Assume that an individual consumes three goods, X, Y, and Z. The marginal utility (assumed measurable) of each good is independent of the rate of consumption of other goods. The prices of X, Y, and Z are, respectively, $1, $3, and $5. The total income of the consumer is $65, and the marginal utility schedule is as follows:
Units of good
Marginal utility of X (units)
Marginal utility of Y (units)
Marginal utility of Z (units)
1
12
60
70
2
11
55
60
3
10
48
50
4
9
40
40
5
8
32
30
6
7
24
25
7
6
21
18
8
5
18
10
9
4
15
3
10
3
12
1
(a)Given a $65 income, how much of each good should the consumer purchase to maximize utility?
(b)Suppose income falls to $43 with the same set of prices; what combination will the consumer choose?
(c)Let income fall to $38; let the price of X rise to $5 while the prices of Y and Z remain at $3 and $5. How does the consumer allocate income now? What would you say if the consumer maintained that X is not purchased because he or she could no longer afford it?
Assume that an individual consumes three goods, X, Y, and Z. The marginal utility (assumed measurable) of each good is independent of the rate of consumption of other goods. The prices of X, Y, and Z are, respectively, $1, $3, and $5. The total income of the consumer is $65, and the marginal utility schedule is as follows:
| Units of good | Marginal utility of X (units) | Marginal utility of Y (units) | Marginal utility of Z (units) |
| 1 | 12 | 60 | 70 |
| 2 | 11 | 55 | 60 |
| 3 | 10 | 48 | 50 |
| 4 | 9 | 40 | 40 |
| 5 | 8 | 32 | 30 |
| 6 | 7 | 24 | 25 |
| 7 | 6 | 21 | 18 |
| 8 | 5 | 18 | 10 |
| 9 | 4 | 15 | 3 |
| 10 | 3 | 12 | 1 |
(a)Given a $65 income, how much of each good should the consumer purchase to maximize utility?
(b)Suppose income falls to $43 with the same set of prices; what combination will the consumer choose?
(c)Let income fall to $38; let the price of X rise to $5 while the prices of Y and Z remain at $3 and $5. How does the consumer allocate income now? What would you say if the consumer maintained that X is not purchased because he or she could no longer afford it?
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Related questions
Assume that an individual consumes two goods, X and Y. The total utility (assumed measurable) of each good is independent of the consumption rate of other goods.
The price of X and Y are respectively $40 and $60. Use the following table of total utilities to answer the following questions.
| Good | Total utility X | Total utility Y |
| Ā 1 | Ā 20 | Ā 45 |
| Ā 2 | Ā 38 | Ā 78 |
| Ā 3 | Ā 54 | Ā 108 |
| Ā 4 | Ā 68 | Ā 135 |
| Ā 5 | Ā 80 | Ā 159 |
| Ā 6 | Ā 90 | Ā 180 |
a. The marginal utility of the fourth unit of Y is__________.
b. The marginal utility of the fifth unit of X is___________.
c. The marginal utility per dollar spent on the third unit of X is__________.
d. The marginal utility per dollar spent on the second unit of Y is__________.
e. If the consumer has $420 to spend, ______ unit of X and _______units of Y maximize utility subject to the budget constraint. Explain.
f. If the consumer has $220 to spend, _______ units of X and_______ units of Y maximize utility subject to the budget constraint. Explain.
g. If the consumer wanted 4 units of X and 6 units of Y, what would have to be his/her budget constraint in order to maximize his/her utility? Explain.
A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility that 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.
(a) Complete the following table by computing the marginal utility per dollar for successive units of X, Y, and Z to one or two decimal places.
(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.
(c) Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z?
| Product X | Product Y | Product Z | ||||||||
|
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 | Ā | 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 | _____ |
||
