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 goods. The prices of X and Y are, respectively, $5 and $10. Units of the GoodTotal Utility of XTotal Utility of Y 150400 295750 3135950 41701,100 52001,220 62251,320 72451,400 82601,450 Given the above, if the consumer has $65 to spend on X and Y, the utility-maximizing bundle is

Given that the consumer has $65 to spend on X and Y, the utility-maximizing bundle is 1 unit of good X and 6 units of good Y with a total expenditure of $65

Explanation:

Note: See the attached excel file for the calculation of the marginal utility of X and Y as well as the ratios of marginal utility of each of X and Y to the price of each.

The utility maximization theory states that the utility of two or more commodities bundle are maximized when the ratios of the marginal utility to price of each commodity are equal.

For two commodities bundle in this case, utility is maximized where

MUX/PX = MUY/PY (1)

Where,

MUX = Marginal utility of X

MUY = Marginal utility of Y

PX = Price of X

PY = Price of Y

Marginal utility is the additional utility obtained by consuming one unit of commodity. It is calculated by deducting the previous total utility from the new total utility.

From the attached excel file, equations (1) holds:

(a) When Unit of good X = 1, and Unit of good Y = 6.

At this point,

MUX/PX = MUY/PY = 10,

Total expenditure = (Units of good X * Price of good X) +  (Units of good Y * Price of good Y) = (1 * $5) + (6 * $10) = $65

and;

(b) When Unit of good X = 6, and Unit of good Y = 8.

At this point,

MUX/PX = MUY/PY = 5,

Total expenditure = (Units of good X * Price of good X) +  (Units of good Y * Price of good Y) = (6 * $5) + (8 * $10) = $110

Conclusion

Given that the consumer has $65 to spend on X and Y, the utility-maximizing bundle is 1 unit of good X and 6 units of good Y with a total expenditure of $65 as calculated in (a) above.