Econ. 410
Spring 2008
Tauchen/Biglaiser
Practice Problems
The Consumer Theory Model, Consumer Surplus, and the Edgeworth Box Model
1. In class, we determined the income and substitution effects of the increase in the price of
good X and will now work through the income and substitution effects for a price decrease. We
will need the handout on income and substitution effects which was distributed in class. (If you
no longer have the sheet, a copy is available at
http://www.unc.edu/courses/2008spring/econ/410/007/HO-Reading.html ) The fourth page of the
handout deals with the income and substitution effects of a price decrease.
a. Show the substitution effect on the top graph. Is the substitution effect necessarily in the
direction of more of good X? Suppose that both X and Y are normal goods. Show more of the
individual’s indifference map consistent with both goods being normal goods. What is the
direction of the income effect on the consumption of good X for this case?
b. Show the substitution effect on the bottom graph. Now assume that X is an inferior good and
that Y is a normal good. Show more of the individual’s indifference map consistent with X being
an inferior good and Y being a normal good. What is the direction of the income effect on the
consumption of good X for this case?
c. Complete the chart which is on the last page of the handout and which is repeated below.
Effect of a Decrease in the Price of Good X on the Optimal Consumption of Good X
Normal Good
Inferior Good
Substitution Effect
Income Effect
Total Effect
2. Determine the intercepts of the budget lines for the following cases. Also determine the "no
borrow, no lend" bundle.
Case A: I1 = $10,000, I2 = $12,000, p1 = $1, p2 = $1, r = .10
Case B: I1 = $10,000, I2 = $13,800, p1 = $1, p2 = $1.15, r = .265
Case C: I1 = $10,000, I2 = $13,800, p1 = $1, p2 = $1.15, r = .10
3. Explain how an increase in the interest rate affects the budget line (ceteris paribus). Explain
how an increase in period 1 income affects the budget line (ceteris paribus).
4. We define consumption in period 1 as a normal good if an individual chooses to consume
more in period 1 when there is a parallel shift out in the budget line. Consumption in period 1 is
an inferior good if an individual chooses to consume less in period 1 when there is a parallel shift
out in the budget line. (The corresponding definitions for consumption in period 2 are exactly
analogous.)
a. What changes in incomes, prices, or interest rates would cause a parallel shift out in the budget
line?
b. Construct a graph in which you show consumption in period 1 as a normal good and an
example in which you show consumption in period 1 as an inferior good.
4. Construct a graph to show that with an increase in the interest rate, ceteris paribus, an
individual may switch from being a borrower in period 1 to being a saver in period 1.
5. Given her initial two-period, intertemporal budget line, Bernadette chooses to save some of
her income in period 1.
a. Construct a graph on which you show Bernadette’s budget line and an indifference curve
consistent with her initial choice. Determine the income and substitution effects of an increase in
the interest rate. [Hint: Determine the income and substitution effects in the same way as for two
goods consumed in the same period. Construct a hypothetical budget line with the same slope as
the new budget line and tangent to the initial indifference curve.] Is the substitution effect for
consumption in period 1 positive or negative? Is the substitution effect for saving in period 1
positive or negative? Is the substitution effect for consumption in period 2 positive or negative?
b. Rather than finding the income and substitution effects of an interest rate increase for a
specific example, we want to think about whether or not the direction of the income and
substitution effects is unambiguous for all preferences satisfying the usual assumptions about
preferences. As in part a., we assume that the individual was a saver in the initial period. We
will also assume that consumption in period 1 is measured on the horizontal and consumption in
period 2 on the vertical axis.
b1. Is the new budget line steeper or flatter than the initial budget line? Is the hypothetical
budget line steeper or flatter than the initial budget line? Given the curvature of the
indifference curve does the tangency of the hypothetical budget line with the initial
indifference curve occur down and to the right of the initial optimum or up and to the left of
the initial optimum? What is the direction of the substitution effect on consumption in period
1? on savings (which is income minus consumption in period 1)? on consumption in period
2?
b2 From the individual’s point of view, is the new budget set better or worse than the initial
budget set? Does the shift from the hypothetical budget line represent a shift out in the budget
line or a shift back in the budget line? If consumption in period 1 is a normal good, then how
does such a shift affect consumption in period 1? How does such a shift affect consumption
in period 2?
b3. Let’s assume that consumption in period 1 and in period 2 are normal goods. Does
consumption in period 1 necessarily increase when the interest rate increases (assuming that
the individual was a saver in period 1)? Use the income and substitution effects from parts b1
and b2 to explain your answer.
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6. Thus far we have assumed that individuals may borrow and lend at the same interest rate.
Assume instead that the interest rate at which individuals borrow is higher than the interest rate
that individuals receive on their savings. Specifically, I1 = $20,000, I2 = $10,000, p1 = $1, p2 =
$1. Jan can borrow at r = .8 and receives r = .1 on savings.
a. Construct the budget line. [Hint: Construct the budget lines for r=.8 and the budget line for
r=.1 . For each budget line, identify the segment for which the individual is a borrower and the
segment for which the individual is a saver. As a borrower, the relevant BL is the one for which
r=.8. As a saver, the relevant BL is the one for which r=.1 .
b. Show an example of Jan's indifference map (i) for which she chooses to neither borrow nor
lend since she receives only r = .1 on her savings but (ii) for which she would have saved if she
received the higher rate r = .8.
7. The rate of inflation is defined as (p2 - p1 )/p1 . Macroeconomists commonly use the symbol
π for the rate of inflation. The real interest rate is defined as (r-π)/(1+π). For the intertemporal
problems with only one good in each period, we often think of the price as being a measure of the
general price level. The real income in each period is the income for the period divided by the
price level.
a. Compute the rate of inflation for each case in question 2. Also compute the real income for
each period and the real interest rate. Determine the budget line for each case.
b. Is the individual better off for Case A in which there is no inflation or Case B in which there is
inflation? Explain.
c. In comparing Cases A and B, note that the real incomes and the real interest rate are the same
for the two cases although the dollar incomes and the interest rates differ. The budget lines are
also the same. We want to show that all cases for which the real incomes and the real interest
rates are the same yield the same budget lines.
Consider the following two cases and show that the budget lines are the same.
Case with No Inflation: The incomes in the two periods are Iˆ1 and Iˆ2 . The prices in both period
are p̂ and the interest rate at which the individual borrows and lends is r̂ .
Case with Inflation at Rate ̂ : The income in period 1 is Iˆ1 and the income in period 2 is
Iˆ (1+ ̂ ). The price in period 1 is p̂ and the price in period 2 is p̂ (1+ ̂ ). The interest rate
2
for this case is r̂ + ̂ + r̂ ̂ .
8. The market for good X is competitive and the price is determined by supply and demand.
Construct a graph and show the effect of a per unit tax on the equilibrium price and quantity.
Then show the effect on consumer surplus. Does the effect of the tax on consumer surplus
depend upon whether producers pay the tax out of their revenues or consumers pay the tax (in
addition to paying the seller for the good)? Use the graph to explain your answer.
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9. The next page shows the graph that we used in the class discussion of compensating and
equivalent variation. The individual’s income is $32/time period and the price of good Y is $4.
The price of good X is initially $1 and then increases to $4. The individual has quasilinear
preferences and the indifference curves are parallel relative to the horizontal axis.
The budget line for the initial prices is denoted BL:LP and for the higher price of good X
is denoted BL:HP. The hypothetical budget lines used to determine the compensating and
equivalent variation are denoted BL:CV and BL:EV respectively.
a. The individual initially chose the bundle (10, 5.5). What is the slope of the indifference curve
at that point? What is the slope of the indifference curve that goes through every other bundle
with 10 units of good X? [Hint: Remember that the individual has quasilinear preferences.]
b. At the higher price for good X, the individual selected the bundle (1.5,6.7). What is the slope
of the indifference curve that goes through this point? What is the slope of the indifference curve
that goes through any other bundle which contains 1.5 units of good X?
c. The budget line BL:CV is constructed for the new prices and is tangent to the indifference
curve that the individual attained with the initial lower price for good X. Let’s refer to this
indifference curve as IC1. What is the slope of IC1 at the tangency to the budget line BL:CV?
Given that the preferences are quasilinear and the indifference curves are parallel, what is the
amount of X at the bundle where the budget line BL:CV is tangent to IC1?
d. The budget line BL:EV is constructed for the initial prices and is tangent to the indifference
curve that the individual attained with the new higher price for good X. Let’s refer to this
indifference curve as IC2. What is the slope of IC2 at the tangency to the budget line BL:EV?
Given that the preferences are quasilinear and the indifference curves are parallel, what is the
amount of X at the bundle where the budget line BL:EV is tangent to IC2?
e. Given the parallel indifference curves what can you conclude regarding the distance between
BL:HP & BL:CV versus the distance between BL:LP & BL:EV? What does your answer to this
question imply regarding the relative values of the compensating and the equivalent variation?
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Y
10
9
CC
C1 is the optimum at the initial price of good X
C2 is the optimum at the higher price of good X
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CC is the optimum for the BL used in finding CV
CE is the optimum for the BL used in finding EV
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C2
6
C1
5
4
CE
3
BL:LP
2
1
BL-HP
BL-EV
BL:CV
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
X
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10. Condoleezza consumes two goods – X and Y. Her income is $27 and the price of good Y is
$3. The budget line for px=$1 is denoted BL1 and the budget line for px=$3 is denoted BL2.
a. Use the graph below to determine the compensating variation for the price increase. To
do so, first construct a hypothetical budget line for the new higher price of good X and for
which Condoleezza obtains the same level of well-being as for the initial price of good X, but
no higher level of well-being. How much additional income would be required for
Condoleezza to have this budget line? The compensating variation is the change in income
required for her to have the hypothetical budget line that you constructed.
b. Suppose that you provided Condoleezza with enough additional income to purchase C1,
which is the optimum for the initial lower price for good X, rather than the income required
for her to obtain the same level of well-being, but no greater level of well-being, as at the
initial lower price. Would she be better off, equally well off, or better off than with the
compensating variation income increase? Explain.
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c. Use the graph below to determine the equivalent variation for the price increase. To do
so, first construct the budget line that is for the initial prices and by which Condoleezza
achieves the same level as well-being as with BL2 (the budget line with the higher price for
good X) but no higher level of well-being. The equivalent variation is the reduction in
income that would give Condoleezza the budget line that you have just constructed.
d. Compare the compensating and equivalent variation.
11. Use the graphs above to identify the compensating and equivalent variation for a decrease in
the price of good X from $3 to $1.
12. An individual’s endowment is 6 units of good X and 6 units of good Y. The individual may
buy and sell the goods at the market prices. Construct the budget lines for the following
combinations.
a.
b.
c.
d.
e.
px =$1 and py=$1
px =$10 and py=$10
px =$2 and py=$1
px =$10 and py=$40
px =$10 million and py=$40 million
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13. Mitt’s endownment is 6 units of good X and 6 units of good Y.
a. What prices generate
the budget line shown on
the graph? How much
of good X does Mitt
supply or demand at
these prices? good Y?
b. Construct an
example in which Mitt
supplies one unit of good
X for px=$2 and py=$1.
14. A quote by A. K. Sen, Economics, who won the Sveriges Riksbank (Bank of Sweden) Prize
in Economic Sciences in Memory of Alfred Nobel, appears below. (The Economics award is
judged and administered by the Nobel Foundation in the same way as the five original Nobel
prizes but was not part of Nobel’s will. The prize is often referred to as the Nobel Prize in
Economics.)
"An allocation of resources may be Pareto efficient even when some people are rolling in
luxury and others are near starvation as long as the starvers cannot be made better off
without cutting into the pleasures of the rich. In short, a society can be Pareto optimal and
still be perfectly disgusting."
Use the Edgeworth Box to provide an example of an allocation that is “Pareto optimal and
perfectly disgusting.” Is the description of an allocation as “perfectly disgusting” a positive or
normative economic statement?
15. You have $100 (or 100 candies) to divide among the members of our class, each of whom
prefers more to less.
a. Identify the efficient allocations of the $100 among the members of the class.
b. Identify the fair allocations of the $100 among the members of the class.
16. Consider a simple two-person, two-good exchange economy. Assume that both individuals'
preferences satisfy the usual assumptions. Construct an Edgeworth box diagram and show an
example of an initial endowment for which there are no benefits to trade. Be sure to label the
axes of your graph.
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17. Consider a two person economy with two individuals (George and Al) and two goods (X
and Y). Both George and Al have linear indifference curves. The units of good X per time period
are measured on the horizontal axis and the units of good Y per time period on the vertical axis.
For any bundle, Al’s indifference curves have slope equal to –1 and George’s have slope equal to
–2. Each individual has an allocation of two units of each good.
a. Construct a graph on which you show Al’s initial allocation and several of his indifference
curves.
b. Do the same for George.
c. Construct an Edgeworth Box with the origin of George’s graph in the lower left and Al’s in
the upper right. Show indifference curves for each individual. Identify the efficient allocations.
18. Answer the question above assuming that George’s linear indifference curves also have slope
equal to -1.
19. John K. and John D. each consume two goods – X and Y. Each of their preferences satisfy
the usual assumptions. Each individual has a positive initial allocation of each good. At his
initial allocation, John K’s Marginal Rate of Substitution between X and Y is 12 whereas John
D’s is 2. Can they benefit from trade? Explain.
20. Circle all of the correct answers.
a. A good learning strategy at the university is to do only work for which there is an immediate
reward such as credit for a homework problem.
b. Working on homework problems is pointless unless they are collected and graded.
c. When I am professionally employed, my supervisor will immediately give me a cookie for
every task that I perform well.
d. The best study strategy for practice problems is to buy a big cup of coffee (and some candy)
the night before the exam and then start the problems.
e. The best way to understand how to apply economic principles is to memorize pages of the text
along with the author’s middle name.
f. It requires valuable academic discipline to do practice problems in a timely way rather than
postponing work on them until the day before an exam.
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