Chapter 4: Comparative Statics: Analysis of Individual Demand and
Labor Supply
Utility Maximization
Household’s
Supply of Labor
Comparative Statics
Comparative Statics
Changes
in Income
Changes
in Price
Changes
in Wages
Engel
Curves
Slutsky
Equation
Adjusted
Slutsky Equation
Normal
Goods
Luxury
Goods
Household’s
Demand for Commodities
Inferior
Goods
Necessary
Goods
Income
Effect
Giffen
Goods
Substitution
Effect
Ordinary
Goods
Income
Effect
Substitution
Effect
Backward Bending
Labor Supply
Curve
Outline and Conceptual Inquiries
Deriving Household Demand
Shifting Demand versus a Change in Quantity Demanded
How to Inverse Demand Curves
Is quantity dependent on price or is price dependent on quantity?
Application: How to Bid on eBay
Generalizing for k Commodities
Understanding Comparative Statics
What are Homogeneous of Degree Zero Demand Functions?
Do you have money illusion?
Application: Do Criminals Suffer from Money Illusions?
Changing Income
What kind of preferences do deeply religious individuals have?
How do you measure poverty?
Changing Price
When the price of a commodity decreases, why do you consume more of it?
Application: Why Did Marshall Introduce the Giffen Paradox?
Why Consider Compensated Price Changes?
Should your grandmother receive the full cost of living increase in her monthly Social
Security check?
Application: Consider a Gasoline Tax with an Income Tax Rebate
© Michael E. Wetzstein, 2012
How do we decrease our dependence on foreign oil?
Changing the Price of another Commodity
Appendix to Chapter 4
How the Slutsky Equation relates to the Household’s Supply of Labor
Considering Both Time and Income Constraints
Is time a constraint on happiness?
Doing Utility Maximization
Understanding Substitution and Income Effects
Using the Adjusted Slutsky Equation
Deriving Household’s Supply of Labor Curve
Application: Empirical Estimates of an Income/Leisure Indifference Function
Summary
1. An individual household’s demand function for a commodity is a theoretical relation between
price and quantity. This demand function is derived from utility maximization and is based
on commodity prices, the household’s preferences, and income.
2. A household’s demand curve illustrates how much of a commodity a household is willing
and able to purchase at a given price.
3. A shift in demand results when one of the parameters being held constant changes. For
example, a change in income will result in the demand curve shifting. In contrast, a change in
quantity demanded occurs when the own price of the commodity, which is not being held
constant, changes.
4. An inverse demand curve represents price being dependent on quantity, and a direct demand
curve represents quantity being dependent on price.
5. Comparative statics associated with a household’s demand function investigates the effect on
the demand for a commodity when price and/or income change.
6. Homogeneous of degree zero demand functions result when demands are not affected by
pure inflation. For example, doubling all prices and income does not affect a household’s
demand for commodities.
7. Engel curves relate income to the demand for each commodity, holding prices constant.
Positively sloped Engel curves are associated with normal goods and negatively sloped
curves with inferior goods.
8. The total effect of a price change may be decomposed into the substitution and income
effects, with the aid of the Slutsky equation. The own substitution effect is always negative
and if the commodity is a normal good, the income effect will also be negative. In this case,
© Michael E. Wetzstein, 2012
the Law of Demand states a commodity will always change opposite to any change in its own
price. If the income effect is positive, the sign of the total effect is indeterminate.
9. The compensated price effect indicates the effect from a pure price change, where the level
of consumption is adjusted for a change in real income. The two major types of
compensation are Hicks and Slutsky compensations. Hicks compensation compensates a
household to the point where its level of satisfaction is unaffected by a price change. Slutsky
compensation keeps a household’s purchasing power constant.
10. The Slutsky equation can also be expressed for a change in the consumption of a commodity
resulting in a change in the price of a related commodity. Substitutes and complements for a
commodity are then determined by the associated substitution and income effects.
11. (Appendix) There is a Slutsky-type equation for determining a household’s labor supply
curve. A labor supply curve is derived by households maximizing utility subject to a limited
amount of time as well as an income constraint.
12. (Appendix) A household’s labor supply curve illustrates how much labor a household is
willing and able to supply at a given wage rate.
Key Concepts
adjusted Slutsky equation
backward-bending labor supply curve
ceteris paribus
comparative statics analysis
Consumer Price Index
direct demand function
Engel’s Law
Giffen good
Hicksian substitution
homothetic preference
household’s demand curve
Key Equations
Demand functions are homogeneous of
degree zero in prices and income.
© Michael E. Wetzstein, 2012
household’s labor supply curve
inferior good
inverse demand functions
luxury good
necessary good
normal good
numeraire price
ordinary good
Slutsky equation
substitution effect
Slutsky equation, which states that the total
effect can be decomposed into the
substitution and income effects.
Adjusted Slutsky equation for the effect that
wages have on leisure.
© Michael E. Wetzstein, 2012
TEST YOURSELF
Multiple Choice
1. A household’s quantity demanded for commodity x1 depends on
a. Its preferences
b. Its income
c. Commodity prices
d. All of the above.
2. A household’s demand for x1 will generally be a function of
a. Its utility level
b. The consumption of other commodities
c. Its income and the price of x1
d. All of the above.
3. A price consumption curve
a. Traces out a household’s optimal bundles as the price of a commodity varies
b. Has a negative slope
c. Illustrates how quantity demanded increases as price declines
d. Indicates whether a good is normal or inferior.
4. Demand curves are derived by tracing out a household’s utility-maximizing choices as
a. Income varies
b. The own price varies
c. Preferences vary
d. Costs vary.
5. A demand function is homogeneous of degree zero in prices and income. This indicates that
if
a. All prices and income are doubled, demand will double
b. Prices are doubled but income is not, demand will remain unchanged
c. All prices and income are doubled, demand will remain unchanged
d. Income doubles, but prices remain unchanged, demand will also remain unchanged.
6. The income consumption path
a. Can be used to derive the Engel curve
b. Illustrates how a household’s optimal bundle changes as the price of a commodity
changes
c. Indicates whether a good is normal or inferior
d. Both a and c.
7. If a commodity is a normal good, its Engel curve will be
a. Horizontal
b. Upward sloping
c. U-shaped
© Michael E. Wetzstein, 2012
d. Downward-sloping.
8. If ∂x1/∂I > 0, then
a. Commodity x1 is a Giffen good
b. The household’s preferences are homothetic
c. Commodity x1 is a normal good
d. Commodity x1 is a luxury good.
9. If Sean has homothetic preferences for ice cream,
a. His purchases of ice cream will rise as his income falls
b. The proportion of his income spent on ice cream will remain the same at all levels of his
income
c. Ice cream is a Giffen good
d. He will buy the same quantity of ice cream at all income levels.
10. If x1 is an inferior good, then
a. ∂x1/∂I < 0
b. ∂x1/∂p1 < 0
c. ∂x1/∂x1 > 0
d. ∂x1/∂I > 0.
11. Engel’s law suggests that as income rises, the proportion of income spent on
a. Food will fall
b. All commodities will remain the same
c. Food will rise
d. Luxuries will fall.
12. If ∂x1/∂p1 < 0 and ∂x1/∂I > 0, then x1 is a(n)
a. Ordinary and normal good
b. Ordinary and inferior good
c. Luxury and normal good
d. Giffen and normal good.
13. The substitution effect measures
a. The change in purchasing power resulting from a price change
b. How a household’s optimal choice is affected by a decline in income
c. The incentive to purchase more of a lower-priced commodity and less of a higher-priced
commodity
d. The incentive to purchase more of a normal good when income rises.
© Michael E. Wetzstein, 2012
14. Which of the following is the Slutsky equation?
a.
b.
c.
d.
15. Suppose x1 is an inferior good. In order for x1 to also be an ordinary good, in absolute value
a. The income effect must be equal to the substitution effect
b. Commodity x1 must be a necessary good
c. The income effect must be greater than the substitution effect
d. The income effect must be smaller than the substitution effect.
16. Hicks compensation differs from Slutsky compensation because Hicks compensation holds
a. Purchasing power constant while Slutsky compensation holds utility constant
b. Utility constant while Slutsky compensation holds purchasing power constant
c. When p1 rises, it allows the individual to reach a higher level of utility
d. Income constant while Slutsky compensation holds purchasing power constant.
17. The Laspeyres Index
a. Is a Hicks compensation index
b. Allows purchasing power to change
c. Is used to calculate the Consumer Price Index
d. Holds utility constant.
18. If ∂x1/∂p2 < 0, then
a. Commodities x1 and x2 are gross substitutes
b. Commodities x1 and x2 are independent goods
c. The cross-substitution effect is negative
d. Commodities x1 and x2 are gross complements.
© Michael E. Wetzstein, 2012
Short Answer
1. Explain the difference between a shift in demand and a change in quantity demanded.
2. Illustrate graphically how the demand for x1 can be derived.
3. A demand curve that is displayed in a graph is sometimes called an inverse demand function.
Explain.
4. What does it mean for the demand function to be homogeneous of degree zero in all prices
and income?
5. Comment on the following: “If a household’s preferences for x1 are homothetic, the Engel
curve for x1 will be linear with a positive slope.”
6. Explain how we can use the shape of the Engel curve to determine if the commodity is
normal or inferior. Can we also determine if a normal good is a necessary or a luxury?
7. Comment on the following: “A Giffen good may be either normal or inferior.”
8. Darien spends his income of $200 per week on soda with a price of $8 per case and tickets to
the basketball game at $20 each.
a. Illustrate Darien’s budget constraint (with basketball tickets on the horizontal axis).
b. If Darien chooses to buy 6 tickets to the game, use an indifference curve to illustrate
Darien’s choice.
c. Suppose the price of tickets declines to $10 each. Graph Darien’s new budget line.
d. Show the income and substitution effects of this price decrease. Assume that game
tickets are a normal good.
9. Use the Slutsky equation to explain why ∂x1/∂p1 < 0 if x1 is a normal good.
10. Assume that x1 is an inferior good. Explain the necessary condition for x1 to also be a Giffen
good.
11. Illustrate graphically the difference between Hicks and Slutsky compensations.
12. Use the Slutsky equation to explain the difference between gross substitutes and gross
complements. What condition must hold for two goods to be independent?
13. (Appendix) Explain why a household’s labor supply curve may be backward-bending.
© Michael E. Wetzstein, 2012
Problems
1. Suppose a household’s utility function is
for x1 and x2.
Derive the demand functions
2. Suppose the demand function for x1 can be represented by x1 = 12I/(25p1). Calculate the
compensated price change for a change in p1.
3. Suppose Harrison’s utility function is U(x1, x2) = x1x2. Solve for Harrison’s demand functions
for x1 and x2. Are x1 and x2 normal or inferior? Explain. Are x1 and x2 gross substitutes or
gross complements? Explain.
Solve for his demand functions for x1 and x2. Do
4. Suppose Ethan’s utility function is
these demand functions have a negative slope?
5. (Appendix) Brian receives utility from leisure h and from income I according to his utility
function U(h, I) = h1/2I1/2. If his wage is $12 per hour and he has nonlabor income of $16 per
day, how many hours will Brian work?
© Michael E. Wetzstein, 2012