Note: These problems focus on the material from the Appendix to Chapter 1. Hence they are primarily numerical.
1.1The following data represent 5 points on the supply curve for orange juice:
PRICE
($ PER GALLON)
QUANTITY (MILLIONS OF GALLONS)
1 100
2 300
3 500
4 700
5 900
and these data represent 5 points on the demand curve for orange juice:
PRICE
($ PER GALLON)
QUANTITY (MILLIONS OF GALLONS)
1 700
2 600
3 500
4 400
5 300
a. Graph the points of these supply and demand curves for orange juice. Be sure to put price on the vertical axis and quantity on the horizontal axis.
b. Do these points seem to lie along two straight lines? If so, figure out the precise algebraic equation of these lines. (Hint: If the points do lie on straight lines, you need only consider two points on each of them to calculate the lines.) c. Use your solutions from part b to calculate the
‘‘excess demand’’ for orange juice if the market price is zero.
d. Use your solutions from part b to calculate the
‘‘excess supply’’ of orange juice if the orange juice price is $6 per gallon.
1.2 Marshall defined an equilibrium price as one at which the quantity demanded equals the quantity supplied.
a. Using the data provided in problem 1.1, show that P¼3 is the equilibrium price in the orange juice market.
b. Using these data, explain whyP¼2 andP¼4 are not equilibrium prices.
c. Graph your results and show that the supply- demand equilibrium resembles that shown in Figure 1.3.
d. Suppose the demand for orange juice were to increase so that people want to buy 300 mil- lion more gallons at every price. How would that change the data in problem 1.1? How would it shift the demand curve you drew in part c?
e. What is the new equilibrium price in the orange juice market, given this increase in demand?
Show this new equilibrium in your supply- demand graph.
f. Suppose now that a freeze in Florida reduces orange juice supply by 300 million gallons at every price listed in problem 1.1. How would this shift in supply affect the data in problem 1.1? How would it affect the algebraic supply curve calculated in that problem?
g. Given this new supply relationship together with the demand relationship shown in pro- blem 1.1, what is the equilibrium price in this market?
h. Explain whyP¼3 is no longer an equilibrium in the orange juice market. How would the participants in this market knowP¼3 is no longer an equilibrium?
i. Graph your results for this supply shift.
1.3The equilibrium price in problem 1.2 isP¼3. This is an equilibrium because at this price, quantity demanded is precisely equal to quantity supplied (Q¼500). One might ask how the market is to reach this equilibrium point. Here we look at two ways:
a. Suppose an auctioneer calls out prices (in dol- lars per gallon) in whole numbers ranging from
$1 to $5 and records how much orange juice is demanded and supplied at each such price. He or she then calculates the difference between quantity demanded and quantity supplied.
You should make this calculation and then describe how the auctioneer will know what the equilibrium price is.
b. Now suppose the auctioneer calls out the var- ious quantities described in problem 1.1. For each quantity, he or she asks, ‘‘What will you demanders pay per gallon for this quantity of orange juice?’’ and ‘‘How much do you sup- pliers require per gallon if you are to produce this much orange juice?’’ and records these dollar amounts. Use the information from problem 1.1 to calculate the answers that the auctioneer will get to these questions. How
will he or she know when an equilibrium is reached?
c. Can you think of markets that operate as described in part a of this problem? Are there markets that operate as described in part b?
Why do you think these differences occur?
1.4In several places, we have warned you about the decision of Marshall to ‘‘reverse the axes’’ by putting price on the vertical axis and quantity on the horizon- tal axis. This problem shows that it makes very little difference how you choose the axes. Suppose that quantity demanded is given by QD¼ Pþ10, 0 P10, and quantity is supplied by QS¼P2, P2.
a. Why are the possible values forPrestricted as they are in this example? How do the restric- tions onPalso impose restrictions onQ?
b. Graph these two equations on a standard (Marshallian) supply-demand graph. Use this graph to calculate the equilibrium price and quantity in this market.
c. Graph these two equations with price on the horizontal axis and quantity on the vertical axis. Use this graph to calculate equilibrium price and quantity.
d. What do you conclude by comparing your answers to parts a and b?
e. Can you think of any reasons why you might prefer the graph part a to that in part b?
1.5 This problem involves solving demand and supply equations together to determine price and quantity.
a. Consider a demand curve of the form QD¼ 2Pþ20
whereQDis the quantity demanded of a good and P is the price of the good. Graph this demand curve. Also draw a graph of the supply curve
QS¼2P4
whereQSis the quantity supplied. Be sure to put Pon the vertical axis andQon the hor- izontal axis. Assume that all theQSandPs are nonnegative for parts a, b, and c. At what values ofPandQdo these curves intersect—
that is, where doesQD¼QS?
b. Now suppose at each price that individuals demand four more units of output—that the demand curve shifts to
QD0¼ 2Pþ24
Graph this new demand curve. At what values ofPandQdoes the new demand curve inter- sect the old supply curve—that is, where does QD0¼QS?
c. Now, finally, suppose the supply curve shifts to QS0¼2P8
Graph this new supply curve. At what values of PandQ doesQD’¼QS’? You may wish to refer to this simple problem when we discuss shifting supply and demand curves in later sec- tions of this book.
1.6 Taxes in Oz are calculated according to the formula
T¼:01I2
whereTrepresents thousand of dollars of tax liability and I represents income measured in thousands of dollars. Using this formula, answer the following questions:
a. How much in taxes is paid by individuals with incomes of $10,000, $30,000, and $50,000?
What are the average tax rates for these income levels? At what income level does tax liability equal total income?
b. Graph the tax schedule for Oz. Use your graph to estimate marginal tax rates for the income levels specified in part a. Also show the average tax rates for these income levels on your graph.
c. Marginal tax rates in Oz can be estimated more precisely by calculating tax owed if persons with the incomes in part a get one more dollar. Make this computation for these three income levels. Compare your results to those obtained from the calculus-based result that, for the Oz tax function, its slope is given by .02I.
1.7 The following data show the production possi- bilities for a hypothetical economy during one year:
OUTPUT OFX OUTPUT OFY
1000 000
0800 100
0600 200
0400 300
0200 400
0000 500
a. Plot these points on a graph. Do they appear to lie along a straight line? What is that straight line’s production possibility frontier?
b. Explain why output levels of X¼400, Y¼200 orX¼300,Y¼300 are inefficient.
Show these output levels on your graph.
c. Explain why output levels of X¼500, Y¼350 are unattainable in this economy.
d. What is the opportunity cost of an additional unit ofXoutput in terms ofYoutput in this economy? Does this opportunity cost depend on the amounts being produced?
1.8Suppose an economy has a production possibility frontier characterized by the equation
X2þ4Y2¼100
a. In order to sketch this equation, first compute its intercepts. What is the value ofXifY¼0?
What is the value ofYifX¼0?
b. Calculate three additional points along this production possibility frontier. Graph the frontier and show that it has a general elliptical shape.
c. Is the opportunity cost of X in terms of Y constant in this economy, or does it depend on the levels of output being produced?
Explain.
d. How would you calculate the opportunity cost ofXin terms ofYin this economy? Give an example of this computation.
1.9Suppose consumers in the economy described in problem 1.8 wished to consume X and Y in equal amounts.
a. How much of each good should be produced to meet this goal? Show this production point on a graph of the production possibility frontier.
b. Assume that this country enters into interna- tional trading relationships and decides to pro- duce only goodX. If it can trade one unit ofX for one unit ofYin world markets, what possi- ble combinations ofXandYmight it consume?
c. Given the consumption possibilities outlined in part b, what final choice will the consumers of this country make?
d. How would you measure the costs imposed on this country by international economic sanc- tions that prevented all trade and required the country to return to the position described in part a?
1.10Consider the functionY¼XÆZ,X,Z0.
a. Graph theY¼4 contour line for this function.
How does this line compare to the Y¼2 con- tour line in Figure 1A.5? Explain the reasons for any similarities.
b. Where does the lineXþ4Z¼8 intersect the Y¼4 contour line? (Hint: Solve the equation forXand substitute into the equation for the contour line. You should get only a single point.)
c. Are there any points on theY¼4 contour line other than the point identified in part b that satisfy this linear equation? Explain your reasoning.
d. Consider now the equation Xþ4Z¼10.
Where does this equation intersect the Y¼4
contour line? How does this solution compare to the one you calculated in part b?
e. Are there points on the equation defined in part d that would yield a value greater than 4 forY?
(Hint: A graph may help you explain why such points exist.)
f. Can you think of any economic model that would resemble the calculations in this problem?