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18
Elasticities, Price Distorting Policies and
Non-Price Rationing
Solutions for Microeconomics: An Intuitive
Approach with Calculus (International Ed.)
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created for the companion book Microeconomics: An Intuitive Approach.
• Solutions to Within-Chapter Exercises are provided in the student Study Guide.
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Elasticities, Price Distorting Policies and Non-Price Rationing
18.1 Consider, as we did in much of the chapter, a downward sloping linear demand curve.
A: In what follows, we will consider what happens to the price elasticity of demand as we approach
the horizontal and vertical axes along the demand curve.
(a) Begin by drawing such a demand curve with constant (negative) slope. Then pick the point
A on the demand curve that lies roughly three quarters of the way down the demand curve.
Illustrate the price and quantity demanded at that point.
Answer: This is depicted in panel (a) of Graph 18.1
Graph 18.1: Price Elasticity as we Approach the Axes
(b) Next, suppose the price drops by half and illustrate the point B on the demand curve for that
lower price level. Is the percentage change in quantity from A to B greater or smaller than the
absolute value of the percentage change in price?
Answer: Since the price drops by half, we know that the absolute value of the percentage
change in price is 50%. It should be obvious from the graph that the percentage increase
in quantity from A to B is smaller than 50% — because the starting quantity at A is large
relative to the change in quantity from A to B .
(c) Next, drop the price by half again and illustrate the point C on the demand curve for that new
(lower) price. The percentage change in price from B to C is the same as it was from A to B . Is
the same true for the percentage change in quantity?
Answer: Since we again halved the price, the percentage change in price is 50%. The percentage change in quantity from B to C , however, is now smaller than it was from A to B
— because the starting quantity at B is higher than it was at A while the absolute change
from B to C is smaller. For the same percentage change in price, we therefore get smaller
percentage changes in quantity as we move down the demand curve.
(d) What do your answers imply about what is happening to the price elasticity of demand as we
move down the demand curve?
Answer: Price elasticity of demand is the percentage change in quantity over the percentage
change in price. We have held the percentage change in price constant as we moved from
A to B and then from B to C — implying the denominator of the price elasticity formula
was kept the same. But we concluded that the percentage change in quantity (for the same
percentage change in price) is less when we move from B to C than when we moved from A
to B — implying that the absolute value of price elasticity gets smaller as we move down the
demand curve.
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640
(e) Can you see what will happen to the price elasticity of demand as we get closer and closer to
the horizontal axis?
Answer: If you imagine repeating what we have done again — i.e. cutting price by half and
checking what happens to the percentage change in quantity, we will keep getting the same
result: The percentage change in quantity for the same percentage change in price will get
smaller and smaller. Thus, the numerator of the price elasticity formula gets smaller and
smaller while the denominator stays the same — implying that the fraction that represents
price elasticity gets smaller and smaller — and closer and closer to zero — as we move toward the endpoint of the linear demand curve.
(f) Next, start at a point A ′ on the demand curve that lies only a quarter of the way down the
demand curve. Illustrate the price and quantity demanded at that point. Then choose a
point B ′ that has only half the consumption level as at A ′ . Is the percentage change in price
from A ′ to B ′ greater or less than the absolute value of the percentage change in quantity?
Answer: This is illustrated in panel (b) of Graph 18.1. In moving from A ′ to B ′ , we know that
quantity drops by half — i.e. changes by 50%. But it should be obvious from the graph that
price increases by less than 50%. This is because the beginning price at A ′ is already relatively high — and the incremental change in price from A ′ to B ′ is relatively low compared
to its starting point.
(g) Now pick the point C ′ (on the demand curve) where the quantity demanded is half what it
was at B ′ . The percentage change in quantity from A ′ to B ′ is then the same as the percentage
change from B ′ to C ′ . Is the same true of the percentage change in price?
Answer: The percentage change is the same because we again dropped quantity by 50%.
But the percentage change in price is now less as we go from B ′ to C ′ than it was when we
went from A ′ to B ′ . This is because the starting price is higher at B ′ than it was at A ′ while
the incremental increase in price from B ′ to C ′ is less than it was from A ′ to B ′ . Thus, as
we move up the demand curve, the percentage change in price gets smaller and smaller (for
the same percentage decrease in quantity).
(h) What do your answers imply about the price elasticity of demand as we move up the demand
curve? What happens to the price elasticity as we keep repeating what we have done and get
closer and closer to the vertical intercept?
Answer: We have shown that, for the same percentage decrease in quantity, the percentage
increase in price gets smaller as we move up the demand curve. Price elasticity of demand is
the percentage change in quantity divided by the percentage change in price. We have held
the percentage change in quantity fixed — i.e. we have held the numerator of the price elasticity formula fixed; but we have concluded that the denominator becomes smaller as we
move up the demand curve. This implies that we are dividing the same number by smaller
and smaller numbers — which means the overall fraction is increasing in absolute value.
The price elasticity of demand therefore gets larger and larger in absolute value as we move
up the demand curve — and gets closer and closer to (negative) infinity as we get closer to
the vertical intercept.
B: Consider the linear demand curve described by the equation p = A − αx.
(a) Derive the price elasticity of demand for this demand curve.
Answer: First, we write the demand curve as a demand function by solving for x — i.e.
x(p) =
A−p
.
α
(18.1)
The price elasticity of demand is then
εd =
µ
¶
dx p
p
1
−p
=−
.
=
d p x(p)
α (A − p)/p
A−p
(18.2)
(b) Take the limit of the price elasticity of demand as price approaches zero.
Answer: As p approaches zero, the numerator goes to zero while the denominator goes to A
— implying a limit of zero. Thus, the price elasticity of demand converges to zero as price
converges to zero — i.e. as we approach the horizontal axis.
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Elasticities, Price Distorting Policies and Non-Price Rationing
(c) Take the limit of the price elasticity as price approaches A.
Answer: As p approaches A, the numerator converges to −A while the denominator approaches zero — implying that the price elasticity approaches minus infinity. Thus, as we
approach the vertical axis along the demand curve, price elasticity approaches negative infinity.
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18.2 In this exercise, we explore the concept of elasticity in contexts other than own-price elasticity of
(uncompensated) demand. (In cases where it matters, assume that there are only two goods).
A: For each of the following, indicate whether the statement is true or false and explain your answer:
(a) The income elasticity of demand for goods is negative only for Giffen goods.
Answer: This is false. The income elasticity of demand is positive for normal goods (because
the quantity demanded increases as income increases for normal goods) and negative for
inferior goods (because quantity decreases as income increases for inferior goods). Thus,
although it is true that the income elasticity of demand is negative for Giffen goods, Giffen
goods are not the only goods for which this is true. In particular, inferior goods that are not
Giffen goods (i.e. “regular inferior goods”) also have negative income elasticities of demand.
(b) If tastes are homothetic, the income elasticity of demand must be positive.
Answer: This is true — because homothetic tastes are tastes for normal goods, and all normal goods have the property that quantity demanded increases as income increases — thus
giving rise to positive income elasticity of demand.
(c) If tastes are quasilinear in x, the income elasticity of demand for x is zero.
Answer: This is also true. Quasilinear goods don’t give rise to income effects — thus, if
income goes up, quantity demanded remains unchanged. Thus, the percentage change in
quantity over the percentage change in income is zero.
(d) If tastes are quasilinear in x 1 , then the cross-price elasticity of demand for x 1 is positive.
Answer: This is true. If x 1 is quasilinear, then there are no income effects relative to x 1
and only substitution effects. If p 2 increases, this then implies that the consumer will consume less of what has become more expensive (x 2 ) and more of what has become relatively cheaper (x 1 ). Thus, the price of x 2 moves in the same direction as the consumption
quantity of x 1 — implying a positive relationship between the two, and therefore a positive
cross-price elasticity of demand for x 1 .
(e) If tastes are homothetic, cross price elasticities must be positive.
Answer: This is false. Suppose, for instance, that two goods are perfect complements —
which implies tastes are homothetic. Then as p 2 increases, consumption of x 1 falls (since
the two goods have to be consumed together). Thus, we have identified a case of homothetic tastes for which p 2 and consumption of x 1 move in opposite direction — implying a
negative cross price elasticity of demand. Of course, if the two goods are sufficiently substitutable, the reverse will be true — as p 2 increases, consumption of x 1 will increase —
implying a positive cross-price elasticity of demand. Thus, for homothetic tastes, the sign
of the cross-price elasticity of demand will depend on the degree of substitutability between
the goods.
(f) The price elasticity of compensated demand is always negative.
Answer: Compensated demand curves — or what we also called marginal willingness to
pay curves — incorporate only substitution (and not income) effects and are thus always
downward sloping. Thus, the price elasticity of compensated demand is always negative —
and the statement is true.
(g) The more substitutable two goods are for one another, the greater the price elasticity of compensated demand is in absolute value.
Answer: The more substitutable two goods are, the greater the substitution effect that is incorporated in compensated demand curves — i.e. the shallower, or more elastic, the compensated demand curves are. The statement is therefore true. Put differently, the more
substitutable goods are, the greater will be the responsiveness to price if we compensate the
consumer to reach the same indifference curve as before.
B: Consider first the demand function x = αI /p that emerges from Cobb-Douglas tastes.
(a) Derive the income elasticity of demand and explain its sign.
Answer: The elasticity is
µ
¶
dx
I
I
α
=
= 1.
d I x(p, I ) p αI /p
(18.3)
This elasticity is positive because goods represented by Cobb-Douglas tastes must be normal — implying that income and consumption move in the same direction,
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Elasticities, Price Distorting Policies and Non-Price Rationing
(b) We know Cobb-Douglas tastes are homothetic. In what way is your answer to (a) simply a
property of homothetic tastes.
Answer: Homothetic tastes are such that, as income increases by a certain percentage, consumption always increases by the same percentage. Put differently, homothetic tastes represent tastes over goods that are neither luxuries nor necessities. Since income elasticity of
demand is the percentage change in quantity over the percentage change in income, this
implies that, for homothetic tastes, the numerator and denominator of the income elasticity formula are always the same — which must mean that the income elasticity of demand
is 1.
(c) What is the cross-price elasticity of demand? Can you make sense of that?
Answer: Since the price of the other good does not appear in the demand functions for
Cobb-Douglas tastes, the cross-price elasticity is zero. This is because, for Cobb-Douglas
tastes, the income and substitution effects relative to x 1 exactly offset each other when p 2
changes.
(d) Without knowing the precise functional form that can describe tastes that are quasilinear in
x, how can you show that the income elasticity of demand must be zero?
Answer: If tastes are quasilinear in x, then the demand function for x is not a function of
income — i.e. it takes the form x 1 (p 1 , p 2 ). When we then take the derivative with respect to
I as we derive the income elasticity, we know that this derivative must be zero — and thus
the income elasticity of demand must be zero.
(e) Consider the demand function x 1 (p 1 , p 2 ) = (αp 2 /p 1 )β . Derive the income and cross-price
elasticities of demand.
Answer: The income elasticity of demand is
!
Ã
d x1 I
I
= 0.
=0
d I x1
(αp 2 /p 1 )β
(18.4)
The cross-price elasticity of demand is
!
Ã
µ
¶
α β (β−1)
p2
d x1 p2
= β.
=β
p2
d p2 x1
p1
(αp 2 /p 1 )β
(18.5)
(f) Can you tell whether the tastes giving rise to this demand function are either quasilinear or
homothetic?
Answer: Yes, you can tell they are quasilinear in x 1 because the income elasticity of demand
is zero. (Note also that the cross-price elasticity is positive as we concluded in A(d) must be
true for quasilinear tastes.)
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18.3 In the labor market, we can also talk about responsiveness — or elasticity — with respect to wages
(and other prices) on both the demand and supply sides.
A: For each of the following statements, indicate whether you think the statement is true or false (and
why):
(a) The wage elasticity of labor supply must be positive if leisure and consumption are normal
goods.
Answer: This is false. As wages change, there are off-setting substitution and wealth effects
relative to leisure when leisure is a normal good. This implies that the impact of a change in
wage on labor supply is ambiguous when leisure is normal — and the relationship between
wage and the quantity of labor supplied may be positive or negative (which further implies
that this elasticity may be positive or negative.)
(b) In end-of-chapter exercise 9.5, we indicated that labor supply curves are often “backwardbending”. In such cases, the wage elasticity of labor supply is positive at low wages and negative at high wages.
Answer: This is true. The backward bending labor supply that labor economists have identified as empirically important is upward sloping for low wages and downward sloping for
high wages — implying a positive relationship (and thus a positive wage elasticity) between
wage and quantity of labor supplied at low wages and a negative relationship at high wages.
(c) The wage elasticity of labor demand is always negative.
Answer: This is true — as we illustrated in our treatment of firms, labor demand is always
downward sloping — which implies a negative relationship between wage and the quantity
of labor demanded (and thus a negative wage elasticity of labor demand).
(d) In absolute value, the wage elasticity of labor demand is at least as large in the long run as it
is in the short run.
Answer: This is also true. In Chapter 13, we showed that labor demand curves are shallower
in the long run than in the short run — except for one special case where they have the same
slope. Thus, labor demand is more responsive — or more elastic — in the long run than in
the short run, implying a higher wage elasticity in absolute value.
(e) (The compensated labor supply curve, which we will cover more explicitly in Chapter 19, is
the labor supply curve that would emerge if we always insured you reached the same indifference curve regardless of the wage rate.) The wage elasticity of compensated labor supply must
always be negative.
Answer: This is false. Compensated labor supply curves only incorporate substitution effects (and not wealth effects). The substitution effect is unambiguous — if wage increases,
leisure becomes relatively more expensive and is thus consumed in smaller amounts (absent wealth effects). This implies that, as wage increases, the compensated labor supply also
increases — i.e. the relationship between wage and compensated labor supply is positive.
Thus, the wage elasticity of compensated labor supply is positive.
(f) The (long run) rental rate (of capital) elasticity of labor demand (which is a cross-price elasticity) is always positive.
Answer: This is false. We showed in Chapter 13 that the relationship between the rental rate
of capital and the quantity of labor demanded may be positive or negative depending on
whether labor and capital are relatively substitutable or relatively complementary in production. Thus, the elasticity may be positive or negative.
(g) The output price elasticity of labor demand is positive and increases from the short to the long
run.
Answer: This is partly true and partly false. As output price increases, firms want to produce
more and thus hire more of all inputs — including labor. Thus, the relationship between
output price and the quantity of labor demanded is positive, both in the short and long run.
But we showed in Chapter 13 that such responses may be larger or smaller in in the long run
than in the short run depending again on the degree of substitutability of labor and capital.
(It may be that the firm will hire more labor in the short run but in the long run lets some
labor go and substitutes into capital instead.)
B: Suppose first that tastes over consumption and leisure are Cobb-Douglas.
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Elasticities, Price Distorting Policies and Non-Price Rationing
(a) Derive the functional form of the labor supply function.
Answer: Define c as dollars of consumption and ℓ as hours of labor. With L denoting leisure
endowment, leisure consumption is (L − ℓ) and the utility maximization problem is
max c α (L − ℓ)(1−α) subject to wℓ = c.
c,ℓ
(18.6)
Solving this in the usual way, we get the labor supply function ℓ = αL.
(b) What is the wage elasticity of labor supply in this case? Explain how this relates to the implicit
elasticity of substitution in Cobb-Douglas tastes.
Answer: Since w does not appear in the labor supply function ℓ = αL that we derived in (a),
the wage elasticity of labor supply is zero (since d ℓ/d w = 0) — i.e. the labor supply curve is
perfectly inelastic. This is a direct result of the fact that the elasticity of substitution between
leisure and consumption is 1. We know that substitution and wealth effects point in opposite directions with respect to leisure (and thus labor supplied) — and with this elasticity of
substitution, the wealth effect (which suggests the worker will work less as wage increases)
is exactly offset by the substitution effect (which suggests the worker will work more as wage
increases). If the elasticity of substitution goes above 1, the substitution effect will dominate
the wealth effect — implying a positive wage elasticity of labor supply; and if the elasticity
of substitution is less than 1, the wealth effect outweighs the substitution effect — implying
a negative wage elasticity of labor supply.
(c) Next, suppose that the decreasing returns to scale production process takes labor and capital
as inputs and is also Cobb-Douglas. Derive the long run wage elasticity of labor demand.1
Answer: The wage elasticity of labor demand is
Ã
dℓ w
−(1 − β) p Aα(1−β) ββ
=
dw ℓ
1−α−β
rβ
=
!1/(1−α−β)
w
³
−(1−β)
−1
1−α−β
´




w


µ

¶
 p Aα(1−β) ββ 1/(1−α−β) 
w (1−β) r β
−(1 − β)
< 0.
1−α−β
(18.8)
Since the production process has decreasing returns to scale, we know that (1 − α − β) > 0
and (1 − β) > 0. Thus, the above expression is negative — i.e. the wage elasticity of labor
demand is negative.
(d) Derive the rental rate elasticity of labor demand. Is it positive or negative?
Answer: The rental rate elasticity of labor demand is
dℓ r
−β
=
dr ℓ 1− α − β
Ã
!1/(1−α−β) ³
p Aα(1−β) ββ
r
w (1−β)
−β
−1
1−α−β
´




r


µ

 p Aα(1−β) ββ ¶1/(1−α−β) 
w (1−β) r β
−β
=
< 0.
1−α−β
(18.9)
Since β > 0 and (1−α−β) > 0, the above expression is negative — i.e. the rental rate elasticity
of labor demand is negative.
1 It may be helpful to recall that, for Cobb-Douglas functions that take the form f (ℓ,k) = Aℓα k β , the
labor demand function is
ℓ(w,r, p) =
.
Ã
p Aα(1−β) ββ
w (1−β) r β
!1/(1−α−β)
(18.7)
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646
(e) Derive the long run output price elasticity of labor demand. Is it positive or negative?
Answer: The output price elasticity of labor demand is
dℓ p
1
=
dp ℓ 1−α−β
=
Ã
Aα(1−β) ββ
w (1−β) r β
!1/(1−α−β) ³
´




1
p
−1 

p 1−α−β
µ

¶
 p Aα(1−β) ββ 1/(1−α−β) 
w (1−β) r β
1
> 0.
1−α−β
(18.10)
Since (1 − α − β) > 0, this expression is positive — implying a positive output price elasticity
of labor demand.
(f) In the short run, capital is fixed. Can you derive the short run wage elasticity of labor demand
and relate it to the to long run elasticity you calculated in part (c)?
α β
Answer: If we start with the long run production
h
i function f (ℓ,k) = Aℓ k , we can write the
short run production function as f (ℓ) = Ak β ℓα or simply f (ℓ) = B ℓα where B is defined
as B = Ak β . The short run profit maximization problem is then
max pB ℓα − wℓ
(18.11)
ℓ
which solves in the usual way to give us the short run labor demand function
ℓ(p, w) =
µ
¶
αpB 1/(1−α)
.
w
(18.12)
The short run wage elasticity of labor demand is then
−1
dℓ w
=
(αpB )1/(1−α) w
dw ℓ
1−α
³
−1
1−α −1
´


³

−1
w

.
=
´
1−α
αpB 1/(1−α)
(18.13)
w
In (c), we calculated the long run elasticity as −(1 − β)/(1 − α − β) — which is greater in
absolute value than −1/(1−α). (To prove this, suppose (1−β)/(1−α −β) ≤ 1/(1−α). Crossmultiplying, this implies (1 − β)(1 − α) ≤ 1 − α − β, and multiplying through on the left hand
side, we get 1 − α − β + αβ ≤ 1 − α − β — which has to be false since αβ > 0. Thus, we have a
contradiction which proves that (1 − β)/(1 − α − β) > 1/(1 − α).)
(g) Can you derive the short run output price elasticity of labor demand and compare it to the
long run elasticity you calculated in part (e)?
Answer: Using the short run labor demand function in (18.12), we can calculate the short
run output price elasticity of labor demand as


³
´
µ
¶
1 −1
1
αB 1/(1−α) 1−α
1
p
dℓ p


p
=
.
=
³
´
dp ℓ 1−α w
1−α
αpB 1/(1−α)
(18.14)
w
In (e) we calculated the long run output price elasticity of labor demand as 1/(1 − α − β)
which is larger than the short run elasticity; i.e. labor demand responds more to output
price in the long run than in the short run.
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Elasticities, Price Distorting Policies and Non-Price Rationing
18.4 In this exercise, treat the real interest rate r as identical to the the rental rate on capital.
A: We will now consider the responsiveness — or elasticity — of savings and borrowing behavior with
respect to changes in the interest rate (and other prices). Suppose that tastes over consumption now
and in the future are homothetic, and further suppose that production frontiers (that use labor and
capital as inputs) are homothetic.
(a) Can you tell whether the interest rate elasticity of savings (or capital supply) is positive or
negative for someone who earns income now but not in the future?
Answer: When tastes over consumption now and in the future are homothetic and all income is earned now (rather than in the future), we have offsetting wealth and substitution
effects on consumption now as the interest rate changes. In particular, an increase in the
interest rate causes consumption now to become relatively more expensive — implying a
substitution effect that suggests less consumption now, and thus more savings. At the same
time, an increase in the interest rate causes real wealth to increase — implying a wealth effect that suggests more consumption now, and thus less savings. Whether the relationship
between the interest rate and savings is positive or negative then depends on the degree of
substitutability of consumption across time — and thus we cannot generally say whether
the interest rate elasticity of savings (or capital supply) for someone who earns all income
now is positive or negative.
(b) Can you tell whether the interest rate elasticity of borrowing (or capital demand) is positive
or negative for someone who earns no income now but will earn income in the future?
Answer: For someone who earns all income in the future, an increase in the interest rate
results in an increase of the relative price of consuming now as well as a decrease in real
wealth. As a result, the substitution effect tells us that an increase in the interest rate for this
person will cause him to consume less now — and thus borrow less, and the wealth effect
will similarly tell us that this person will consume less now (and in the future) — and thus
will again borrow less. Since both effects point in the same direction, there is no ambiguity
— an increase in the interest rate causes a decrease in borrowing. The interest rate elasticity
of borrowing (or capital demand) is therefore negative.
(c) Is the interest rate elasticity of demand for capital by firms positive or negative?
Answer: Input demand curves for firms always slope down — which implies that the quantity of capital demanded by firms declines as the interest rate increases. The interest rate
elasticity of capital demand by firms is therefore negative.
(d) Is the wage elasticity of demand for capital by firms positive or negative?
Answer: We showed in Chapter 13 that the direction of the demand response for capital from
a change in the wage rate depends on the degree of substitutability of capital and labor in
production. If capital and labor are highly substitutable, then an increase in the wage rate
can result in an increased demand for capital (even though the firm produces less output);
but if labor and capital are relatively complementary, then an increase in the wage rate will
result in a decrease of capital demanded. Thus, the wage elasticity of capital demand by
firms may be positive or negative.
(e) Is the output price elasticity of demand for capital positive or negative?
Answer: As output price increases, firms will produce more and hire more inputs. Thus, as
price increases, capital demanded increases — which implies that the output price elasticity
of capital demand by firms is positive.
B: Suppose that intertemporal tastes over consumption are Cobb-Douglas. Furthermore, suppose
that production technologies (which take capital and labor as inputs) have decreasing returns to
scale and are Cobb-Douglas.
(a) Suppose that your income this period is e 1 and your income in he future is e 2 . Set up your
intertemporal utility maximization problem and derive your demand for consumption c 1
now.
Answer: The utility maximization problem is
max c 1α c 2(1−α) subject to (1 + r )e 1 + e 2 = (1 + r )c 1 + c 2 .
c 1 ,c 2
(18.15)
Elasticities, Price Distorting Policies and Non-Price Rationing
648
(If the budget constraint in this problem does not make sense, review the section on intertemporal budget constraints in part B of Chapter 3.) Solving this in the usual way, we get
demand for consumption now as
α [(1 + r )e 1 + e 2 ]
.
(18.16)
(1 + r )
(b) Suppose all your income occurs now (i.e. e 2 = 0). What is your savings (or capital supply)
function, and what is the interest rate elasticity of savings?
Answer: With e 2 = 0, equation (18.16) becomes c 1 = αe 1 . Since r does not appear in this
equation, the derivative of c 1 with respect to r is zero — and the interest rate elasticity of
savings is zero; i.e. savings behavior is not responsive to interest rate changes. (This is,
of course, because substitution and wealth effects exactly offset one another in the case of
Cobb-Douglas tastes when income all falls into the current period.)
c1 =
(c) Suppose instead that all your income happens next period (i.e. e 1 = 0). What is the interest
rate elasticity of borrowing (or capital demand)?
Answer: When e 1 = 0, equation (18.16) becomes
αe 2
(18.17)
(1 + r )
which is also the equation that defines how much the individual borrows (since he has no
income now). The interest rate elasticity of borrowing is then
c1 =
d c1 r
−αe 2
=
d r c 1 (1 + r )2
Ã
r
αe 2
(1+r )
!
=
−r
< 0.
(1 + r )
(18.18)
Thus, the interest rate elasticity of borrowing (or capital demand) is negative. (This is because in this case, substitution and wealth effects unambiguously point in the same direction so long as consumption is a normal good.)
(d) Next, derive the interest rate elasticity of capital demand by firms. Is it positive or negative?2
Answer: The interest rate elasticity of capital demand is
dk r
−(1 − α)
=
dr k 1− α − β
=
Ã

!1/(1−α−β) ³
´

−(1−α)
p Aαα β(1−α)
−1 
1−α−β
r
µ
α

w
r
p Aαα β(1−α)
w α r (1−α)
−(1 − α)
< 0.
1−α−β



¶1/(1−α−β) 

(18.20)
This is negative because (1 − α) > 0 and (1 − α − β) > 0 (since the production process has
decreasing returns to scale).
(e) Repeat this for the wage elasticity of capital demand as well as the output price elasticity of
capital demand for firms.
Answer: The wage elasticity of capital demand is
Ã
dk w
−α
p Aαα β(1−α)
=
dw k
1−α−β
r (1−α)
−α
< 0.
=
1−α−β
!1/(1−α−β)
³
´



−α −1 
w


w 1−α−β

µ
¶
 p Aαα β(1−α) 1/(1−α−β) 
w α r (1−α)
(18.21)
2 It will be helpful to know that, for Cobb-Douglas functions that take the form f (ℓ,k) = Aℓα k β , the
capital demand function is
Ã
!1/(1−α−β)
p Aαα β(1−α)
k(w,r, p) =
(18.19)
.
w α r (1−α)
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Elasticities, Price Distorting Policies and Non-Price Rationing
Finally, the output price elasticity of capital demand is
1
dk p
=
dp k
1−α−β
=
Ã
Aαα β(1−α)
w α r (1−α)
1
> 0.
1−α−β
!1/(1−α−β) ³
p
1
−1
1−α−β
´




p



µ
 p Aαα β(1−α) ¶1/(1−α−β) 
w α r (1−α)
(18.22)
Elasticities, Price Distorting Policies and Non-Price Rationing
650
18.5 In our treatment of price floors, we illustrated the case of a government program that purchases any
surplus produced in the market. Now consider a price ceiling — and the analogous case of the government
addressing disequilibrium shortages through purchases on international markets.
A: Suppose, for instance, that the U.S. demand and supply curves for coffee intersect at p ∗ which is
also the world price of coffee.
(a) Suppose that the government imposes a price ceiling p c below p ∗ for domestic coffee sales.
Illustrate the disequilibrium shortage that would emerge in the domestic coffee market.
Answer: This is illustrated in panel (a) of Graph 18.2.
Graph 18.2: Coffee Price Ceiling
(b) In the absence of any further interference in the market, what would you expect to happen?
Answer: Consumers would have to compete for the quantity x s — expending effort that,
in equilibrium, will equal the bold vertical distance in panel (a) of Graph 18.2. Thus, the
effective price consumers will end up paying (including their effort cost) is p ′ .
(c) Next, suppose that, as part of the price ceiling policy, the government purchases coffee in the
world market (at the world market price p ∗ ) and then sells this coffee at p c domestically to
any consumer that is unable to purchase coffee from a domestic produce. What changes in
your analysis?
Answer: Now consumers no longer have to compete with each other for the limited amount
of coffee supplied domestically at the price ceiling — because the government supplies anything that domestic producers do not supply. Thus, consumers will actually only have to pay
the price ceiling p c (rather than p ′ ).
(d) Illustrate — in a graph with the domestic demand and supply curves for coffee — the deadweight loss from this government program (assuming that your demand curve is a good approximation of marginal willingness to pay).
Answer: This is illustrated in panel (b) of Graph 18.2. Consumers now get surplus (a + b +
c + d + e + f ) while domestic producers get surplus (h). The government has to buy the
difference between x d and x s at the world market price p ∗ — and then sells this quantity
at p c . Thus, the government loses (p ∗ − p c ) on the quantity (x d − x s ). This is equal to area
(e + f + g ). Adding up producer and consumer surplus — and subtracting the net cost of
651
Elasticities, Price Distorting Policies and Non-Price Rationing
the government program, we then get overall surplus of (a + b + c + d + h − g ). Were these
programs not in existence, overall surplus would be (a+b+c +d +e +h). Thus, we lose (e +g )
as a result of this government program — i.e. the deadweight loss is (e + g ).
B: Suppose demand and supply are given by x d = (A − p)/α and x s = (p − B )/β (and assume that
demand is equal to marginal willingness to pay).
(a) Derive the equilibrium price p ∗ that would emerge in the absence of any interference.
Answer: Setting demand equal to supply and solving for p, we get (as in the text),
p∗ =
βA + αB
.
α+β
(18.23)
(b) Suppose the government imposes a price ceiling p c that lies below p ∗ . Derive an expression
for the disequilibrium shortage.
Answer: Substituting p c into the demand and supply equations and then subtracting x s
from x d , we get
Disequilibrium Shortage = x d (p c ) − x s (p c ) =
βA + αB − (α + β)p c
.
αβ
(18.24)
(c) Suppose, as in part A, that the government can purchase any quantity of x on the world market for p ∗ and it implements the program described in A(c). How much will this program cost
the government?
Answer: The program will cost the government (p ∗ − p c ) times the disequilibrium shortage
calculated above. By replacing p ∗ with our expression from part (a), we can derive
¸
βA + αB − (α + β)p c
=
αβ
·
¸
¸·
βA + αB
βA + αB − (α + β)p c
=
− pc
=
α+β
αβ
£
¤2
βA + αB − (α + β)p c
.
=
αβ(α + β)
Cost to Government = (p ∗ − p c )
·
(18.25)
(d) What is the deadweight loss from the combination of the price ceiling and the government
program to buy coffee from abroad and sell it domestically at p c ?
Answer: From panel (b) of Graph 18.2 and our answer to A(d) we can see that the deadweight loss area (e + g ) is equal to half the cost (e + f + g ) of the government program. Thus,
deadweight loss is
DWL =
£
¤2
βA + αB − (α + β)p c
2αβ(α + β)
.
(18.26)
Elasticities, Price Distorting Policies and Non-Price Rationing
652
18.6 Everyday Application: Scalping College Basketball Tickets: At many universities, college basketball
is intensely popular and, were tickets sold at market prices, many students who wish to attend games
would not be able to afford to do so. As a result, universities have come up with non-price rationing
mechanisms to allocate basketball tickets.
A: Suppose throughout this exercise that demand curves are equal to marginal willingness to pay
curves and no one would ever pay more than $250 for a basketball ticket.
(a) First, suppose only students care about basketball. Draw a demand and supply curve for
basketball tickets (to one game) assuming the stadium capacity is 5,000 seats and assuming
that supply and (student) demand intersect at $100.
Answer: This is illustrated in panel (a) of Graph 18.3.
Graph 18.3: Basketball Tickets
(b) Suppose students have an opportunity cost of time equal to $20 per hour. The university gives
away tickets to the game for free to anyone with a valid student ID, but only the first 5,000
students who line up will get a ticket. In equilibrium, how long will the line for basketball
tickets be; i.e. how long will students have to wait in line to get a ticket?
Answer: Since the equilibrium price is $100 and students value their time at $20, the equilibrium line length must be 5 hours.
(c) What is the deadweight loss from the free ticket policy in (b)? (You can show this on your
graph as well as arrive at a dollar figure).
Answer: The deadweight loss DW L is indicated in panel (a) of the graph. It is equal to the
value of time spent by students standing in line — which is a cost for students but of no
benefit to anyone else. The dollar value is 5,000(100) = $500,000.
(d) Now suppose that faculty care about basketball every bit as much as students. Unlike students, however, faculty have an opportunity cost of time equal to $100 per hour. Will any
faculty attend basketball games under the policy in (b) (assuming students are not allowed to
sell tickets to the faculty)?
Answer: With just students lining up, the line is 5 hours long. Faculty value their time at
$100 per hour — and the problem states that no one is willing to spend more than $250 for
a ticket. A 5 hour line costs $500 for faculty — which is more than any of them are willing to
pay to attend a game. Thus, no faculty would line up, and no faculty would attend the game.
(e) Now suppose anyone can sell, or “scalp”, his ticket at any price if he obtained one standing
in line. Draw a new supply and demand graph — but this time let this be the market for
653
Elasticities, Price Distorting Policies and Non-Price Rationing
tickets after the university has allocated them using their zero price/waiting-in-line policy.
The suppliers are therefore those who have obtained tickets by standing in line, and the supply
curve is determined by the willingness of those people to sell their tickets. What would this
supply curve look like? Who would be the demanders?
Answer: This is illustrated in panel (b) of Graph 18.3. The supply curve is composed of
the 5,000 students who got tickets in panel (a) — and who value them between $100 and
$250 dollars. The demand curve is the demand for tickets by faculty. These will intersect at
some price p ∗ between $100 and $250 — with x ∗ tickets sold to faculty and the remaining
(5,000 − x ∗ ) tickets remaining with students.
(f) A market such as the one you have just illustrated is called a secondary market — i.e. a
market where previous buyers now become sellers. The common policy (often enshrined into
law) of not permitting “scalping” of tickets is equivalent to setting a price ceiling of zero in this
market. Under this policy, how many tickets will be sold in the secondary market?
Answer: A price ceiling of zero will result in no tickets being sold in the secondary market.
(The real price ceiling imposed by no-scalping laws is often the “face value” of the ticket —
which would usually be below a value that facilitates many sales. In our example, no sales
will happen in the secondary market so long as the face value is below $100 per ticket.)
(g) How much surplus is being lost through the “no scalping” policy? Is anyone made worse off
by allowing scalping of tickets?
Answer: The no scalping policy results in a loss of surplus equal to a plus b in panel (b) of
Graph 18.3. The a portion is surplus lost by faculty who are kept from going to the game,
and the b portion is surplus lost by students who would prefer to sell their ticket to a faculty
member at price p ∗ rather than go to the game. High demanding faculty and low demanding students would be better off if scalping would be allowed — and no one else would be
affected. Thus, no one would be made worse off by allowing scalping, and some people
would be made better off.
(h) In the absence of this policy, how would the mix of people attending the game change?
Answer: With the no-scalping policy, only students attend the games. Without the noscalping policy, both faculty and students will attend the games.
B: Suppose that the students’ aggregate demand curve for tickets x is p = 250 − 0.03x and assume
throughout that there are no relevant income effects to worry about. Suppose further that the aggregate demand for tickets by faculty is the same as that for students and, as in part A, 5000 seats are
available.
(a) What is the aggregate demand function for students and faculty jointly? If the tickets were
allocated through a market price, what would be the price?
Answer: Solved for x, the aggregate demand curve for students (and that for faculty) becomes x = (250 − p)/0.03. Since they are the same for students and faculty, we need to
multiply this by 2 — which gives us x = (500 − 2p)/0.03 as the aggregate demand function
for faculty and students together. Solved for p, we get the aggregate demand curve (which,
in the absence of income effects, is also the aggregate marginal willingness to pay curve) of
p = 250−0.015x. If the 5,000 tickets were allocated through a market price, that price would
then be
p = 250 − 0.015(5000) = $175.
(18.27)
(b) Suppose that the university only sold tickets to students. What would the equilibrium price
be then?
Answer: If the university limits sales to students, we only use the demand curve for students
to calculate the equilibrium price
p = 250 − 0.03(5000) = $100.
(18.28)
(c) Now suppose the tickets were allocated to those students who waited in line. Do you have
to know anything about students’ value of time to calculate the deadweight loss from this
allocation mechanism?
Elasticities, Price Distorting Policies and Non-Price Rationing
654
Answer: No, you only have to know their value of time to calculate the length of the line.
Whatever their value of time, they will spend $100 worth of time in line in equilibrium —
which means that they will each incur a cost of $100 that benefits no one else and is therefore
deadweight loss. Given there are 5,000 tickets, this implies a deadweight loss of $500,000.
(d) Suppose again that students are the only ones who are allocated tickets — and suppose they
are prohibited from selling, or “scalping”, them to faculty. Derive the demand and supply
curves in the secondary market where students are potential suppliers and faculty are potential demanders.
Answer: The students who obtained tickets are those with marginal willingness to pay of
$100 and above. Thus, the supply curve has vertical intercept of $100 (since no student who
has a ticket would sell it for less than that) and slope of 0.03 (just as the demand curve) that
leads to the supply curve pictured in panel (b) of Graph 18.3. The supply curve is therefore
p = 100 + 0.03x. The demand curve, on the other hand, is simply the demand curve for
tickets by faculty — i.e. p = 250 − 0.03x.
(e) What would be the price for tickets in this secondary market if it were allowed to operate?
Answer: Writing the supply and demand curves we just derived with x on the left hand side,
we get a supply function of x = (p − 100)/0.03 and a demand function of x = (250 − p)/0.03.
Setting these equal to one another and solving for p, we get the equilibrium price p ∗ = 175.
(f) What fraction of the attendees at the game will be faculty?
Answer: At p ∗ = 175, faculty will buy x ∗ = (250 − 175)/0.03 = 2,500 tickets — i.e. exactly half
the available seats will be taken by faculty.
(g) How large is the deadweight loss from the no-scalping policy? Does this depend on whether
students bought the tickets as in (c) or waited in line as in (d)?
Answer: The area a in panel (b) of Graph 18.3 is equal to (250 − 175)(2500)/2 = $93,750.
The area b in the graph is similarly equal to (175 − 100)(2500)/2 = $93,750. The deadweight
loss from not permitting students and faculty to trade is equal to the sum of these — i.e.
$187,500.
(h) Compare the outcome in (a) and (e). Would the composition of the crowd at the basketball
game differ between the scenario in which everyone can buy tickets at the market price as
opposed to the scenario where students get tickets by waiting in line but can then sell them?
Answer: No, the composition of the crowd would be identical because the price that rations
tickets ends up being $175 in both cases. In the first scenario, the tickets are immediately
allocated through this price, with the university collecting the revenues and with half of all
purchasers being faculty and half being students. In the second scenario, the tickets are
given away to students first and then re-allocated in the secondary market at the price of
$175 per ticket — with half the students who originally got tickets holding onto them and
the remainder selling them to faculty.
655
Elasticities, Price Distorting Policies and Non-Price Rationing
18.7 Business and Policy Application: Minimum Wage Laws: Most developed countries prohibit employers from paying wages below some minimum level w . This is an example of a price floor in the labor market — and the policy has an impact in a labor market so long as w > w ∗ (where w ∗ is the equilibrium
wage in the absence of policy-induced wage distortions.)
A: Suppose w is indeed set above w ∗ , and suppose that labor supply slopes up.
(a) Illustrate this labor market — and the impact of the minimum wage law on employment.
Answer: This is illustrated in panel (a) of Graph 18.4 where ℓ∗ is the pre-minimum wage
level of employment and ℓ A is the post-minimum wage level of employment. Thus, employment falls by (ℓ∗ − ℓ A ).
Graph 18.4: Minimum Wage Laws
(b) Suppose that the disequilibrium unemployment caused by the minimum wage gives rise to
more intense effort on the part of workers to find employment. Can you illustrate in your
graph the equilibrium cost of the additional effort workers expend in securing employment?
Answer: The disequilibrium unemployment is the difference between ℓB and ℓ A in panel
(a) of Graph 18.4. If this disequilibrium is resolved by workers competing more intensely
for jobs, then workers will require a higher wage in order to cover these additional costs.
Thus, the labor supply curve shifts up as workers incur additional costs and point A does
not become an equilibrium until the labor supply curve has shifted up by (w − w ′ ) which is
the equilibrium increase in effort costs on the part of workers.
(c) If leisure were quasilinear (and you could therefore measure worker surplus on the labor supply curve), what’s the largest that deadweight loss from the minimum wage might become?
Answer: The largest possible deadweight loss is composed of two parts (both represented in
panel (a) of Graph 18.4): First, there is a deadweight loss from the decrease in employment
(from ℓ∗ to ℓ A ). This deadweight loss is equal to area (c +d ). The second part of deadweight
loss emerges from the increased effort costs of workers to secure employment. These costs
sum to (a + b) — and how much of this is deadweight loss depends on how much of it is
recouped by someone else in the economy. If none of it is recouped by anyone else (as
when workers simply run from place to place applying frantically for jobs), the entire area is
deadweight loss. Thus the largest possible area of deadweight loss is (a + b + c + d ).
(d) How is the decrease in employment caused by the minimum wage (relative to the non-minimum
wage employment level) related to the wage elasticity of labor demand? How is it related to
the wage elasticity of labor supply?
Answer: The more inelastic the demand curve, the smaller the distance between ℓ∗ and
ℓ A — i.e. the smaller the impact of the minimum wage law on employment. The elasticity
Elasticities, Price Distorting Policies and Non-Price Rationing
656
of the supply curve plays no role in determining ℓ A — the post-minimum wage employment level — and therefore is not relevant for determining the employment impact of the
minimum wage law. This can be seen in panels (b) and (c) of Graph 18.4. In panel (b), the
demand curve D is more inelastic than the demand curve D ′ — with the former leading to
a reduction in employment from ℓ∗ to ℓ A and the latter leading to a larger reduction from
′
ℓ∗ to ℓ A . In panel (c), the supply curve S ′ is more elastic than the curve S — but for both,
the new equilibrium moves to A which is unaffected by the elasticity of the supply curve.
(e) Define unemployment as the difference between the number of people willing to work at a
given wage and the number of people who can find work at that wage. How is the size of
unemployment at the minimum wage affected by the wage elasticities of labor supply and
demand?
Answer: This definition if unemployment is the difference between ℓB and ℓ A in panel (a) of
the graph. In panels (b) and (c), it is obvious that greater elasticity of demand or supply will
widen the gap between ℓ A — the number of workers that can work under the minimum
wage, and ℓB — the number of workers who would like to work at the minimum wage.
When labor demand curves get more wage elastic, the increase unemployment therefore
comes from a larger reduction in actual employment; and when the labor supply curve becomes more elastic, the increase in unemployment comes from an increase in the number
of workers that would like to work at the higher wage.
(f) How is the equilibrium cost of effort exerted by workers to secure employment affected by the
wage elasticities of labor demand and supply?
Answer: Again, we can read the answers off the graphs in panels (b) and (c) of Graph 18.4.
In panel (b), demand becomes more elastic from D to D ′ — leading to an increase in the
′
equilibrium effort cost from (w − w A ) to (w − w A ). In panel (c), on the other hand, the
supply curve becomes more elastic from S to S ′ — this time leading to a decrease in the
′
effort cost from (w − w B ) to (w − w B ). An increase in the wage elasticity of labor therefore
leads to an increase in the effort cost — while an increase in the elasticity of supply leads to
a decrease in that cost.
B: Suppose that labor demand is given by ℓD = (A/w)α and labor supply is given by ℓS = (B w)β .
(a) What is the wage elasticity of labor demand and labor supply?
Answer: The wage elasticity of labor demand is equal to
!
µ
¶Ã
Aα
w
d ℓD w
= −α.
= −α
Aα
d w ℓD
w (α+1)
α
(18.29)
w
The wage elasticity of labor supply is
¶
µ
d ℓS w
w
= β.
= βB β w (β−1)
β
d w ℓS
(B w)
(18.30)
(b) What is the equilibrium wage in the absence of any distortions?
Answer: Setting ℓD equal to ℓS and solving for w, we get
w∗ =
µ α ¶1/(α+β)
A
.
(18.31)
Bβ
(c) What is the equilibrium labor employment in the absence of any distortions?
Answer: Plugging our answer for w ∗ back into either the labor demand or supply function,
we get
ℓ∗ = (AB )αβ/(α+β) .
(18.32)
(d) Suppose A = 24,500, B = 500 and α = β = 1. Determine the equilibrium wage w ∗ and labor
employment ℓ∗ .
Answer: Plugging these into our expression for w ∗ and ℓ∗ , we get
ℓ∗ = 3,500 and w ∗ = 7.00.
(18.33)
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Elasticities, Price Distorting Policies and Non-Price Rationing
Wage Elasticities and Minimum Wages
Demand and Supply Parameters
w∗
ℓ∗
ℓ A (ℓ∗ − ℓ A )
A = 24, 500, B = 500, α = β = 1 $7.00 3,500 2,450
1,050
A = 11, 668, B = 500, α = 1.1, β = 1 $7.00 3,500 2,364
1,136
A = 24, 500, B = 238.1, α = 1, β = 1.1 $7.00 3,500 2,450
1,050
U
2,550
2,636
2,731
Table 18.1: Table for Exercise 18.7 — Minimum Wage=$10
(e) Suppose that a minimum wage of $10 is imposed. What is the new employment level ℓ A —
and the size of the drop in employment (ℓ∗ − ℓ A )?
Answer: To find the new employment level ℓ A , we simply plug the minimum wage into the
labor demand function to get ℓ A = 2,450, a drop of (ℓ∗ − ℓ A ) = 1,050.
(f) How large is unemployment under this minimum wage — with unemployment U defined
as the difference between the labor that seeks employment and the labor that is actually employed at the minimum wage?
Answer: We have already found the actual employment level ℓ A = 2,450 under the minimum wage. To find the level of employment desired by workers when the wage is 10, we
simply plug this minimum wage into the labor supply function to get ℓB = 5,000. Thus,
unemployment, defined as U = (ℓB − ℓ A ), is 2,550.
(g) If the new equilibrium is reached through workers expending increased effort in securing employment, what is the equilibrium effort cost c ∗ ?
Answer: The equilibrium effort cost is the equal to the difference between the minimum
wage and w ′ as depicted in panel (a) of Graph 18.4. We can find w ′ by first solving the labor
supply function for the labor supply curve — i.e. solving for p to get w = ℓ1/β /B — and then
plugging ℓ A = 2450 in for ℓ. This gives us w ′ = 4.9 — which implies c ∗ = 10 − 4.9 = 5.1.
(h) Create a table with w ∗ , ℓ∗ , ℓ A , (ℓ∗ −ℓ A ), U and c ∗ along the top. Then fill in the first row for
the case you have just calculated — i.e. the case where A = 24,500, B = 500 and α = β = 1.
Answer: This is done in Table 18.1.
(i) Next consider the case where A = 11,668, B = 500, α = 1.1 and β = 1. Fill in the second row
of the table for this case — and explain what is happening in terms of the change in wage
elasticities.
Answer: This is done in Table 18.1. The table shows that we have changed the labor demand
parameters in such a way as to not change the equilibrium in the absence of wage distortions. Since we know from part (a) that the wage elasticity of labor demand is −α, we know
that what we have done is to make labor demand more elastic (by raising it in absolute value
from 1 to 1.1). In panel (b) of Graph 18.4, we illustrated that this should lead to a larger drop
in employment, an increase in unemployment and an increase in the effort cost for securing
work. This is indeed what the figures in the second row of the table show.
(j) Finally, consider the case where A = 24,500, B = 238.1, α = 1 and β = 1.1. Fill in the third row
of the table for this case — and again explain what is happening in terms of the change in
wage elasticities.
Answer: This is done in Table 18.1. This time the labor supply parameters are changed in
such a way as to keep the no-distortion equilibrium unchanged. By increasing β, however,
we have now increased the wage elasticity of labor supply — just as we did in panel (c) of
Graph 18.4. In that graph, we concluded that this should lead to no change in the drop of
employment from the minimum wage (because firm demand is unchanged), an increase in
unemployment (because more workers want to find work), and a decrease in the effort cost
for securing work. This is again consistent with what the table indicates.
c∗
$5.10
$5.27
$4.92
Elasticities, Price Distorting Policies and Non-Price Rationing
658
18.8 Business and Policy Application: Usury Laws: The practice of charging interest on money that is
lent by one party to another, while commonplace now, has been historically controversial. Major religions
have prohibited the charging of interest in the past (and some do so today), and governments have often
codified this moral objection to interest in what is known as usury laws that limit the amount of interest
that individuals can charge one another.
A: Usury laws are thus simply an example of a price ceiling in the market for financial capital.
(a) Illustrate a demand and upward sloping supply curve in the market for financial capital
(with the interest rate on the vertical axis). Denote the equilibrium interest rate in the absence of distortions as r ∗ .
Answer: This is done in panel (a) of Graph 18.5.
Graph 18.5: Usury Laws and Elasticities
(b) If usury laws prohibit interest rates above r ∗ , will they have any impact?
Answer: No, they would not. If the price ceiling is set above the equilibrium price, then the
equilibrium price is in fact legal — and thus will emerge in the market.
(c) Suppose the highest legal interest rate r is set below r ∗ . Explain what will happen to the
amount of financial capital provided by suppliers of such capital.
Answer: This is also illustrated in panel (a) of Graph 18.5 where the amount of financial
capital transacted in the market falls from k ∗ to kS as a result of the usury law.
(d) In light of the fact that financial capital is essential for an economy to grow, what would you
predict will happen to economic growth as a result of such a usury law?
Answer: Since the usury law reduces the amount of financial capital available, a number of
projects will not take place in the economy and economic growth will suffer.
(e) How is the decrease in financial capital from usury laws related to the interest rate elasticity
of demand? How is it related to the interest rate elasticity of supply?
Answer: This is treated in panels (b) and (c) of Graph 18.5. In panel (b), the interest rate
elasticity of demand increases (in absolute value) from D ′ to D ′′ — i.e. demand becomes
more elastic. Since the amount of financial capital that is transacted under the usury law
is determined by the supply (and not the demand) curve, kS is the same for both demand
curves and is therefore unaffected by the interest rate elasticity of demand. In panel (c), the
interest rate elasticity of supply increases from S ′ to S ′′ — and as the supply curve becomes
more elastic, the drop in financial capital increases from the initial drop to kS′ to kS′′ . Thus, as
the interest rate elasticity of supply increases, the impact of usury laws on financial capital
increases.
659
Elasticities, Price Distorting Policies and Non-Price Rationing
(f) Consider how a new equilibrium is likely to be reached in the financial market after the imposition of such a usury law. In addition to the dampening effect of less capital on economic
growth, can you think of another related factor that may dampen such growth?
Answer: When usury laws interfere with the price signal that rations financial capital, some
other mechanism has to ration the lower amount of financial capital in the market. The
most natural assumption is that investors will now have to expend greater effort in securing financial capital for their projects — with the equilibrium effort level per unit of capital
equal to (r ′ −r ) in panel (a) of Graph 18.5. Such effort could have gone into more productive
uses and may therefore further dampen economic growth.
(g) How is this factor (relating to the effort expended on securing financial capital) affected by
the interest rate elasticity of demand and supply?
Answer: This is again treated in panels (b) and (c) of Graph 18.5. In panel (b), the demand
curve becomes more elastic from D ′ to D ′′ — and with it the per unit equilibrium effort cost
falls from (r ′ − r ) to (r ′′ − r ). Thus, the more elastic the demand curve, the less effort will
have to be exerted by investors to get to the reduced level of capital. In panel (c), the supply
curve becomes more elastic from S ′ to S ′′ — and the effort cost nowincreases from (r ′ − r )
to (r ′′ − r ). Thus, the per unit effort cost of getting to the reduced level of capital increases
with the elasticity of supply.
B: Suppose that demand and supply curves are similar to those used in exercise 18.7, with demand
given by kD = (A/r )α and supply by kS = (B w)β .
(a) Derive the interest rate elasticity of capital demand and supply.
Answer: The interest rate elasticity of capital demand is equal to
!
µ
¶Ã
Aα
r
d kD r
= −α.
= −α
α
A
d r kD
r (α+1)
α
(18.34)
r
The interest rate elasticity of capital supply is
¶
µ
d kS r
r
= β.
= βB β r (β−1)
β
d r kS
(B r )
(18.35)
(b) What is the equilibrium interest rate in the absence of price distortions?
Answer: Setting kD equal to kS and solving for r , we get
r∗ =
µ α ¶1/(α+β)
A
Bβ
.
(18.36)
(c) What is the equilibrium level of financial capital transacted in the absence of any price distortions.
Answer: Plugging our answer for r ∗ back into either the capital demand or supply function,
we get
k ∗ = (AB )αβ/(α+β) .
(18.37)
(d) Suppose A = 24,500, B = 500 and α = β = 1. Determine the equilibrium interest rate r ∗ and
the equilibrium level of financial capital k ∗ .
Answer: Plugging these into our expression for r ∗ and k ∗ , we get
k ∗ = 3,500 and r ∗ = 7.00.
(18.38)
(e) Suppose the usury law sets a maximum interest rate r = 5. What is the new level of financial
capital k ′ transacted — and how big is the drop (k ∗ −k ′ ) in financial capital as a result of the
usury law?
Answer: To find the new level of financial capital transacted, we simply need to plug the
interest rate ceiling of 5 into the supply function to get k ′ = (500(5))1 = 2,500 — implying a
drop in financial capital of 1,000 units of capital (from k ∗ of 3,500).
Elasticities, Price Distorting Policies and Non-Price Rationing
Interest Rate Elasticities and Usury Laws
Demand and Supply Parameters
r∗
k∗
k′
A = 24, 500, B = 500, α = β = 1 7.00 3,500 2,500
A = 11, 668, B = 500, α = 1.1, β = 1 7.00 3,500 2,500
A = 24, 500, B = 238.1, α = 1, β = 1.1 7.00 3,500 2,417
660
(k ∗ − k ′ )
1,000
1,000
1,083
c∗
4.80
4.51
5.14
Table 18.2: Table for Exercise 18.8 — Interest Rate Ceiling = 5
(f) If the new equilibrium is reached by investors expending additional effort to get to financial
capital, what is the equilibrium effort cost c ∗ ?
Answer: To find this, we first have to find the value for r ′ as depicted in panel (a) of Graph
18.5. This is read off the demand curve — so we first solve the demand function for r to
give us the expression for the demand curve; i.e. r = A/(k 1/α ). Then we plug in the new
level of financial capital — k ′ = 2,500 — to get r ′ = 9.8. The effort cost is then the difference
between r ′ and the interest rate ceiling r = 5 — i.e. c ∗ = 4.8.
(g) Create a table with r ∗ , k ∗ , k ′ , (k ∗ − k ′ ) and c ∗ at the top. Then fill in the first row for the case
you just calculated — i.e. A = 24,500, B = 500 and α = β = 1.
Answer: This is done in Table 18.2.
(h) Next consider the case where A = 11,668, B = 500, α = 1.1 and β = 1. Fill in the second row of
the table for this case — and explain what is happening in terms of the change in interest rate
elasticities.
Answer: This is done in Table 18.2. The change from the first row is that we have made
the demand curve more elastic while adjusting A such that the non-distortionary equilibrium remains unchanged. The table shows that this has no impact on the size of the drop
in financial capital but it does lower the equilibrium effort cost for getting to the financial
capital. This is exactly what we can see in panel (b) of Graph 18.5 where the increase in the
absolute value of the interest rate elasticity of demand from D ′ to D ′′ has no impact on kS ,
the level of financial capital transacted under the usury law, and decreases the effort cost
from (r ′ − r ) to (r ′′ − r ).
(i) Finally, consider the case where A = 24,500, B = 238.1, α = 1 and β = 1.1. Fill in the third row
of the table for this case — and again explain what is happening in terms of the change in
interest rate elasticities.
Answer: This is done in Table 18.2. Now we are changing the interest rate elasticity of supply
while altering B to keep the no-distortion equilibrium the same. As a result of making supply more elastic, we see that the drop in financial capital is more severe and the effort cost
of getting to the financial capital increased. This is exactly what we can see in panel (c) of
Graph 18.5 where the supply curve becomes more elastic from S ′ to S ′′ . As a result, we see
that the level of financial capital drops from kS′ to kS′′ (as it does in the table), and the effort
cost now increases from (r ′ − r ) to (r ′′ − r ) (as it does in the table).
661
Elasticities, Price Distorting Policies and Non-Price Rationing
18.9 Business and Policy Application: Subsidizing Corn through Price Floors: Suppose the domestic demand and supply for corn intersects at p ∗ — and suppose further that p ∗ also happens to be the world
price for corn. (Since the domestic price is equal to the world price, there is no need for this country to
either import or export corn.) Assume throughout that income effects do not play a significant role in the
analysis of the corn market.
A: Suppose the domestic government imposes a price floor p that is greater than p ∗ and it is able to
keep imports of corn from coming into the country.
(a) Illustrate the disequilibrium shortage or surplus that results from the imposition of this price
floor.
Answer: This is illustrated in panel (a) of Graph 18.6 where domestic supply and demand
intersect at p ∗ and the price floor p is imposed above p ∗ . This results in a disequilibrium
surplus, with x S supplied but only x D demanded.
Graph 18.6: Price Floor in Corn Market
(b) In the absence of anything else happening, how will an equilibrium be re-established and
what will happen to producer and consumer surplus?
Answer: Consumer surplus will fall from (a +b +e) to a while producer surplus will fall from
(c +d + f ) to d . This is because, in equilibrium, producers will have to exert additional effort
— i.e. incur additional costs — to compete for the limited number of consumers — which
will cause the effective price they receive to fall to p ′ . (The additional marginal cost of effort
on the part of producers must be (p −p ′ ) in order to make point A in panel (a) of Graph 18.6
the new equilibrium in which the disequilibrium shortage has been eliminated.)
(c) Next, suppose the government agrees to purchase any corn that domestic producers cannot
sell at the price floor. The government then plans to turn around and sell the corn it purchases
on the world market (where its sales are sufficiently small to not affect the world price of corn).
Illustrate how an equilibrium will now be re-established — and determine the change in domestic consumer and producer surplus from this government program.
Answer: This is illustrated in panel (b) of Graph 18.6 where the difference between x S and
x D — previously labeled a “disequilibrium surplus” in panel (a) — now becomes the quantity of corn purchased by the government. In essence, the government purchasing program causes the equilibrium to settle at B rather than A (as in panel (a) of the graph) —
Elasticities, Price Distorting Policies and Non-Price Rationing
662
because producers no longer have an incentive to expend additional effort to attract consumers since the government is guaranteeing it will purchase what cannot be sold at the
price floor. Consumer surplus is then again a (since consumers purchase x D at p as before;
producer surplus, however, now increases to (b + c + d + e + f + g ) as producers supply x S at
the price p.
(d) What is the deadweight loss from the price floor with and without the government purchasing
program?
Answer: The greatest possible surplus achievable in this market is (a +b +c +d +e + f ). With
just the price floor (and no government purchasing program), we concluded above that total
surplus will be at most (a +b +c +d ) — implying a deadweight loss of (e + f ). When the price
floor is supplemented with the government purchasing program, the sum of consumer and
producer surplus becomes (a + b + c + d + e + f + g ). However, we now need to take into
account that the government is also having to spend resources in order to buy the surplus
at the price floor p and then sell it at a loss at p ∗ . It will therefore cost (e + f + g + h + i + j )
to buy the surplus corn and, when sold at p ∗ , it will raise revenues of ( f + i + j ) — leaving a
government loss of (e + g + h). The total surplus is then the sum of producer and consumer
surplus minus the government loss — which comes to (a + b + c + d + e + f + g ) − (e + g +
h)=(a + b + c + d + f − h). Compared to the most possible surplus of (a + b + c + d + e + f ),
this implies a deadweight loss of (e + h).
(e) In implementing the purchasing program, the government notices that it is not very good at
getting corn to the world market — and all of it spoils before it can be sold. How does the
deadweight loss from the program change depending on how successful the government is at
selling the corn on the world market?
Answer: The government loss now becomes (e + f + g + h + i + j ) — which gives us total
surplus of (a +b +c +d +e + f + g )−(e + f + g +h +i + j )=(a +b +c +d −h −i − j ). Compared
to the maximum possible surplus of (a + b + c + d + e + f ), this gives us a deadweight loss of
(e + f + h + i + j ).
(f) Would either consumers or producers favor the price floor on corn without any additional
government programs?
Answer: As illustrated in part (b) of the question, both producers and consumers lose surplus under the price floor policy without additional government programs. Thus, neither
would favor such a program.
(g) Who would favor the price floor combined with the government purchasing program? Does
their support depend on whether the government succeeds in selling the surplus corn? Why
might they succeed in the political process?
Answer: As illustrated in part (c) of the question, producers gain substantial amounts of surplus when the government program is added to the price floor — and the amount of surplus
they gain does not depend on what the government does with the surplus corn that was purchased. Thus, producers would favor the price floor when combined with the government
purchasing program — and they might succeed in the political process because they are a
relatively small group (compared to consumers and tax payers) experiencing concentrated
benefits. This gives them an incentive to expend resources to lobby for such a program —
and the diffuse nature of the costs (spread over many consumers and taxpayers) makes it
unlikely that those who lose from the program will politically organize against it.
(h) How does the deadweight loss from the price floor change with the price elasticity of demand?
Answer: It decreases as demand becomes more inelastic.
B: Suppose the domestic demand curve for bushels of corn is given by p = 24−0.00000000225x while
the domestic supply curve is given by p = 1 + 0.00000000025x. Suppose there are no income effects
to worry about.
(a) Calculate the equilibrium price p ∗ (in the absence of any government interference). Assume
henceforth that this is also the world price for a bushel of corn.
Answer: Re-writing the demand and supply curves as demand and supply functions (i.e.
solving for x to be on one side), we get
663
Elasticities, Price Distorting Policies and Non-Price Rationing
xD =
24 − p
p −1
and x S =
.
0.00000000225
0.00000000025
(18.39)
Setting these equal to one another and solving for p, we get the equilibrium price p ∗ = 3.3
per bushel.
(b) What is the quantity of corn produced and consumed domestically? (Note: The price per
bushel and the quantity produced is roughly equal to what is produced and consumed in the
U.S. in an average year.)
Answer: Plugging the equilibrium price of 3.3 into either the demand or supply function in
equation (18.39), we get x ∗ = 9,200,000,000 or 9.2 billion bushels.
(c) How much is the total social (consumer and producer) surplus in the domestic corn market?
Answer: Calculating these as the relevant triangles above the supply and below the demand
curves, we get
(24 − 3.3)(9,200,000, 000)
= 95,220,000,000 and
2
(3.3 − 1)(9,200,000,000)
PS =
= 10,580,000,000
2
CS =
(18.40)
for a total social surplus of $105,800,000,000 or $105.8 billion.
(d) Next suppose the government imposes a price floor of p = 3.5 per bushel of corn. What is the
disequilibrium shortage or surplus of corn?
Answer: Plugging this price floor into the demand and supply functions of equation (18.39),
we get
x D = 9,111,111,111 and x S = 10,000,000,000
(18.41)
bushels of corn — giving us a disequilibrium surplus of (x S − x D ) = 888,888,889 bushels of
corn.
(e) In the absence of any other government program, what is the highest possible surplus after
the price floor is imposed — and what does this imply about the smallest possible size of the
deadweight loss?
Answer: The consumer surplus under the price floor is easy to calculate as just the area
under the demand curve down to the price floor p = 3.5 — i.e.
CS =
(24 − 3.5)(9,111,111, 111)
= 93,388,888,889.
2
(18.42)
To calculate the producer surplus given that producers will have to incur additional costs
in order to compete for the lower quantity demanded by consumers requires the additional
step of calculating p ′ in panel (a) of Graph 18.6 — which we get by plugging in the quantity
demanded into the supply curve equation; i.e.
p ′ = 1 + 0.00000000025(9, 111, 111, 111) = 3.27777... ≈ 3.278.
(18.43)
The producer surplus triangle (equivalent to the triangle d in panel (a) of Graph 18.6) is then
PS =
(3.27777778 − 1)(9, 111,111,111)
= 10,376,543,210.
2
(18.44)
Consumer and producer surplus together then sum to $103,765,432,099 or approximately
$103.765 billion. If we want to arrive at the highest possible figure for social surplus, we
need to assume that the costs paid by producers to compete for consumers were not socially wasteful — and these costs (equivalent to area (b+c) in panel (a) of Graph 18.6) is (3.5−
3.278)(9,111,111,111) = 2,024,691,358. Added to the sum of producer and consumer surplus, we therefore get the highest possible social surplus as approximately $105,790,123,457.
Compared the original surplus of $105,800,000,000, we therefore get a deadweight loss of
$9,876,543.
Elasticities, Price Distorting Policies and Non-Price Rationing
664
(f) Suppose next that the government purchases any amount that corn producers are willing to
sell at the price floor p but cannot sell to domestic consumers. How much does the government
have to buy?
Answer: To determine the amount the government has to buy, we need to subtract the
amount that consumers demand — i.e. 9,111,111,111 bushels — from the amount that producers will supply at the price floor. To determine the latter, we simply plug the price floor
of 3.5 into the supply function to get 10,000,000,000. Thus, the difference is 888,888,889
bushels of corn — which is the disequilibrium surplus previously calculated in (d).
(g) What happens to consumer surplus? What about producer surplus?
Answer: Consumer surplus stays the same as before — because consumers continue to buy
the same amount at the same price floor. Producer surplus, however, is now equal to the
triangle (b + c + d + e + f + g ) in panel (b) of Graph 18.6 — which is
PS =
(3.5 − 1)(10,000, 000,000)
= 12,500,000,000.
2
(18.45)
(h) What happens to total surplus assuming the government sells the corn it buys on the world
market at the price p ∗ ?
Answer: The total surplus is now the sum of consumer and producer surplus minus the loss
the government takes by buying corn at the price floor of 3.5 and selling it at the world price
of 3.3. This gives us
Social Surplus = 93,388,888,889 + 12, 500, 000, 000 − (3.5 − 3.3)(888, 888, 889) =
= 105,711,111,111.
(18.46)
In the absence of any program, the total surplus was $150,800,000,000.
This fell to $105,790,123,457 with the imposition of just the price floor, and we have now
shown it falls further to $105,711,111,111 if the government purchasing program is added to
the price floor. This implies that the deadweight loss jumps from $9,876,543 under just the
price floor to $88,888,889 when the government purchasing program is added — an increase
of $79,012,346.
(i) How much does deadweight loss jump under just the price floor as well as when the government purchasing program is added if p = 4 instead of 3.5? What if it is 5?
Answer: Going through steps similar to those above, the overall surplus falls from the original $105,800,000,000 to $105,679,012,346 under the price floor of p = 4.
It furthermore falls to $104,711,111,111 if the government purchasing program is added.
This implies a deadweight loss under just the price floor of $120,987,654 which increases
to $1,088,888,889 when the purchasing program is added. If the price floor is raised to p =
5, the overall social surplus falls from $105,800,000,000 to $105,086,419,753 under just the
floor and $99,377,777,778 if the purchasing program is added. This implies a deadweight
loss of $713,580,247 under just the price floor and $6,422,222,222 when the government
purchasing program is added.
665
Elasticities, Price Distorting Policies and Non-Price Rationing
18.10 Business and Policy Application: Corn Subsidies through Price Floors (continued): Consider the
same set-up as in exercise 18.9.
A: Suppose again that a price floor p greater than the equilibrium price p ∗ has been imposed and
that the government has committed to purchase the difference between what is supplied at the price
floor and what is demanded.
(a) If you have not done so in exercise 18.9, illustrate the smallest possible deadweight loss in the
absence of the government purchasing program as well as the deadweight loss if the government purchases the excess corn and then sells it at the world price p ∗ .
Answer: Using the logic of exercise 18.9, the smallest possible deadweight loss of just the
price floor is (e+ f ) in panel (a) of Graph 18.7; and the deadweight loss when the government
purchases the excess corn produced and sells it at the world market price is (e + h).
Graph 18.7: Corn Subsidies through Price Floors (cont)
(b) How would the deadweight loss change if the government found a way to give the corn it
purchases to those consumers that place the highest value on it.
Answer: In this case, the consumers who purchase corn at the price floor still get consumer
surplus equal to area a; and among the remaining consumers, those who value the corn the
most are those whose demand falls on the demand curve just below the price floor. If the
government purchases (x S −x D ), this implies the consumers on the demand curve between
x D and x S will receive the corn the government gives away — getting consumer surplus of
(e + f + k + m) (since they pay nothing for the corn). Thus, total consumer surplus is now
(a + e + f + k + m). Producers get surplus (b + c + d + e + f + g ) as they sell x S at the price
floor p, and the government incurs a cost of (e + f + g + h + i + k + m). Thus, the total social
surplus is
Total Surplus = (a + e + f + k + m) + (b + c + d + e + f + g ) − (e + f + g + h + i + k + m)
= a +b +c +d +e + f −h −i.
(18.47)
Given that the maximum social surplus in the absence of price distortions is (a + b + c + d +
e + f ), this implies a deadweight loss of (h + i ).
(c) What happens to the deadweight loss if the government instead sets a price at which all the
excess corn gets sold assuming it can keep those who purchased at the price floor from buying
at the lower government price.
Elasticities, Price Distorting Policies and Non-Price Rationing
666
Answer: Consumer surplus still remains a for those who purchase at the price floor. In
order to sell the excess corn, the government would have to set the price p in panel (a) of
Graph 18.7 — implying that consumers who did not buy at the price floor will get consumer
surplus of (e + f + k). Adding the two consumer surplus areas together, overall consumer
surplus is then (a +e + f +k). Producer surplus would still be (b +c +d +e + f + g ) as before;
but the government’s cost is now (e + f + g + h + i + k) (and no longer included m because
this amount is collected as the excess corn is sold). Thus, the overall social surplus is
Total Surplus = (a + e + f + k) + (b + c + d + e + f + g ) − (e + f + g + h + i + k)
= a +b +c +d +e + f −h −i.
(18.48)
Thus, the deadweight loss is again (h + i ).
(d) Compare your answers to (b) and (c) — they should be the same. Can you explain intuitively
why this is the case?
Answer: The reason the two answers are the same is that, in both cases, the excess corn is
consumed by those who value it most. In (b), this is done because the government is somehow able to magically find the people who value the corn the most and give it to them; in
(c), the government sets the price at which it can sell all the excess corn — and lets that price
determine who gets the corn. If it sets the price correctly, the same consumers will get it as
if the government somehow identified those who value it the most. In (b), the government
loses m in revenue but consumers pick it up as surplus; in (c) the government picks up m in
revenue and the consumers lose it in surplus.
(e) Consider the policy as described in (c). After the initial set of consumers purchase corn at the
price floor, illustrate the demand curve for the remaining consumers — and the supply curve
for corn from the government. What’s the elasticity of supply of government corn — and at
what price must this supply curve cross the demand curve of the consumers who did not buy
at the price floor?
Answer: This is done in panel (b) of Graph 18.7. The demand curve of those who remain after the initial purchases at the price floor happen, denoted D r , has to begin at p — because
anyone with marginal willingness to pay above p has already bought at the price floor. The
rest of the demand curve then has the same slope as in panel (a) since it is simply the part of
the original demand curve below p. The government supply curve, denoted S g , is perfectly
inelastic at the quantity the government committed to purchase. Supply and demand must
intersect at p from panel (a).
(f) Finally, suppose that everyone (including those with marginal willingness to pay the exceeds
the price floor) wants to buy at the lower government price but the government still agrees
to buy any amount of corn that producers are willing to supply at the price floor. What will
happen — and how will it affect the deadweight loss?
Answer: Now no one except the government buys at the price floor — and all consumers buy
at p. Producer surplus is unchanged at (b+c+d +e+ f +g ) (since they are still selling x S to the
government at p.) Since all consumers now purchase at p from the government, consumer
surplus is (a +b +c +d +e + f +ℓ+k). And the government now purchases the entire amount
x S at p and sells it at p — implying a government loss of (b + c + d + e + f + g + h + i + ℓ + k).
Total social surplus is then
Social Surplus = (a + b + c + d + e + f + ℓ + k) + (b + c + d + e + f + g )
− (b + c + d + e + f + g + h + i + ℓ + k)
= a +b +c +d +e + f −h −i.
(18.49)
Given that the maximum surplus under no price distortions is (a + b + c + d + e + f ), the
deadweight loss is again (h + i ).
(g) Why is your answer again the same as under the previous policies?
Answer: Again, given the quantity x S that is produced under the combination of the price
floor and government purchase guarantee, the corn is allocated to those who value it the
667
Elasticities, Price Distorting Policies and Non-Price Rationing
most. The government now incurs a much greater cost than before because it purchases all
the corn instead of having some consumers buy at the price floor — but this is exactly offset
by additional consumer surplus. In essence, we can think of this as a two-stage process:
In the first stage, the government buys x S at p from producers — giving rise the producer
surplus. It then inelastically supplies x S to consumers — with the government supply therefore vertical at x S . This implies that demand and government supply cross at p — with those
who value the corn the most purchasing it from the government.
B: Consider again, as in exercise 18.9, a demand curve p = 24−0.00000000225x supply curve is given
by p = 1 + 0.00000000025x.
(a) Calculate consumer surplus, producer surplus and deadweight loss under the scenario described in A(b) assuming a price floor of p = 3.5.
Answer: The supply function corresponding to the this supply curve (which is just the equation given in the problem solved for x) is x S = (p − 1)/(0.00000000025). Plugging the price
floor of 3.5 into this equation, we get that x S = 10,000,000,000 bushels of corn. Producer
surplus is then (and will be throughout the exercise)
PS =
(3.5 − 1)(10,000,000, 000)
= 12,500,000,000.
2
(18.50)
We can similarly determine the quantity that consumers will purchase at a price of 3.5 by
solving the demand curve for x and plugging in p = 3.5 to get 9,111,111,111 bushels of corn.
Thus, consumers who purchase at the price floor get area a in panel (a) of Graph 18.7 —
which is equal to (24 − 3.5)(9,111,111,111)/2 = 93,388,888,889. Consumers who are given
the corn get surplus of (e + f +l +m) in the graph. To calculate this area, we can first calculate
p by plugging x S = 10,000,000,000 into the equation of the demand curve to get
p = 24 − 0.00000000225(10, 000, 000, 000) = 1.5.
(18.51)
Thus, area m is equal to 1.5(888,888,889) = 1,333,333,333 and area (e + f + k) is (3.5 −
1.5)(888,888,889)/2 = 888,888,889. Summing these, and adding the consumer surplus area
a from those who purchase at the price floor, we get
C S = 1,333,333,333 + 888,888,889 + 93,388,888, 889 = 95,611,111,111.
(18.52)
Finally, the government incurs a cost of 3.5(888,888,889) = 3,111,111,111. Adding P S and
C S and subtracting the government cost, we then get
Social Surplus = 12,500,000,000 + 95, 611, 111, 111 − 3, 111, 111, 111 =
= $105,000,000,000
(18.53)
or $105 billion. The maximum possible surplus in the absence of price distortions (calculated in exercise 18.9) is $800,000,000 higher than this, giving us a deadweight loss of $800
million. You can easily verify that this is exactly equal to areas (h + i ) in the graph — just as
we concluded in part A of the exercise.
(b) Consider the scenario described in A(c). Derive the demand curve that remains once the consumers who are willing to purchase at the price floor.
Answer: The demand curve will have the same slope as the original demand curve, but the
intercept changes to p = 3.5. Thus, the remaining demand curve is p = 3.5−0.00000000225x.
(c) Given the quantity supplied to the remaining demanders by the government, what is the price
the government has to charge to sell all the excess corn. Calculate consumer and producer
surplus and verify that the deadweight loss is the same as in (a).
Answer: The price will emerge from the intersection of the inelastic government supply of
888,888,889 bushels of corn with the remaining demand curve of p = 3.5 − 0.00000000225x.
Plugging the government supply quantity into this demand curve, we get p = 1.5. We can
now calculate the consumer surplus of these remaining consumers as the area above p up
to the demand curve D r in panel (b) of Graph 18.7 — which is (3.5 − 1.5)(888,888,889)/2 =
Elasticities, Price Distorting Policies and Non-Price Rationing
668
888,888,889. Added to the consumer surplus of those who purchase at the price floor (which
is unchanged from what we calculated in part (a)), we get total consumer surplus of
C S = 93,388,888,889 + 888, 888, 889 = 94,277,777,778.
(18.54)
The government’s cost for purchasing the excess corn is $3,111,111,111 as in part (a), but
now the government raises revenue of 1.5(888,888,889) = $1,333,333,333 — implying a net
government cost of $1,777,777,778. Adding producers surplus (which is unchanged from
part (a)) to consumer surplus and subtracting this net government cost, we get total surplus
Social Surplus = 12,500,000,000 + 94, 277,777,778 − 1, 777, 777, 778 =
= $105,000,000,000
(18.55)
or $105 billion as in part (a). The deadweight loss is then also again $800 million.
(d) Finally, consider the scenario in A(f). Verify that the price the government has to charge to
sell all its corn is the same as in (c). Then calculate consumer surplus, producer surplus and
deadweight loss.
Answer: The government now purchases all 10,000,000,000 bushels of corn that producers
are still selling at the same price floor —i.e. now no consumers purchase no corn at the price
floor and the government purchases all that is produced. This implies that the government
supply of corn is perfectly inelastic at 10,000,000,000 bushels. Plugging this into the demand
curve, we get
p = 24 − 0.00000000225(10, 000, 000, 000) = 1.5.
(18.56)
Consumer surplus is then the area under the demand curve down to the consumer price of
1.5 — which is
CS =
(24 − 1.5)(10,000,000, 000)
= 112,500,000,000.
2
(18.57)
The government purchases 10,000,000,000 bushels at $3.5 per bushel and sells at $1.5 per
bushel — implying a net government cost of $20,000,000,000. Adding the (unchanged) producer surplus to consumer surplus and subtracting the government cost, we then get
Social Surplus = 112,500,000,000 + 12, 500,000,000 − 20,000,000,000 =
= $105,000,000,000
(18.58)
or $105 billion as before. This again implies a deadweight loss unchanged at $800 million.
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18.11 Policy Application: Rent Control: A portion of the housing market in New York City (and many
other cities in the world) is regulated through a policy known as rent control. In essence, this policy puts a
price ceiling (below the equilibrium price) on the amount of rent that landlords can charge in the apartment buildings affected by the policy.
A: Assume for simplicity that tastes are quasilinear in housing.
(a) Draw a supply and demand graph with apartments on the horizontal axis and rents (i.e.
the monthly price of apartments) on the vertical. Illustrate the “disequilibrium shortage” that
would emerge when renters believe they can actually rent an apartment at the rent-controlled
price.
Answer: This is illustrated in panel (a) of Graph 18.8.
Graph 18.8: Rent Controlled Apartments
(b) Suppose that the NYC government can easily identify those who get the most surplus from
getting an apartment. In the event of excess demand for apartments, the city then awards
the right to live (at the rent-controlled price) in these apartments to those who get the most
consumer surplus. Illustrate the resulting consumer and producer surplus as well as the deadweight loss from the policy.
Answer: This is illustrated in panel (a) of Graph 18.8. The area (a + b) would now be consumer surplus — because these “high demanders” are the ones who get the apartments and
only have to pay the price ceiling. Producer surplus is just area c — and deadweight loss is
area d that no one receives because these apartments are not put on the market under rent
control.
(c) Next, suppose NYC cannot easily identify how much consumer surplus any individual gets —
and therefore cannot match people to apartments as in (b). So instead, the mayor develops
a “pay-to-play” system under which only those who pay monthly bribes to the city will get to
“play” in a rent-controlled apartment. Assuming the mayor sets the required bribe at just the
right level to get all apartments rented out, illustrate the size of the monthly bribe.
Answer: This is illustrated as the distance B in panel (a) of Graph 18.8. By charging a bribe
of this size, the mayor is able to collect a bribe from everyone who values these apartments
at least as much as the price ceiling plus B — which is exactly the number of people we can
fit into the rent-controlled apartments that are available.
(d) Will the identity of those who live in rent-controlled apartments be different in (c) than in (b)?
Will consumer or producer surplus be different? What about deadweight loss?
Answer: In both cases, only the high demanders get into the apartments — so the identity
of those living in the apartments is the same under both policies. The producer surplus and
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670
deadweight loss remains similarly the same. But now part of what is consumer surplus in (b)
becomes revenue from bribes in (c); i.e. consumer surplus is (a + b) in (b) but only a in (c)
— because now each consumer has to pay the bribe B on top of the price ceiling. The area b
then becomes the bribe revenue for the mayor. Since this is a pure transfer from consumers
to the mayor, it is not deadweight loss.
(e) Next, suppose that the way rent-controlled apartments are allocated is through a lottery.
Whoever wants to rent a rent-controlled apartment can enter his/her name in the lottery, and
the mayor picks randomly as many names as there are apartments. Suppose the winners can
sell their right to live in a rent-controlled apartment to anyone who agrees to buy that right at
whatever price they can agree on. Who do you think will end up living in the rent-controlled
apartments (compared to who lived there under the previous policies)?
Answer: When all is said and done, the same people should once again end up in the apartments — whether they won in the lottery or not. This is because they value the apartments
the most. If they win a ticket, they won’t find someone that will buy it for an amount that
exceeds how much they value the apartment. If they don’t win a ticket, they will find someone that does not value the apartment as much as they do — and will thus buy the right to
the apartment.
(f) The winners in the lottery in part (e) in essence become the suppliers of “rights” to rentcontrolled apartments while those that did not win in the lottery become the demanders.
Imagine that selling your right to an apartment means agreeing to give up your right to occupy the apartment in exchange for a monthly check q. Can you draw a supply and demand
graph in this market for “apartment rights” and relate the equilibrium point to your previous
graph of the apartment market?
Answer: The lottery will have made winners of some that really value the apartments highly
and some that really don’t value it very much at all. Thus, some of the winners will be willing
to sell their rights at relatively low prices while others will demand higher prices. This results
in a supply curve of the form in panel (b) of Graph 18.8 where the supply curve effectively
ends — or becomes vertical — at x s , the number of apartments that were raffled off in the
lottery. The demand curve is made up of those who did not win in the lottery. We can’t tell
precisely what each of these curves will look like because of the randomness of the lottery —
but they will intersect at an equilibrium price that allocates apartments to those who value
them most. That price has to be equal to the size of the bribe B we identified in panel (a) as
“clearing the market” — i.e. in equilibrium, those who get an apartment will again pay the
price ceiling plus B = q ∗ .
(g) What will be the equilibrium monthly price q ∗ of a “right” to live in one of these apartments
compared to the bribe charged in (c)? What will be the deadweight loss in your original graph
of the apartment market? How does your answer change if lottery winners are not allowed to
sell their rights?
Answer: As already explained in the answer to (f), q ∗ = B . The end result of the lottery
combined with the market in “rights” will therefore again be the same as before — output is
still limited to x s , with high demanders living in the apartments. Those high demanders that
won the lottery get a surplus equal to the difference between their marginal willingness to
pay and the price ceiling; those who had to buy a right to an apartment because they did not
win in the lottery only get a surplus of the difference between their MW T P and the rental
price inclusive of q ∗ . And those who won but sold their rights get a surplus q ∗ . Overall,
consumers therefore get surplus a in panel (a) of Graph 18.8, and the sum of all the q ∗
surpluses — whether made by those who won and chose to live in an apartment or by those
who won and sold their rights — is equal to area b. Landlords still get c — but no one gets d .
Thus, d continues to be the deadweight loss. If, however, the lottery winners are not allowed
to sell their rights, the deadweight loss will be larger because of the effective price ceiling of
zero in the “rights” market where surplus is lost. Put differently, the “wrong” people will
live in the apartments — in the sense that some who value the apartments more would be
willing to pay these people an amount at which they would prefer not to live there and let
others move in. The only way that the deadweight loss would not increase if we prohibit
trade in the rights to apartments is if the lottery magically chose only the high demanders
as winners — which would make everything identical to the case where the mayor magically
knew who the high demanders were and allocated the apartments to them.
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(h) Finally, suppose that instead the apartments are allocated by having people wait in line. Who
will get the apartments and what will deadweight loss be now? (Assume that everyone has the
same value of time.)
Answer: The same people will again live in the apartments — but they will now pay the cost
B in the form of waiting in line. Since no one benefits from this, that implies that area b in
Graph 18.8 now becomes part of deadweight loss.
B: Suppose that the aggregate monthly demand curve is p = 10000 − 0.01x while the supply curve is
p = 1000 + 0.002x. Suppose further that there are no income effects.
(a) Calculate the equilibrium number of apartments x ∗ and the equilibrium monthly rent p ∗ in
the absence of any price distortions.
Answer: When written in terms of x rather than p, the demand and supply functions become x = 1,000,000 − 100p and x = 500p − 500,000. Setting these equal to each other and
solving for p, we get p ∗ = 2,500. Plugging this back into either the demand or supply function, we get x ∗ = 750,000.
(b) Suppose the government imposes a price ceiling of $1,500. What’s the new equilibrium number of apartments?
Answer: Plugging this into the supply function x = 500p − 500,000, we get x = 250,000.
(c) If only those who are willing to pay the most for these apartments are allowed to occupy them,
what is the monthly willingness to pay for an apartment by the person who is willing to pay
the least but still is assigned an apartment?
Answer: We can get this by plugging in the number of apartments under rent control —
i.e. 250,000 — into the demand curve (or inverse demand function — which is equal to
the marginal willingness to pay function when there are no income effects). We then get
p = 10,000 − 0.01(250,000) = $7,500.
(d) How high is the monthly bribe per apartment as described in A(c)?
Answer: This would simply be the difference between the value placed on the apartments
by the marginal occupant — i.e. $7,500 — and the amount this person has to pay in rent
under rent control — i.e. $1,500. Thus, the monthly bribe is $6,000.
(e) Suppose the lottery described in A(e) allocates the apartments under rent control, and suppose that the “residual” aggregate demand function by those who did not win in the lottery is
given by x = 750,000 − 75p. What is the demand function for y — the “rights to apartments”
(described in A(f))? What is the supply function in this market? (Hint: You will have to determine the marginal willingness to pay (or inverse demand) curves for those who did not win to
get the demand for y and for those who did win to get the supply for y . And remember to take
into account the fact that occupying an apartment is more valuable than having the right to
occupy an apartment at the rent controlled price.)
Answer: The residual demand curve (or inverse residual demand function) is the function
x = 750,000 − 75p solved for p — i.e. p = 10,000 − (x/75). This is (in the absence of income
effects) the marginal willingness to pay for these apartments by those who did not win the
lottery. If someone who values an apartment at $5,000 were asked his highest monthly price
he would be willing to pay for a ticket that allows him to rent the apartment for the rentcontrolled price of $1,500, he would have to say he is willing to pay $3,500 for such a ticket.
Thus, the marginal willingness to pay for “rights” y to rent at the rent controlled price is
$1,500 lower than the marginal willingness to pay for the apartments. Letting q denote the
marginal willingness to pay for y , we thus get q = 8,500 − (y /75) — which is also the inverse
demand function for y . Solving for y , we get the demand function
y d = 637,500 − 75q.
(18.59)
The supply function comes from those who did win the lottery. The original aggregate demand function was x = 1,000,000 − 100p while the residual demand function by those who
did not win was x = 750,000 − 75p. Subtracting the latter from the former gives us the aggregate demand for apartments from those who won the lottery — i.e. x = 250,000 − 25p
which gives us a demand (and marginal willingness to pay) curve p = 10,000−(1/25)x. This
Elasticities, Price Distorting Policies and Non-Price Rationing
672
is how much those who won tickets value the apartments. Someone who values an apartment at $5,000 and owns a “right” to the apartment would then be willing to sell that right y
for $3,500 — because he gets that much consumer surplus from exercising his “right”. The
supply curve for these rights is then q = −1500 + (y /25). (To be slightly more accurate, we
should specify this curve as flat at zero for those who value apartments at less than $1,500,
but, since the equilibrium will lie at a price q ∗ above zero, this does not matter for the math).
Solving this for y , we get the supply function
y s = 37,500 + 25q.
(18.60)
(f) What is the equilibrium monthly price of a right y to occupy a rent-controlled apartment?
Compare it to your answer to (c).
Answer: Now, all we have to do is set demand in the market for y equal to supply; i.e. y d = y s .
Using functions derived above, we therefore have
637,500 − 75q = 37,500 + 25q
(18.61)
which solves to an equilibrium price q ∗ = $6,000. Plugging this back into the demand (or
supply) function for y , we get that 187,500 rights get traded after the lottery — with only
72,500 of the original lottery winners choosing to live in the rent-controlled apartment they
won.
(g) Calculate the deadweight loss from the rent control for each of the scenarios you analyzed
above.
Answer: In each of the scenarios above, the deadweight loss is simply equal to the lower
bound of the deadweight loss from a price ceiling — because in each case, those who live
in the apartments are those who value them the most. (The producer price in each case
is the price ceiling $1,500 while the actual price necessary to clear the market for 250,000
apartments available at that price is $7,500 — leaving a difference of $6,000 that someone
gets per apartment in each of the scenarios.) You can calculate this in a number of different
ways by simply calculating the areas of the relevant triangles — and you should get that the
deadweight loss is $1,500,000,000 or $1.5 billion.
(h) How much would the deadweight loss increase if the rationing mechanism for rent-controlled
apartments were governed exclusively by having people wait in line? (Assume that everyone
has the same value of time.)
Answer: This would imply that no one is now getting the $6,000 times the 250,000 apartments — which implies the deadweight loss will increase by 6000(250, 000) = $1,500,000,000
— i.e. the deadweight loss would double.
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18.12 Policy Application: NYC Taxi Cab Medallions: In New York City, you are allowed to operate a taxi
cab only if you carry a special taxi “medallion” made by the Taxi Commission of New York. Suppose 50,000
of these have been sold, and no further ones will be put into circulation by the Taxi Commission. We
will see that restricting supply in this way is another way in which governments can inefficiently distort
price.
A: Suppose for simplicity that there are no income effects of significance in this problem. We will
analyze the demand and supply of a day’s worth of cab rides — which we will call “daily taxi rides”.
(a) On a graph with daily taxi rides on the horizontal axis and dollars on the vertical, illustrate
the daily aggregate demand curve for NYC taxi rides. Given the fixed supply of medallions,
illustrate the supply curve under the medallion system.
Answer: This is illustrated in Graph 18.9 where the supply curve is perfectly inelastic at
50,000 daily taxi rides because of the limit imposed by the medallion system.
Graph 18.9: Taxi Cab Medallions
(b) Illustrate the daily revenue a cab driver will make. (Since we are denoting quantity in terms
of “daily cab rides”, the price of one unit of the output is equal to the daily revenue.)
Answer: The equilibrium price for a day’s worth of cab rides is then p M as indicated in the
graph. This is the daily revenue collected by a cab driver.
(c) In the absence of the medallion system, taxi cabs would be free to enter and exit the cab business. Assuming that everyone faces the same cost to operating a cab, what would the long
run supply curve of cabs look like? Illustrate this on your graph under the assumption that
removal of the medallion system would result in an increase in the number of cab rides. Indicate the long run daily price of a cab and the number of cabs operating in the absence of the
medallion system.
Answer: If everyone faces the same cost of operating a cab, then the long run supply curve
should be horizontal at that marginal cost of operating a cab — because new cabs can always enter and old ones exit as demand shifts. This is illustrated in Graph 18.9 with a flat
line that lies below p M and thus results in more daily cab rides x ∗ sold at the long run competitive price p ∗ .
(d) Suppose you own a medallion and you can rent it out to someone else. Indicate in your graph
the equilibrium daily rental fee you could charge for your medallion. How much profit are
those who rent a medallion in order to operate a cab making? Is that different from how much
profit those who own a medallion and use it to operate a cab are making?
Answer: As a cab driver, you make zero profit if you simply make p ∗ per day. Thus, under
the medallion system, your profit is (p M − p ∗ ) if you pay nothing for the medallion. You
would therefore be willing to pay up to that amount in a daily rental fee for a medallion
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674
— which is indicated as distance R in the graph. This is the equilibrium daily rental fee
for medallions. As a result, those who rent medallions are making zero profit. Those who
own medallions and choose to operate a cab are also making zero profit — because the
opportunity cost of using the medallion to drive a cab is equal to the rental fee R that the
owner could have gotten by letting someone else rent the medallion. Of course, if you are
a cab driver, you would still prefer to own a medallion rather than rent one — but that is
because as the owner you get to collect the equilibrium rent R whether you drive the cab
(and don’t have to rent a medallion) or whether you rent it to someone else.
(e) True or False: The only individuals who would be made worse off if medallions were no longer
required to operate a cab are the owners of medallions.
Answer: This is true. Owners would lose something that has value if medallions are no
longer required. Cab drivers make zero profit with and without the medallion system. Consumers, of course, are better off without the medallion system where they get to ride around
in cabs at cheaper rates.
(f) Illustrate in your graph the daily deadweight loss from the medallion system. Can you think
of a policy proposal that would make everyone better off?
Answer: Referring to areas labeled in Graph 18.9, consumer surplus in the absence of the
medallion system is (a + b + c). This shrinks to just area a under the medallion system —
implying that consumers are made worse off by (b +c) under the medallion system. Owners
of medallions earn rents equal to area b under medallion system — which is surplus to
them. Thus, only area c is lost — and thus becomes the deadweight loss under the medallion
system. Any policy that makes everyone better off would have to compensate owners of
medallions for the loss of the value of their medallions. For instance, the city could buy
back all the medallions at a price equal to the present discounted value of all the income
that the owners could have made from the medallions. Since this is less than the present
discounted value of additional consumer surplus, this would be a deal that would create
more benefit than cost for the city — so long as the city can find a way of raising the money
to buy off the owners of medallions in a relatively non-distortionary way.
B: Let x denote a day’s worth of cab rides and suppose the demand curve for x was given by p =
2500 − (x/100).
(a) Given the fixed supply of 50,000 medallions, what is the price of a day’s worth of cab rides?
Answer: Plugging the fixed supply of x = 50,000 into the demand curve equation, we get
p M = 2500 − (50000/100) = $2,000.
(b) Suppose that the daily cost of operating a cap is $1,500 (in the absence of having to pay for a
medallion). What is the equilibrium daily rental fee for a medallion?
Answer: The equilibrium rental fee is then 2000 − 1500 = $500 per day. (The reasoning for
this is explained in the answer to part A(d).)
(c) Suppose that everyone expects the rental value of a medallion to remain the same into the
future. How much could you sell a medallion for — assuming a daily interest rate of 0.01%?
Answer: Recall that the value of an asset that produced y in income in perpetuity is y /r
where r is the periodic interest rate expressed as a decimal. Here, the owner of a medallion gets $500 per day and the daily interest rate is 0.01% — or just 0.0001. The value of a
medallion is therefore 500/0.0001 = $5,000,000.
(d) How many more cabs would there be on NYC streets if the medallion system were eliminated
(and free entry and exit into the cab business is permitted)?
Answer: Since the marginal cost of operating a cab is $1,500 per day, we know the long run
supply curve is perfectly elastic at $1,500. This further implies that the long run equilibrium
price for a day’s of cab rides must be $1,500. Substituting $1,500 in for p in the demand
curve, we get 1500 = 2500 − (x/100). Solving for x, we determine that the total number of
cabs in the city will double from 50,000 to 100,000.
(e) What is the daily deadweight loss of the medallion system?
Answer: Referring to Graph 18.9, the deadweight loss area c is (p M −p ∗ )(x ∗ −50,000)/2. We
have calculated p M = 2,000, p ∗ = 1,500 and x ∗ = 100,000. Substituting these in, we get
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Elasticities, Price Distorting Policies and Non-Price Rationing
DWL =
(2000 − 1500)(50,000)
= $12,500,000 per day.
2
(18.62)
(f) What do you think is the biggest political obstacle to eliminating the system?
Answer: We estimated the value of each medallion (given the parameters of this example) to
be $5 million. The biggest political obstacle to eliminating the medallions would therefore
seem to be the concentrated losses to those who own the medallions — who have a deep
interest in keeping the system in place.
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676
18.13 Policy Application: Kidney Markets: A large number of patients who suffer from degenerative kidney disease ultimately require a new kidney in order to survive. Healthy individuals have two kidneys but
usually can live a normal life with just a single kidney. Thus, kidneys lend themselves to “live donations”
— i.e. unlike an organ like the heart, the donor can donate the organ while alive (and live a healthy life
with a high degree of likelihood). It is generally not permitted for healthy individuals to sell a kidney —
kidney’s can only be donated for free (with only the medical cost of the kidney transplant covered by the
recipient or his insurance). In effect, this amounts to a price ceiling of zero for kidneys in the market for
kidneys.
A: Consider, then, the supply and demand for kidneys.
(a) Illustrate the demand and supply curves in a graph with kidneys on the horizontal axis and
the price of kidneys on the vertical. Given that there are some that in fact donate a kidney for
free, make sure your graph reflects this.
Answer: This is done in Graph 18.10 where the supply curve has to contain a flat spot on the
horizontal axis in order to account for the fact that some give one of their kidneys away for
free.
Graph 18.10: Kidney Market
(b) Illustrate how the prohibition of kidney sales results in a “shortage” of kidneys.
Answer: In the graph, the quantity of kidneys supplied at a price of zero is x S while the
quantity demanded at that price is x D . The difference is the shortage.
(c) In what sense would permitting the sale of kidneys eliminate this shortage? Does this imply
that no one would die from degenerative kidney disease?
Answer: If the equilibrium price p ∗ were to be permitted to ration kidneys in this market,
the quantity demanded at that price would equal the quantity supplied at that price. In
this sense, there is no shortage. However, this does not imply that no one would die from
kidney failure — as those not willing (or able) to pay the equilibrium price would still not
get kidneys.
(d) Suppose everyone has the same tastes but people differ in terms of their ability to generate
income. What would this imply about how individuals of different income levels line up
along the kidney supply curve in your graph? What does it imply in terms of who will sell
kidneys?
Answer: If everyone has the same tastes, then the income of kidney “suppliers” increases
along the supply curve as we move to the left; i.e. poorer individuals would be willing to
accept lower prices for one of their kidneys than richer people.
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(e) How would patients who need a kidney line up along the demand curve relative to their income? Who would not get kidneys in equilibrium?
Answer: If tastes were again the same, higher income patients would be willing to pay more
than lower income patients — thus lower income patients would be more likely not to get a
kidney than higher income patients.
(f) Illustrate in your graph the lowest that deadweight loss from prohibiting kidney sales might
be assuming that demand curves can be used to approximate marginal willingness to pay.
(Hint: The lowest possible deadweight loss occurs if those who receive donated kidneys under
the price ceiling are also those that are willing to pay the most.)
Answer: This is illustrated as the area (c + d ). This is because, under the price ceiling, only
x S kidneys are donated. If they are donated to those willing to pay the most, total surplus is
(a + b). If the equilibrium price p ∗ rationed kidneys in the market, the number of kidneys
transplanted would rise to x ∗ — giving total surplus of (a + b + c + d ). We lose (c + d ) by
prohibiting the selling of kidneys.
(g) Does the fact that kidneys might be primarily sold by the poor (and disproportionately bought
by well-off patients) change anything about our conclusion that imposing a price ceiling of
zero in the kidney market is inefficient?
Answer: No, it does not. By prohibiting sales that make both seller and buyer better off,
we are prohibiting mutually beneficial trades from occurring — which is the source of the
inefficiency. The fact that sellers might be relatively poor does not take away from the fact
that they value their kidney less than they value the compensation the receive. And the fact
that the kidneys might be disproportionately bought by the rich does not take away from
the fact that they value the kidneys they buy more than the amount they pay for it — and
more than sellers value the kidneys they are selling.
(h) In the absence of ethical considerations that we are not modeling, should anyone object to a
change in policy that permits kidney sales? Why do you think that opposition to kidney sales
is so wide-spread?
Answer: In principle, it is difficult to rationalize objections to permitting kidney sales in
the absence of ethical considerations. To the extent that live kidney donations occur in
the absence of kidney markets, these donations are typically among relatives who would
likely still donate to their loved ones if kidney markets operated. (To the extent that kidney
donations come from donors who have died, this is not the case — and some who would
receive kidneys in this way may not be able to afford to buy a kidney if such kidneys could be
sold). In principle, there may be some who lose in the transition to a market for kidneys —
but the increase in the number of kidneys would likely lead to a large increase in the number
of lives saved. Still, there are many reasons why one might object to permitting kidney sales
— even though this would save many lives. Some might be concerned that some low income
patients who might have received a kidney under the current system might not have the
same chance in a kidney market. Others might be concerned that individuals who sell their
kidneys might not always make such decisions in a rational state of mind. Yet others might
point to potential abuses — with those in hierarchical power relationships able to coerce
participation in kidney markets.
(i) Some people might be willing to sell organs — like their heart — that they cannot live without in order to provide financially for loved ones even if it means that the seller will die as a
result. Assuming that everyone is purely rational, would our analysis of deadweight loss from
prohibiting such sales be any different? I think opposition to permitting such trade of vital
organs is essentially universal. Might the reason for this also, in a less extreme way, be part of
the reason we generally prohibit trade in kidneys?
Answer: In principle, the analysis would be no different at all. To the extent to which there
are potential sellers of their heart who would value the compensation their heirs receive
more than they value their life, we are prohibiting trades that are mutually beneficial — and
are therefore eliminating social surplus that could be generated. The natural opposition to
the operation of such markets is founded in large part on the fact that those who might participate in such markets are likely to not be fully “rational” in the sense that their motives
might emerge from psychological difficulties. Those contemplating suicide, for instance,
would be natural candidates for sellers in this market, and we generally do not consider
Elasticities, Price Distorting Policies and Non-Price Rationing
678
suicide as a rational act. In a less extreme way, concerns over the psychological issues that
might lead one to participate in kidney markets may also play a role in the general opposition to permitting such markets.
B: Suppose the supply curve in the kidney market is p = B + βx.
(a) What would have to be true in order for the phenomenon of kidney donations (at zero price)
to emerge?
Answer: It would have to be the case that B < 0 — implying a negative vertical intercept for
the supply curve (as illustrated through the dashed portion in Graph 18.10).
(b) Would those who donate kidneys get positive surplus? How would you measure this — and
how can you make intuitive sense of it?
Answer: Yes, those who donate at a zero price would still receive surplus — which would
be measured as the triangle that emerges under the horizontal axis. In essence, those who
donate kidneys would in fact be willing to pay to give their kidney away. This is not difficult
to imagine in kidney donations where loved ones are involved.
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18.14 Policy Application: Oil Shocks and Gasoline Prices: In 1973, the OPEC countries sharply reduced
the supply of oil in the world market — raising the price of oil and thus the marginal cost of producing
gasoline in domestic refineries. In 2008, uncertainties over the stability of oil supplies and increasing
demand from developing countries (as well as from oil speculators) also caused sharp increases in the price
of oil — again dramatically increasing the marginal cost of producing gasoline in domestic refineries.
While the causes of higher oil prices differed, the impact on domestic gasoline refineries was similar. Yet
in 1973, vast gasoline shortages emerged, leading cars to line up for miles at gasoline stations and causing
governments to ration gasoline — but in 2008 no such shortages emerged. In this exercise, we explore the
difference between these experiences.
A: The difference is attributable to the following policy intervention used in 1973: In 1973, the government imposed price controls — i.e. price ceilings — in order to combat inflationary pressures, but
in 2008 the government did no such thing.
(a) Consider first the experience of 1973. Begin by drawing the equilibrium in the gasoline market
prior to the oil shock.
Answer: This is done in panel (a) of Graph 18.11 where the initial supply curve is S and the
initial equilibrium price is p ∗ .
Graph 18.11: 1973 vs. 2008
(b) Now illustrate the impact of the OPEC countries’ actions on the domestic gasoline market.
Answer: The increase in the price of oil resulting from the actions of OPEC caused an increase in the marginal cost of producing gasoline — which in turn shifted the supply curve
for gasoline from S to S ′ . In the absence of any government interference, this would lead to
an increase in the price of gasoline from p ∗ to p ′ .
(c) As gasoline prices began to rise, the government put in place a price ceiling between the precrisis price and the price that would have emerged had the government not interfered. Illustrate this price ceiling in your graph.
Answer: This is also illustrated in panel (a) of Graph 18.11 where p represents the government imposed price ceiling. This would make no difference had there been no shift in supply — because the original equilibrium price p ∗ fell below p. But after the shift in supply, the
price ceiling binds in the sense that it falls below what would otherwise be the equilibrium
price p ′ .
(d) If we take into account the cost of time spent in gasoline lines, what was the effective price of
gasoline that consumers faced?
Answer: In the presence of the price ceiling, consumers have to spend additional effort
procuring gasoline, in our case by waiting in line. The point A in panel (a) of Graph 18.11
Elasticities, Price Distorting Policies and Non-Price Rationing
680
does not become a new equilibrium until the equilibrium effort cost on the part of consumers is sufficient to push demand down such that it intersects S ′ at A. This requires a
cost of waiting in line equal to (p ′′ − p) — implying that the effective price, which includes
both the dollar price plus the cost of waiting, rose to p ′′ .
(e) Now consider 2008 when the government did not impose a price ceiling as gasoline prices
nearly quadrupled over a short period. Illustrate the change in equilibrium — and the reason
no shortage emerged.
Answer: This is illustrated in panel (b) of Graph 18.11 where the shift in supply simply moves
us from the original equilibrium B to the new equilibrium C — with price increasing from
p ∗ to p ′ .
(f) Suppose that the 1973 and 2008 shocks to the marginal costs of refineries were identical as
were initial supply and demand curves. If we take into account the cost of waiting in lines for
gasoline in 1973, in which year did the real price of gasoline faced by consumers rise more?
Answer The real cost faced by consumers in 1973 was p ′′ while the real cost faced by consumers in 2008 was p ′ . Since p ′′ > p ′ , the real cost to consumers was higher in 1973 than in
2008 despite the fact that the government actively tried to keep prices down in 1973 but not
in 2008.
(g) When the government compiles statistics on inflation, in which year would it have shown a
larger jump in inflation due to the increase in the price of gasoline?
Answer: When compiling statistics on inflation, the government would not include the cost
of waiting in line, only the cash prices. The cash price of gasoline rose to p in 1973 and to p ′
in 2008. Since p ′ > p, government statistics on inflation due to the price of gasoline would
show a greater increase in 2008 than in 1973 (even though the real cost to consumers rose
more in 1973 than in 2008).
B: Suppose that the demand curve for gasoline in both years is given by p = A−αx while the pre-crisis
supply curve is given by p = B + βx.
(a) Derive the pre-crisis equilibrium price p ∗ .
Answer: Solving the two curves to get demand and supply functions with x in terms of p —
and then setting them equal to one another and solving for p, we get
p∗ =
βA + αB
.
α+β
(18.63)
(b) Suppose the crises in both years cause the supply curve to change to p = C + βx where C > B .
Derive the new equilibrium price p ′ that emerged in 2008.
Answer: Using the same steps as in (a), we get
p′ =
βA + αC
.
α+β
(18.64)
(c) Now consider 1973 when the government imposed a price ceiling p between p ∗ and p ′ . Derive
the real price p ′′ paid by consumers (taking into account the effort cost of waiting in line).
Answer: First, we have to determine the quantity of gasoline sold under the price ceiling.
This is derived by simply plugging the price ceiling p into the supply function (which we get
from the supply curve by solving for x in terms of p). Thus, the quantity of gasoline sold is
x = (p −C )/β. The real cost of gasoline is then derived by plugging x into the demand curve;
i.e.
p ′′ = A − αx = A −
(d) Can you show that p ′′ > p ′ ?
α(p −C ) βA + αC αp
=
−
.
β
β
β
(18.65)
Answer: We can show this by showing that this holds so long as p < p ′ — i.e. so long as the
price ceiling is set below what would otherwise be the equilibrium price. Begin by assuming
that p ′′ > p ′ ; i.e.
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Elasticities, Price Distorting Policies and Non-Price Rationing
βA + αC
βA + αC αp
−
>
.
β
β
α+β
(18.66)
Add (αp)/β to both sides and subtract (βA + αC )/(α + β) from both sides to re-write this as
αp
βA + αC βA + αC
−
>
.
β
α+β
β
(18.67)
Then multiply both numerator and denominator of terms with only β in the denominator
by (α + β), and similarly multiply both numerator and denominator of the term with only
(α + β) in the denominator by β. We can then multiply both sides by β(α + β) to get
(α + β)(βA + αC ) − β(βA + αC ) > α(α + β)p
(18.68)
which further simplifies to
α(βA + αC ) > α(α + β)p.
(18.69)
Dividing both sides by α(α + β), we then get
βA + αC
> p.
α+β
(18.70)
Note that the left-hand side is just equal to p ′ — which implies that the statement (18.66)
we started with is true so long as the price ceiling is set below what would otherwise be the
equilibrium price; i.e. p ′′ > p ′ so long as p < p ′ .