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Eco 301
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Problem Set 3 Key
21 September 2007
1. Suppose that the U.S. demand and supply for soft drinks are given by
QD = 300 - 2 P
QS = 4 P
The price is in cents per can and quantity is in millions of cans per day.
a. Using a ruler, graph the demand and supply lines for this market. Indicate at least two points on each
line (such as their intercepts along the axes). (Let each line in the grid represent 10 cents along the price
axis and 20 million cans along the quantity axis.)
b. Solve algebraically for the equilibrium price and quantity. How much do consumers spend on soft
drinks per day (in millions of dollars)?
QD = 300 - 2 P
Equilibrium: QD = QS
and
QS = 4 P
300 - 2 P = 4 P
300 = 6 P
P = 50
We can use either the formula for QD or QS to solve for equilibrium Q. For example,
Q = 4 P = 4 (50) = 200
Total Expenditure = P Q = ($0.50)(200) = $100 million per day
c. Calculate the price elasticity of demand at the equilibrium (using the equilibrium price and quantity as
the “base values”). Is demand elastic or inelastic? Explain briefly.
The straight-line demand curve is QD = 300 - 2 P . Thus,
dQ D
= − 2 at all points along it.
dP
The price elasticity of demand at the equilibrium is given by
ηD =
dQD
P
− 2 × 50
1
×
=
=−
dP
QD
(200)
2
Demand is inelastic, since the elasticity is between zero and -1.
d. Suppose that supply now becomes QS = 60 + 4 P. Has supply increased or decreased? Explain
briefly. Calculate algebraically the new equilibrium price, quantity, and consumer expenditure.
Supply has increased. At every price, QS is higher by 60 million cans per day.
Equilibrium: QD = QS
300 - 2 P = 60 + 4 P
240 = 6 P
P = 40
Q = 60 + 4 P = 60 + 4 (40) = 220
Total Expenditure = P Q = ($0.40)(220) = $88 million per day
e. Has consumer expenditure gone up or down? Is this consistent with your finding about the price
elasticity of demand at the initial equilibrium? Explain briefly.
Consumer expenditure has gone down. This is consistent with the earlier finding that demand is
inelastic at the initial equilibrium. If demand is inelastic, then a decrease in price will lead to a
decrease in expenditure: quantity demanded does not increase by much relative to the decrease
in price.
f. Now assume that the original supply and demand curves apply once more, but suppose that the
government decides to levy an excise tax of 30 cents per can on soft drinks. Solve algebraically for the
new equilibrium supply price (PS), demand price (PD), quantity transacted, and tax revenues. (You may
wish to examine your diagram to see if your calculations make sense.)
In the case of an excise tax, the relationship between the supply and demand prices becomes
PS = PD - T ,
and in this case the tax per unit, T, is equal to 30. So now we have
QD = 300 - 2 PD (PD is the price that demanders pay)
QS = 4 P S
(PS is the price that suppliers receive)
Substituting for PS , and imposing our usual equilibrium condition (QD = QS), we get
300 – 2 PD = 4 (PD – 30)
Q = 4 PS = 4 (40) = 160
420 = 6 PD
PD = 70
PS = PD – 30 = 40
Tax revenues = T Q = ($0.30)(160) = $48 million per day
g. Explain intuitively why the supply price, demand price, and quantity have changed as you found in part
f. In particular, does the price paid by consumers go up by the amount of the tax per unit? Explain why
or why not.
Imposition of an excise tax puts a wedge between the price that suppliers receive and the price
that demanders pay. Because demand slopes downward and supply slopes upward in this case, the
burden of the tax will be shared between demanders and suppliers: each side of the market has
alternatives, and thus can avoid the tax to some extent. As a consequence, quantity transacted
goes down, the demand price goes up, and the supply price goes down.
Extra moral credit: (No worries: questions of this sort will not appear on examinations). Find the tax per
can that would maximize tax revenues for the government. Find the supply price, demand price, quantity
transacted, and tax revenues, given that tax per can.
We proceed as in part f above, except that now we do not set T = 30 but rather leave T as a
variable to be solved for. Thus, the equilibrium condition becomes
300 – 2 PD = 4 (PD – T)
300 = 6PD – 4T
PD = (300 + 4T)/6
We can use this to solve for PD as a function of T:
PD = (300 + 4 T)/6.
This then allows us to solve for quantity as a function of T:
Q = 300 – 2 PD = 300 – 2 (300 + 4 T)/6 = 200 – (8/6) T
Tax revenues will equal R = T Q = 200 T – (4/3) T2.
Take the derivative of R with respect to T and set it equal to 0:
dR/dT = 200 – (8/3) T = 0
T = 600/8 = 75
Thus, the tax per can that maximizes tax revenues is equal to 75 cents per can. We can use our
expression above to solve for the demand price:
PD = (300 + 4 (75))/6 = 100
PS = PD – T = 100 – 75 = 25
Q = 4 PS = 4 (25) = 100
Total tax revenues becomes R = T Q = ($0.75)(100) = $75 million per day, which is higher than the
revenues we calculated in part f.
Since the supply of tickets is fixed at 34,000, the supply curve is completely inelastic, i.e.
vertical. Assuming a normal, downward sloping demand curve, a tax on consumers shifts the
demand curve down by a vertical distance equal to the amount of the tax—here that is $5.
2.
Without the tax, the equilibrium price would be P*--this is what consumers would pay and what
producers would receive. With the tax, the demand curve shifts down by $5 and the amount
producers receive is reduced to PP. However, the amount consumers must pay is PC, which is
exactly what they would pay without the tax, P*. Hence, the burden on consumers, PC - P*, is
zero, while the burden on producers is P* - PP= $5. This results from the fact that supply is
completely inelastic—the Red Sox cannot alter supply in response to changes in price. So they
absorb the entire tax in lower ticket prices.
3. Harding Enterprises has developed a new product called the Gillooly Shillelagh. The market demand
for this product is given as follows:
Q = 240 - 4P
a. At what price is the price elasticity of demand equal to zero?
The price elasticity of demand is zero where the price is zero. A zero price implies that
quantity demanded is 240.
b. At what price is demand infinitely elastic?
The price elasticity of demand is infinite where the quantity demanded is zero. An infinite
price elasticity occurs at a price of $60.
c. At what price is the price elasticity of demand equal to one?
For a linear demand curve the price elasticity of demand is equal to one (absolute value) at
the midpoint on the demand curve. Quantity demanded is 120 and price is $30.
d. If the shillelagh is priced at $40, what is the point price elasticity of demand?
At a price of $40 consumers buy 80 Gillooly Shillelaghs per year. The point elasticity is
determined from the formula
(dq/dp)×(P/Q) = -4×(40/80) = 4(0.5) = -2
4. Suppose the demand curve for a product is given by Q = 10 - 2P + PS where P is the price of the product
and PS is the price of a substitute good. The price of the substitute good is $2.00.
(a) Suppose P = $1.00. What is the price elasticity of demand? What is the cross-price elasticity of
demand?
First you need to find the quantity demanded at the price of $1.00. Q = 10 - 2(1) + 2
= 10.
Price elasticity of demand = η D =
dQD
P
− 2 ×1
2
×
=
=−
= − 0 .2
dP
QD
(10)
10
Cross-price elasticity of demand = ηx =
dQD
Ps
2
2
×
=
× (1) =
= 0 .2
dPs
QD
10
10
(b) Suppose the price of the good P increases to $2.00. Now what is the price elasticity of demand, and
what is the cross-price elasticity of demand?
First you need to find the quantity demanded at the price of $2.00. Q = 10 - 2(2) + 2
= 8.
Price elasticity of demand = η D =
dQD
P
2
4
×
= × ( − 2 ) = − = − 0 .5
dP
QD
8
8
Cross-price elasticity of demand = ηx =
dQD
Ps
2
2
×
= × (1) = = 0.25
dPs
QD
8
8
5. The inverse demand curve for product X is given by:
PX = 25 - 0.005Q + 0.15PY,
where PX represents price in dollars per unit, Q represents rate of sales in pounds per week, and PY
represents selling price of another product Y in dollars per unit. The inverse supply curve of product X is
given by:
PX = 5 + 0.004Q.
a. Determine the equilibrium price and sales of X. Let PY = $10.
Equate supply to demand to calculate Q.
25 - 0.005Q + 0.15(10) = 5 + 0.004Q
21.5 = 0.009Q
Q = 2,388.9 units per week
At Q = 2,388.9, P = 25 - .005(2,388.9) + 0.15(10) = $14.56 per unit.
b. Determine whether X and Y are substitutes or complements.
Since we can solve for quantity demanded as a function of prices,
Q = (25 + 0.15Py – Px) ÷ 0.005
we see that there is a direct, positive relationship between Q and Py.
Increases in the price of good Y raise the demand for good X. This implies that goods Y
and X are substitutes.
6. The market for digestive biscuits is described by the following equations: QD = 150 - P and
QS = 2P. The Digestives Minister has decided that he wants to reduce the consumption of
digestive biscuits as much as possible. The minister is considering either imposing a $40 per
unit price ceiling or a per unit tax of $15 on consumers of digestives.
a. What is the equilibrium price and quantity before any policy is imposed?
Set supply equal to demand and solve for P or Q.
QD = QS
150 – P = 2P
3P = 150
P = $50
Therefore in equilibrium P = $50 and Q = 100.
b. Which of the two policies better achieves the minister’s objective and why?
Since the price ceiling is lower than the equilibrium price, we know it is binding. If the price
ceiling is imposed, then QS = 2*40 = 80 and QD = 110. QS will be the quantity transacted, and
there will be excess demand in the market.
If the tax is imposed, then the consumer demand curve will shift left by the amount of the tax, so
that the after-tax demand curve is QD = 150 – P – tax = 135 – P. Set the after-tax demand equal to
the supply and you get
135 – P = 2P
3P = 135
P = $45 Implying that Q = 90 under the tax.
Since the goal is to reduce consumption as much as possible, the minister should impose
the price ceiling.
c. Suppose the minister implements the $15 per unit sales tax on consumers. What is the
government’s revenue?
Revenue will equal quantity sold times the tax, or 90* $15 = $1350.
d. Suppose the minister imposes a $15 per unit sales tax on producers instead of consumers.
Compared to the tax on consumers, does the market price and quantity change? Explain.
If the tax is imposed on producers the supply curve will shift left, and QS = 2 (P – tax) or QS = 2
(P – 15). Next, set supply equal to demand—
150 – P = 2 (P-15)
3P = 180
P = $60 Q = 90
This price is different than the price when the tax is on consumers, as in part b. However, we
know that shifting this tax from consumers to producers has no effect on the effective prices paid
by consumers and received by producers. When the tax is on the consumer, the effective or total
price paid is the market price plus the tax that the consumer must pay to the government. In part
b, then, the price paid by the consumer is the market price of $45 plus the $15 tax for a total
price of $60, just as it is here.
7. Consider the market for rental housing in Boston. Suppose that all houses are equal in Boston
and that the mayor wishes to assure that its citizens can afford housing. Consider the following
two ways of pursuing this goal. For each method, predict what effect each proposal would have
on the price and quantity of rental housing. Explain your answers carefully and illustrate
graphically the changes that occur.
a. The mayor offers a subsidy to all builders of homes.(Note: homes built are used for rental
housing.)
A subsidy to builders of homes reduces the cost of building homes and shifts the supply curve to
the right. This results in lower prices and an increase in the number of rental units. The new
equilibrium price (rent) is P2* and the new equilibrium quantity is Q2*. Producers (i.e: landlords)
receive P2* plus the subsidy (which is higher than in the initial situation) and consumers pay P2*
(which is lower than in the initial situation).
b. The mayor provides a subsidy directly to renters.
The subsidy to renters shifts the demand curve for apartments to the right resulting in higher
rents and a greater number of apartments. The new equilibrium price (rent) is P2* and the new
equilibrium quantity is Q2*. Producers (i.e: landlords) receive P2* (which is higher than in the
initial situation) and consumers pay P2* minus the subsidy (which is lower than in the initial
situation).
c. Who gains and who looses from the implementation of policy a. and b.?
Subsides to builders or renters lead to an increase in the number of rental units available.
Effective rents paid by consumers decrease and effective rents received by landlords increase.
Note that if we did a general equilibrium analysis we should also have considered how the
governments raised the money to finance the subsidies and who benefits and who gains from the
government’s financing decision.