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 P = 3 is the equilibrium price in the orange juice market.
B. Using these data, explain why P = 2 and P = 4 are not equilibrium prices.22
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 million 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 problem 1.1, what is the equilibrium price in this market?
h. Explain why P = 3 is no longer an equilibrium in the orange juice market. How would the participants in this market know P =3is no longer an equilibrium?
I. Graph your results for this supply shift.
1.3
The equilibrium price in problem 1.2 is P =3. This is an equilibrium because at this price, quantity demanded is precisely equal to quantity supplied (Q = 500). 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 dollars 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 various 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?
1.5
This problem involves solving demand andsupply equations together to determine price and quantity.
a. Consider a demand curve of the form
QD= -2P + 20
Where QD is 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
Qs= 2P - 4
Where Qs is the quantity supplied. Be sure to put P on the vertical axis and Q on the horizontal axis. Assume that all the Qs and Ps are nonnegative for parts a, b, and c. At what values of P and Q do these curves intersect- that is, where does QD= Qs?
b. Now suppose at each price that individuals demand four more units of output-that the demand curve shifts to
QD'=-2P + 24
1.8
Suppose an economy has a production possibility frontier characterized by the equation
X2+ 4Y2= 100
a. In order to sketch this equation, first compute its intercepts. What is the value of X if Y = 0?
What is the value of Y if X = 0?
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 of X in terms of Y in this economy? Give an example of this computation.
1.10
Consider the function Y = X * Z, X, Z > 0.
a. Graph the Y ¼ 4 contour line for this function. How does this line compare to the Y = 2 contour line in Figure 1A.5? Explain the reasons for any similarities.
b. Where does the line X + 4Z = 8 intersect the Y = 4 contour line? (Hint: Solve the equationfor X and substitute into the equation for the contour line. You should get only a single point.)
c. Are there any points on the Y = 4 contour line other than the point identified in part b that satisfy this linear equation? Explain your reasoning.
d. Consider now the equation X + 4Z = 10.
Where does this equation intersect the Y = 4 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 for Y?
(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?
2.1
Suppose a person has $8.00 to spend only on apples and bananas. Apples cost $.40 each, and bananas cost $.10 each.
a. If this person buys only apples, how many can be bought?
b. If this person buys only bananas, how many can be bought?
c. If the person were to buy 10 apples, how many bananas could be bought with the funds left over?
d. If the person consumes one less apple (that is, nine), how many more bananas could be bought? Is this rate of trade-off the same no matter how many apples are relinquished?
e. Write down the algebraic equation for this person's budget constraint, and graph it showing the points mentioned in parts a through d (using graph paper might improve the accuracy of your work).
2.2
Suppose the person faced with the budget constraint described in problem 2.1 has preferences for apples (A) and bananas (B) given by
Utility = √ A . b
a. If A = 5 and B = 80, what will utility be?
b. If A = 10, what value for B will provide the same utility as in part a?
c. If A = 20, what value for B will provide the same utility as in parts a and b?
d. Graph the indifference curve implied by parts a through c.
e. Given the budget constraint from problem 2.1, which of the points identified in parts a through c can be bought by this person?
f. Show through some examples that every other way of allocating income provides less utility than does the point identified in part b. Graph this utility-maximizing situation.
2.5
Ms. Caffeine enjoys coffee (C) and tea (T) according to the function U(C, T)= 3C t 4T.
a. What does her utility function say about her MRS of coffee for tea? What do her indifference curves look like?
b. If coffee and tea cost $3 each and Ms. Caffeine has $12 to spend on these products, how much coffee and tea should she buy to maximize her utility?
c. Draw the graph of her indifference curve map and her budget constraint, and show that the utility-maximizing point occurs only on the T-axis where no coffee is bought.
d. Would this person buy any coffee if she had more money to spend?
e. How would her consumption change if the price of coffee fell to $2?
2.7
Assume consumers are choosing between housing services (H) measured in square feet and consumption of all other goods (C) measured in dollars.
a. Show the equilibrium position in a diagram.
b. Now suppose the government agrees to subsidize consumers by paying 50 percent of their housing cost. How will their budget line change? Show the new equilibrium.
c. Show in a diagram the minimum amount ofincome supplement the government wouldhave to give individuals instead of a housing subsidy to make them as well-off as they were in part b.
d. Describe why the amount shown in part c is smaller than the amount paid in subsidy in part b.
2.10
A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X, Y)= XαYβ where α and β are decimal exponents that sum to 1.0 (that is, for example, 0.3 and 0.7).
a. Explain why the utility function used in problem 2.2 and problem 2.3 is a special case of this function.
b. For this utility function, the MRS is given by MRS = MUx=MUy = αY/βX. Use this fact together with the utility-maximizing condition (and that α+ β = 1) to show that this person will spend the fraction of his or her income on good X and the fraction of income on good Y-that is, show PxX/I=α, PyY/I =β.
c. Use the results from part b to show that total spending on good X will not change as the price of X changes so long as income stays constant.
d. Use the results from part b to show that a change in the price of Y will not affect the quantity of X purchased.
e. Show that with this utility function, a doubling of income with no change in prices of goods will cause a precise doubling of purchases of both X and Y.
