Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 1 of 8
4.1.
In problem 3.3 of PS1(chapter 3), we considered Julie’s preferences for food F and clothing C. Her
utility function was U(F,C) = FC. Her marginal utilities were MUF = C and MUC = F. You were
asked to draw the indifference curves U = 12, U = 18, and U = 24, and to show that she had a
diminishing marginal rate of substitution of food for clothing. Suppose that food costs $1 a unit and
that clothing costs $2 a unit. Julie has $12 to spend on food and clothing.
a) Using a graph (and no algebra), find the optimal (utility-maximizing) choice of food and
clothing. Let the amount of food be on the horizontal axis and the amount of clothing be vertical
axis.
b) Using algebra (the tangency condition and the budget line), find the optimal choice of food and
clothing.
c) What is the marginal rate of substitution of food for clothing at her optimal basket? Show this
graphically and algebraically.
d) Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget (instead
of the optimal basket). Would her marginal utility per dollar spent on food be grater than or less
than her marginal utility per dollar spent on clothing. What does this tell you about how she
should substitute food for clothing if she wants to increase her utility without spending any more
money?
4.2.
The utility that Ann receives by consuming food F and clothing C is given by U(F,C) = FC + F. The
marginal utilities of food and clothing are MUF=C＋1 and MUC=F. Food costs $1 a unit, and
clothing costs $2 a unit. Ann’s income is $22.
a) Ann is currently spending all of her income. She is buying 8 units of food. How many units of
clothing is she consuming?
b) Graph her budget line. Place the number of units of clothing on the vertical axis and the number
of units of food on the horizontal axis. Plot her current consumption basket.
c) Draw the indifference curve associated with a utility level of 36, and another indifference curve
associated with a utility level of 72. Are the indifference curves bowed in toward the origin?
d) Using a graph (and no algebra), find the utility-maximizing choice of food and clothing.
e) Using algebra, find the utility-maximizing choice of food and clothing.
f)
What is the marginal rate of substitution of food for clothing when utility is maximized? Show
this graphically and algebraically.
Does Ann have a diminishing marginal rate of substitution of food for clothing? Show this
graphically and algebraically.
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 2 of 8
4.4.
This problem will help you understand what happens if the marginal rate of substitution is not
diminishing. Dr. Strangetaste buys only French fries (F) and hot dogs (H) out of his income. He has
positive marginal utilities for both goods, and his MRS H,F is increasing. The price of hot dogs is P H,
and the price of French fries is P F.
a) Draw several of Dr. Strangetaste’s indifference curves, including one that is tangent to his
budget line.
b) Show that the point of tangency does not represent a basket at which utility is maximized, given
the budget constraint. Indicate on your graph where the optimal basket is located.
4.5.
Toni likes to purchase round trips between the cities of Pulmonia and Castoria and other goods out
of her income of $10,000.
Fortunately, Pulmonian Airways provides air service and has a
frequent-flyer program. A round trip between the two cities normally costs $500, but any customer
who makes more than ten trips a year gets to make additional trips during the year for only $200 per
round trip.
a)
On a graph with round trips on the horizontal axis and “other goods” on the vertical axis, draw
Toni’s budget line. (Hint: This program demonstrates that a budget line need not always be a
straight line.)
b)
On the graph you drew in part (a), draw a set of indifference curves that illustrates why Toni
may be better off with the frequent flyer program.
c)
On a new graph draw the same budget line you found in part (a). Now draw a set of
indifference curves that illustrates why Toni might not be off with the frequent flyer program.
4.6.
A consumer has preferences between two goods, hamburgers (measured by H) and milkshakes
(measured by M). His preferences over the two goods are represented by the utility function. For this
utility function and function U 
H  M . For this utility function MU H  1 /( 2 H ) and
MU C  1 /( 2 M ) .
a) Determine if there is a diminishing MRSH,M for this utility function.
b) Draw a graph to illustrate the shape of a typical indifference curve. Label the curve U1. Does the
indifference curve intersect either axis? On the same graph, draw a second indifference curve U2,
with U2＞U1.
c) A consumer has an income of $24 per week. The price of a hamburger is $2 and the price of a
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 3 of 8
milkshake is $1. How many milkshakes and hamburgers will he buy each week if he maximizes
utility? Illustrate your answer on a graph.
4.7.
A student consumes root beer and a composite good whose price is $1.Cureently the government
imposes an excise tax of $0.50 per six pack of root beer. The student now purchases 20 six packs of
root beer per month. (Think of the excise tax as increasing the price of root beer by $0.50 per six
pack over what the price would be without the tax.) The government is considering eliminating the
excise tax on root beer and, instead, requiring consumers to pay $10.00 per month as a lump sum tax
(i.e., the student pays a tax of $10.00 per month, regardless of how much root beer is consumed).
If the new proposal is adopted, how will the student’s consumption pattern (in particular, the amount
of root beer consumed) and welfare be affected? (Assume that the student’s marginal rate of
substitution of root beer for other goods is diminishing.)
4.8.
When the price of gasoline is $2.00 per gallon, Joe consumes 1000 gallons per year. The price
increases to $2.50, and to offset the harm to Joe, the government gives him cash transfer of $500 per
year. Will Joe be better off or worse off after the price increase and cash transfer than he was before?
What will happen to his gasoline consumption? (Assume that Joe’s marginal rate of substitution of
gasoline for other goods is diminishing.)
4.9.
Sally consumes housing (denote the number of units of housing by h) and other goods (a composite
good whose units are measured by y), both of which she likes. Initially she has an income of $100,
and the price of a unit of housing (
) is $10. At her “initial” choice basket she consumes 2 units of
housing. A few months later her income rises to $120; unfortunately, the price of housing in her city
also rises to $15. The price of the composite good does not change. At her “final” choice basket she
consumes 1 unit of housing. Using revealed preference analysis (without drawing indifference
curves), what can you say about how she ranks her initial and final baskets?
4.10.
A consumer buys two goods, food and housing, and likes both goods. When she has budget line BL1,
her optimal choice is A. Given budget line BL2, she chooses B, and with BL3, she chooses C (see
Figure 4.21).
a) What can you infer about how the consumer ranks baskets A, B, and C? If you can infer a
ranking, explain why not.
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 4 of 8
b) On the graph, shape in (and clearly label) the areas that are revealed to be less preferred to
basket B, and explain why you indicated these areas.
c) On the graph, shade in (and clearly label) the areas that are revealed to be (more) preferred to
basket B, and explain why you indicated these areas.
A
Housing
B
C
BL1
BL２
BL３
Food
5.3.
Show that the following statements are true:
a) An inferior good has a negative income elasticity of demand.
b)
A good whose income elasticity of demand is negative will be an inferior good.
5.5.
Suzie purchases two goods, food and clothing. She has the utility function U(x,y) = xy, where x
denotes the amount of food consumed and y the amount of clothing. The marginal utilities for this
utility function are MUx = y and MUy = x.
a) Show that the equation for her demand curve for clothing is y = I/(2Py).
b) Is clothing a normal good? Draw her demand curve for clothing when the level of income is I =
200. Label this demand curve D1. Draw the demand curve when I = 300 and label this demand curve
D2.
c) What can be said about the cross-price elasticity of demand of food with respect to the price of
clothing?
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 5 of 8
5.6.
Rich purchases two goods, food and clothing. He has a diminishing marginal rate of substitution of
food for clothing. Let x denote the amount of food consumed, and y the amount of clothing. Suppose
the price of food increases from Px to Py. On a clearly labeled graph, illustrate the income and
substitution effects of the price change on the consumption of food. Do so for each of the following
cases:
a) Case 1: Food is a normal good.
b) Case 2: The income elasticity of demand for food is zero.
c) Case 3: Food is an inferior good, but not a Giffen good.
d) Case 4: Food is a Giffen good.
5.7.
Some texts define a “luxury good” as a good for which the income elasticity of demand is greater
than 1. Suppose that a consumer purchases only two goods. Can both goods be luxury goods?
Explain.
5.10.
Suppose that a consumer’s utility function is U(x,y)=xy＋10y. the marginal utilities for this utility
function are MUx=y and MUy=x＋10. The price of x is Px and the price of y is Py, with both prices
positive. The consumer has income I. (this problem shows that an optimal consumption choice need
not be interior, and may be at a corner point.)
a) Assume first that we are at an interior optimum. Show that the demand schedule for x can be
written as x=I／(2Px)－5.
b) Suppose now that I=100. Since x must never be negative, what is the maximum value of Px for
which this consumer would ever purchase any x?
c) Suppose Py=20 and Px=20. On a graph illustrating the optimal consumption bundle of x and y,
show that since Px exceeds the value you calculated in part (b), this corresponds to a corner
point at which the consumer purchases only y. (In fact, the consumer would purchase
y=I／Py=5 units of y and no units of x.)
d) Compare the marginal rate of substitution of x for y with the ratio (Px／Py) at the optimum in
part (c). Does this verify that the consumer would reduce utility if she purchased a positive
amount of x?
e) Assuming income remains at 100, draw the demand schedule for x for all values of Px. Does its
location depend on the value of Py?
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 6 of 8
5.11.
Suppose rental cars have two market segments, business travelers and vacation travelers. The
demand curve for rental cars by business travelers is Qb = 35 – 0.25P, where Qb is the quantity
demanded by business travelers (in thousands of cars) when the rental price is P dollars per day. No
business customers will rent cars if the price exceeds $140 per day.
The demand curve for rental cars by vacation is Qv = 120 – 1.5P, where Qv is the quantity
demanded by vacation travelers (in thousands of cars) when the rental price is P dollars per day. No
vacation customers will rent cars if the price exceeds $80 per day.
a) Fill in the table to find the quantities demanded in the market at each price in the table.
Price
Business
Vacation
Market Demand
($ / day)
(000cars / day)
(000cars / day)
(000cars / day)
100
90
80
70
60
50
b) Graph the demand curves for each segment, and draw the market demand curve for rental cars.
c)
Describe the market demand curve algebraically. In other words, show how the quantity
demanded in the market Qm depends on P. Make sure that your algebraic equation for the
market demand is consistent with your answers to parts (a) and (b).
If the price of a rental car is $60, what is the consumer surplus in each market segment?
6.2.
Suppose that the production function for floppy disks is given by
Q = KL2 – L3
where Q is the number of disks produced per year, K is machine-hours of capital, and L is man-hours
of labor flow of capital service.
a) Suppose K=600. Find the total product function and graph it over the range L=0 to L=500. Then
sketch the graph of the average and marginal product function. At what level of labor L does the
average product curve appear to reach its maximum? At what level does the marginal product
curve appear to reach its maximum?
b) Replicate the analysis in (a) for the case in which K=1200.
c) When either K=600 or K=1200, does the total product function have a range of increasing
marginal returns?
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 7 of 8
6.3.
Are the following statements correct or incorrect?
a) If average product is increasing, marginal product must be less than average product.
b) If marginal product is negative, average product must be negative.
c) If average product is positive, total product must be rising.
d) If total product is increasing, marginal product must also be increasing.
6.5.
Suppose the production function is given by the equation
Q=L
Graph the equation of the isoquant corresponding to Q=10, Q=20, and Q=50. Do these isoquants
exhibit diminishing marginal rate of technical substitution?
6.6.
Consider again the production function for floppy disks
Q=K
－
a) Sketch a graph of the isoquants for this production function.
b) Does this production function have an uneconomic region? Why?
6.7.
Suppose the production function is given by the equation (where a and b are positive constants)
Q = aL + bK
What is the marginal rate of technical substitution of labor for capital (MRTSN,k) at any point along
an isoquant ?
6.8.
Let B be the number of bicycles produced from F bicycle frames and T tires. Every bicycle needs
exactly two tires and one frame .
a) Draw the isoquants for bicycle production.
b) Write a mathematical expression for the production function for bicycles.
6.9.
A firm produces a quantity Q of breakfast cereal using labor L and material M with production
function
Q = 50 ML + M + L
The marginal product functions for this production function are
Microeconomics: Problem set 2 (Due on May 18-19 in the TA recitation)
Instructor: Toshihiro Ichida
Page 8 of 8
MPL = 25
L
+1
M
MPM = 25
L
+1
M
a) What is the nature of returns to scale (increasing, constant, or decreasing) for this production
function?
b) Is the marginal product of labor ever diminishing for this production function? If so, when? Is it
ever negative, and if so, when?
6.10.
Consider a CES production function given by
]2
a) What is the elasticity of substitution for this production function ?
b) Does this production function exhibit increasing , decreasing , or constant returns to scale?
c) Suppose that the production function took the form
Q=500[L＋3K]
Does this production function exhibit increasing , decreasing , or constant returns to scale?
6.11.
Suppose initially a firm’s production function took the form Q=1000[0.5L＋10K]
a) Show that the innovation has resulted in technological progress in the sense defined in the next.
b) Is the technological progress neutral, labor saving, or capital saving?