sells the same soft drinks for $1.25 per bottle. The convenience store decides to run a special and lowers the price of soft drinks to $1.00 per bottle. As a result, your sales decrease to 1,800 bottles per week. You didn’t change the vending machine price for soft drinks, but your sales decreased due to the convenience store’s sale. The cross-price elasticity of demand will tell you how responsive your vending machine soft drink sales are to the change in price at the convenience store.

Here’s what you do to determine how much the convenience store’s sale affects your demand:

1. Because $1.25 is the initial price of soft drinks at the convenience store (good y), and 2,000 is quantity of soft drinks sold in vending machines (good x), put $1.25 into Py0 and 2,000 into Qx0.

2. Because $1.00 and 1,800 are the new price for good y (convenience stores) and quantity for good x (vending machines), put $1.00 into Py1 and 1,800 into Qx1.

3. Divide the expression on top of the equation.

(Qx1 – Qx0) equals –200, and (Q1 + Q0) equals 3,800. Dividing –200 by 3,800 equals –¹⁄₁₉.

4. Divide the expression in the bottom of the equation.

(Py1 – Py0) equals –$0.25, and (Py1 + Py0) equals $2.25. Dividing –$0.25 by $2.25 equals –¹⁄₉.

5. Divide the top result, –¹⁄₁₉, by the bottom result, –¹⁄₉.

You get the cross-price elasticity of demand ⁹⁄₁₉ or 0.474.

So the cross-price elasticity of demand for soft drinks equals

The cross-price elasticity of demand tells you a 1 percent decrease in the price of good y, the convenience store soft drink, causes a 0.474 percent decrease in demand for soft drinks from vending machines. Vending machine sales are not affected very much by changes in convenience store prices.

Identifying substitutes and complements

Substitutes are goods that are used interchangeably — one is used in the place of another. Think of potato chips and pretzels. Thus, an increase in the price of one good, good y, causes an increase in the quantity consumed of the second good, good x. This change occurs because customers will tend to switch to the lower priced good. So, an increase in the price of potato chips, good y, means customers will switch and purchase more of good x, pretzels. Thus, a direct relationship exists between the price of good y and the demand for good x, and they are substitutes.

Complements are goods that are used together, such as coffee and cream. For complements, an inverse relationship exists between good y’s price and good x’s demand; if good y’s price increases, the demand for good x decreases and vice versa. So, if the price of coffee increases, good y, you drink less coffee, and your demand for cream, good x, decreases.

Finally, the larger the value, either positive or negative, for the cross-price elasticity of demand, the stronger the relationship between the two goods.

Finishing Up with the Advertising Elasticity of Demand

The advertising elasticity of demand measures the responsiveness of a good’s demand to changes in spending on advertising. The advertising elasticity of demand measures the percentage change in demand that occurs given a 1 percent change in advertising expenditure.

The advertising elasticity of demand is calculated using the following formula:

The symbol ηA represents the advertising elasticity of demand. In the formula, the symbol Q0 represents the initial demand or quantity purchased that exists when spending on advertising equals A0. The symbol Q1 represents the new demand that exists when advertising expenditures change to A1.

The advertising elasticity of demand should be positive. (A negative value would indicate the more you spend on advertising, the lower your sales. That is a really bad ad! You should probably fire whomever is in charge of advertising.)

As with all elasticity values, the larger the number, the more responsive the good’s demand is to a change in advertising.

Your vending machine company starts a new ad campaign, “Vend for Yourself.” (Okay, I’ll stick to economics rather than advertising.) Currently, your company sells soft drinks at $1.50 per bottle, and at that price, customers purchase 2,000 bottles per week. Initially, you spend $400 per week on advertising. After a month, you’re spending $500 per week on advertising and, without changing the price of soft drinks, sales have increased to 3,000 bottles per week. The advertising elasticity of demand tells you how responsive your vending machine soft drink sales are to the change in advertising expenditures.

To determine the advertising elasticity of demand, follow the customary steps:

1. Because $400 and 2,000 are the initial advertising expenditures and quantity sold, put $400 into A0 and 2,000 into Q0.

2. Because $500 and 3,000 are the new spending on advertising and sales, put $500 into A1 and 3,000 into Q1.

3. Divide the expression on top of the equation.

(Q1 – Q0) equals 1,000 and (Q1 + Q0) equals 5,000. Dividing 1,000 by 5,000 equals ¹⁄₅.

4. Divide the expression in the bottom of the equation.

(A1 – A0) equals $100, and (A1 + A0) equals $900. Dividing $100 by $900 equals ¹⁄₉.

5. Divide the top result, ¹⁄₅, by the bottom result, ¹⁄₉.

You get the advertising elasticity of demand equal to ⁹⁄₅ or 1.8. Thus, the advertising elasticity of demand for soft drinks equals

You can conclude that a 1 percent increase in advertising expenditures increases demand 1.8 percent.

Calculating Elasticity with Calculus (If You Must)

The formulas throughout this chapter determine average elasticities for a range of values. For example, in the section “Determining the price elasticity of demand: Formulas are your friend,” you calculate the price elasticity of demand for the range of values between P0 and P1. Similarly, you calculate the income elasticity of demand for the income range between I0 and I1 in the section “Determining the income elasticity of demand: Yet another formula friend.” However, sometimes you need a more precise elasticity value. In these cases, you need to determine what is called the point elasticity, and calculus comes to your rescue.

The most important point elasticity is the point price elasticity of demand. This value is used to calculate marginal revenue, one of the two critical components in profit maximization. (The other critical component is marginal cost, which I introduce in Chapter 8.) As you can see in Chapter 9, profits are always maximized when marginal revenue equals marginal cost.

The formula to determine the point price elasticity of demand is

In this formula, ∂Q/∂ P is the partial derivative of the quantity demanded taken with respect to the good’s price, P0 is a specific price for the good, and Q0 is the quantity demanded associated with the price P0.

The following equation represents soft drink demand for your company’s vending machines:

In the equation, Q represents the number of soft drinks sold weekly, P is the price per bottle from the vending machine in dollars, I is weekly income in dollars, PC is the price at a convenience in dollars, and A is weekly advertising expenditures in dollars. Assume initially that P is $1.50, I is $600, PC is $1.25, and A is $400. Substituting those values into the demand equation indicates that 2,000 bottles will be sold weekly.

To determine the point price elasticity of demand given P0 is $1.50 and Q0 is 2,000, you need to take the following steps:

1. Take the partial derivative of Q with respect to P, ∂ Q/∂ P.

For your demand equation, this equals –4,000.

2. Determine P0 divided by Q0.

Because P is $1.50, and Q is 2,000, P0/Q0 equals 0.00075.

3. Multiply the partial derivative, –4,000, by P0/Q0, 0.00075.

The point price elasticity of demand equals –3.

Therefore, at this point on the demand curve, a 1 percent change in price causes a 3 percent change in quantity demanded in the opposite direction (because of the negative sign).

In order to maximize profits, you need to know how much each additional unit you sell adds to your revenue, or in other words, you need to know marginal revenue. If you know the point price elasticity of demand, η, the following formula can enable you to quickly determine marginal revenue, MR, for any given price

Assume your company charges a $1.50 per bottle of soft drink, and the point price elasticity of demand is –3. To determine how much revenue you add by selling an additional bottle:

1. Determine (1 + 1/η).

Substituting –3 for η gives (1 + 1/[–3]) or (1 – ¹⁄₃) or ²⁄₃.

2. Multiply the price, $1.50, by ²⁄₃.

The marginal revenue equals $1.00.

So the marginal revenue received when an additional bottle is sold is

If your cost of providing the extra bottle is less than $1.00, you will increase your profits by selling it.

Similarly, you can calculate point elasticities for the income elasticity of demand, cross-price elasticity of demand, and advertising elasticity of demand using the following formulas:

The point income elasticity of demand:

In this formula, ∂Q/∂I is the partial derivative of the quantity taken with respect to income, I is the specific income level, and Q is the quantity purchased at the income level I.

The point cross-price elasticity of demand:

In this formula, ∂Qx/∂Py is the partial derivative of good x’s quantity taken with respect to good y’s price, Py is a specific price for good y, and Qx is the quantity of good x purchased given the price Py.

The point advertising elasticity of demand:

In this formula, ∂Q/∂A is the partial derivative of the quantity demanded taken with respect to advertising expenditures, A is the specific amount spent on advertising, and Q is the quantity purchased.