Intermediate Microeconomics
Spring 2008
Prof. David Bjerk
Problem Set 2 (Answer Key)
1 – Consider each of the following different utility functions.
(i) U(q1,q2) = q14q22
(ii) U(q1,q2) = 5q1 + 2q2
(iii) U(q1,q2) = (5q1 + 2q2)3
(iv) U(q1,q2) = 5(q1)8(q2)4
(a) Derive the expression for the Marginal Rate of Substitution (MRS) for each of the
above utility functions.
Answer:
4q13 q 22
q
= −2 2
4 1
q1
2q1 q 2
5
(ii) MRS = −
2
3(5q1 + 2q 2 ) 2 5
5
(iii) MRS = −
=−
2
2
(5q1 + 2q 2 ) 2
(i) MRS = −
5 * 8 * (q1 ) * (q 2 )
7
(iv) MRS = −
4
5 * 4 * (q1 ) * (q 2 )
8
3
= −2
q2
q1
(b) What is the MRS evaluated at the bundle {q1=1, q2 = 9}and at the bundle {q1=4, q2 =
4} for each of the above utility functions?
Answer:
(i) MRS(1,9) = -18
MRS(4,4) = -2
(ii) MRS(1,9) = -5/2 MRS(4,4) = -5/2
(iii) MRS(1,9) = -5/2 MRS(4,4) = -5/2
(iv) MRS(1,9) = -18 MRS(4,4) = -2
(c) Which of the above utility functions exhibit diminishing MRS?
Answer: U(q1,q2) = q14q22 and U(q1,q2) = 5(q1/4)4(q2/4)2
2 – Bill and Ted both like chicken wings and martinis. Bill’s preferences are captured by
the utility function UB(qc,qm) = qc0.8qm0.2, where qc is ounces of chicken wings and qm is
ounces of martinis. Ted’s preferences are captured by the utility function UT(qc,qm) =
qc0.5qm0.5. Suppose both guys initially were given 4 chicken wings and 6 oz of martini. To
get one more chicken wing, who would be willing to give up more martini, Bill or Ted?
Answer:
Bill and Ted’s MRS of martinis for chicken wings will be given by
∂U (q c , q m )
∂q c
MRS (q c , q m ) = −
∂U (q c , q m )
∂q m
0.8q c−0.2 q m0.2
q
= −4 m , which means at the bundle {4,6}, his
So for Bill, MRS (q c , q m ) = −
0.8 − 0.8
qc
0.2q c q m
MRS equals -4(6/4) or -6.
0.5q c−0.5 q m0.5
q
For Ted, MRS (q c , q m ) = −
= − m , which means at the bundle {4,6}, his
0.5 − 0.5
qc
0.5q c q m
MRS equals -(6/4) or -3/2.
Graphically, this means
qm
qm
6
6
slope = -6
Bill
slope = -3/2
4
5
qc
Ted
4
5
qc
Since the slope of Bill’s Indifference curve at {3,6} is substantially steeper than the slope
of Ted’s Indifference curve at {3,6}, we know that Bill will be willing to give up more
oz. of martini than Ted for 1 more chicken wing (see downward pointing arrows).
3 – Assume Jane’s preferences over q1 and q2 are given by U(q1,q2) = q1 + q2.
(a) If the p1 = $2 and p2 = $3, draw Jane’s Income expansion path and corresponding
Engle curve for q1 for incomes ranging from $6 to $12.
Answer: BCs are thinnest lines, ICs are thicker lines, Income Expansion Path is thickest.
q2
income expansion path is on
horizontal axis
q1
income
Engel Curve between incomes of
$6 and $12
$12
$6
3
4 5 6
q1
(b) Suppose instead that p1 = $4 and p2 = $3. Re-draw Jane’s income expansion path and
Engle curve for q1 for incomes ranging from $6 to $12.
Answer: BCs are thinnest lines, ICs are thicker lines, Income Expansion Path is thickest.
q2
income expansion path is on
vertical axis
income
q1
$12
$6
Engel Curve between incomes of
$6 and $12
q1
4 - Anna’s preferences are characterized by the utility function U(q1,q2) = q1q2 and she
has an endowment of $100. The price for each unit of q1 is $5 and the price for each unit
of q2 is $4.
(a) Suppose the government imposes a sales tax of $1 per unit on q2. Derive Anna’s
optimal affordable bundle.
Answer:
Anna’s general problem is to find q1* & q2* to solve
subject to p1q1 + p2q2 ≤ m (where m is her endowment)
max q1q2
We can plot a couple of Anna’s indifference curves and recognize that they will look like
below, and also recognize that this means that her optimal bundle will always be on her
budget constraint (meaning p1q1* + p2q2* = m) and will be such that MRS(q1*, q2*) = p1/p2.
q2
q2*
q1*
q1
Solving the B.C. equation for q2 we get q2* = m/p2 – p1/p2 q1* (denote this as equation 1).
The MRS for these preferences will equal -q2/q1. Therefore, plugging equation 1 into the
second condition for the optimal bundle we get:
- (m/p2 – p1/p2 q1*)/q1* = - p1/p2
Solving this for q1* we get q1* = m/2p1. Plugging this back into equation 1, we get
q2* = m/2p2.
(Note: Instead of doing all of this to derive the above demand functions, you could also
just remember that for a Cobb-Douglas utility function of the form U(q1,q2) = q1aq2b and a
budget constraint of the form p1q1 + p2q2 ≤ m, the demand functions will be given by q1*
= a/(a+b) m/p1 and q2* = b/(a+b) m/p2. Given a = 1 and b = 1 in this question, we then get
q1* = ½ m/p1 and q2* = ½ m/p2, identical to what was derived above).
Finally, note that the tax on q2 means the effective price Anna must pay for each unit of
q2 is its own price ($4) plus the tax ($1), or $5 per unit. Therefore, with an endowment of
$100 and effective prices of $5 for each unit of each good, her optimal bundle under this
tax is
q1* = 100/(2)5 = 10
q2* = 100/(2)5 = 10
How much revenue does the government collect given Anna behaves optimally?
Total Revenue = tax*q2* = 1*10 = $10.
(b) Suppose instead of taxing only q2, the government imposes a lump sum tax of $10 to
be paid out of Anna’s endowment. Derive Anna’s optimal bundle under this tax scheme.
Answer:
Given the demand functions derived above, and taking the original prices of p1 = 5 and p2
= 4, and the new endowment of m – t = 100 – 10 = 90, we get
q1* = 90/(2)5 = 9
q2* = 90/(2)4 = 11.25
(c) Which tax scheme makes Anna better off? Why do you give the answer that you do?
(hint: think about her utility under each bundle)
Answer:
Under the tax just on q2, Anna’s utility from her optimal affordable bundle is U(10,10) =
10*10 = 100. Under the lump sum tax, Anna’s utility from her optimal affordable bundle
is U(9,11.25) = 9*11.25 = 101.25. Therefore, here utility is higher under the lump sum
tax, meaning she is better off under the lump sum tax.
6 - Given an individual’s preferences are captured by a utility function U(q1, q2) = q110q2.
(a) Will q2 be a normal good, an inferior good, or neither for this individual?
Answer: We know that the demand function for q1 given Cobb-Douglas preferences will
be of the form
q2(p1,p2,m) = b/(a+b) m/p2,
So given the functional form above, we know this individual’s demand for q2 will be
given by
q2(p1,p2,m) = 1/11 m/p2
Since this function is increasing in m, we know q2 will be a normal good for this
individual.
(b) Will q1 be a gross substitute, a gross complement, or neither for q2?
Answer: From above we know that the demand function for q1 given Cobb-Douglas
preferences will be of the form
q2(p1,p2,m) = 1/11 m/p2
Since this is not a function of the price of the other good (p1), we know that if an
individual has Cobb-Douglas preferences over two goods, the two goods will be neither
complements or substitutes (i.e. the derivative of q1(p1,p2,m) with respect to p2 is zero
which is not positive or negative).
7 – Menesh’s preferences over q1 and q2 are depicted by the indifference curves below.
q2
a1
a2
2
3 a3
5
q1
(a) Sketch Menesh’s Demand Curve for q1 between p1 = 1 and p1 = 3, given p2 = 3 and he
has $12. (Note: use your best guess as to the exact coordinates of each optimal bundle)
Answer: Given m = 12, and p1 = 1 and p2 = 3, Menesh’s BC is shown by line (a1) above.
This gives an optimal bundle that contains q1 = 5. If p1 rises to 2, Menesh’s BC moves to
line (a2) giving him an optimal bundle containing q1 = 3. Finally, when p1 rises to 3,
Menesh’s BC moves to line (a3), giving him an optimal bundle containing q1 = 2. Given
this information, we can draw Menesh’s demand curve for q1 which is labeled “a” below.
price
3
2
1
a
b
c
q1
(b) On the same graph you drew for (a), show how Menesh’s Demand curve changes if
his income increased to $6. (The indifference curve map from above is shown again
below)
q2
b2
b3
b1
1.3 1.6
4
q1
Answer: Given m = 6, and p1 = 1 and p2 = 3, Menesh’s BC is shown by line (b1) above.
This gives an optimal bundle that contains q1 = 4. If p1 rises to 2, Menesh’s BC moves to
line (b2) giving him an optimal bundle containing q1 = 1.6. Finally, when p1 rises to 3,
Menesh’s BC moves to line (b3), giving him an optimal bundle containing q1 = 1.3.
Given this information, we can draw Menesh’s demand curve for q1 given this lower
income which is labeled “b” above.
(c) Re-draw your graph from (a) and then show how Menesh’s Demand curve changes
when p2 = 4 (but he still has $12). (The indifference curve map from above is shown
again below)
q2
c2
c3
c1
2.6 3.3
6
q1
Answer: Given m = 12, and p1 = 1 and p2 = 4, Menesh’s BC is shown by line (c1) above.
This gives an optimal bundle that contains q1 = 6. If p1 rises to 2, Menesh’s BC moves to
line (c2) giving him an optimal bundle containing q1 = 3.3. Finally, when p1 rises to 3,
Menesh’s BC moves to line (c3), giving him an optimal bundle containing q1 = 2.6.
Given this information, we can draw Menesh’s demand curve for q1 given this new p2
which is labeled “c” above.