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- 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 ) = 5 X + 2 Y b. U ( X , Y ) = X 0.33 Y 0. c. U ( X , Y ) = 10 X 0.5^ + 5 Y
- a. U ( X , Y ) = 5 X + 2 Y
MU (^) X =
∂_ U ( X , Y )
∂ X
MU Y =
∂_ U ( X , Y )
∂ Y
Marginal utility is constant for each good.
MRS (^) XY =
MU _ X
MU Y
= _^5
MRS is constant so indifference curves will have a constant slope (i.e., they are linear). For
__
U = 100,
U ( X , Y ) =
__
U = 100 = 5 X + 2 Y
2 Y = 100 – 5 X
Y = 50 – 2.5 X
x
y
50
(^020)
b. U ( X , Y ) = X 0.33 Y 0.
MUX =
∂_ U ( X , Y )
∂ X
= 0.33 X –^ 0.67^ Y 0.
MUY =
∂_ U ( X , Y )
∂ Y
= 0.67 X 0.33 Y –^ 0.
Chapter 4 Appendix:
The Calculus of Utility Maximization
and Expenditure Minimization
45
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 =
_ MYX
MUY
= 0.33 X^
– 0.67 Y 0.
__
0.67 X 0.33^ Y –^ 0.^
= _ Y
2 X
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 0.33 Y 0.
1,000,000 = XY^2
Y^2 = _1,000,
X
Y = 1,000 X –^ 0.
x
y
100
(^0100)
c. U ( X , Y ) = 10 X 0.5^ + 5 Y
MU (^) X =
∂_ U ( X , Y )
∂ X
= (0.5)10 X –^ 0.5^ = 5 X –^ 0.
MUY =
∂_ U ( X , Y )
∂ Y
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.
_
= X –^ 0.
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 = 10 X 0.5^ + 5 Y
5 Y = 100 – 10 X 0.
Y = 20 – 2 X 0.
x
y
20
(^0100)
Note : This type of utility function is known as a “quasi-linear” utility function. The indifference curves for quasi- linear utility functions are parallel. In other words, the slopes of the indifference curve are the same, given a value of X.
48 Part 2 Consumption and Production
From the first two conditions,
λ = 0.25 x –^ 0.5 Y 0.5^ = 0.5 X 0.5 Y –^ 0.
Y = 2 X
Substituting into the third FOC, we get
100 – 2 X – 2 X = 0
X (^) A = 25
Then YA = 50. For B , max X , Y X^
0.8 Y 0.2 (^) s.t. 300 = 2 X + Y
max X , Y , λ ^ =^ X^
0.8 Y 0.2 (^) + λ (300 – 2 X – Y )
FOC:
∂_
∂ X
= 0.8 X –^ 0.2 Y 0.2^ – 2 λ = 0
_^ ∂
∂ Y
= 0.2 X 0.8 Y –^ 0.8^ – λ = 0
_^ ∂
∂ λ
= 300 – 2 X – Y = 0
From the first two conditions,
λ = 0.4 X –^ 0.2 Y 0.2^ = 0.2 X 0.8 Y –^ 0.
y = 40.5 x
Substituting into the third FOC, we get
300 – 2 X – 40.5 X = 0
X (^) B = 120
Then YB = 60. b. The fi rst terms in the first two FOCs are MU (^) X and MU (^) Y , respectively. Therefore,
MRS XYA^ =
_ MU^ X
MUY
= 0.5 X
– 0.5 Y 0.
_
0.5 X 0.5 Y –^ 0.^
_ YA
X A
= _^50
MRS XYB^ =
MU _ X
MU Y
= 0.8 X^
– 0.2 Y 0.
_
0.2 X 0.8^ Y –^ 0.^
4 _ YB
X B
_4(60)
c. First, notice that A and B both have MRS equal to 2, even though their utility functions and their incomes are dif- ferent. C ’s MRS will be equal to 2, just like A and B. In fact, the MRS for all consumers will be equal to 2 as long as all consumers consume both goods (i.e., if they have an interior solution). This is because all consumers face the same prices and all consumers maximize their utilities where their MRS is equal to the price ratio.
- Katie likes to paint and sit in the sun. Her utility function is U ( P , S ) = 3 PS + 6 P , where P is the number of paint brushes and S is the number of straw hats. The price of a paint brush is $1 and the price of a straw hat is $5. Katie has $50 to spend on paint brushes and straw hats. a. Solve Katie’s utility-maximization problem using a Lagrangian. b. How much does Katie’s utility increase if she receives an extra dollar to spend on paint brushes and straw hats?
- a. Write out the maximization problem and the Lagrangian: max P , S^3 PS^ +^6 P^ s.t. 50^ =^ P^ +^5 S
max P , S , λ ^ =^3 PS^ +^6 P^ +^ λ (50^ –^ P^ –^^5 S^ )
Appendix: The Calculus of Utility Maximization and Expenditure Minimization Chapter 4 49
FOC:
_^ ∂
∂ P
= 3 S + 6 – λ = 0
_^ ∂
∂ S
= 3 P – 5 λ = 0
_^ ∂
∂ λ
= 50 – P – 5 S = 0
From the fi rst two conditions,
λ = 3 S + 6 = 0.6 P
S = 0.2 P – 2
Substituting into the third FOC, we get
50 – P – 5 S = 50 – P – 5(0.2 P – 2) = 60 – 2 P = 0
P = 30
Then S = 4. b. We need to solve for the Lagrange multiplier λ. From above,
λ = 3 S + 6 = 0.6 P
Substituting for the optimal values of S or P gives λ = 18. Therefore, Katie’s level of utility would increase by 18 units if she receives an extra dollar to spend.
- Suppose that a consumer’s utility function for two goods ( X and Y ) is
U ( X , Y ) = 10 X 0.5^ + 2 Y
The price of good X is $5 per unit and the price of good Y is $10 per unit. Suppose that the consumer must have 80 units of utility and wants to achieve this level of utility with the lowest possible expenditure. a. Write a statement of the constrained optimization problem. b. Use a Lagrangian to solve the expenditure-minimization problem.
- a. min (^) X , Y 5 X + 10 Y s.t. 80 – 10 X 0.5^ – 2 Y = 0 b. Write out the Lagrangian and the first-order conditions:
min X , Y , λ ^ =^5 X^ +^10 Y^ +^ λ (80^ –^^10 X^
0.5 – 2 Y )
FOC:
_^ ∂
∂ X
= 5 – 0.5 λ 10 X –^ 0.5^ = 5 – 5 λX –^ 0.5^ = 0
_^ ∂
∂ Y
= 10 – 2 λ = 0
_^ ∂
∂ λ
= 80 – 10 X 0.5^ – 2 Y = 0
Solve for λ in the fi rst two conditions and set these two expressions equal to one another:
λ = X 0.5^ and λ = 5
X 0.5^ = 5
X = 25
Substituting 25 for X in the third constraint yields Y = 15. Then the minimum expenditure is $5(25) + $10(15) = $275.
