Calculus of Utility Maximization and Expenditure Minimization PDF

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The Calculus of Utility Maximization and Expenditure Minimization

4

Appendix

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 MRSXY 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) = X0.33Y 0.67 c. U(X,Y) = 10X0.5 + 5Y

Solution 1. a.

U(X,Y) = 5X + 2Y ∂U(X,Y) MUX = _ = 5 ∂X ∂U(X,Y) MUY = _ = 2 ∂Y Marginal utility is constant for each good. MUX MRSXY = _ = 5_ MU Y 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 y 50

0

b.

20

x

U(X,Y) = X 0.33Y 0.67 ∂U(X,Y) MUX = _ = 0.33X – 0.67Y 0.67 ∂X ∂U(X,Y) MUY = _ = 0.67X 0.33Y  – 0.33 ∂Y 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

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Part 2

Consumption and Production

MRSXY 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 = X0.33Y 0.67 1,000,000 = XY2 Y2 =

1,000,000 _ X

Y = 1,000X –0.5 y

100

0

c.

x

100

U(X,Y) = 10X0.5 + 5Y ∂U(X,Y) MUX = _ = (0.5)10X – 0.5 = 5X – 0.5 ∂X ∂U(X,Y) MUY = _ = 5 ∂Y The marginal utility of good X decreases as more X is consumed. The marginal utility of good Y is constant: MUX 5X  – 0.5 MRSXY = _ = _ = X – 0.5 5 MU Y 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 = 10X0.5 + 5Y 5Y = 100 – 10X0.5 Y = 20 – 2X 0.5 y 20

0

100

x

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.

2 S

ose that Maggie ca es o l abo t chai a d bagels He

tilit f

ctio is U

CB

he e C is the

Appendix: The Calculus of Utility Maximization and Expenditure Minimization

Chapter 4

c. Write a statement of Maggie’s constrained optimization problem. d. Solve Maggie’s constrained optimization problem using a Lagrangian.

Solution 2. a. max CB C,B

b. Income = PCC + PBB or 6 = 3C – 1.5B or 6 – 3C – 1.5B = 0 c. max CB s.t. 6 – 3C + 1.5B C,B

d. Write out the Lagrangian for the problem in part (c): max (C,B,λ) = CB + λ(6 – 3C – 1.5B) C,B,λ

FOC: ∂ _ = B – 3λ = 0

∂C ∂ _ = C – 1.5λ = 0 ∂C ∂ _ = 6 – 3C – 1.5B = 0 ∂λ From the first two conditions, λ=

B C _ =_ 3

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 UA(X,Y) = X 0.5Y0.5 UB(X,Y) = X0.8Y 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?

Solution 3. a. For A,

max  X0.5Y 0.5 s.t. 100 = 2X + Y X,Y

max   = X0.5Y 0.5 + λ(100 – 2X – Y) X,Y,λ

FOC: ∂ _ = 0.5X – 0.5Y 0.5 – 2λ = 0 ∂X ∂ _ = 0.5X 0.5Y – 0.5 – λ = 0 ∂Y ∂ _ = 100 – 2X – Y = 0 ∂λ

44

Part 2

Consumption and Production

From the first two conditions, λ = 0.25x– 0.5Y 0.5 = 0.5X0.5Y – 0.5 Y = 2X Substituting into the third FOC, we get 100 – 2X – 2X = 0 XA = 25 Then YA = 50. For B, max X 0.8Y 0.2 s.t. 300 = 2X + Y X,Y

max   = X 0.8Y0.2 + λ(300 – 2X – Y) X,Y,λ

FOC: ∂ _ = 0.8X – 0.2Y0.2 – 2λ = 0 ∂X ∂ _ = 0.2X 0.8Y – 0.8 – λ = 0 ∂Y ∂ _ = 300 – 2X – Y = 0 ∂λ From the first two conditions, λ = 0.4X – 0.2Y0.2 = 0.2X0.8Y – 0.8 y = 40.5x Substituting into the third FOC, we get 300 – 2X – 40.5X = 0 X B = 120 Then YB = 60. b. The first terms in the first two FOCs are MUX and MUY, respectively. Therefore, A

MRSXY =

MUX 0.5X  – 0.5Y0.5 _ =_ MUY

0.5

0.5X Y

 – 0.5

YA 50 = _= 2 =_ 25 XA

4(60) MUX 4YB _ 0.8X  – 0.2Y0.2 _ B =2 =_ = = MRSXY =_ 0.8 – 0.8 120 MUY X Y B 0.2X c. First, notice that A and B both have MRS equal to 2, even though their utility functions and their incomes are different. 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.

4. Katie likes to paint and sit in the sun. Her utility function is U(P, S) = 3PS + 6P, 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?

Solution

4. a. Write out the maximization problem and the Lagrangian: max 3PS + 6P s.t. 50 = P + 5S P,S

Appendix: The Calculus of Utility Maximization and Expenditure Minimization

Chapter 4

FOC: ∂ _ = 3S + 6 – λ = 0

∂P ∂ _ = 3P – 5λ = 0 ∂S ∂ _ = 50 – P – 5S = 0 ∂λ From the first two conditions, λ = 3S + 6 = 0.6P S = 0.2P – 2 Substituting into the third FOC, we get 50 – P – 5S = 50 – P – 5(0.2P – 2) = 60 – 2P = 0 P = 30 Then S = 4. b. We need to solve for the Lagrange multiplier λ. From above, λ = 3S + 6 = 0.6P 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.

5. Suppose that a consumer’s utility function for two goods (X and Y) is U(X,Y) = 10X0.5 + 2Y 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.

Solution

5. a. minX,Y 5X + 10Y s.t. 80 – 10X0.5 – 2Y = 0 b. Write out the Lagrangian and the first-order conditions: min   = 5X + 10Y + λ(80 – 10X0.5 – 2Y) X,Y,λ

FOC: ∂ _ = 5 – 0.5λ 10X  – 0.5 = 5 – 5λ X

 – 0.5 =0 ∂X ∂ _ = 10 – 2λ = 0 ∂Y ∂ _ = 80 – 10X 0.5 – 2Y = 0 ∂λ Solve for λ in the first two conditions and set these two expressions equal to one another:

λ = X0.5 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....