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

#Short Answer# Given a utility function U(x,y) with independent marginal utilities and diminishing marginal utility, we can find that the Marginal Rate of Substitution (MRS) is diminishing. Through analysis, we showed that under the condition of diminishing marginal utility, the derivative of MRS concerning x is negative when marginal utilities are independent, which implies diminishing MRS. However, the converse is not true, as demonstrated by a Cobb-Douglas utility function example where the MRS is diminishing, but marginal utilities are not independent.

Step-by-step solution

Write down the utility function and find the partial derivatives

Let's consider a utility function U(x, y) for two goods x and y. To find the marginal utilities, we need to compute the partial derivatives of the utility function with respect to x and y: \[MU_x = \frac{\partial U(x,y)}{\partial x}\] \[MU_y = \frac{\partial U(x,y)}{\partial y}\]

Write down the condition for independent marginal utilities

The given condition for independent marginal utilities states that \[\frac{\partial^2 U}{\partial y\partial x} = \frac{\partial^2 U}{\partial x\partial y} = 0\]

Find the MRS

The MRS is defined as the ratio of the marginal utilities. That is, \[MRS = \frac{MU_x}{MU_y}\]

Differentiate MRS w.r.t. good x and use the condition of independent marginal utilities

Now we differentiate the MRS with respect to the quantity of one good (x), and use the condition of independent marginal utilities to simplify it: \[\frac{\partial MRS}{\partial x} = \frac{\partial (\frac{MU_x}{MU_y})}{\partial x}\] \[\frac{\partial MRS}{\partial x} = \frac{\frac{\partial^2 U}{\partial x^2} \cdot MU_y - \frac{\partial^2 U}{\partial y\partial x} \cdot MU_x}{MU_y^2}\] Since we have \[\frac{\partial^2 U}{\partial y\partial x} = 0\], the equation becomes \[\frac{\partial MRS}{\partial x} = \frac{\frac{\partial^2 U}{\partial x^2} \cdot MU_y}{MU_y^2}\]

Show diminishing MRS under diminishing marginal utility condition

Now, we need to show that under the condition of diminishing marginal utility (negative second derivative), \(\frac{\partial MRS}{\partial x}\) is negative. Assuming that \(\frac{\partial^2 U}{\partial x^2}<0\), we have: \[\frac{\partial MRS}{\partial x} = \frac{\frac{\partial^2 U}{\partial x^2} \cdot MU_y}{MU_y^2} (<0)\] Since both \(MU_y\) and \(\frac{\partial^2 U}{\partial x^2}\) are negative, \(\frac{\partial MRS}{\partial x}\) becomes negative, meaning that the MRS is diminishing when marginal utilities are independent and diminishing.

Provide an example showing the converse is not true

Let's consider an example utility function to show that the converse is not true. Suppose a Cobb-Douglas utility function: \[U(x,y) = x^a \cdot y^b\] with \(0 < a, b < 1\). In this case, the MRS is diminishing, which can be shown as: \[MRS = \frac{a\cdot x^{a-1}\cdot y^b}{b\cdot x^a\cdot y^{b-1}} = \frac{a}{b}\cdot \frac{y}{x}\] However, the marginal utilities for this utility function are not independent: \[\frac{\partial^2 U}{\partial x\partial y} = \frac{\partial^2 U}{\partial y\partial x} = ab\cdot x^{a-1}\cdot y^{b-1}\neq 0\] Therefore, the converse of the statement is not true: the MRS can be diminishing even if the marginal utilities are not independent.