Chapter 3: Consumer Preferences
Chapter 3: Utility Maximization
Axiom
Based
Revealed
Preference
Preference
Ordering
Budget
Constraint
Utility
Maximization
Outline and Conceptual Inquiries
Consider a Household’s Budget Constraint: Feasible Grocery Carts
Consuming Discrete Commodities
Can you consume one-third of an airline ticket?
Assuming Continuous Choice
Nonlinear Budget Constraints
Do you bargain with grocery clerks?
How to choose a Household Utility Maximizing Bundle
What is the Tangency Condition?
What have you done when you are in a grocery checkout line?
Understanding the Marginal Utility per Dollar Condition
Application: Understanding Marginal Analysis
Should you study or spend time with your significant other?
Problems with Nonconvexity
Possible Corner Solutions
Why will you only purchase apples when you could purchase a combination of apples
and oranges?
The Invaluable Lagrangian Tool
Implications of the FOCs
At a grocery store, why does it not matter which commodity you purchase with the small
change you found on the floor?
Application: Why not Ration Commodities Instead?
Application: Which is Preferable an Excise Tax or an Income Tax?
Application: Which is Preferable a Subsidized Price or Income?
Why is there a black market for food stamps?
© Michael E. Wetzstein, 2012
Appendix to Chapter 3
Revealed Preference: Observed Grocery Cart Ordering
Principle of Rationality: Observing Rational Choices
Is your observed market behavior rational?
Summary
1. The major determinant of utility maximization is the satisfaction a household receives from a
commodity bundle. Unfortunately, the choice of the utility-maximizing bundle is constrained
by the budget set.
2. The budget set is the set of all commodity bundles that can be purchased at a given level of
income and a fixed set of prices.
3. A boundary of the budget set is the budget line whose slope measures the opportunity cost of
increasing the consumption of one commodity by the resulting decrease in consumption of
another commodity.
4. A household will maximize utility for a given budget constraint by shifting to higher
indifference curves until the point of tangency is reached between an indifference curve and
the budget constraint. At this tangency point, the slope of the indifference curve is equal to
the slope of the budget line. The household’s utility is then maximized for the given budget
constraint. This corresponds to where the marginal rate of substitution is equal to the
economic rate of substitution.
5. At the utility-maximizing bundle for a given budget constraint, the marginal utilities per
dollar for all commodities a household purchases are equal. A household is in equilibrium
when the last dollar spent on one commodity yields the same additional utility (marginal
utility) as the last dollar spent on another commodity.
6. The utility-maximizing conditions can be solved mathematically by using the Lagrangian.
This results in the identical tangency condition between the budget line and an indifference
curve, with the further result of the marginal utility per dollar of each commodity equaling
the Lagrange multiplier. In this case, the multiplier is the marginal utility of income.
7. An application of utility maximization subject to a budget constraint is government rationing
of commodities. Such a mechanism design allocates a fixed quantity of a commodity to each
household. Unfortunately, rationing generally results in decreasing households’ satisfaction.
This creates incentives for households to enter the black market in order to increase their
satisfaction.
8. In terms of increasing households’ satisfaction, a purchasing power tax (income tax) is
generally more efficient than taxes imposed on individual commodities. No distortions in the
market prices occur under a general purchasing power tax, so the utility-maximizing
commodity bundle can be obtained.
© Michael E. Wetzstein, 2012
9. Similar to a tax, a direct income grant (subsidy) will generally increase households’ utility
more than providing commodities at subsidized prices.
10. (Appendix) Revealed preference theory is an alternative to an axiom-based approach for
characterizing consumer preference ordering of commodity bundles. Revealed preference
determines a household’s preference ordering for bundles based on the observed market
choices it makes.
Key Concepts
budget constraint
budget line
corner solution
income subsidy
interior optimum
lump-sum income tax
marginal utility of income (MUI)
Key Equations
p1x1 + p2x2 = I
Total expenditure is the price per unit of
commodity 1 times the amount consumed
plus the price per unit of commodity 2 times
the amount of commodity 2 consumed.
Total expenditure is exhausted if it is equal
to the household’s income.
MRS(x2 for x1) = p1/p2
A condition for maximizing utility subject to
a given budget constraint. The marginal rate
of substitution is equal to the economic rate
of substitution.
MU1/p1 = MU2/p2 = … = MUk/pk
When deciding what commodities to spend
its income on, a household attempts to
equate the marginal utility per dollar for all
the commodities it purchases.
© Michael E. Wetzstein, 2012
opportunity cost
optimal choice
price takers
quantity tax
rationing
revealed preference theory
subsidized price
TESTING YOURSELF
Multiple Choice
1. A household is a price-taker when
a. It has the ability to bargain with sellers
b. It can purchase all it wants at the current market price
c. It faces an upward-sloping supply curve
d. Quantity supplied is greater than quantity demanded.
2. With x2 on the vertical axis, the slope of the budget constraint p1x1 + p2x2 = I is equal to
a. −p1/p2
b. −p2/p1
c. p1/p2
d. p2/p1
3. With x2 on the vertical axis, the slope of the budget constraint represents
a. The opportunity cost of consuming x2
b. The price of x2
c. The opportunity cost of consuming x1.
d. The price of x1.
4. Given p1x1 + p2x2 = I, when the price of x2 rises,
a. The budget constraint shifts parallel inward
b. The budget constraint tilts outward with the x1 intercept unchanged
c. The budget constraint shifts parallel outward
d. The budget constraint tilts inward with the x1 intercept unchanged.
5. Given p1x1 + p2x2 = I, when the individual’s income falls,
a. The budget constraint shifts parallel inward
b. The budget constraint tilts outward with the x1 intercept unchanged
c. The budget constraint shifts parallel outward
d. The budget constraint tilts inward with the x1 intercept unchanged.
6. Suppose Bernice’s income is $100 per week and she spends all of her income purchasing x1
and x2. If the price of x1 is $2 and the price of x2 is $4, Bernice’s budget constraint is
a. 4x1 + 2x2 = 100
b. x1 + 2x2 = 50
c. 2x1 + 4x2 = 100
d. Both b and c.
© Michael E. Wetzstein, 2012
7. Suppose Paul’s income is $500 per week and he spends all of his income purchasing x1 and
x2. The price of x1 is $10 and the price of x2 is $15. With x2 on the vertical axis, the slope of
Paul’s budget constraint is
a. −3
b. −2/3
c. −50
d. −3.33.
8. The household’s optimal choice occurs at the commodity bundle where
a. The slope of the indifference curve equals the price ratio
b. the slope of the budget constraint equals the MRS
c. The slope of the budget constraint equals the slope of the indifference curve
d. All of the above.
9. Given p1x1 + p2x2 = I, at the household’s optimal choice
a. The rate at which the consumer is willing to trade x1 for x2 is equal to p1/p2
b. The household is at global bliss
c. The economic rate of substitution is equal to the household’s marginal rate of substitution
d. All of the above.
10. Suppose Hazel spends all of her income on hairspray and bubble gum. The price of hairspray
is $2 per can and the price of a package of bubble gum is $1. At her current consumption
choice, the marginal utility of hairspray is 15 and the marginal utility of bubble gum is 10.
Given this information, we can determine that Hazel
a. Should purchase less hairspray and more bubble gum
b. Is maximizing her utility
c. Should purchase more hairspray and less bubble gum
d. Should purchase less bubble gum and less hairspray.
11. Given p1x1 + p2x2 = I, at the corner solution,
a. MRS(x1 for x2) = p1/p2
b. The household chooses to purchase only x1 or x2
c. MU1/p1 = MU2/p2
d. The household has concave indifference curves.
12. If two goods are perfect substitutes,
a. A corner solution is likely
b. MRS is constant
c. The individual will purchase equal amounts of the two goods
d. Both a and b.
13. The Lagrange multiplier at its optimal value, λ*, is equal to the
a. Maximum level of utility for a given budget
© Michael E. Wetzstein, 2012
b. Partial of the constraint with respect to the objective function
c. Local bliss
d. The marginal utility of income.
14. If MUj/pj < λ* the household
a. Will choose to purchase more of commodity j
b. Will choose to purchase less of commodity j
c. Will not purchase any more units of commodity j
d. Is at a corner solution.
15. Compared to a commodity tax on commodity xk, a lump-sum income tax will
a. Provide the household with a higher level of utility
b. Reduce the household’s level of utility
c. Reduce the household’s real income
d. Cause a greater change in the slope of the household’s budget constraint.
© Michael E. Wetzstein, 2012
Short Answer
1. Explain what it means for a household to be a price-taker.
2. Describe a household’s budget line for commodities x1 and x2. You should describe the
intercepts and the slope.
3. Anthony spends his monthly income of $2000 on movies and books. If the price of a movie
is $10 and the price of a book is $2, graph Anthony’s budget constraint and indicate the slope
and intercepts.
4. Graph a household’s budget constraint
constraint under the following circumstances:
Show what happens to the budget
a. The price of x1 falls to
b. Income rises to I′
c. The price of x2 rises to
5. Graph a consumer’s optimal choice between x1 and x2. What conditions must be met?
6. Marj uses all of her income to purchase toothpaste and bottles of lotion. She currently
purchases 10 tubes of toothpaste and 3 bottles of lotion. The price of a tube of toothpaste is
$4 and the price of a bottle of lotion is $3. At her current consumption choice, the marginal
utility of toothpaste is 24 and the marginal utility of lotion is 21. Is Marj maximizing her
utility? Explain. If not, how should Marj alter her consumption in order to raise her utility
level?
7. Explain why a consumer may end up at a corner solution.
8. Suppose Douglas considers apples and pears as perfect substitutes. If the price of pears is
lower than the price of apples, graph Douglas’ optimal choice.
9. In the Lagrangian equation for utility maximization, what does λ represent?
10. Explain why a lump-sum income tax reduces a consumer’s utility by a lesser extent than an
equally-sized commodity tax on a product purchased by the household.
11. (Appendix) Sandra purchases two commodities (x1 and x2). When p1 = 3 and p2 = 5, she
consumes where x1 = 4 and x2 = 3. When the prices are p1 = 4 and p2 = 4, she consumes at x1
= 3 and x2 = 4. Is Sandra rational and attempting to maximize utility? Explain.
© Michael E. Wetzstein, 2012
Problems
1. Suppose Judy spends all of her income I = 84 on x1 and x2. Let p1 = 4 and p2 = 6. Write an
equation to represent her budget constraint. What is the slope of the constraint? What are the
two intercepts?
2. Suppose Russell’s preferences can be represented by the utility function U = 4x1+ 9x2. Given
p1 = 8 and p2 = 12, how will Russell allocate his income of 184? Explain.
3. Earl has the utility function
Let p1 = 2, p2 = 5, with I = 60. Derive the
Lagrangian and the FOCs. Use the FOCs to solve for Earl’s utility-maximizing bundle.
Let p1 = 8, p2 = 6, and I = 96.
4. Kimberly has the utility function
Derive the Lagrangian and derive the FOCs. Use the FOCs to solve for Kimberly’s utilitymaximizing bundle.
5. Mark has the utility function
for Mark’s utility-maximizing bundle.
Let p1 = 5, p2 = 8, and I = 130. Solve
6. Erika has the utility function
Let p1 = 5, p2 = 5, and I = 300. Derive the
Lagrangian and solve for the FOCs. Use the FOCs to solve for Erika’s utility-maximizing
bundle. How much utility does she receive? Suppose, because of a government rationing
program, she is limited to consuming only 20 units of x2. Will she be better or worse off?
Explain.
7. Consider Problem 3. Suppose the government places a tax on each unit of x2 consumed of $1.
What will happen to Earl’s utility-maximizing bundle? Is he better or worse off? Explain.
What is Earl’s utility level if a lump-sum income tax (with the same level of tax revenue) is
employed instead?
© Michael E. Wetzstein, 2012