EC270 Lecture Notes - Lecture 1: Quotient Rule, Product Rule
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11. A decision maker wishes to maximize the total benefit associated with three activities, X, Y, and Z. The price per unit of activities X, Y, and Z is $1, $2, and $3, respectively. The following table gives the ratio of the marginal benefit to the price of the activities for various levels of each activity:
Level of activity | MBXPX | MBYPY | MBZPZ |
1 | 10 | 22 | 14 |
2 | 9 | 18 | 12 |
3 | 8 | 12 | 10 |
4 | 7 | 10 | 9 |
5 | 6 | 6 | 8 |
6 | 5 | 4 | 6 |
7 | 4 | 2 | 4 |
8 | 3 | 1 | 2 |
a. If the decision maker chooses to use one unit of X, one unit of Y, and one unit of Z, the total benefit that results is $______
b. For the fourth unit of activity Y, each dollar spent increases total benefit by $______. The fourth unit of activity Y increases total benefit by $______.
c. Suppose the decision maker can spend a total of only $18 on the three activities. What is the optimal level of X, Y, and Z? Why is this combination optimal? Why is the combination 2X, 2Y, and 4Z not optimal?
d. Now suppose the decision maker has $33 to spend on the three activities. What is the optimal level of X, Y, and Z? If the decision maker has $35 to spend, what is the optimal combination? Explain.
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?