Final answer:
The optimal choice for a consumer with a perfect substitute preference utility function, U(x, y) = x + y, is to spend the entire budget on the good with the lower price per unit of utility. If one good is cheaper, all the budget goes to that good. If both goods have equal prices, the budget can be divided in any proportion.
Step-by-step explanation:
To mathematically derive the optimal choice that maximizes a consumer's utility subject to his/her budget constraint with a perfect substitute utility function U(x, y) = x + y, we need to consider the budget constraint, which can be represented as Pxx + Pyy = I, where Px and Py are the prices of goods x and y, respectively, and I is the income available to the consumer.
Since the goods x and y are perfect substitutes, the consumer will spend their entire budget on the good that provides the highest utility per unit of money, which is determined by the ratio of the marginal utility to the price of the goods. The marginal utility of good x and good y is 1 since the utility function U(x, y) = x + y implies that the increase in utility for a one-unit increase in x or y is 1.
If Px < Py, the optimal choice for the consumer is to spend all their income on x, buying I/Px units. If Px > Py, then the optimal choice is to purchase I/Py units of y. If the prices are equal, the consumer can split their budget in any way between x and y without affecting the total utility.