Homework #3
Due: Tuesday, June 9, 2015
1. Suppose initially two people, persons A and B, have identical budget constraints involving goods
x1 and x2 where x2 is the Numeraire for both of them. Each has $100 as their money constraint
and p1= $1.25. In addition to the money budget constraint, the government has a ration limit of
an average of 40 units of x1 per person. So, initially both people receive 40 coupons from the
government. They must present one coupon to the seller for each unit of x1 they wish to buy. If
they run out of coupons, they cannot buy additional units of x1. This means that each person
can choose to buy up to 40 units of xx by both paying $1.25 for each unit AND by giving up one
coupon for each unit.
a. On graph paper, using a straight edge, draw the budget constraints for each person.
Draw one for each person. Label everything carefully and completely on your diagrams.
b. Suppose that the ration coupons can be bought and sold (for money) and that Person A
sells 10 ration coupons to Person B for a total of $20. Using the same diagrams as you
used in part (a), draw the new carefully- labeled budget constraint for each person.
[Note: while the initial budget constraints were identical for both A and B, the new
budget constraints will be different.]
c. Finally, for each person draw a set of indifferent curves that shows that each person is
better off after they have traded the coupons than if the government prohibited the
ration coupons from being bought and sold. Label the indifference curves so it is very
clear which represents the greater level of utility. [Note: the indifference curves will be
different for the two people.]
d. If the preference maps (indifference curves) for A and B were identically the same, and
their initial budgets were as you show in part (a), would they ever find it mutually
beneficial for one to purchase ration coupons from the other? Show and explain.
2. If goods x1 and x2 are complements in consumption, as p2 increases, the consumption of x1
decreases. If the two goods are substitutes in consumption, as p2 increases, the consumption of
x1 increases. [See section 6.7 on pp. 111-112 of your text as well.]
a. Demonstrate (with a well-labeled graph) these effects if the goods are
i. Perfect substitutes (with indifference curves similar to those in figure 5.5).
ii. Perfect complements (with indifference curves similar to those in figure 5.6).
[Hints: draw two different graphs—one for complementary goods and the other
for substitute goods. Use the graphs to explain—in words—how the graph
demonstrates these effects. Also, make sure you put x1 on the horizontal axis
and x2 on the vertical axis.]
b. Using the criteria stated above, two goods can still be substitutes or complements even
if they are not perfect substitutes or perfect complements, but have smooth, convex,
bowed (convex to the origin) indifference curves. Demonstrate these effects using
curved indifference curves.
0.4
0.6
c. If the consumer has a Cobb-Douglass utility function in the form of x1 x2 , are x1 and x2
substitutes, complements, or neither? Show and explain each step of your answer.
[Hint: Use the properties of Cobb-Douglass utility functions that are described on p. 83.]
d. If consumers have quasilinear preferences (as shown in figure 4.4 on p. 63), are x1 and x2
substitutes, complements, or neither? Graph this situation and explain your answer
using the graph.
3. Figures 6.12 and 6.13 on page 108 show the derivation of the demand curves for perfect
substitutes and perfect complements.
a. On figure 6.12, demonstrate and explain (using your own words!)
i. Why there is a horizontal section of the demand curve.
ii. Why the downward sloping part of the demand curve is NOT a straight line.
b. On figure 6.13, demonstrate and explain (using your own words!) why the demand
curve is NOT a straight line.