Question

Suppose the marginal utility to David Jones of product \(\mathrm{A}\) is defined by the function, \(\mathrm{MU}_{\mathrm{A}}=10-\mathrm{x}\), where \(\mathrm{x}\) is the number of units of A produced. For B, MU \(_{B}=21-2 y\), where \(y\) is the number of units of B produced. Assume that the price of \(\mathrm{A}=\) price of \(\mathrm{B}=\$ 1\) How much of \(\mathrm{A}\) and \(\mathrm{B}\) would David buy if he had \(\$ 7\) to spend?

Short answer

David would buy \(7\) units of product A and \(0\) units of product B with his \(\$7\) budget.

Step-by-step solution

Calculate the utility per dollar spent for each product

First, we need to find the utility per dollar spent for each product Product A: \(\frac{\mathrm{MU}_{\mathrm{A}}}{Price_{\mathrm{A}}}= \frac{10-x}{1} \Rightarrow 10-x\) Product B: \(\frac{\mathrm{MU}_{\mathrm{B}}}{Price_{\mathrm{B}}}= \frac{21-2y}{1} \Rightarrow 21-2y\)

Determine the optimal proportion of spending for each product

To find the best allocation of spending for David, we must equalize the marginal utility per dollar spent for each product, that is: \[10-x = 21-2y\] By solving for either x or y, we'll determine the relationship between the products bought. \[x = 11+2y\]

Calculate the number of units of each product to be purchased with the given budget

We can now use the provided budget of \(7 and the relationship we found between x and y to find the optimal number of units to buy. We know the total price of A and B combined should equal \)7, so: \[1(x)+1(y)=7\] Now substitute the expression for x, which is \(x = 11+2y\), into the equation above: \[1(11+2y)+1(y) = 7\] Solve for y: \[3y = -4\Rightarrow y = -\frac{4}{3}\] Since we can't buy a negative number of units, it means we've reached the point where allocating any more budget on B will result in decreasing marginal utility. Thus, David will spend all of his $7 on product A: \[x = 7\] \[y = 0\] So, David would buy 7 units of product A and 0 units of product B.

Key concepts

Budget Allocation

When approaching budget allocation, think of it as a way to distribute limited financial resources across different goods and services to maximize satisfaction. For an individual like David Jones in our exercise, he must decide how to allocate his $7 budget between products A and B to get the most utility, or satisfaction, from his purchase.

Each decision on allocation hinges on the cost of each item and the personal value or utility it offers. Since both products in the exercise cost the same, the focus shifts to the utility gained from purchasing additional units of each. The very essence of budget allocation in this context is about striking a balance between the cost and the expected enjoyment or usefulness of the products.

Marginal Utility Per Dollar

Marginal utility per dollar is a pivotal concept that helps in understanding how consumers like David make spending choices. It's a way to quantify the additional satisfaction received from spending one more dollar on a particular good.

To determine the marginal utility per dollar for products A and B, we use the marginal utility functions given in the exercise. It's obtained by dividing the marginal utility of each product by its price. In the problem, this calculation shows diminishing satisfaction with each additional unit purchased. It suggests that with every unit of product A or B David buys, the less additional satisfaction he gets from each subsequent unit. When comparing products, consumers should aim to get the highest marginal utility per dollar from their purchases, which leads to the utility-maximizing combination of goods.

Utility Maximization

Utility maximization is the goal of consumers like David to get the greatest amount of satisfaction from their purchases within the limits of their budget. It involves allocating funds in such a way that the last dollar spent on each good provides the same level of marginal utility.

In our exercise, this principle guides us to the conclusion that David should spend all his money on product A because the marginal utility per dollar of product A remains higher than that of product B for each unit he could afford. This outcome is based on the equality between the marginal utilities per dollar spent, which helps ensure that every dollar is used in the most efficient way to achieve maximum overall satisfaction.