"Elasticity of Demand "

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If we take the long view, meanwhile, we would expect to see demand for autos rise more than demand for food as a country grows. This is why you hear so much about the growth in auto demand in the world's poorer countries that are growing rapidly - why environmentalists are terrified about the prospect of hundreds of millions on Chinese turning in their bicycles for cars as income in the nation continues to grow rapidly.

As important as income elasticity may be, attention is usually focused on the own-price elasticity of demand, or in its shortened version, the price elasticity of demand, or the even shorter version, elasticity of demand. The price elasticity of demand is a measure of how demand responds to price changes and it would be calculated by dividing the percentage change in demand by the percentage change in price. As we will see later, the size of the resulting number will matter when we look at the relationship between revenue and price. If demand is unresponsive, we would talk about inelastic demand while a responsive demand would be referred to as elastic demand.

The formulas for demand and supply elasticities are:

Income elasticity of demand
- definition: responsiveness of demand to income change
- formula: e_y = %DQ/%DY
Cross-price elasticity of demand
- definition: responsiveness of demand (good A) to change in price of another good (price of B)
formula: e_c = %DQ_A/%DP_B _B
Own-Price elasticity of demand
- definition: responsiveness of demand to price change
- formula: e_p = %DQ/%DP
- possibilities
  - inelastic: unresponsive = | e_p| < 1
  - elastic: responsive | e_p| > 1
  - unitary elastic: | e_p| = 1
Own-Price elasticity of supply
- definition: responsiveness of supply to price change
- formula: e_ps = %DQ_s/%DP

Now that you have looked at the formulas, what are you to do with them? A suggestion is to think of the formulas as an equation with three unknowns where you are given two of the unknowns and you must solve for the third. As an example, consider the following question which is very similar to review question number 5.

If the price elasticity of demand is -2 and you wanted the output sold by a company to increases 10 percent, what would you suggest as a price strategy?

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You need to start by recognizing we are talking about the relationship between price and quantity demanded - we are talking about own-price elasticity. The formula is: e_p = %DQ/%DP. In this problem we have the figure for elasticity (2) and for the desired change in Q (6). If you now plug this into the equation you get:

e_p = %DQ/%DP

-2 = 10/(%DP )

where you need to solve for %DP. With the help of some algebra you can get %DP = 10/-2 = -5. If you are to increase demand by 6 percent, you will need to decrease the price by 5 percent.

To try one more, assume that in the last recession you observed that when income went down by 6 percent, demand went down by 3 percent. What is the income elasticity of demand?

You need to start by recognizing that we are talking about the relationship between income and quantity demanded - we are talking about income elasticity. The formula is: e_y = %DQ/%DY . In this problem we have the figure for percentage change in income (-6) and for the percentage change in demand (-3). If you now plug this into the equation you get:

e_y = %DQ/%DY

e_y = -6/-3 = 2

where you need to solve for e_y. The solution is that the income elasticity is 2.

Graphical Representation

Slope is not the same as elasticity, but it does give us some insight into the concept. To see the nature of the relationship let's compare the two demand curves that intersect at one point in the diagram below. Put your pen at the point where the two demand curves intersect and then move it up to signify an increase in price. At the higher price there will be less demand - but how much does demand fall? If you now move your pen to the left your pen hits the green line (demand B) first. If you keep moving further to the left your pen finally comes across the blue line (demand A). The change in demand caused by the price increase was greater when the curve was flatter so we would say that demand curve A is more elastic.

Slope and Responsiveness

wpe2.jpg (9614 bytes)

What generalizations can we make? All other things equal, an increase in price elasticity of demand will show up as a flattening of the demand curve. In the left-side diagram below, a small increase in price will produce a large decrease in demand. For this reason if you thought demand was responsive to price changes (elastic), then you would draw a flat demand curve.

If on the other hand you thought demand was unresponsive to price (inelastic), if demand was inelastic, then you would want to draw a steep demand curve. When demand is inelastic, a large increase in price will produce only a small decrease in demand (right-side diagram).

The story is similar when we talk about price elasticity of supply. If quantity supplied is responsive to price changes, if supply is elastic, then the supply curve would be flat. An inelastic supply curve, meanwhile, would be represented by a steep supply curve.

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Determinants of Price Elasticity of Demand

Given the significance of price elasticity of demand, it is important to know how responsive demand is to price changes. For our gas station operator considering a price increase, it would matter a good deal if the elasticity was -.5 rather than -2. As we will see in the section on revenue, if the elasticity were -.5, then the price hike would raise total revenue while if it were -2, the price hike would lower revenues. In one case the strategy is a win - in the other it is a loss.

The bad news is that it is not easy to estimate elasticity of demand. The good news is that there are some rules of thumb that we can use to get a handle on elasticity.

Availability of substitutes affects price elasticity of demand. In general we will find demand becomes more responsive as the number of substitutes increases. If we were to look at the demand for Hood milk we would expect it to be elastic since there are many substitutes for Hood milk. When we look at the demand for milk, however, we may expect to find demand to be less responsive since there are not many very close substitutes to milk - unless you see soda or juice as a good substitute.

This would be where we would see the influence of loyalty. When a firm successfully creates brand loyalty what it is doing is effectively reducing the availability of substitutes in the minds of buyers. Because the buyers no longer believe the other products are close substitutes, the increase in price is less likely to lower demand for the product.

Size matters. Price elasticity of demand is also likely to be smaller when your total expenditure on the item is small - when it is a small ticket item. For example, if the price of paper clips rose substantially it is unlikely that you would reduce your demand since total expenditures on paper clips is still very small and the change would have virtually no impact on your total budget.

There is also the question of durability. If you have an item that is durable, you would expect demand to be more responsive to price changes. When an item is durable you will not need to buy it if the price rises because you can put off your purchase and still receive the benefits from your previous purchase. If the price of garden tools rose, you could use your old tools because they are durable, but if the price of milk rose, it is unlikely that you will be able to get by with using milk that you bought last month.

Time matters. If you give people longer to adjust their behavior, they will be able to make larger adjustments. For example, if the price of gas rises rapidly then your initial reaction is likely to be minimal. You will still buy the same amount of gas because your driving habits will not have changed. If you are given a longer period to adjust, however, you may find that you buy a new, more gas efficient car which will lower your demand for gas. The generalization to be drawn from this is that elasticity increases with the time horizon.

Here is where habit enters the picture. Because habits are slow to break, a change in price is not likely to have a large immediate impact on demand. If you give people a chance to adjust, however, they may very well be able to 'break' their habits and thus we could expect a larger response to the price change in the long-run.

When we switch to supply, we can expect that time horizon will have a significant effect on the price elasticity of supply. For example, if the price of rentals increased in the Kingston area in September as URI students returned to school, you would expect to see little increase in the supply of rental housing since the housing stock is fixed in the short-term. By next Fall, however, if landlords believed that the price increases were going to last and they could make money on new construction, then the supply of housing would tend to increase as a result of the new construction. The same would be true if the price of gasoline rose rapidly and consumers drove up the price of small, gas efficient cars. In the short-term the auto companies could not shift assembly lines to produce the small cars so supply would tend to be inelastic. Once there was enough time to shift the assembly lines, the supply increase associated with the price increase would be greater and supply elasticity would be higher.

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availability of substitutes
- more substitutes - more responsive
importance
- smaller expense - less responsive
durability
- more durable - more responsive
time
- more time to adjust - more responsive

Now that you have the determinants of price elasticity of demand, let's look at a problem similar to question 4 in the review quiz.

For which of the following products do you expect to see the lowest price elasticity of demand?

socks
homes
leather jackets
automobiles

To answer this we need to return to the discussion of the determinants of demand elasticity. Automobiles and homes would, all other things equal, tend to have higher elasticities because they are big ticket items. If the price went up it would have a potentially big impact on your expenses so you might be 'persuaded' by the price increase to cut back on your purchases. The same would be true of leather jackets. Demand for homes and autos would also tend to be more responsive because they are both durable, and maybe the same could be said of leather jackets. This would tend to make demand for these items more responsive to price changes because you could continue to use what you have if the price went up - you could forestall consumption. The winner in terms of inelasticity would be socks, a small ticket item on which people probably do not even check the price.

Relationship Between Revenue and Elasticity

If you return to the opening questions, many of them had to do with the link between elasticity and revenue. Why would McDonald's lower the price of a Big Mac to raise revenues? Why would some politicians argue for lower tax rates as a way to increase tax revenue? Why would Massachusetts' attempt to raise tax revenues by raising the sin taxes on booze and butts? Why would Brazil, one of the world's largest producers of coffee, burn some of their coffee harvest as a way of increasing the value of coffee exports? Why would the OPEC countries lower production if their goal was greater income? Why does farm income fall in years of a good harvest? In each case the focus was on this link.

The relationship between elasticity and revenue can be described three ways - graphically, algebraically, and verbally. We will begin with the verbal approach and then shift to the graphical before we close with the algebra.

The words

To understand the relationship between revenue and elasticity, let's return to our gas station operator who is considering raising the price of gas and the Brazilian officials considering destroying some of the coffee crop. If demand for gasoline is inelastic, then the price increase will have little effect on demand. In this situation the price increase would put upward pressure on revenue, while the small decrease in quantity would put only limited downward pressure on revenues. Given that revenue = price times quantity ( R = P * Q), the net effect would likely be an increase in revenue since the increase in price (Ý P) will not be offset completely by a decrease in quantity (ßQ). They will be selling slightly less gas at a higher price so that total sales revenue will increase.

In the coffee example, what could we expect when Brazilian officials reduce the supply of coffee. Coffee drinkers seem to need their coffee and they can be expected to pay whatever they need to pay to get their coffee fix. In this situation the reduction in supply will lead to a substantial increase in price as the demanders compete for the smaller supply. The net effect on revenue will be positive with the increase in price (Ý P) more than compensating for the decreased quantity (ßQ). This is precisely what we will find out with the graphs and the algebra.

The graphics

Let's assume there is an increase in supply - the supply curve shifting to the right. Total revenue is by definition equal to the price times the quantity sold (P*Q). If you sell 20 units at a price of $5, total revenue will be $100 [$5*20]. In the diagrams below the initial situation is described by the black supply curve (inner curve). The revenue earned from selling the output is the areas A + B. After the increase in supply shifts the supply curve to the right (red line), revenue equals the area B + C. Revenue will increase as a result of the increase in supply if (area C) > (area A). In the diagrams below we see that this happens when the demand curve is flat - when demand is elastic. When demand is elastic, revenue will increase if we decrease the price or increase supply. Revenue and output move in the same direction while revenue and price move in opposite directions when demand is elastic. When demand is inelastic, revenue will decrease if we decrease the price or increase supply. Revenue and output move in opposite directions while revenue and price move in the same direction when demand is inelastic.

Elastic Demand Inelastic Demand

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The algebra

In this example you will be spared the derivation, although you should be able to derive the equations if you have some algebra skills. There are two important inputs into the algebra - the formula for the change in revenue (DR) and the formula for elasticity (e_p).

The first formula simply states that the change in revenue equals changes in price and changes in quantity, while the second defines elasticity as the ratio of percentage changes in quantity and price.

DR = DQ*P + DP*Q or %DR = %DQ + %DP

e_p = %DQ/%DP

Now you can combine these equations and derive the relationship between the change in revenue (

DR) and the change in price (DP), or quantity (DQ). These equations are specified in both change and percentage change terms.

DR = Q*(1+e_p)*DP

DR = P*(1+1/e_p)*DQ

or in percentage terms

%DR = (1+e_p)*%DP

%DR =(1+1/e_p)*%DQ

Now that you have made it through the tough part, we can interpret the results in the situation where demand is inelastic and where it is elastic by plugging in values for elasticity. You would use equations 1 and 3 if you were attempting to determine the impact of a price change on revenue while you would use equations 2 and 4 if you wanted to know about the impact of a change in output on revenue. For example, let us assume that demand elasticity is -2 [demand is elastic]. If we plug this value into equation 3, the term in parentheses becomes -1 [(1 + -2)] and we get the equation:

%DR = (-1)%DP or %DR = -%DP

With a value of the term in the parentheses at -1, we find that revenue and price change in opposite directions. When demand is elastic, revenue and price change in opposite directions. If you wanted to raise revenue and demand was elastic, then you would lower the price.

Similarly, if we plug the value -2 into equation 4 the term in the parentheses becomes .5 [(1 + 1/(-2)) = (1 - .5) = .5].

%DR = (.5)*%DQ or DR = .5*%DQ

When demand is elastic you will find revenue and output moving in the same direction. If you wanted to raise revenue and demand were elastic, then you would increase supply.

Guidelines

We can now come up with some guidelines that tell us what to do with price or output if our goal is to raise revenue. The general rules appear below.

Inelastic Demand |

Output and revenue are negatively related: to raise revenue you would lower output

Price and revenue are positively related: to raise revenue you would raise price

Elastic Demand | e_p| > 1

Output and revenue are positively related: to raise revenue you would raise output

Price and revenue are negatively related: to raise revenue you would lower price

e_p| < 1

To test your self on the relationship between elasticity and revenue, let's look at a question similar to question # 3 in the Review Quiz. The OPEC countries once controlled the supply of oil and they were meeting to decide what to do about their levels of oil production. Some wanted to raise output while others wanted to lower output. The strategy to lower output would be most effective when:

income elasticity of demand was high
cross price elasticity was low
price elasticity of demand is low
price elasticity of supply is high

Let's begin with the basics - Revenue = P*Q. The change in revenue will depend upon the changes in price and quantity. The decision to restrict output (decrease in Q) as a means to increase revenue works when we have reason to believe that revenue and output tend to move in opposite directions (Revenue increases when Quantity falls). We could attempt to answer this question using graphs, algebra, or words. Let's try the words. If we cut production, the only way that this will increase revenue is if the price rises substantially. This will happen if we are talking about a product where price does not have much of an effect on demand - a product where demand is inelastic. To convince yourself about this using the algebra you could plug -.5 into equation 2 just as you did in the equation above and you would find that %DR = (1+1/-2)*%DQ. With a little rearranging we get %DR/%DQ = (1+1/-.5) = (1-2) = -1. In this case we find that revenue (R) and quantity (Q) move in opposite directions.

Now let's look at the graph. The two options appear below. Because demand is inelastic, the curve is steep so the appropriate diagram is the one on the right. The original equilibrium is where the supply curve and demand curve intersect [price = P1 and the quantity = Q1]. Total revenue is equal to the area A + B. If the supply is increased, the supply curve shifts out, then the new equilibrium will generate revenue equal to the area B + C. If we compare the revenues you see that the decision to expand output will lower revenue when demand is inelastic. In this case, if OPEC though that demand was inelastic, the group should agree to restrict output which is exactly what they did.