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CHAPTER 2
THE BASICS OF SUPPLY AND DEMAND
EXERCISES
1. Consider a competitive market for which the quantities demanded and supplied (per year) at various prices are given as follows:
Price ($)
Demand (millions)
Supply (millions)
60 22 14 80 20 16 100 18 18 120 16 20
a. Calculate the price elasticity of demand when the price is $80. When the price is $100. We know that the price elasticity of demand may be calculated using equation 2.1 from the text:
E
Q
Q
P
P
P
Q
Q
P
D
D D D
= = D
With each price increase of $20, the quantity demanded decreases by 2. Therefore,
∆ Q D
∆ P
At P = 80, quantity demanded equals 20 and
ED =^
Similarly, at P = 100, quantity demanded equals 18 and
ED =^
b. Calculate the price elasticity of supply when the price is $80. When the price is $100. The elasticity of supply is given by:
E
Q
Q
P
P
P
Q
Q
S P
S S S
= = S
With each price increase of $20, quantity supplied increases by 2. Therefore,
∆ Q S
∆ P
At P = 80, quantity supplied equals 16 and
ES =^
Similarly, at P = 100, quantity supplied equals 18 and
ES =^
c. What are the equilibrium price and quantity?
The equilibrium price and quantity are found where the quantity supplied equals the quantity demanded at the same price. As we see from the table, the equilibrium price is $100 and the equilibrium quantity is 18 million.
d. Suppose the government sets a price ceiling of $80. Will there be a shortage, and, if so, how large will it be? With a price ceiling of $80, consumers would like to buy 20 million, but producers will supply only 16 million. This will result in a shortage of 4 million.
2. Refer to Example 2.4 on the market for wheat. At the end of 1998, both Brazil and Indonesia opened their wheat markets to U.S. farmers. (Source: http://www.fas.usda.gov/) Suppose that these new markets add 200 million bushels to U.S. wheat demand. What will be the free market price of wheat and what quantity will be produced and sold by U.S. farmers in this case?
The following equations describe the market for wheat in 1998: QS = 1944 + 207 P and QD = 3244 - 283 P.
If Brazil and Indonesia add an additional 200 million bushels of wheat to U.S. wheat
demand, the new demand curve Q′D , would be equal to QD + 200, or
Q ′D = (3244 - 283 P ) + 200 = 3444 - 283 P.
Equating supply and the new demand, we may determine the new equilibrium price, 1944 + 207 P = 3444 - 283 P , or 490 P = 1500, or P* = $3.06 per bushel. To find the equilibrium quantity, substitute the price into either the supply or demand equation, e.g., QS = 1944 + (207)(3.06) = 2,577. and QD = 3444 - (283)(3.06) = 2,577.
3. A vegetable fiber is traded in a competitive world market, and the world price is $9 per pound. Unlimited quantities are available for import into the United States at this price. The U.S. domestic supply and demand for various price levels are shown below.
Price U.S. Supply U.S. Demand (million lbs.) (million lbs.)
3 2 34 6 4 28 9 6 22 12 8 16 15 10 10
To find the free market price for apartments, set supply equal to demand: 100 - 5 P = 50 + 5 P , or P = $500, since price is measured in hundreds of dollars. Substituting the equilibrium price into either the demand or supply equation to determine the equilibrium quantity: QD = 100 - (5)(5) = 75 and QS = 50 + (5)(5) = 75.
We find that at the rental rate of $500, 750,000 apartments are rented. If the rent control agency sets the rental rate at $100, the quantity supplied would then be 550,000 ( QS = 50 + (5)(1) = 55), a decrease of 200,000 apartments from the free market equilibrium. (Assuming three people per family per apartment, this would imply a loss of 600,000 people.) At the $100 rental rate, the demand for apartments is 950,000 units, and the resulting shortage is 400,000 units (950,000-550,000). The city population will only fall by 600,000 which is represented by the drop in the number of apartments from 750.000 to 550,000, or 200,000 apartments with 3 people each. These are the only people that were originally in the City to begin with.
Appartments
(10,000’s)
Rent
Excess
Demand
Demand Supply
Figure 2.
b. Suppose the agency bows to the wishes of the board and sets a rental of $900 per month on all apartments to allow landlords a “fair” rate of return. If 50 percent of any long-run increases in apartment offerings comes from new construction, how many apartments are constructed? At a rental rate of $900, the supply of apartments would be 50 + 5(9) = 95, or 950, units, which is an increase of 200,000 units over the free market equilibrium. Therefore, (0.5)(200,000) = 100,000 units would be constructed. Note, however, that since demand is only 550,000 units, 400,000 units would go unrented.
5. Much of the demand for U.S. agricultural output has come from other countries. From Example 2.4, total demand is Q = 3244 - 283 P****. In addition, we are told that domestic demand is Qd = 1700 - 107 P****. Domestic supply is QS = 1944 + 207 P****. Suppose the export demand for
wheat falls by 40 percent.
a. U.S. farmers are concerned about this drop in export demand. What happens to the free market price of wheat in the United States? Do the farmers have much reason to worry? Given total demand, Q = 3244 - 283 P , and domestic demand, Qd = 1700 - 107 P , we may subtract and determine export demand, Qe = 1544 - 176 P.
The initial market equilibrium price is found by setting total demand equal to supply: 3244 - 283 P = 1944 + 207 P , or P = $2.65. The best way to handle the 40 percent drop in export demand is to assume that the export demand curve pivots down and to the left around the vertical intercept so that at all prices demand decreases by 40 percent, and the reservation price (the maximum price that the foreign country is willing to pay) does not change. If you instead shifted the demand curve down to the left in a parallel fashion the effect on price and quantity will be qualitatively the same, but will differ quantitatively. The new export demand is 0.6Qe=0.6(1544-176P)=926.4-105.6P. Graphically, export demand has pivoted inwards as illustrated in figure 2.5a below.
Q e
P
Figure 2.5a Total demand becomes QD = Qd + 0.6 Qe = 1700 - 107 P + (0.6)(1544 - 176 P ) = 2626.4 - 212.6 P.
Equating total supply and total demand, 1944 + 207 P = 2626.4 - 212.6 P , or P = $1.63, which is a significant drop from the market-clearing price of $2.65 per bushel. At this price, the market-clearing quantity is 2280.65 million bushels. Total revenue has decreased from $6614.6 million to $3709.0 million. Most farmers would worry.
b. Now suppose the U.S. government wants to buy enough wheat each year to raise the price to $3.50 per bushel. With this drop in export demand, how much wheat would the government have to buy each year? How much would this cost the government?
ED = − b
P *
Q *
. Here ED = -0.4 (the long-run price elasticity), P* = 0.75 (the
equilibrium price), and Q* = 7.5 (the equilibrium quantity). Solving for b ,
−0 .4 = − b
- 75
, or b = 4.
To find the intercept, we substitute for b , QD (= Q *), and P (= P *) in the demand equation: 7.5 = a - (4)(0.75), or a = 10.5. The linear demand equation consistent with a long-run price elasticity of -0.4 is therefore QD = 10.5 - 4 P.
b. Using this demand curve, recalculate the effect of a 20 percent decline in copper demand on the price of copper. The new demand is 20 percent below the original (using our convention that quantity demanded is reduced by 20% at every price):
Q ′D = (^ 0.8)^ (10 .5^ − 4 P )^ = 8 .4 − 3 .2 P.
Equating this to supply, 8.4 - 3.2 P = -4.5 + 16 P , or P = 0.672. With the 20 percent decline in the demand, the price of copper falls to 67.2 cents per pound.
8. Example 2.8 analyzes the world oil market. Using the data given in that example,
a. Show that the short-run demand and competitive supply curves are indeed given by
D = 24.08 - 0.06 P SC = 11.74 + 0.07 P****.
First, considering non-OPEC supply: Sc = Q * = 13.
With ES = 0.10 and P * = $18, ES = d ( P */ Q *) implies d = 0.07.
Substituting for d , Sc , and P in the supply equation, c = 11.74 and Sc = 11.74 + 0.07 P.
Similarly, since QD = 23, ED = - b ( P */ Q *) = -0.05, and b = 0.06. Substituting for b , QD = 23, and P = 18 in the demand equation gives 23 = a - 0.06(18), so that a = 24.08. Hence QD = 24.08 - 0.06 P.
b. Show that the long-run demand and competitive supply curves are indeed given by
D = 32.18 - 0.51 P SC = 7.78 + 0.29 P****.
As above, ES = 0.4 and ED = -0.4: ES = d ( P / Q ) and _ED = -b(P/Q)_ , implying 0.4 = d (18/13) and -0.4 = - b (18/23). So d = 0.29 and b = 0.51. Next solve for c and a : Sc = c + dP and QD = a - bP , implying 13 = c + (0.29)(18) and 23 = a - (0.51)(18).
So c = 7.78 and a = 32.18.
c. During the late 1990s, Saudi Arabia accounted for 3 billion barrels per year of OPEC’s production. Suppose that war or revolution caused Saudi Arabia to stop producing oil. Use the model above to calculate what would happen to the price of oil in the short run and the long run if OPEC’s production were to drop by 3 billion barrels per year.
With OPEC’s supply reduced from 10 bb/yr to 7 bb/yr, add this lower supply of 7 bb/yr to the short-run and long-run supply equations: Sc ′ = 7 + Sc = 11.74 + 7 + 0.07 P = 18.74 + 0.07 P and S ″ = 7 + Sc = 14.78 + 0.29 P.
These are equated with short-run and long-run demand, so that:
18.74 + 0.07 P = 24.08 - 0.06 P ,
implying that P = $41.08 in the short run; and
14.78 + 0.29 P = 32.18 - 0.51 P ,
implying that P = $21.75 in the long run.
9. Refer to Example 2.9, which analyzes the effects of price controls on natural gas.
a. Using the data in the example, show that the following supply and demand curves did indeed describe the market in 1975: Supply: Q = 14 + 2 PG + 0.25 PO
Demand: Q = -5 PG + 3.75 PO
where PG and PO are the prices of natural gas and oil, respectively. Also, verify that if the price of oil is $8.00, these curves imply a free market price of $2.00 for natural gas. To solve this problem, we apply the analysis of Section 2.6 to the definition of cross-price elasticity of demand given in Section 2.4. For example, the cross-price-elasticity of demand for natural gas with respect to the price of oil is:
EGO =^
∆ QG ∆ PO
PO
QG
.
∆ Q G
∆ PO
is the change in the quantity of natural gas demanded, because of a small
change in the price of oil. For linear demand equations,
∆ Q G
∆ PO
is constant.^ If we
represent demand as: QG = a - bPG + ePO
(notice that income is held constant), then
∆ Q^ G
∆ PO
=^ e.^ Substituting this into the cross-
price elasticity, EPO = e
PO
QG
, where^ PO
We know that PO^ * = $8 and QG^ * = 20 trillion cubic feet (Tcf). Solving for e ,
Equating supply and demand and solving for the equilibrium price, 18 + 2 PG = 60 - 5 PG , or PG = $6.
The price of natural gas would have tripled from $2 to $6.
10. The table below shows the retail price and sales for instant coffee and roasted coffee for 1997 and 1998.
Retail Price of Sales of Retail Price of Sales of Instant Coffee Instant Coffee Roasted Coffee Roasted Coffee Year ($/lb) (million lbs) ($/lb) (million lbs)
1997 10.35 75 4.11 820 1998 10.48 70 3.76 850
a. Using this data alone, estimate the short-run price elasticity of demand for roasted coffee. Also, derive a linear demand curve for roasted coffee. To find elasticity, you must first estimate the slope of the demand curve: ∆ Q ∆ P^
=
820 − 850 4.11 (^) − 3.
= −
= −85 .7.
Given the slope, we can now estimate elasticity using the price and quantity data from the above table. Since the demand curve is assumed to be linear, the elasticity will differ in 1997 and 1998 because price and quantity are different. You can calculate the elasticity at both points and at the average point between the two years:
Ep^97 =
P
Q
∆ Q ∆ P^
=
( (^) −85.7) (^) = −0.
Ep
98
P
Q
∆ Q ∆ P
=
(−85 .7) = −0.
EpAVE^ =
P 97 + P 98
Q 97 + Q 98
∆ Q ∆ P^
=
(−85 .7) (^) = −0..
To derive the demand curve for roasted coffee, note that the slope of the demand curve is -85.7=-b. To find the coefficient a, use either of the data points from the table above so that a=830+85.74.11=1172.3 or a=850+85.73.76=1172.3. The equation for the demand curve is therefore Q=1172.3-85.7P.
b. Now estimate the short-run price elasticity of demand for instant coffee. Derive a linear demand curve for instant coffee. To find elasticity, you must first estimate the slope of the demand curve: ∆ Q ∆ P
=
75 − 70 10 .35 − 10.
= −
= −38 .5.
Given the slope, we can now estimate elasticity using the price and quantity data from the above table. Since the demand curve is assumed to be linear, the elasticity will differ in 1997 and 1998 because price and quantity are different. You can calculate the elasticity at both points and at the average point between the two years:
Ep^97 =
P
Q
∆ Q ∆ P^
=
(−38 .5) (^) = −5.
Ep^98 =
P
Q
∆ Q ∆ P^
=
(−38 .5) (^) = −5.
Ep
AVE
P 97 +^ P 98
Q 97 + Q 98
∆ Q ∆ P
=
(−38 .5) = −5..
To derive the demand curve for instant coffee, note that the slope of the demand curve is -38.5=-b. To find the coefficient a, use either of the data points from the table above so that a=75+38.510.35=473.1 or a=70+38.510.48=473.1. The equation for the demand curve is therefore Q=473.1-38.5P.
c. Which coffee has the higher short-run price elasticity of demand? Why do you think this is the case? Instant coffee is significantly more elastic than roasted coffee. In fact, the demand for roasted coffee is inelastic and the demand for instant coffee is elastic. Roasted coffee may have an inelastic demand in the short-run as many people think of coffee as a necessary good. Instant coffee on the other hand may be viewed by many people as a convenient, though imperfect substitute for roasted coffee. Given the higher price per pound for instant coffee, and the preference for roasted over instant coffee by many consumers, this will cause the demand for roasted coffee to be less elastic than the demand for instant coffee. Note also that roasted coffee is the premium good so that the demand for roasted coffee lies to the right of the demand for instant coffee. This will cause the demand for roasted coffee to be more inelastic because at any given price there will be a higher quantity demanded for roasted versus instant coffee, and this quantity difference will be large enough to offset the difference in the slope of the two demand curves.
