Econ 3070
Prof. Barham
Problem Set – Chapter 4 Questions
1. Below is the equation the budget constraint for Joe for his coffee (C) and
doughnuts (D) consumption. His income is 10, the price of doughnuts is $2 and
the price of coffee is $1.
Income= 1C + 2D
Doughnuts
7.5
5
BL3
BL1
BL2
5 7.5
10
Coffee
a. On the graph above with doughnuts on the vertical axis and coffee of the
horizontal axis draw the budget line. Hint: figure out where the budget line hits
each axis (so if D is zero what is C, if C is zero what is D), then connect the
points. Label this BL1 on your graph and give the numbers where the budget line
hits both axis.
The budget line is 10= 1C + 2D, see above graph for line. Note when D is zero, C is 10.
When C is zero D is 5. This tells us where the budget line hits the two axis.
b. Now the price of coffee goes up to $2, draw a second budget line on the graph
that reflects the price change. Label it BL2. You can use the same graph but make
sure you label your budget lines!
See above graph for line. BL2 is rotated in as only the price of coffee changes. The
budget line hits the D axis at the same spot and the line rotates in because Joe can now
buy less coffee with his $10.
c. Now Joe receives a pay raise and his income is now $15. The price of doughnuts
and coffee is still $2 each. Draw a third budget line to reflect the income raise.
Label this BL3.
BL2 shifts out in a parallel fashion to BL3, see above graph.
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Econ 3070
Prof. Barham
2. Mona’s preferences for food, F, and clothing, C, are derived by a utility function
U(F,C)=FC. Food costs $1 a unit and clothing costs $2 a unit. Julie has $12 to
spend on food and clothing.
.
a. Write the equation for Mona’s budget line. If food is on the vertical axis, what
is the slope of the budget line?
Budget equation is: F + 2C = 12, slope = - −
Pc
= −2
Pf
b. Graph Mona’s budget line. Place food on the vertical axis and clothing on the
horizontal axis.
F
12
6
C
c. On the same graph, draw an indifference curves that is tangent to her budget
line.
See above point b.
d. Mona is a utility maximizer, write the objective function.
Her objective function is her utility function U(F,C)=FC.
e. Write down the full optimization problem with the objective function and the
constraint.
The full optimization problem is
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Econ 3070
Prof. Barham
Max U (F, C) =FC
C,F
s.t. F+2C=12
f. Using calculus and algebra, find the basket of food and clothing that
maximizes Mona’s utility (i.e. solve the maximization problem you wrote
down in e) (Assume Mona can purchase fractional amounts of both goods.)
Do not solve this using the tangency condition taught in the book solve it like in
class.
Max U (F, C) =FC
C,F
s.t. F+2C=12
Step 1: using the budget constraint solve for F or C: F=12-2 C
Step 2: Substitute the budget constraint into the utility function so the utility function is a
function of one good.
MAX U(C)=(12-2C)C=12C-4C 2
C
Step 3: Now we need to maximize U with respect to C.
∂U
= 12 − 4C = 0
∂C
C=3
Step 4: Sub C back into budget constraint to figure out optimal F
F+2(3)=12 so F=6
The basket which optimizes Mona’s utility function is C*=3 and F*=6.
3. Petra consumes only two goods, pizza (P) and hamburgers (H) and considers
them to be perfect substitutes, as shown by his utility function: U(P, H) = P + 4H.
The price of pizza is $3 and the price of hamburgers is $6, and Paul’s monthly
income is $300.
a. Write the equation for Petra’s budget line.
The budget line is: 3P + 6H=300
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Econ 3070
Prof. Barham
b. If hamburgers are on the vertical axis, what is the slope of the budget line?
The slope is: -1/2
Rewrite the budget equation with H on the RHS.
H=300/6 – 3/6p
c. Graph Petra’s budget line. Place the hamburgers servings on the vertical axis
and pizza on the horizontal axis. Make sure to indicate the values of where
the budget line hits each axis.
Note the budget line is the solid straight line in the graph below.
H
50
0
IC at maximization point
BL1
100
P
d. On the same graph, draw several of Petra’s indifference curves, including one
that show where Petra will maximize his utility. Make sure to clearly indicate
which indifference curve that maximizes utility.
The indifference curves are the dotted line. The indifference curve which maximizes
utility is the one that is the highest, where there is a corner solution and Petra only eats
hamburgers.
e. Petra is a utility maximizer. Write down the full optimization problem with
the objective function and the constraint.
Max U (H, P) = P + 4H
H ,P
s.t. 3P + 6H = 300
f. Solve for the values of P and H that maximizes Petra’s utility.
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Econ 3070
Prof. Barham
The is an example of perfect substitutes. Because both the indifference curves at the
budget line are straight we will have a corner solution (i.e. Petra will choose to eat all
of one good). To figure out the answer these types of problems we determine which
good gives a greater “bang for the buck” by comparing:
MUh ∂U / ∂H 4 2
=
= =
Ph
Ph
6 3
MUp ∂U / ∂p 1
=
=
Pp
Pp
3
Because the indifference curve is linear the MRS is the same for any value of and H. The
“bank for buck” analysis shows us that for each dollar spent, the utility is higher from
hamburgers than pizza.
So Petra will spend her money only on hamburgers, she will not eat pizza. Using the
budget constraint, we know she has 300 dollars and it costs $6 for a hamburger so she
can afford 50 hamburgers.
So after utility maximizing: H*=50 and P*=0
4. Ch 4, Problem 4.6
Jorge likes hamburgers (H) and milkshakes (M). His indifference curves are
bowed in and toward the origin and do not intersect the axes. The price of a
milkshake is $1 and the price of a hamburger is $3. He is spending all her
income at the basket he is currently consuming, and his marginal rate of
substitution of hamburgers for milkshakes is 2. Is he at an optimum? If so,
show why. If not, should he buy fewer hamburgers and more milkshakes, or the
reverse?
NO Jorge is not at an optimum. Jorge can increase her total utility by reallocating her
spending to purchase fewer hamburgers and more milkshakes. See below for why.
Note: here we have to use the tangency condition since we are not given enough
information for consumer optimization.
From the given information, we know that PH = 3, PM = 1, and MRSH,M = 2. Comparing
the MRSH,M to the price ratio,
MRS H , M = 2 <
PH 3
= .
PM 1
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Econ 3070
Prof. Barham
Since these are not equal, Jorge is not currently at an optimum. In addition, we can say
that
PH
MU H
,
> MRS H , M =
PM
MU M
which is equivalent to
MU M MU H
.
>
PM
PH
That is, the “bang for the buck” from milkshakes is greater than the “bang for the buck”
from hamburgers. So Jorge can increase her total utility by reallocating her spending to
purchase fewer hamburgers and more milkshakes.
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