Bob's utility functioin over goos X and Y is U(X,Y) = 10X + 5Y. His income is 100 and the price of X is 2 and price of Y is 5
Answers
Answer:
1. For the following utility functions,
• Find the marginal utility of each good.
• Determine whether the marginal utility decreases as consumption of each good increases (i.e., does the utility
function exhibit diminishing marginal utility in each good?).
• Find the marginal rate of substitution.
• Discuss how MRS XY changes as the consumer substitutes X for Y along an indifference curve.
• Derive the equation for the indifference curve where utility is equal to a value of 100.
• Graph the indifference curve where utility is equal to a value of 100.
a. U(X,Y ) = 5X + 2Y
b. U(X,Y ) = X 0.33Y 0.67
c. U(X,Y ) = 10X 0.5 + 5Y
1. a. U(X,Y ) = 5X + 2Y
MU X =
∂
_
U(X,Y )
∂X
= 5
MU Y =
∂
_
U(X,Y )
∂Y
= 2
Marginal utility is constant for each good.
MRS XY =
MU
_ X
MU Y
=
_5
2
MRS is constant so indifference curves will have a constant slope (i.e., they are linear).
For __
U = 100,
U(X,Y ) =
__
U = 100 = 5X + 2Y
2Y = 100 – 5X
Y = 50 – 2.5X
x
y
50
20 0
b. U(X,Y ) = X 0.33Y 0.67
MU X =
∂
_
U(X,Y )
∂X
= 0.33X – 0.67Y 0.67
MU Y =
∂
_
U(X,Y )
∂Y
= 0.67 X 0.33Y – 0.33
Chapter 4 Appendix:
The Calculus of Utility Maximization
and Expenditure Minimization
45
Goolsbee2e_Solutions_Manual_Ch04_Calc.indd 45 oolsbee2e_Solutions_Manual_Ch04_Calc.indd 45 22/09/15 11:19 AM 2/09/15 11:19 AM
46 Part 2 Consumption and Production
The marginal utility of X decreases as the quantity of X increases, holding the quantity of Y constant. Also, the
marginal utility of Y decreases as the quantity of Y increases, holding the quantity of X constant. You can get this
result by inspecting the marginal utilities or by checking the signs of the derivatives of these marginal utilities.
MRS XY =
MY
_ X
MU Y
=
0.33X – 0.67Y __ 0.67
0.67 X 0.33Y – 0.33 =
_Y
2X
MRS XY decreases as the consumer increases consumption of X along an indifference curve so the indifference
curves are convex.
For __
U = 100,
U(X,Y ) =
__
U = 100 = X 0.33Y 0.67
1,000,000 = XY 2
Y 2 =
_1,000,000
X
Y = 1,000X – 0.5
x
y
100
100 0
c. U(X,Y ) = 10X 0.5 + 5Y
MU X = _
∂U(X,Y )
∂X
= (0.5)10 X – 0.5 = 5 X – 0.5
MU Y = _
∂U(X,Y )
∂Y
= 5
The marginal utility of good X decreases as more X is consumed. The marginal utility of good Y is constant:
MRS XY =
MU
_X
MU Y
=
5 X _
– 0.5
5
= X – 0.5
MRS decreases as the consumer increases consumption of X along an indifference curve so the indifference curves
are convex.
For __
U = 100,
U(X,Y ) =
__
U = 100 = 10X 0.5 + 5Y
5Y = 100 – 10X 0.5
Y = 20 – 2X 0.5
x
y
20
100 0
Note: This type of utility function is known as a “quasi-linear” utility function. The indifference curves for quasilinear utility functions are parallel. In other words, the slopes of the indifference curve are the same, given a value
of X.
Goolsbee2e_Solutions_Manual_Ch04_Calc.indd 46 oolsbee2e_Solutions_Manual_Ch04_Calc.indd 46 22/09/15 11:19 AM 2/09/15 11:19 AM
Appendix: The Calculus of Utility Maximization and Expenditure Minimization Chapter 4 47
2. Suppose that Maggie cares only about chai and bagels. Her utility function is U = CB, where C is the number of cups
of chai she drinks in a day, and B is the number of bagels she eats in a day. The price of chai is $3, and the price of
bagels is $1.50. Maggie has $6 to spend per day on chai and bagels.
a. What is Maggie’s objective function?
b. What is Maggie’s constraint?
c. Write a statement of Maggie’s constrained optimization problem.
d. Solve Maggie’s constrained optimization problem using a Lagrangian.
2. a. max
C, B CB
b. Income = P C C + P B B or 6 = 3C – 1.5B or 6 – 3C – 1.5B = 0
c. max
C, B CB s.t. 6 – 3C + 1.5B
d. Write out the Lagrangian for the problem in part (c):
max
C, B, λ (C, B, λ) = CB + λ(6 – 3C – 1.5B)
FOC:
_∂
∂ C
= B – 3λ = 0
_∂
∂ C
= C – 1.5λ = 0
_∂
∂λ
= 6 – 3C – 1.5B = 0
From the fi rst two conditions,
λ =
_
B
3
=
_C
1.5
B = 2C
Substituting into the third FOC, we get
6 – 3C – 1.5B = 6 – 3C – 1.5(2C ) = 6 – 6C = 0
C* = 1
Then B* = 2.
So, Maggie buys 1 cup of chai and 2 bagels per day.
3. Suppose that there are two goods (X and Y ). The price of X is $2 per unit, and the price of Y is $1 per unit. There are
two consumers (A and B). The utility functions for the consumers are
U A(X, Y ) = X 0.5Y 0.5
U B(X, Y ) = X 0.8Y 0.2
Consumer A has an income of $100, and Consumer B has an income of $300.
a. Use Lagrangians to solve the constrained utility-maximization problems for Consumer A and Consumer B.
b. Calculate the marginal rate of substitution for each consumer at his or her optimal consumption bundles.
c. Suppose that there is another consumer (let’s call her C ). You don’t know anything about her utility function or her
income. All you know is that she consumes both goods. What do you know about C’s marginal rate of substitution
at her optimal consumption bundle? Why?
Explanation: