Final answer:
The optimal quantities of goods A and B that a person will consume to maximize utility, considering a utility function U = 300√AB, are found by setting up the equality of marginal utilities to prices and solving the budget constraint. The calculation reveals that the utility-maximizing quantities are 20 units of A and 60 units of B.
Step-by-step explanation:
The question asks how to determine the optimal quantities of goods A and B that a person will consume to maximize utility, given their utility function U = 300√AB, budget of $240, and the prices of goods A and B. To find the utility-maximizing quantities of A and B, we must consider the two conditions for utility maximization: the budget constraint and the condition that the ratios of marginal utilities to prices (MU/P) for both goods must be equal at the optimal point.The utility function given is U = 300A^0.5B^0.5. The marginal utility of A (MUA) is 150B^0.5/A^0.5, and the marginal utility of B (MUB) is 150A^0.5/B^0.5. The budget constraint is 6A + 2B = 240. By setting up the equality of MU/P ratios, we get MUA/PA = MUB/PB. Substituting the marginal utilities and prices, we have (150B^0.5/A^0.5)/6 = (150A^0.5/B^0.5)/2. Simplifying this, we get B/A = 3. Finally, we can plug this ratio into the budget constraint to solve for the quantities of A and B that maximize utility.Using the budget constraint, we substitute B = 3A into 6A + 2B = 240 to find the values of A and B. 6A + 2(3A) = 240 simplifies to 12A = 240, giving us A = 20. Using B = 3A, we find that B = 60. Thus, the utility-maximizing quantities are 20 units of A and 60 units of B.