Managerial Economics (Chapter 5)

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  • Topic: Indifference curve, Consumer theory, Marginal rate of substitution
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Chapter 5:
Question 3:
Suppose that two units of X and eight units of Y give a consumer the same utility as four units of X and two units of Y. Over this range: a. If the consumer obtains one more unit of X, how many units of Y must be given up in order to keep utility constant

∆Y∆X=2-84-2= - 62= -3

~ Utility unchanged, if consumer exchanges 3 units of Y for 1 unit of X.

b. If the consumer obtains one more unit of Y, how many units of X must be given up in order to keep utility constant?

∆Y∆X= 4-22-8= 26= -13

~ Utility unchanged, if consumer exchanges 1/3 units of X for 1 unit of Y.

c. What is the marginal rate of substitution?

X = 2, Y = 8
X = 4, Y = 2
∆Y∆X=MRS

-( 8-2 )( 2-4 )= - 62=3

8

6

4

2

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Question 7:
Suppose a customer has indifference map shown below. The relevant budget line is LZ. The price of Y is $10.

60
50L
40 B
30
20
10 AM
01020304050607080

a. What is the consumer’s income?

Quantity Y = 5050 × $10 = $500
Price Y = $10

b. What is the price of X?

Quantity X = 4040 × Px = 500
Price X = ? Px = 500 / 40
Income = 500 Px = $12.50

c. Write the equation for the budget line LZ.

Budget line LZ
Py = $10
Px = $12.50
M = $ 500

PxX + PyY = y
12.50x + 10y = 500ory = 50 – 1.25x

d. What combination of X and Y will the consumer choose? Why?

Consumer choose 20 units of X and 25 unit of Y.
Indifferent curve II is tangent to budget line LZ.
No other combination costing $500 provides more utility than X = 20, Y = 25

e. What is the marginal rate of substitution at this combination?

MRS= PxPy= $12.50$10=1.25

f. Explain in terms of the MRS why the consumer would not choose combinations designated by A or B.

Combination A : MRS is the consumer can give up 1 unit of X in return for MRS more units of Y and the consumer’s utility will not change. Market prices Px and Py make the consumer can buy Px / Py ( = 1.25) more units of Y if 1 less unit of X is purchased and remain on the budget line. Slopes of the indifference curve and budget line at point A shows that Px / Py > MRS at combination A. The consumer can buy Px / Py more Y if 1 fewer unit of X is purchased, remain indifferent. Therefore, this will increase the utility, and combination A would not be chosen by the consumer.

Combination B : Consumer could trade MRS units of Y to get 1 more unit of X and the consumer’s utility would be unchanged (i.e., MRS units of Y). Since the consumer gives up less Y than the amount that would leave the consumer indiffrent, trading Px/Py units of 1 more X must increase utility, and combination B would not be chosen by the consumer.

g. Suppose the budget pivots to LM, money income remaining constant. What is the new price of X? What combination of X and Y is now chosen?

80 × Px = $500; thus Px = $6.25.
The consumer will choose 30 units of Y and 32 units of X
( $10 × 30) + ( $6.25 × 32 ) = $500, where the indifferent curve III is tangent to budget line LM.

h. What is the new MRS?

MRS=∆Y∆X=$6.2510=0.625

Question 9:
The following graph shows a portion of a consumer’s indifference map. The consumer faces the budget line LZ, and the price of X is $20.

a. The consumer’s income = $600.

30 × 20 = $600

b. The price of Y is $$20 .

c. The equition for the budget line LZ is 20X + 20Y = 600, or Y = 30 - X.

d. What combination of X and Y does the consumer choose? Why?

The consumer chooses 10X and 20Y.

e. The marginal rate of substitution for this combination is
MRS=1=PxPy= $20$20 .

f. Explain in terms of MRS why the consumer does not choose either combination A or B.
At point A, MRS > 1. The consumer is willing to give up more...
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