Question

Chapter 6 Problem A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below. Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively, and that the consumer has an income of $74 to spend. Product X (Price $10) Product Y (Price $2) Product Z (Price $8) Quantity Utility Marginal Utility per $ Quantity Utility Marginal Utility per $ Quantity Utility Marginal Utility per $ 1 42 1 14 1 32 2 82 2 26 2 60 3 118 3 36 3 84 4 148 4 44 4 100 5 170 5 50 5 110 6 182 6 54 6 116 7 182 7 56.4 7 120 (a) Complete the table by computing the marginal utility per dollar for successive units of X, Y, and Z to one or two decimal places. Remember the marginal utility per dollar would be calculated by first getting the marginal utility which is the change in utility as quantity increases and then dividing it by the price. When doing Quantity 1 you are going from 0 units to 1 unit. The utility for 0 units would be $0. (b) How many units of X, Y, and Z will the consumer buy when maximizing utility and spending all income? Show this result using the utility maximization formula. (Meaning they need to spend all of their income of $74) (c) Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z?

Answer 1

	X(Price of X=$10)
Quantity	Utility	MU	MU/$
1	42
2	82	40	4
3	118	36	3.6
4	148	30	3
5	170	22	2.2
6	182	12	1.2
7	182	0	0

	Y(Price of Y=$2)
Quantity	Utility	MU	MU/$
1	14
2	26	12	6
3	36	10	5
4	44	8	4
5	50	6	3
6	54	4	2
7	56.4	2.4	1.2

	Z(Price of Z=$8)
Quantity	Utility	MU	MU/$
1	32
2	60	28	3.5
3	84	24	3
4	100	16	2
5	110	10	1.25
6	116	6	0.75
7	120	4	0.5

Utility is maximized when MUx/Px=MUy/Py=MUz/Pz

We have only combination satisfying this condition is

X=4, Y=5 and Z=3 Total Utility is 282 ( Maximum Possible)

Expenditure	Possible Combinations	Total Utility
74	(1,4,7)	206
74	(2,7,5)	248.4
74	(2,3,6)	270
74	(3,2,5)	254
74	(3,6,4)	272
74	(4,1,4)	262
74	(4,5,3)	282 ( Maximum Utility)
74	(5,0,3)	266
74	(5,4,2)	274
74	(6,3,1)	250

Ans C)

For Combination X=2, Y=4 and Z=1 we are at inefficient allocation that means we can still increase our consumption and total utility because this point is inside Budget line hence we will never have highest utility at this combination.