Econ. 410
Spring 2008
Tauchen
Practice Problems -- Deriving Demand Curves and Other Applications
Y
1. Jennifer consumes two goods – X and Y. Her income is $100/ time period and the price of good X is
$1. Use her indifference map as shown below to construct her demand curve for good Y for prices $1,
$2, and $3.
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2. Shown to the right is Ben’s demand curve
for good Y. Construct an indifference map
consistent with the amount of good Y he
demands at prices of $1, $2, and $3 for good Y.
3. We can apply the consumer theory model to understand the decisions faced by government agencies
and public officials. Governments have budget constraints determined by their funding sources and
governments (or government officials) make choices given their preferences. We will use this model to
explain some of the opposition to across-the-board budget cuts.
Assume that the government allocates its funds between education and law enforcement expenditures. (In
defining the goods in terms of expenditures, we are assuming that the prices of goods used in education
and law enforcement are not changing during the time period for which we are considering the
government budgeting.) The government’s preferences for law enforcement and education expenditures
satisfy the usual properties of the consumer theory model.
a. The government has a fixed budget amount B that it can allocate between the two goods. Use a graph
to describe the choice problem and the characteristics of the optimum allocation.
b. Suppose that the budget falls by 25 % and that the legislature reduces the previous expenditures on all
services by 25%. Use the consumer theory model to explain whether this is the best reaction to the
budget reduction.
4. Ben consumes two goods, X and Y. His preferences satisfy the usual assumptions. The market price
of each good is $1.
Ben works for a firm that produces and sells good X. Ben’s salary is $1000/week. In addition, the
firm allows its employees to purchase good X at a discount of 50%. Given his budget and his
preferences, Ben purchases 1000 units of X per week.
a. Construct a graph and show an indifference map consistent with the information given above. There
are many indifference maps consistent with this information. You are asked to show one such
indifference map.
b. Ben’s employer proposes eliminating its employee discount program and increasing salaries. Since
employees purchased, on average, 1000 units of X per week under the discount program, the firm
proposes compensating employees for the elimination of the discount by increasing each worker’s salary
by $500. Does the elimination of the discount program, along with the salary increase, make Ben better
off or worse off? Use the graph to explain your answer.
c. Would all workers’ well-being be affected in the same direction as Ben’s? Use a graph to support your
answer.
d. Use the graph that you constructed for parts a. and b. to determine the minimum salary increase
(approximately)required to compensate Ben for the loss of the discount program.
e. One of the firm’s accountants advises that the firm re-consider the proposed salary increase. She points
out that employees are required to shop at off-peak times and need very little customer assistance. Hence,
the employee discount program costs the firm less than $500/employee (on average). The accountant
estimates that the cost to the firm of the 50% employee discount is only $.25 per unit sold to an employee
rather than $.50. Thus, she suggests that the firm increase salaries by only $250 per person if it eliminates
the discount program.
Would eliminating the discount and increasing his salary by $250 necessarily make Ben worse off than
with the original discount program? Use a graph to support your answer.
g. Provide an economic justification for the employee discounts offered by many firms.
5. The initial and new budget lines are labeled BL0 and BLN respectively.
a. What change in income and/or prices would cause this rotation in the budget line?
b. Determine the income and substitution effects for this change in the budget line. Is good X a normal
or an inferior good (for this part of the indifference map)? Is good Y a normal or an inferior good (for
this part of the indifference map)?
Y
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BLN
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6. A household consumes two goods: educational quality and good Y. The household’s preferences
satisfy the usual assumptions. The household has income I; educational quality and good Y can be
purchased at market prices. Public education is not available. The household decides how much
educational quality and how much of good Y to purchase. We’ll assume that the household has only one
child so that we do not have to consider the possibility of different quality for different children.
a. Construct a graph on which you show school quality on the horizontal axis and good Y on the vertical
axis. Use the standard consumer theory model to describe the household’s choice problem. You are not
given numerical values for income or the prices of the good but can apply the consumer theory model to
describe the choice problem that the family faces.
b. The government now offers free,
public education of quality level Qg.
The government selects the quality
of the public education. (We are
ignoring taxes used to support the
schools.) A household’s choices are
to (i) purchase private educational
quality as in part a or (ii) send the
child to the public school of quality
Qg for free.
Keep in mind that if the household
selects public education, then it can
spend all of its income on good Y. If
you have any difficulty with the
budget set, then try making up
simple numbers. Once you
understand the budget set for a
simple numerical example, you can
often figure out the general budget
set.]
b1. For each family, identify the
optimum education quality choice if
the free public education were not
available.
b2. For each family, determine the
optimum choice with free public
schools of quality Qg available.
b3. For each family, compare the
quality of education chosen (i) when
no free education is available and-(ii)
when free public education of quality
Qg is available.
c. Suppose now that the government
offers households another option.
They may select the public education
for free or they may have a voucher.
The voucher can be used only to
purchase private education. The
value of the voucher equals the
expenditure required to purchase private education of quality Qg. For each of the households, answer the
following questions.
c1. Construct the household’s budget line with the voucher. [Again, make up simple numbers if
you have difficulty identifying the budget line. For example, suppose that the household’s income is
$1000 per time period. The prices of good Y and of school quality are $1 (with the price of $1 chosen to
reduce the algebra and focus on the economic question). The quality of the public school is 200. The
household may send the child to the public school of quality 200 for free. Or, the household may receive
a $200 voucher which it may use, along with funds of its own, to purchase private education. Identify
combinations of good Y and educational quality that the household could afford.]
c2. Does introducing the voucher program result in higher, lower, or the same quality of
education in comparison with part b where free public education of quality Qg was available but there
were no vouchers
c3. Does introducing the voucher program make households better off, worse off, or equally well
off than in part b? Explain..
7.
Consider the standard labor-leisure choice model and assume that Jo’s preferences satisfy the
usual assumptions. The standard graph for this model shows leisure per time period on the horizontal
axis and the consumption of good Y on the vertical axis. For simplicity, assume that Jo has no unearned
income. She may select the number of hours to work at the market wage.
a Construct a graph and show an example for which Jo chooses to work fewer hours (consume more
leisure) at a high wage than a low wage. Determine the income and substitution effects of the change in
the age.
b
Construct another example for which she chooses to work more hours at a higher wage.
c Suppose that the government imposes a tax of 50% on wage income. Do individuals necessarily
work fewer hours? Explain.
8. Let’s again consider Jo’s choices concerning work and leisure.
a
Jo’s wage is $8 per hour; she has no unearned income. Construct a graph and show Jo’s budget line.
b Jo is now eligible to receive a Social Security payment of $64 per day if her earned income is below
$32 per day. Construct Jo’s new budget line.
i
Show an indifference map for which Jo selects to earn less than $32 per day and thus to
receive Social Security.
ii
Also show a preference map for which she selects to work more than four hours per day and
thus forfeit the social security payment. If her wage increased would she necessarily
continue working more than four hours per day? Use the graph to explain your answer.
9. Let’s again consider the standard labor leisure choice model. The wage that Howard receives
depends upon whether he works part-time or full-time. If he works part-time, defined as six hours per day
or less, he earns a wage of $6 per hour. If he works full time, defined as more than six hours per day, he
earns a wage of $12 per hour. Construct his budget set. Show an example of preferences for which he
works part time and preferences for which he works full-time.
10. Eli, Peyton and Tom each consume two goods – X and Y. Their utility functions are
U(x,y) = xy, U(x,y) = x2y2 , and U(x,y) = x.5y.5 respectively. They each have income I and the prices of
the two goods are px and py. Derive the demand function for the quarterback of the winning 2008 Super
Bowl. Can you answer this question before the game begins?
11. Holden’s utility function is U(x,y) = a ln x + y where a is a positive number. He has income I and the
prices of the two goods are px and py. Derive his demand curve. For this case, you may assume that I-apy
is positive. [Hint: When you first begin the problem, ignore the assumption that I-apy is positive. Then,
as you work through the problem, you will understand why the assumption is necessary.]
12. Suppose instead that Holden’s utility is U(x,y) = a x1/4 +y1/4. Derive his demand curve. [Math
reminder: Suppose that Az1/4 = w1/4 . You can solve for w by raising both sides of the equality to the
4th power. If you raise both the left and the right sides to the 4th power, the equality still holds or A4 z=
w.]
13. An individual consumes two goods – X and Y.
a a. Suppose that an individual has Leontief preferences. Determine the income and substitution
effects of an increase in the price of good X on the individual’s consumption of good X and of good Y.
b Now suppose that the individual has quasilinear preferences and that the indifference curves are
parallel in the vertical direction, as shown in the graph below. Determine the income and substitution
effects of an increase in the price of good X on the individual’s consumption of good X and of good Y.
Y
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14. In the early 1900s, many large cities did not meter water consumption. The costs of the water
systems were paid through taxes rather than user fees. As the technology improved, meters were installed
and most of the costs of urban water systems were born by the users. Answer the following questions in
order to determine whether the switch to metering made the typical individual better off, equally well off,
or worse off.
An individual consumes two goods – water and good Y. The individual’s indifference map is
shown below. The individual’s well-being is strictly increasing in the consumption of good Y. The
individual eventually becomes satiated in the amount of water. For example, leaving the faucet running
and listening to the water would eventually become irritating. Leaving the hose running outside would
eventually produce mud.
The individual’s income is $80 per time period. The price of good Y is $1. The cost to the city
of providing water is $1 per unit.
a. The city levies a tax of $40. There is no fee for using water. Identify the consumer’s optimum. Does
the tax cover the city’s cost of providing the water consumed by the individual?
b. The city eliminates the tax and instead charges $1 per unit for the water. Could the individual continue
to consume the optimum bundle from part a? Does the individual select this bundle? Is the individual
better off, equally well off, or worse off than in part a? Use the graph to explain your answer.
c. In part b. we ignored any cost of metering. Suppose now that the metering costs the city $5 per time
period. As in part b., the city charges $1 per unit for water. In addition, the city charges a fixed fee of $5
that does not depend upon the amount of water used. Identify the new budget line and the new optimum.
d. As in part c., the city charges a fixed fee equal to the metering cost. Explain how an increase in the
metering cost affects the budget line. What is the maximum metering cost (approximately) for which it is
better for the city to charge the fee plus $1 for each unit of water rather than the set up in part a.
e. At the beginning of the spring 2008 semester, the student bookstore began charging for blue books and
the computer answer sheets. Apply the reasoning above to explain whether or not the change makes the
typical student better or worse off.
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Grading notes:
Question 4. One point for each of the following subquestions answered correctly up to a maximum of 3
points for the question

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Show the bundle and provide a valid explanation of how the individual’s well-being is affected
Explain why not all workers are not necessarily better off.
A valid argument for how to determine the minimum salary for which Ben is equally well-off
A valid argument to explain whether or whether not $250 is adequate to compensate Ben for the
loss of the discount
Question 8: Three points
Question 10: One point
The demand function is required for full credit.
Question 14: Three points