Problem Set No. 2
The demand schedule (or demand function) for a good shows the total quantities (q) that buyers are willing to buy at various alternative prices (p) in some period of time. For example, here is a demand function illustrating the very special but convenient case of a linear demand (with q measured in some physical unit of quantity such as bushels or tons and with p measured in dollars per unit):
q = 2,100 - 50p.
Sometimes it is convenient to express this in the inverse form showing the prices that buyers are willing to pay for various quantities. This is called a demand-price function.
1. State the demand-price function corresponding to the given demand function.
2. Plot the corresponding demand schedule on graph paper with q on the horizontal axis (up to 3,000 units) and p on the vertical axis (up to $60 per unit). Label this demand schedule D1.
I. The Case of Fixed Supply
A supply schedule (or function or curve), defined analogously, shows the total quantities (q) that sellers are willing to sell at various alternative prices (p) in the given period of time. One very special case is that of a "fixed supply" where the quantity supplied is a constant, independent of the price, such as:
q = 1,200.
(This type of supply may apply, for example, in the short period when a given quantity of a perishable commodity is brought to market and must be sold at any price or go to waste. In a slightly different meaning of supply, again in the short run, it may apply to a service such as housing or, even in the long run, to the services of a permanent resource such as land.)
3. Plot this supply schedule on your diagram and label it S1. Determine the equilibrium price where demand (which you have already plotted on your diagram and labelled D1) equals supply:
4. Suppose that the demand now increases suddenly to:
q = 2,700 - 50p.
State the corresponding demand-price function, and plot it on your diagram, labeling it D2.
5. Determine the new equilibrium price:
6. Suppose that consumers, indignant about this price increase, persuade their government to institute price control and roll back the price to its former level. Distinguishing quantities demanded and supplied by the superscripts d and s show that these differ:
7. Be prepared to discuss the following statements:
"In a market where prices are determined by demand and supply, a price ceiling below the unregulated equilibrium price helps consumers in one respect but hurts them in another."
"Prices are a device for rationing available supplies. If prices are prevented from discharging this function, then the market must turn to some other form of rationing whether formal or informal, orderly or disorderly."
II. The Case of Constant Cost
As another special case, assume that, in the long run, any relevant quantity of the good can be produced at the constant cost of $18 per unit. (This unit cost is a "full cost," i.e., the minimum cost per unit that producers must be able to cover if they are to be willing to go on producing indefinitely.) This implies that the industry's supply-price function is given by:
p = 18.
8. Plot this on your diagram, labeling it S2, and note the equilibrium quantity (again using demand curve D1) where demand price equals supply price:
9. If the demand now increases as before to D2 (q = 2,700 - 50p), calculate the new equilibrium price and quantity:
10. If, alternatively, the government prevents this increase of output by requiring a license or permit for each unit of output, how much will a license be worth per unit of output per period of time if only the original output is allowed (How much would you pay for the right to produce a unit?):
Value of license per unit of output per period =
III. The Case of Increasing Cost
As a somewhat more general case, assume that the supply function is positively slopedthough still linear for convenience:
q = -600 + 100p where p > 6.
11. State the inverse of this function, i.e., the minimum prices that producers must receive if they are to be willing to go on producing various alternative quantities:
Supply curves may be positively sloped in the long run because, as the industry's output expands, (a) the prices of needed labor, raw material, etc., are bid up or (b) the outputs of higher cost producers are needed. In the short run, supply curves are almost certainly positively sloped as the industry's output approaches its current capacity.
12. Plot this supply schedule on your diagram with the label S3. Again using the demand curve D1, calculate the equilibrium price and quantity:
13. If the demand now increases as before to D2 (q = 2,700 - 50p), calculate the new equilibrium price and quantity:
14. Be prepared to discuss this statement:
"In a market governed by the law of supply and demand, an increase in demand may increase (a) the price but not the quantity bought, (b) the quantity bought but not the price, (c) both the price and quantity, (d) neither the price nor the quantity."
IV. The Effects of Specific Taxes
A specific tax is a tax of so many dollars or cents per physical units of the good. If a specific tax of amount t is imposed, the new equilibrium condition is:
pd - ps = t.
That is to say, the demand price paid by buyers, inclusive of the tax (pd), must exceed the sellers' net supply price (ps) by the amount of the tax.
With the original demand (D1) in each of the foregoing three cases, assume that a tax of t = $6 per unit is imposed.
15. In constant-cost Case II, calculate the effect on the gross price paid by buyers (pd), the net price received by sellers (ps), the quantity bought and sold (q), the total tax revenue (T), and the net change in price paid by buyers (Dpd)(how much did the buyers pay before the tax, how much after, the difference is Dpd:)
16. Do the same in the increasing-cost Case III:
17. Do the same in the fixed-supply Case I:
18. Be prepared to discuss the following statements:
"In an industry governed by supply and demand, the imposition of a specific tax may cause the price paid by consumers to increase (a) by an amount equal to the tax, (b) by an amount less than the tax, or (c) not at all."
"Furthermore, the same tax may cause the price paid by consumers, successively, (a) to remain unchanged in a very short run, (b) then to rise by less than the tax in an ordinary short run, and (c) eventually to rise by the full amount of the tax in the long run."