Economics 301
Problem Set 4
Name ______________________________
5 October 2007
Budget Lines and Indifference Curves and the Consumer Optimum
1. Parvez, a pharmacology student, has allocated $120 per month to spend on paperback novels
and used CDs. Novels cost $8 each; CDs cost $6 each. Draw his budget line.
Novels
Novels
Novels
30
15
15
20
Original BL
CDs
CDs
12
Part a
24
CDs
Part b
a. Draw and label a second budget line that shows what happens when the price of a CD rises to
$10.
b. Draw and label a third budget line that shows what happens when the price of a CD rises to
$10 and Parvez's income rises to $240.
2. What happens to a consumer's equilibrium if all prices and income double? [Hint: What
happens to the intercepts of the budget line?]
3. Continental Long Distance Telephone Service offers an optional package for in-state calling whereby
each month the subscriber gets the first 50 minutes of in-state calls free, the next 100 minutes at
$0.25/min, and any additional time at the normal rate of $0.50/min. Draw the budget constraint for instate phone calls and a composite good (whose price equals $1) for a subscriber with an income of
$400/month.
Other
($)
400
375
50
150
900
Minutes
4. Suppose Carmela’s income is $100 per week, which she allocates between sandwiches and books.
Sandwiches cost $2 each. Books cost $10 each if she purchases between 1 and 5 books. If she purchases
more than 5 books in a week, the price falls to $5 for the 6th book and all subsequent books. Draw the
budget constraint. Is it possible that Carmela might have more than one utility-maximizing solution?
Because the price of books falls when Carmela purchases more than 5 books in one week, the budget
constraint is non-linear. Below, the budget line is kinked at a. This non-linearity makes it possible
that a single indifference curve could be tangent to the constraint in two places. In this case, the
consumer is indifferent between purchasing 3 books per week at full price and purchasing 10 books
per week with the discount.
5. In Larry's state, a sales tax of 10% is applied to clothing but not to food. Show the effect of this tax
on Larry's choice between food and clothing using indifference curves and budget lines.
6. Lauren buys pizza slices for $3 a slice and "all other goods" at a price of $1 per unit. Her optimal
bundle is 10 slices of pizza per week and 30 units of other goods per week. Draw her budget constraint.
Label her optimal bundle A and show the relevant indifference curve. The pizzeria raises its price to $4 a
slice. Her parents give her enough extra money that she can buy her original bundle, A. Draw Lauren's
new budget constraint. Can you tell how her new bundle, B, compares to the original one, A?
Lauren is unambiguously
better off. Her original
bundle (previous optimum)
A is still available, but not
chosen, so we know that
Lauren has changed her
consumption in response to
the change in relative
prices.
7. Ann's only income is her annual college scholarship, which she spends exclusively on gallons of ice
cream and books. Last year when ice cream cost $10 and used books cost $20, Ann spent her $250
scholarship on 5 gallons of ice cream and 10 books. This year, the price of ice cream rose to $15 and the
price of books increased to $25. So that Ann can afford the same bundle of ice cream and books that she
bought last year, her college raised her scholarship to $325. Ann has the usual-shaped indifference curves.
Will Ann change the amount of ice cream and books that she buys this year? If so, explain how and why.
Will Ann be better off, as well off, or worse off this year than last year? Why?
Ann is unambiguously better off.
Her original bundle (previous
optimum) is still available, but not
chosen, so we know that Ann has
changed her consumption in
response to the change in relative
prices.
8. Under a welfare plan, poor people are given a lump-sum payment of $L. If they accept this welfare
payment, they must pay half of anything they earn to the government as a tax. If they do not accept the
welfare payment, they do not have to pay a tax on their earnings. Show that whether an individual accepts
welfare depends on the individual's tastes.
9. Suppose that Samantha and Jason both spend $24 per week on video and movie entertainment. When
the prices of videos and movies are both $4, they both rent 3 videos and buy 3 movie tickets. Following a
video price war and an increased cost of movie tickets, the video price falls to $2 and the movie ticket
increases to $6. Samantha now rents 6 videos and buys 2 movie tickets; Jason, however, buys 1 movie
ticket and rents 9 videos.
a. Is Samantha better off or worse off after the price change?
b. Is Jason better off or worse off?
Both Samantha and Jason are
unambiguously better off. Because
original bundles (previous optimum)
are still available, but not chosen, it
must be true that both Samantha
and Jason have moved to higher
indifference curves that were not
previously in their opportunity sets.
10. Last year, the price of heating oil was $4 per gallon, and Jonetta purchased 100 gallons of heating oil.
This year, the price of heating oil falls to $3 per gallon while Jonetta's income is unchanged. Jonetta
decides to share her good fortune by giving her retired father a gift of $100. Consider an indifference
curve-budget line diagram with heating oil on the horizontal axis and "all other goods" on the vertical
axis.
a. Does the price change make Jonetta's budget line flatter or steeper? Justify your choice.
The relative price of heating oil has fallen, so Jonetta's budget line has become flatter.
b. After Jonetta gives the $100 gift, will her new budget line lie above, lie below, or pass through her
initial optimum? Justify your choice.
Last year, Jonetta spent $400 on heating oil. This year, Jonetta could purchase the same amount of
heating oil for $300; the $100 gift would allow her to keep her consumption of all other good
unchanged from last year. After the gift, Jonetta has just enough income to continue purchasing
the same basket as she purchased last year, so her new budget line must pass through her initial
optimum.
c. Sketch an indifference curve-budget line diagram that illustrates this situation. This year, will Jonetta
be better or worse off than she was last year?
As shown in the accompanying diagram, Jonetta reaches a higher indifference curve, so she is
better off this year than she was last year.
11. The Simpsons, a family of four, have a meager annual income of $16000.
a. Draw their budget line on a graph on which the vertical axis shows their quantity consumed of food,
measured in dollars, and the horizontal axis shows their quantity consumed of other goods, also
measured in dollars.
b. Suppose that the Simpsons spend $4000 each year on food. Assuming that the Simpsons are utility
maximizers, draw an indifference curve at their consumption point.
A governmental agency decides that $6000 is the minimum annual expenditure on food needed to
provide proper nutrition for a family of four. Noting that the Simpsons spend less than this minimum
amount, the agency gives them an annual allotment of $2000 worth of food stamps, which can be used
only to buy food. Suppose in addition that the Simpsons' income elasticity of demand for food equals
one.
c. On a new diagram, draw their new budget set. Determine how much the Simpsons will now spend on
food each year. Draw an indifference curve at their new consumption point. Does the government
succeed in its objective of raising their food consumption to the minimum standard?
The Simpson household will increase its food consumption by 12.5 percent, the same percentage
that its income has increased. The Simpsons now consume $4500 worth of food, meaning the
government has not met the objective of food consumption of at least $6000.
Alternatively, with the income elasticity of demand equal to 1, if there is an increase in income,
the new optimum will lie along the ray from the origin shown in the diagram below. Thus, we know
that the new consumption level of food, F, satisfies: 4000/16000 = F/18000
d. Repeat part (c), but assume instead that the agency increases its aid to the Simpsons to $6000 per year
in food stamps.
e. Would the Simpsons prefer to receive the $6000 in cash instead? Would the agency succeed in its
objective in that case? Draw a new diagram to support your answers.
In this case, the family certainly is
going to want to consume at least
$6000 worth of food, but in fact that
is all they will consume. To see this,
note first that maximum food
consumption now increases to $22000.
Then, using the income elasticity
assumption,
4000/16000 = F/22000
Solving, we get F = 5500. This
shows us that the family would like to
consume only $5500 worth of food in
this case, which is shown by point B
along the blue (unattainable) portion of
the new budget set shown to the left.
12. Since 1979, recipients have been given food stamps. Before 1979, however, people bought food stamps
at a subsidized rate. For example, to get $1 worth of food stamps, a household paid about $0.25 (the exact
amount varied by household characteristics and other factors). What is the budget constraint facing an individual if that individual may buy up to $100 per month in food stamps at $0.25 per each $1 coupon?