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196 CHAPTER 5 THE THEORY OF DEMAND

PROBLEMS

5.1. Figure 5.2(a) shows a consumer’s optimal choices a) Derive Karl’s demand curve for beer as a function of
of food and clothing for three values of weekly income: the exogenous variables.
I 1 $40, I 2 $68, and I 3 $92. Figure 5.2(b) illustrates b) Which affects Karl’s consumption of beer more: a one
how the consumer’s demand curve for food shifts as in- dollar increase in P H or a one dollar increase in P B ?
come changes. Draw three demand curves for clothing
(one for each level of income) to illustrate how changes 5.8. David has a quasilinear utility function of the form
in income affect the consumer’s purchases of clothing. U(x, y) 1x y, with associated marginal utility func-
tions MU x 1/(21x ) and MU y 1.
5.2. Use the income consumption curve in Figure 5.2(a) a) Derive David’s demand curve for x as a function of the
to draw the Engel curve for clothing, assuming the price prices, P x and P y . Verify that the demand for x is inde-
of food is $2 and the price of clothing is $4. pendent of the level of income at an interior optimum.
5.3. Show that the following statements are true: b) Derive David’s demand curve for y. Is y a normal
a) An inferior good has a negative income elasticity of good? What happens to the demand for y as P x increases?
demand. 5.9. Rick purchases two goods, food and clothing. He
b) A good whose income elasticity of demand is negative has a diminishing marginal rate of substitution of food
will be an inferior good. for clothing. Let x denote the amount of food consumed
and y the amount of clothing. Suppose the price of food
5.4. If the demand for a product is perfectly price in-
elastic, what does the corresponding price consumption increases from P x 1 to P x 2 . On a clearly labeled graph, illus-
curve look like? Draw a graph to show the price con- trate the income and substitution effects of the price
sumption curve. change on the consumption of food. Do so for each of
the following cases:
5.5. Ann consumes five goods. The prices of all goods a) Case 1: Food is a normal good.
are fixed. The price of good x is p x . She spends 25 per- b) Case 2: The income elasticity of demand for food is zero.
cent of her income on good x, regardless of the size of
her income. c) Case 3: Food is an inferior good, but not a Giffen good.
d) Case 4: Food is a Giffen good.
a) Show that her income elasticity of demand of good x
is the same for any level of income, and determine its 5.10. Reggie consumes only two goods: food and
value. shelter. On a graph with shelter on the horizontal axis
b) Would the value of the income elasticity of demand and food on the vertical axis, his price consumption curve
for x be different if Ann always spends 60 percent of her for shelter is a vertical line. Draw a pair of budget lines
income on good x? and indifference curves that are consistent with this de-
scription of his preferences. What must always be true
5.6. Suzie purchases two goods, food and clothing. She about Reggie’s income and substitution effects as the result
has the utility function U(x, y) xy, where x denotes the of a change in the price of shelter?
amount of food consumed and y the amount of cloth-
2
ing. The marginal utilities for this utility function are 5.11. Ginger’s utility function is U(x, y) x y, with as-
MU x y and MU y x. sociated marginal utility functions MU x 2xy and
2
a) Show that the equation for her demand curve for MU y x . She has income I 240 and faces prices P x
clothing is y I (2P y ). $8 and P y $2.
b) Is clothing a normal good? Draw her demand curve a) Determine Ginger’s optimal basket given these prices
for clothing when the level of income is I 200. Label and her income.
this demand curve D 1 . Draw the demand curve when I b) If the price of y increases to $8 and Ginger’s income
300 and label this demand curve D 2 . is unchanged, what must the price of x fall to in order
c) What can be said about the cross-price elasticity of for her to be exactly as well off as before the change
demand of food with respect to the price of clothing? in P y ?
5.7. Karl’s preferences over hamburgers (H ) and beer 5.12. Ann’s utility function is U(x, y) x y, with as-
(B) are described by the utility function: U(H, B) sociated marginal utility functions MU x 1 and MU y 1.
min(2H, 3B). His monthly income is I dollars, and he Ann has income I 4.
only buys these two goods out of his income. Denote the a) Determine all optimal baskets given that she faces
price of hamburgers by P H and of beer by P B . prices P x 1 and P y 1.