Answer :
Sam should buy 30 bottles of wine and 12 cases of beer.By allocating her budget to purchase 30 bottles of wine and 12 cases of beer, Sam can maximize her utility according to her given utility function.
To maximize her utility, Sam needs to allocate her budget in a way that maximizes the value of her utility function. The utility function is given by U = 390WB, where W represents the number of bottles of wine and B represents the number of cases of beer.
Let's assume Sam buys x bottles of wine and y cases of beer. The cost of wine per month would be 16x, and the cost of beer per month would be 20y. According to the given information, Sam spends $960 per month on wine and beer. So we have the equation:
16x + 20y = 960
To maximize the utility, we need to find the values of x and y that satisfy this equation and maximize the value of U = 390WB.
To simplify the calculation, we can divide the equation by 4:
4x + 5y = 240
Now, we need to find whole number solutions for x and y that satisfy this equation. By trying different values, we find that when x = 30 and y = 12, the equation is satisfied:
4(30) + 5(12) = 120 + 60 = 180 + 60 = 240
Therefore, Sam should buy 30 bottles of wine and 12 cases of beer to maximize her utility.
By allocating her budget to purchase 30 bottles of wine and 12 cases of beer, Sam can maximize her utility according to her given utility function. This allocation of purchases allows her to balance her enjoyment of wine and beer while staying within her budget.
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To maximize utility with a monthly budget of $960, Sam should buy 30 bottles of wine and 24 cases of beer. Calculations are made using the budget constraint 16W + 20B = 960, and maximizing the utility function U = 390WB.
To maximize Sam's utility, we need to determine the number of bottles of wine (W) and cases of beer (B) she should buy given her utility function U = 390WB, the prices of wine and beer, and her monthly budget. The price of wine is $16 per bottle, and the price of beer is $20 per case. Sam's monthly budget for wine and beer is $960.
The problem can be solved using Lagrange multipliers or by setting up the budget constraint equation and finding the optimal bundle that maximizes the utility function subject to this constraint. The budget constraint is given by 16W + 20B = 960.
To find the maximum utility, we solve the system of equations derived from the utility function and the budget constraint:
- 16W + 20B = 960 (budget constraint)
- dU/dW = 390B = 16λ (marginal utility of wine per price = shadow price)
- dU/dB = 390W = 20λ (marginal utility of beer per price = shadow price)
Dividing equation (2) by (3) to eliminate λ we get 390B/390W = 16/20, which simplifies to B/W = 4/5. From the budget constraint, substituting B = 4/5W, we get 16W + 16(O.8W) = 960. This simplifies to W = 30 bottles of wine. Substituting back into B = 4/5W, we get B = 24 cases of beer.
Thus, to maximize utility, Sam should buy 30 bottles of wine and 24 cases of beer given her budget constraint.
