Assume that an individual consumes two goods, X and Y. The total utility (assume

Question

Assume that an individual consumes two goods, X and Y. The total utility (assumed measurable) of each good is independent of the rate of consumption of other good.
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 of X Total Utility of Y
1 20 45
2 38 78
3 54 108
4 68 135
5 80 159
6 90 180

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.

Explanation / Answer

A) Marginal utility of the fourth unit of Y is 27. Marginal utility is the change in utility by producing an additional unit. So, 135 - 108 = 27. B)Marginal utility of the fifth unit of X is 12. 80 - 68 = 12. C) Marginal utility per dollar spent on the 3rd unit of X is $0.40. MU(3rd unit) = 54 - 38 = 16 16 / $40 (price of X) = 2/5 = $0.40 D)Marginal utility per dollar spent on the 2nd unit of Y is $0.55. MU (2nd unit) = 78 - 45 = 33 33 / $60 (price of Y) = $0.55 E) 3 units of X and 5 units of Y F) 1 unit of X and 3 units of Y I'm going to solve for X and Y jointly: Q MU(x) MU(x)/P(x) MU(y) MU(y)/P(y) 1 20 $0.50 45 $0.75 2 18 $0.45 33 $0.55 3 16 $0.40 30 $0.50 4 14 $0.35 27 $0.45 5 12 $0.30 24 $0.40 6 10 $0.25 21 $0.35 Now we need to solve for quantities where MU(x)/P(x) is equal to MU(y)/P(y). Multiply those quantities by the price and sum them to get your budget constraint. They are equal at 1x and 3y, 2x and 4y, 3x and 5y, and 4x and 6y. Q(x)*P(x) + Q(y)*P(y) = Budget Constraint 1*$40 + 3*$60 = $220 (Answer to F) 2*$40 + 4*$60 = $320 3*$40 + 5*$60 = $420 (Answer to E) 4*$40 + 6*$60 = $520 (Answer to G) G) 4*$40 + 6*$60 = $520 is the size of the budget constraint necessary to maximize utility at these respective quantities of X and Y.

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