# Economic Elasticity Problem

Gace814

Chapter 1

Markets

As a segue into the material of intermediate-level microeconomics, we begin with some familiar material from your introductory microeconomics class: market demand, supply, and equilibrium. We cover the same material but utilize algebra in addition to graphs. Then, we take up taxes and subsidies, topics which should also be somewhat familiar to you. Finally, we look at various elasticity concepts in greater detail than is usual in a principles-level class.

1.1 Market Demand and Supply

Consider a single product (say, the market for steel) over a specific geograph- ical area and a relatively short time period, such as a few months.

1.1.1 Plotting a market demand function

A market demand function shows how much is demanded by all potential buyers at different prices and is written generically as Qd = D(p). Here, Qd is the total quantity demanded and is the dependent variable, while the per-unit price, p, is the independent variable. An example of such a market demand function is given by the equation

Qd = 120 − 2p (1.1)

where Qd is measured in thousands of tons and p in dollars per ton. The fact that the derivative dQd/dp is negative means that this market demand

1

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p

Qd0

50

40

30

20

10

60

10080604020 120

Figure 1.1 Market demand

embodies the so-called ‘Law of Demand’: keeping all other factors fixed, as the price of a product increases, its quantity demanded decreases.1

Since an independent variable is measured along the horizontal axis and the dependent variable along the vertical, the variable p ought to be on the horizontal axis and Qd on the vertical. However, economists customarily put p on the vertical axis and Qd on the horizontal axis, thereby depicting the inverse market demand by switching the variables in equation (1.1) and writing the price as a function of the quantity demanded:

p = 60 − Q d

2 . (1.2)

This tradition follows Alfred Marshall’s classic text, Principles of Economics, which was published in 1890 and was very influential in educating genera- tions of economists worldwide over eight editions spanning 30 years. Mar- shall’s interpretation of the inverse demand is that it shows the maximum price (the dependent variable) that someone is willing to pay for a certain quantity (the independent variable). The inverse market demand given by equation (1.2) is therefore linear with a vertical intercept of 60 and slope of –0.5,2 as shown in Figure 1.1.

1Traditionally, the Latin phrase ceteris paribus (sometimes abbreviated as cet. par.) is used instead to mean “keeping all other factors fixed”.

2See section A.1 in the Mathematical appendix. The units of measurement will generally be omitted from the graphs to minimize clutter.

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QUS, QROW, Q d

p

0

50

40

20

45

60

100 20050 150 250 d d

US demand ROW demand

World demand

Figure 1.2 Aggregate demand

1.1.2 Aggregating demand functions

Suppose we are given the market demand curve for steel in the US as

QdUS = 100 − 5 3

p,

while the demand for steel in the rest of the world (ROW) is given by

QdROW = 150 − 10 3

p.

The corresponding inverse demand curves then are

p = 60 − 0.6QdUS and p = 45 − 0.3QdROW ,

shown in Figure 1.2 by the thin blue lines. For a price between $45 and $60, the only demand for steel in the world comes from the US as the ROW demands zero at such a high price. But for 0 ≤ p < 45, there is a positive demand from both the US and the ROW — for instance, at a price of $30, the US demands 50 thousand tons as does the ROW, for a total world demand of 100 thousand tons.

Then in the global market for steel, the quantity demanded by the entire world, Qd, can be graphically derived as the piecewise-linear heavy blue line shown in Figure 1.2. For 45 ≤ p ≤ 60, the world demand follows the US

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demand, but for 0 ≤ p < 45, the aggregate demand is given by the horizon- tal sum of the US and ROW demands:

Qd ≡ QdUS + QdROW = 250 − 5p.

Thus the world demand is found by aggregating the demand functions of the US and the ROW and can be written as

Qd =

{ 250 − 5p if 0 ≤ p < 45 100 − 53 p if 45 ≤ p ≤ 60,

while the corresponding inverse aggregate demand isb ·

p =

{ 60 − 0.6Qd if 0 ≤ Qd ≤ 25 50 − 0.2Qd if 25 < Qd ≤ 250.

Note that the first line of the inverse aggregate demand is the equation for the linear segment that overlaps exactly with the US demand for prices above $45, while the second line is the equation for the flatter linear segment that consists of the horizontal sum of the US and ROW demands for prices below $45. For plotting purposes, note that the vertical intercept of the flatter linear segment of the inverse aggregate demand is at 50, as given by the equation p = 50 − 0.2Qd and shown by the dashed line in Figure 1.2.

1.1.3 Plotting a market supply function

Just as in the case of a market demand, we can write a generic market supply function as Qs = S(p) where Qs is the quantity supplied by all the sellers in this market at the price p. Suppose the world supply curve is given by

Qs = 5p,

from which the inverse world supply is

p = Qs

5 .

Plotting this in Figure 1.3 along with the inverse world demand from Figure 1.2, we see that they intersect at a price of p∗ = $25 and a quantity of Q∗ = 125 thousands of tons.

Algebraically, this intersection point can be found by setting the inverseb · world demand p = 50 − 0.2Qd equal to the inverse market supply p = Qs/5 and solving for Q∗. Substituting Q∗ into either the inverse demand or the inverse supply yields the price p∗.

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Q

p

0

60

30 25

15

25075 125 150 175100

Excess demand

Excess supply

World demand

World supply

Figure 1.3 Market equilibrium

1.1.4 Market equilibrium

A market is said to be in equilibrium if there is a price, p∗, at which the quantity demanded equals the quantity supplied, i.e., if there is a p∗ such that D(p∗) = S(p∗). We refer to p∗ as the equilibrium price and Q∗ as the equilibrium quantity, where Q∗ = D(p∗) = S(p∗).

Even though the notion of market equilibrium is a static one, economists often tack on a dynamic story to drive the intuition that this is a stable equi- librium, i.e., any deviation from equilibrium will be automatically redressed by market forces to restore the price back to its equilibrium level. Suppose, for instance, that the market price is above p∗, say, at $30. At this price, there is excess supply: the 150 units supplied exceeds the quantity demanded of 100. Since the sellers have more of the product on their hands at this price than what buyers are willing to buy, this excess supply exerts downward pressure on the market price back towards the equilibrium price of $25.

Likewise, at a price that is lower than p∗, say $15, there is excess demand because the quantity demanded exceeds the quantity supplied. Here, the shortage of the product exerts upward pressure on the market price towards the equilibrium price, p∗.3

3The presumption is that this price adjustment process works smoothly and that the con- vergence to the equilibrium price happens relatively quickly.

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Q

p

0

60

25

250125

World demand

Consumer surplus

Producer surplus

World supply

Figure 1.4 Consumer and producer surplus

1.1.5 Consumer and producer surplus

In any voluntary transaction between a buyer and a seller, trade takes place at some price in between the maximum price a buyer is willing to pay and the minimum price a seller is willing to accept. The difference between a buyer’s maximum price and the actual price paid measures the buyer’s gain from making this trade and is called the individual consumer surplus. The difference between the price received by a seller and the minimum price this seller is willing to accept is an index of the seller’s gain from the sale and is called the individual producer surplus.

In Figure 1.4, the buyers who purchase steel value it somewhere between $60 and $25 per unit, as reflected by the portion of the world demand curve above p∗ = $25. Since these buyers each pay $25, their total gain from trade or consumer surplus is the blue shaded area below the world demand and above the equilibrium price of $25. Similarly, the sellers who receive $25 for each unit sold value it at somewhere between $0 and $25, as can be seen from the world supply. Therefore, the producer surplus is given by the orange shaded area below the equilibrium price and above the world supply.

In an introductory microeconomics class, the sum of consumer and pro- ducer surpluses is a measure of the gains from trade in this market and re- garded as an index of market efficiency.

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1.2 Determinants of Demand and Supply

From your introductory class, you may recall that the market demand for any product depends on several variables other than the price of that prod- uct. These include (a) the income levels of potential buyers, (b) the prices of other goods, (c) the tastes or preferences of buyers, and (d) the number of buyers. A change in any of these determinants, keeping all other factors fixed, causes a shift in the market demand curve. A rightward shift is called an increase in demand while a leftward shift is a decrease in demand.

When an increase in the income levels of buyers leads to an increase in demand — consumers buy more of the product no matter what the price level is — we say that such a good has a positive income effect. A good with a positive income effect is called a normal good. On rare occasions, the opposite may occur: a good may have a negative income effect. For instance, it is possible that consumers at low levels of income will reduce their purchase of cheap cuts of fatty red meat when their incomes increase, perhaps by buying more expensive lean cuts or switching to chicken instead. A good like this, where the demand shifts to the left when incomes rise, is called an inferior good.

The demand for a product depends on the prices of related goods: sub- stitutes and complements. An increase in the price of a substitute good would make consumers buy more of the good under consideration, thereby increasing its demand. An increase in the price of a complement is likely to cause a decrease in the demand for this product.

On the supply side, the market supply for any product primarily de- pends on (a) the prices of inputs that go into producing the good, (b) the tech- nology that underlies the production process, and (c) the number of firms. A change in any of these determinants causes an increase or decrease in sup- ply. Thus, an innovation in technology that raises the productivity of inputs, or an increase in the number of firms is likely to cause an increase in supply, i.e., a rightward shift of the supply curve. An increase in the price of an input would have the opposite impact, causing a decrease in supply or a leftward supply shift.

1.3 Market Interventions

In this section, we recap some of the material from an introductory microeco- nomics class concerning interventions in markets. These interventions take

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place at either the local, state or federal levels and are of three types: price controls, quantity controls (or quotas), and taxes or subsidies. We illustrate these for a generic product market whose inverse demand and supply curves are given by

p = 24 − Qd and p = 3 + 0.5Qs. (1.3)

Then the (unregulated) market equilibrium is (Q∗, p∗) = (14, 10), the con-b · sumer surplus is $98 and producer surplus is $49.

1.3.1 Price ceilings

A price ceiling (or price cap) is a maximum price imposed on a particular product. For instance, in the US (and many other countries as well), the price per unit of electricity used by residential customers is capped by price regu- lation. For a price ceiling to be effective or binding, this level must be below the equilibrium market price as shown by p̂ = 8 in Figure 1.5. At this price, there is excess demand, i.e., more buyers who are willing to buy than there are sellers willing to sell. Therefore, the sellers have to engage in rationing. Rationing refers to a method of deciding who among the many buyers to sell to. For example, in the case of rent-controlled housing, a landlord may decide that the apartments will be rented on a first come, first served basis.

Note that some of the demand will always remain unmet, i.e., a binding price ceiling leads to a market disequilibrium. Assuming efficient rationing,

Q

p

10 240

24

3

C

B

A

Excess demand

Demand

Supply

p = 8^

Figure 1.5 Price ceiling

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buyers are allocated the good by highest willingness to pay, which results in a consumer surplus of $110 (shown by area A) in Figure 1.5 and a producer ¶ b surplus of $25 (shown by area B). The aggregate gains from trade are now $135, the sum of areas A and B, which falls short of the gains from trade before the price ceiling by $12, the area C. Here area C shows the decrease in the gains from trade as a result of the ceiling and is called the deadweight loss of a price ceiling. A deadweight loss is an indication of market ineffi- ciency.

1.3.2 Price floors

A price floor (or price support) is a minimum price imposed on a particular product. For instance, in the US and EU countries, the price of certain agri- cultural goods cannot fall below a particular price level. For a price support to be binding, it must be set above the equilibrium price as shown by p̄ = 14 in Figure 1.5.

Q

p

0 10 24

24

CB

A Excess supplyp = 14_

3 Demand

Supply

Figure 1.6 Price floor

This also results in a market disequilibrium phenomenon, that of excess supply.4 The new consumer and producer surplus are areas A ($50) and B ($85) in Figure 1.6 and there is market inefficiency as can be inferred from the deadweight loss of area C ($12).

4With agricultural price supports, it is often the case that the government agrees to buy up the excess supply at the support price.

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1.3.3 Quotas

A quota is a maximum quantity limit imposed on a particular product, i.e., the producers collectively cannot sell more than the quantity specified by the quota. Quotas are often imposed on imported items. In Figure 1.7, it is

Q

p

0 10 18 24

24

CB

A

Demand

Original Supply

p = 14

3

Effective Supply

~

Figure 1.7 Quota

assumed that a quota of 10 units has been imposed. Effectively, the inverse supply curve for the product then becomes vertical at the quota. The new market price of p̃ = 14 is given by the intersection of the old inverse demand and the new effective inverse supply, i.e., unlike price controls, the quota results in a new equilibrium relative to the restricted supply. The restriction in the quantity available for trade after the quota prompts the market price to increase from its previous equilibrium level of $10 to $14. As in the case of price controls, quotas also lead to market inefficiency: the new consumer surplus is shown by area A and the new producer surplus by B, which falls short of the original surplus by the area C, the deadweight loss of the quota.

For a quota to be effective, the quantity limit has to be less than the orig- inal market equilibrium quantity. For instance, if the quota were set at 18 units, the effective supply would cross the market demand at a price of $12,! " so this quota would not alter the original market equilibrium.

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1.3.4 Taxes

Taxes may be either per-unit or ad valorem, and imposed on either sellers or buyers. A per-unit tax is a fixed dollar amount for each unit traded to be paid by the responsible party. An ad valorem tax is a tax on the value of a sale, such as a 10 percent sales tax. We will only consider per-unit taxes.

A per-unit tax on sellers

Given the original inverse demand and supply curves in equation (1.3), sup- pose a tax, t, of $6 per unit is imposed on sellers. Then, each seller will raise the minimum price she is willing to accept by the amount of this tax, thereby shifting the inverse supply curve up by $6 at each point as shown in Figure 1.7. In other words, the new vertical intercept of the supply after the tax increases by the amount of the tax while the slope remains unchanged:

pn = 9 + 0.5Qsn. (1.4)

Setting this equal to the original inverse demand in (1.3), the new equilib- ¶ b rium price is p∗n = 14 while the equilibrium quantity is Q∗n = 10.

Note that a $6 per-unit tax raised the equilibrium price from 10 to 14, not 16, i.e., the sellers were not successful in passing on the entire amount of the tax to consumers in terms of a higher price. The price difference of

Q

p

0 24

24

C

B

A

T

Demand

Original Supply

3

9 8

After-tax Supply

t = 6*pn = 14

po = 10*

Qn = 10* 14 = Qo*

Figure 1.8 Per-unit tax on sellers

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14 − 10 = $4 is called the incidence of the tax on buyers — each buyer has to pay $4 more than before to buy one unit of the good after the tax. Sellers earned $10 on each unit sold previously, but now they earn 14 − 6 = $8 net of taxes, i.e., $2 less than before. This $2 is the incidence of the tax on sellers. Thus, buyers bear two-thirds of the tax of $6, while sellers bear the remaining one-third, thereby illustrating a general principle: the incidence of a tax on buyers and sellers must add up to the tax.

Finally, notice that the consumer surplus after the tax is given by the area A in Figure 1.8. The producer surplus is based on the $8 sellers receive after the tax is paid, shown by area B. The green rectangle labeled T is the total tax revenue, which is the per-unit tax of $6 times the new quantity sold, Q∗n = 10. Since the original gains from trade exceed the areas A + B + T by the triangle C, the deadweight loss from a tax, there is market inefficiency.

A per-unit tax on buyers

Suppose the tax of $6 had been imposed on buyers instead of sellers. What changes? Since buyers have to pay the tax after they purchase the product, each buyer will lower her maximum price by the amount of the tax, thereby shifting the demand curve down by $6 at each point. Then the new vertical intercept of the demand after the tax decreases by the amount of the tax while the slope remains unchanged:

pn = 18 − Qdn. (1.5)

Set the new inverse demand equal to the original inverse supply in equa- tion (1.3) to obtain the new market equilibrium quantity of Q∗n = 10 (which isb · the same as when the tax was imposed on sellers) and the new equilibrium price is p∗n = 8. This price, however, does not include the tax that buyers have to pay. Inclusive of tax, buyers have to pay 8 + 6 = $14 to purchase one unit of the good, while sellers receive $8 for each unit sold. Thus the incidence of the tax on buyers is still $4 while that on sellers is still $2.

This example illustrates another general principle: whether a per-unit tax is imposed on buyers or sellers, the new equilibrium quantity is the same, as is the incidence of the tax on buyers and sellers.

1.3.5 Subsidies

A subsidy is a negative tax, i.e., instead of paying the government, the gov- ernment pays the individual buyer or seller as the case may be. Here too,

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subsidies may be per-unit or ad valorem. We will only look at per-unit sub- sidies imposed on sellers.

Q

p

0 24

24

B

A

Demand

Original Supply

3

9

After-subsidy Supplys = 6*po

*pn = 10

Qo = 10* 14 = Qn*

= 14 C

Figure 1.9 Per-unit subsidy

In Figure 1.8, the original inverse demand is po = 24 − Qdo and the origi- nal inverse supply is po = 9 + 0.5Qso which result in an equilibrium quantity ¶ b of Q∗o = 10, an equilibrium price of p∗o = $14, and total gains from trade (i.e., consumer and producer surplus) of $75. A $6 per-unit subsidy moves every point on the original supply down vertically by this amount, resulting in a new equilibrium quantity of 14 and equilibrium price of $10. Here the inci- dence of the subsidy is $4 on buyers (they pay $10 after the subsidy instead of $14) and $2 on sellers (sellers receive $10 from each unit sold plus $6 from the subsidy for a total of $16, as opposed to $14 before the subsidy).

As shown in Figure 1.9, the consumer surplus after the subsidy increases substantially to area A ($98), while the producer surplus shown by area B ($49) is also larger. However, this is not the aggregate gains from trade since this surplus of A + B ($147) does not include the cost of the subsidy to the government of $6 × 14 = $84 shown by the green dashed parallelogram. Subtracting the cost of the subsidy from A + B we obtain the new gains from trade after the subsidy to be equal to $63, which is less than the original gains from trade of $75 by $12. In other words, the aggregate gains from ¶ b trade are smaller than the original gains from trade by the triangle C which is the deadweight loss of the subsidy. Once again, there is market inefficiency.

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1.4 Elasticities

Demand elasticities measure the responsiveness of the quantity demanded to changes in different determinants of demand, such as the price of the product, income, and prices of other goods. On the supply side, the elasticity of supply for a product measures the degree to which the quantity supplied changes with its price.

1.4.1 Price elasticity of demand

The price elasticity of demand, ε,5 is defined as the percentage change in the quantity demanded when there is a percentage change in price. To make this more precise, let po denote an original price level and D(po) the corre- sponding quantity demanded, while the new price is pn and D(pn) is the new quantity demanded. Define the change in quantity demanded as ∆D = D(pn)− D(po) and the corresponding change in price as ∆p = pn − po. Then the percentage change in quantity is (∆D/D(po)) × 100 and, likewise, the percentage change in price is (∆p/po) × 100. Dividing the former by the latter and simplifying, we getb ·

ε = ∆D/D(po)

∆p/po =

∆D ∆p

· po D(po)

. (1.6)

For infinitesimally small changes in price from po, ∆D/∆p is an approxi- mation of the slope of the demand curve passing through the original data point. Writing the slope of the demand function at po as the derivative6

D′(po) and substituting it in (1.6), we get

ε = D′(po) · po

D(po) . (1.7)

Price elasticity for a linear demand

A generic linear inverse demand is written as p = a− bQd, where a indicates the vertical intercept and −b is the slope. Depicting this in Figure 1.9, the vertical intercept a is given by the length of the line segment OA, while the slope is given by the ratio −FA/FB.

We wish to calculate the price elasticity of demand at a single point, B, where the price is OF and quantity is OE. The slope of the demand function,

5Or more precisely, the own-price elasticity of demand. 6See section A.2 in the Mathematical appendix for a review of derivatives.

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QdO

p

A

F B

M

CE

ε = – 1

ε = – ∞

ε = 0

Figure 1.10 Linear demand and price elasticity

dQd/dp, is −1/b = −FB/FA. Using the formula in equation (1.7), the price elasticity at B is then

ε = − FB FA

· OF OE

. (1.8)

Since FB and OE have the same length, we obtain ε = −OF/FA. From the geometry of similar triangles, it follows that

ε = −OF FA

= − EC OE

= − BC AB

, (1.9)

where the final ratio yields some insights into the nature of price elasticities at different points along a linear demand.

First, because the length BC is either positive (for positive prices) or equals zero (when the price is zero), the corresponding price elasticity will always be negative or zero. In particular, the price elasticity at point C can be taken to be zero, so the demand may be said to be perfectly inelastic. At point A, the price elasticity approaches negative infinity, so the demand can be taken to be perfectly elastic in the limit.

Second, ε = −1 at the midpoint of the demand, M, while at B (which lies above the halfway point), |ε| > 1. When |ε| > 1, we say that the demand is elastic or responsive to price changes because the percentage change in the quantity demanded is greater than the percentage change in price in abso- lute terms. Reasoning analogously, |ε| < 1 for any point below the halfway point and the demand is said to be inelastic.

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Thus, different points on a linear demand have different price elastici- ties: from −∞ at the vertical intercept, the price elasticity shrinks in absolute value to −1 at the halfway point, to zero at the horizontal intercept.

In practice, an estimated demand function for a product might be given by a function like

Qd1 = 240 − 0.4p1 + 0.2p2 + 0.001m, (1.10)

where Qd1 is the quantity demanded of good 1, p1 the price of this good, p2 the price of good 2, and m the income level. Suppose we wish to calculate the own-price elasticity of demand for good 1 (written as ε11) when p1 = $200, p2 = $150, and m = $100, 000. For these values of the independent variables on the right hand side of equation (1.10), verify that Qd1 = 290. Thenb ·

ε11 = ∂Qd1 ∂p1

· p1 Qd1

= −0.4 · 200 290

(1.11)

which is essentially the same formula as (1.7), except the slope of the demand is now given by the partial derivative7 with respect to p1. Substituting the values, we obtain ε11 = −0.28, so the demand is inelastic.

Constant price elasticity of demand

Is it possible to have a demand function where the price elasticity of demand does not change as one moves along the demand curve? Indeed, work out the equation (1.7) using the demand function Qd = 100p−2 to verify that it has an elasticity of –2 everywhere. In general, a demand function with ab · constant price elasticity of ε will have an equation of the type

Qd = Apε, (1.12)

where A > 0. When ε = 0, we obtain Qd = A, a vertical inverse demand function that is

perfectly inelastic. When ε = −1, we obtain Qd = A/p or pQd = A, showing a constant expenditure by consumers of A dollars. Hence, any demand curve that has a price elasticity of –1 throughout must consist of quantity-price pairs such that a drop in price raises the quantity demanded by just enough to leave the total expenditure unchanged.

7See section A.5 in the Mathematical appendix for an introduction to partial derivatives.

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Total expenditure and price elasticity◦

How does a change in price affect the total expenditure by consumers on this product? The answer depends on the price elasticity of demand. To see this, let Qd = D(p) be the demand function and TE denote the total expenditure by consumers, where TE(p) = D(p) · p. Differentiate TE with respect to p using the product-of-functions rule8 to get

dTE dp

= D(p) + D′(p)p = D(p) (

1 + D′(p)p D(p)

) .

Note that D′(p)p/D(p) = ε from equation (1.7), so we obtain

dTE dp

= D(p)(1 + ε). (1.13)

An elastic demand implies that (1 + ε) is negative since |ε| > 1. Assuming that the quantity demanded at price p, D(p), is positive, it follows that the right hand side of (1.13) is then negative. Therefore, for an elastic demand, an increase in p reduces TE, while a decrease in p raises TE. Conversely, when the demand is inelastic, (1 + ε) is positive, so an increase in p increases TE, while a decrease in p reduces TE.

The rationale for this is that when the demand is inelastic, an increase in the price does not change the quantity demanded by as much in percentage terms. Since each purchase costs more than before, the consumers end up spending more. Likewise, when the demand is elastic, an increase in price decreases the quantity demanded drastically. So despite the higher price, the expenditure is lower.

This relationship between consumer expenditure and price-elasticity of demand is important for managers because it provides them with a rough rule of thumb: if a product is price-inelastic, raising its price slightly will raise consumer expenditures and result in higher firm revenue. For price- elastic demands, lowering the price slightly will raise consumer expendi- tures and hence firm revenue.

1.4.2 Other elasticities of demand

Two other elasticities of demand can be derived by replacing the price of the product, po, in equation (1.6) with a different determinant of demand, namely, income or the price of some other good.

8See section A.2 in the Mathematical appendix.

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The income elasticity of demand, η, captures the impact of a change in the income level on the quantity demanded, keeping all other determinants, including the price of the product, fixed. Consider the estimated demand function in equation (1.10). Here the income elasticity of demand for good 1, η1, is defined as

η1 = ∂Qd1 ∂m

· m Qd1

. (1.14)

The income elasticity when p1 = $200, p2 = $150 and m = $100, 000 is then

η1 = 0.001 · 100, 000

290 = 0.34.

A normal good has a positive income elasticity which means that an increase in consumers’ incomes leads to an increase in the quantity demanded. While most goods are normal goods, it is possible (though rare) that the quantity demanded decreases when income rises. Such goods are called inferior goods and have a negative income elasticity.

The cross-price elasticity of demand measures the impact of a change in the price of another good on the demand for a particular product. For instance, from equation (1.10), it is possible to define the cross-price elasticity of demand for good 1 when there is a change in the price of good 2 (written as ε12) as follows:

ε12 = ∂Qd1 ∂p2

· p2 Qd1

. (1.15)

The cross-price elasticity when p1 = $200, p2 = $150 and m = $100, 000 is then

ε12 = 0.2 · 150 290

= 0.1.

Goods are said to be substitutes if they have a positive cross-price elastic- ity; a larger magnitude denotes a stronger relationship. Similarly, goods are complements if they have a negative cross-price elasticity. If the cross-price elasticity is close to zero, the goods are essentially unrelated.

1.4.3 Price elasticity of supply

Finally, given a supply function Qs = S(p), we define the price elasticity of supply, εs, as the the degree of responsiveness of the quantity supplied to a change in the price of the product. The formula for this can be derived

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in an analogous manner as that for the price elasticity of demand derived in equation (1.7):

εs = S′(po) · po

S(po) , (1.16)

where (S(po), po) is the quantity-price point on the supply curve at which the supply elasticity is being calculated, and S′(po) denotes the slope of the supply function. Since the slope of the supply is generally positive or zero, the price elasticity of supply is also positive or zero.

Exercises

1.1. Suppose the demand and supply for milk in the European Union (EU) is given by

p = 120 − 0.7Qd and p = 3 + 0.2Qs,

where the quantity is in millions of liters and the price is in cents per liter. Assume that the EU does not import or export milk.

(a) Find the market equilibrium quantity, Q∗, and equilibrium price, p∗.

(b) Find the consumer and producer surplus at the market equilib- rium.

(c) The European farmers successfully lobby for a price floor of p̄ = 36 cents per liter. What will be the new quantity sold in the mar- ket, Q̄?

(d) Find the new consumer and producer surplus after the price floor.

(e) What is the deadweight loss from the price floor?

(f) If the EU authorities were to buy the surplus milk from farmers at the price floor of 36 cents per liter, how much would they spend in millions of euros? (Note: 100 cents = 1€)

1.2. The market for a product has inverse demand and supply functions given by

p = 120 − 0.5Qd and p = 0.5Qs,

where quantity is in thousands of units and the price is in dollars per unit.

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20 Chapter 1

(a) Find the market equilibrium quantity, Q∗, and equilibrium price, p∗.

(b) Suppose the state government levies a tax of $20 on each unit sold, imposed on the sellers. Find the new after-tax equilibrium quan- tity traded in the market, Q∗∗, and the price that consumers pay on the market, p∗∗.

(c) What is the incidence of the tax on buyers?

(d) What is the incidence of the tax on sellers?

(e) What is the tax revenue?

1.3. The world inverse demand for cotton is given by p = 150 − Qd while the inverse supplies of the US and the rest of the world are given by

p = 30 + QsU and p = 30 + Q s R,

where quantity is in thousands of tons and the price is in dollars per ton.

(a) Denote the world supply by Qs = QsU + Q s R. Calculate the world

equilibrium quantity, Q∗, and the world equilibrium price, p∗.

(b) Suppose the US government gives a $30 subsidy to US sellers for each ton sold. The rest of the world has the same inverse supply as before. Find the new world equilibrium quantity, Q∗∗, and the new world equilibrium price, p∗∗.

(c) What is the incidence of the subsidy on US sellers? How does the US subsidy impact sellers from the rest of the world?

(d) What is the total subsidy amount spent by the US government?

1.4. Steel is produced only in the US and the rest of the world (ROW). The inverse demand and supply in the US are

p = 100 − QdU and p = 20 + QsU ,

while in the ROW, they are

p = 80 − QdR and p = QsR.

All quantities are in millions of tons and all prices are in dollars per ton. Since steel is produced more cheaply in the ROW, the US imports

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it from the ROW under international trade. At any price, p, the imports of the US, QM, is the excess demand for steel given by the difference between the quantity demanded and the quantity supplied domesti- cally in the US: QM = QdU − QsU . Similarly, the exports of the ROW, QE, is the excess supply of steel given by the difference between how much they produce and how much they demand: QE = QsR − QdR.

(a) Calculate the world equilibrium price, p∗, at which the quantity exported by the ROW equals the quantity imported by the US. What is the equilibrium quantity traded, Q∗? At p∗, how many millions of tons of steel are sold in each market, in the US and the ROW?

(b) Find the consumer and producer surplus in the US at the price p∗.

(c) The US government imposes a tax of $12 per unit on the ROW’s exports. Find the new world equilibrium price, p∗∗, and new world equilibrium quantity traded, Q∗∗. What are the new quan- tities sold in each market, in the US and the ROW?

(d) What is the tax incidence on buyers and sellers in the US? What is the tax incidence on buyers and sellers in the ROW? Explain briefly.

(e) Find the new consumer and producer surplus in the US at the price p∗∗ and the tax revenue earned by the US government.

1.5. Answer the following elasticity-related questions.

(a) Given the inverse demand curve p = 20− 0.5Qd, what is the own- price elasticity of demand when the price is $15 per unit? $12 per unit?

(b) Revnol, a manufacturer of cosmetics, prices its popular pink lip- stick at $8. On the basis of test-marketing, Revnol believes that women between the ages of 18 and 20 have an own-price elas- ticity of –1.0 and that 60 percent of them are likely to purchase the product. In the age group from 21 to 25 years, the own-price elasticity is –1.2 and 50 percent of them are likely to buy.

(i) In a market with 25, 000 women aged 18 to 20, and 15, 000 aged 21 to 25, how many lipsticks can the firm expect to sell at a price of $8 per unit? Show your calculations!

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22 Chapter 1

(ii) If Revnol were to cut prices by 10 percent, approximately how many more pink lipsticks would it expect to sell? Show your calculations!

(c) On a certain product market, 500 units are demanded at a price of $15. The own-price elasticity is –1.5. What is the equation of a linear inverse demand that passes through the point (500, 15)?

(d) What is the equation of a constant-elasticity demand function that has an own-price elasticity of –2 and passes through the point (500, 10)?

(e) The demand for good x depends on its price, px, the price of good y, py, and the average income level, m. An economist estimates the demand function to be

Qdx = 720 − 1.5px − 2py+ 0.001m.

Suppose px = $200 per unit, py= $100 per unit, and m = $50, 000.

(i) What is the own-price elasticity for good x, εxx? (ii) What is the cross-price elasticity of demand for good x, εxy?

Are x and ycomplements or substitutes or unrelated goods? (iii) What is the cross-price elasticity of demand for good x, ηx? Is

good x normal or inferior?

(f) Show that any linear inverse supply that passes through the origin (i.e., an inverse supply with the functional form p = cQs with c > 0) has a price elasticity of supply equal to one. Show that any linear inverse supply curve with a positive intercept (i.e., having the functional form p = k + cQs with c, k > 0) must be elastic.

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