Question

The utility function \(U=f(x, y)\) is a measure of the utility (or satisfaction) derived by a person from the consumption of two products \(x\) and \(y .\) Suppose the utility function is \(U=-5 x^{2}+x y-3 y^{2}\) (a) Determine the marginal utility of product \(x\). (b) Determine the marginal utility of product \(y\). (c) When \(x=2\) and \(y=3,\) should a person consume one more unit of product \(x\) or one more unit of product \(y\) ? Explain your reasoning. (d) Use a computer algebra system to graph the function. Interpret the marginal utilities of products \(x\) and \(y\) graphically.

Short answer

The marginal utility of product \(x\) is \(MUx = -10x + y\), and the marginal utility of product \(y\) is \(MUy = x -6y\). When \(x=2\) and \(y=3\), the person would prefer to consume more of product \(y\) as it has less negative marginal utility (-16) compared to product \(x\) (-17). The graphical representation of these functions would show the utility changes with the change in consumption of \(x\) and \(y\).

Step-by-step solution

Determine the marginal utility of product \(x\).

The marginal utility of a product is the derivative of the utility function with respect to that product. Therefore, to find the marginal utility of product \(x\) from \(U=-5 x^{2}+x y-3 y^{2}\), we simply take the derivative of \(U\) with respect to \(x\).\nTherefore, the marginal utility of product \(x\) becomes: \(MUx = \frac{dU}{dx} = -10x + y\)

Determine the marginal utility of product \(y\).

Similarly, to find the marginal utility of product \(y\), we take the derivative of \(U\) with respect to \(y\).\nThis results in the marginal utility of product \(y\), \(MUy = \frac{dU}{dy} = x -6y\)

Determine overconsumption of product

When \(x=2\) and \(y=3\), we can substitute these values into the marginal utility functions as \(MUx = -10(2) + 3 = -20 + 3 = -17\) and \(MUy = 2 - 6(3) = 2 - 18 = -16\). As the marginal utility is negative for both the products, the person may not want to consume more of both. However, as the marginal utility for \(y\) (-16) is higher (less negative) than that for \(x\) (-17), if they had to choose, they would rather consume an extra unit of product \(y\).

Graphical Interpretation of Marginal Utilities

Using a computer algebra system (like Mathematica, Maple, or even online tools), the utility function and the marginal utility functions can be graphed. The 3D graph would show changes in utility for changes in \(x\) and \(y\). The marginal utilities indicate the rate at which utility changes with respect to consumption of either goods. The points at which the marginal utilities are zero shows the maximums of the utility.