Question

Two goods have independent marginal utilities if \\[ \frac{\partial^{2} U}{\partial y \partial x}=\frac{\partial^{2} U}{\partial x \partial y}=0 \\] Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing \(M R S\). Provide an example to show that the converse of this statement is not true.

Short answer

Question: Prove that if we assume diminishing marginal utility for each good, any utility function with independent marginal utilities will have a diminishing Marginal Rate of Substitution (MRS). Answer: Using the diminishing marginal utility assumption and the independent marginal utilities condition, we compute the second-order derivative of MRS concerning x: \(\frac{\partial MRS_{xy}}{\partial x}\). We found that \(\frac{\partial MRS_{xy}}{\partial x} < 0\), which shows that any utility function with independent marginal utilities and diminishing marginal utility for each good will have a diminishing MRS.

Step-by-step solution

Define Independent Marginal Utilities

Given that we have two goods x and y, the utility function is U(x, y). Independent marginal utilities are defined as: \[ \frac{\partial^2U}{\partial x \partial y} = \frac{\partial^2U}{\partial y \partial x} = 0 \] This implies that the marginal utility of goods x and y are independent of the quantities of other goods in the basket.

State the assumption

We assume diminishing marginal utility for each good, that is: \[ \frac{\partial^2U}{\partial x^2} < 0 \] and \[ \frac{\partial^2U}{\partial y^2} < 0 \]

Define the Marginal Rate of Substitution (MRS)

The MRS is the rate at which a consumer is willing to trade one good for another, keeping the utility constant. Mathematically, it is defined as: \[ MRS_{xy} = -\frac{\partial U / \partial x}{\partial U / \partial y} \]

Show that any utility function with independent marginal utilities will have diminishing MRS

Using the assumption of diminishing marginal utility and the given condition of independent marginal utilities, let's compute the second-order derivative of MRS with respect to x: \[ \frac{\partial MRS_{xy}}{\partial x} = - \frac{\frac{\partial^3U}{\partial x^3}\frac{\partial U}{\partial y}-\frac{\partial^2U}{\partial y\partial x}\frac{\partial^2U}{\partial x\partial y}}{(\frac{\partial U}{\partial y})^2} \] Since we have \(\frac{\partial^2U}{\partial x\partial y} = \frac{\partial^2U}{\partial y\partial x} = 0\) and \(\frac{\partial^2U}{\partial x^2} < 0\), it follows that: \[ \frac{\partial MRS_{xy}}{\partial x} = -\frac{\frac{\partial^3U}{\partial x^3}\frac{\partial U}{\partial y}}{(\frac{\partial U}{\partial y})^2} < 0 \] This result shows that any utility function with independent marginal utilities and diminishing marginal utility for each good will have a diminishing MRS.

Provide an example to show the converse is not true

Now, we need to find an example where the diminishing MRS does not necessarily imply independent marginal utilities and diminishing marginal utility for each good. Consider the utility function: \[ U(x, y) = \sqrt[3]{x^2y} \] Calculating the MRS: \[ MRS_{xy} = -\frac{2x}{3y} \] Since MRS is decreasing in x, it has a diminishing MRS. However, let's examine the marginal utilities: \[ \frac{\partial U}{\partial x} = \frac{2x}{3\sqrt[3]{x^2y}} \] \[ \frac{\partial U}{\partial y} = \frac{x^2}{3\sqrt[3]{x^2y}} \] The cross-partial derivatives are: \[ \frac{\partial^2U}{\partial x\partial y} = \frac{\partial^2U}{\partial y\partial x} = \frac{x}{3\sqrt[3]{x^2y}} \] As we can see, cross-partial derivatives are not equal to zero, which means the marginal utilities are not independent. Thus, this example demonstrates that having a diminishing MRS does not necessarily imply independent marginal utilities and diminishing marginal utility for each good.