recall that Casper's utility function is 3x + y, where x is his consumption of cocoa and y is his consumption of cheese.If the total cost of x units of cocoa is x², the price of cheese is $4, and Casper's income is $56, how many units of cocoa will he consume? A)6 B)5 C)11 D)3 E)12

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Quizplus · Accepted Answer

Casper's utility function is given as $3x + y$, where $x$ is the consumption of cocoa and $y$ is the consumption of cheese. The total cost of $x$ units of cocoa is given by the expression $x^2$, and the price of cheese is $4. Given that Casper's income is $56, we need to maximize his utility subject to his budget constraint.The budget constraint is given by the cost of cocoa plus the cost of cheese equals Casper's income:$$x^2 + 4y = 56$$To maximize utility, Casper would allocate spending between cocoa and cheese based on their utility contribution. Since the utility function does not directly compare the utility of cocoa to cheese, we look for the most cost-effective way to spend the income. Given the quadratic cost of cocoa, its marginal cost increases with consumption, suggesting a point of optimization where Casper balances his spending.Without loss of generality, let's solve for $x$ when Casper spends all his money on cocoa, ignoring cheese for a moment since the utility from cocoa (3x) increases faster than the utility from cheese (y) per dollar spent, up to a certain point due to the increasing cost of cocoa.Setting $y = 0$ for this scenario and solving for $x$ in the budget constraint gives us:$$x^2 = 56$$$$x = \sqrt{56}$$$$x \approx 7.48$$However, since $x$ must be an integer (as you cannot consume a fraction of a unit of cocoa in this context), and considering the options provided, the closest integer value of $x$ that does not exceed the budget when considering the quadratic cost is 6 (as $6^2 = 36$ leaves room for cheese or additional spending within the $56 budget).Thus, the correct answer is 6 units of cocoa, considering the need to maximize utility within the given budget constraint, and the fact that the utility function and budget constraint suggest a balance between spending on cocoa and cheese.

Recall That Casper's Utility Function Is 3x + Y, Where