Chapter 3, Problem 3.18P

To determine

(a)

Whether the assumption 'more is better' satisfied for both the goods, x and y or not.

Explanation of Solution

The consumer prefers two goods x and y.

The utility function of the consumer is given as follows-

  $\text{U(x,y)}\text{= }{\text{x}}^{\text{0}\text{.4}}{\text{y}}^{\text{0}\text{.6}}$

Also, marginal utility of the two goods is given as follows-

  $\begin{array}{l}{\text{MU}}_{\text{x}}\text{= }\text{0}\text{.4}\left({\text{x}}^{\text{-0}\text{.6}}{\text{y}}^{\text{0}\text{.6}}\right)\left(1\right)\hfill \\ {\text{MU}}_{\text{y}}\text{= }\text{0}\text{.6}\left({\text{x}}^{\text{0}\text{.4}}{\text{y}}^{\text{-0}\text{.4}}\right)\left(2\right)\hfill \end{array}$

The additional utility derived from the consumption of additional unit of good is known as marginal utility.

Since, MU_x > 0 and MU_y> 0. The assumption that more is better is satisfied for both the goods.

To determine

(b)

Whether the marginal utility of good x diminishes, remain constant, or increase as the consumer buys more units of good x.

Explanation of Solution

Differentiate (1) with respect to x to determine whether the marginal utility good x diminishes, remain constant, or increase as the consumer buys more units of good x.

  $\begin{array}{l}\hfill \\ \begin{array}{c}\frac{\partial {\text{MU}}_{\text{x}}}{\partial \text{x}}=\left(\text{0}\text{.4}\right)\left(\text{-0}\text{.6}\right){\text{x}}^{\text{-0}\text{.6-1}}{\text{y}}^{\text{0}\text{.6}}\\ =\text{-}\left(\text{0}\text{.24}\right){\text{x}}^{\text{-1}\text{.6}}{\text{y}}^{\text{0}\text{.6}}\end{array}\hfill \end{array}$

As, $\frac{\partial {\text{MU}}_{\text{x}}}{\partial \text{x}}<0.$

Thus, it can be concluded that marginal utility of good x diminishes as the consumer buys more units of good x.

To determine

(c)

Marginal rate of substitution of the two goods, x and y.

Explanation of Solution

The rate at which consumer is willing to sacrifice some units of good x to get an additional unit of good y is known as marginal rate of substitution (MRS).

MRS_x,y, is equal to the ratio of the marginal utility of good x to the marginal utility of good y.

  $\begin{array}{c}{\text{MRS}}_{\text{x,y}}\text{=}\frac{{\text{MU}}_{\text{x}}}{{\text{MU}}_{\text{y}}}\\ \text{=}\frac{\text{0}\text{.4}\left({\text{x}}^{\text{-0}\text{.6}}{\text{y}}^{\text{0}\text{.6}}\right)}{\text{0}\text{.6}\left({\text{x}}^{\text{0}\text{.4}}{\text{y}}^{\text{-0}\text{.4}}\right)}\\ \text{=}\text{0}\text{.667}\left(\frac{\text{y}}{\text{x}}\right)\left(3\right)\end{array}$

To determine

(d)

Whether marginal rate of substitution of the two goods, x and y diminishes, remain constant, or increase as the consumer substitutes x for y along an indifference curve.

Explanation of Solution

Differentiate (3) with respect to x to determine whethermarginal rate of substitution of the two goods, x and y diminishes, remain constant, or increase as the consumer substitutes x for y along an indifference curve.

  $\begin{array}{l}\hfill \\ \frac{\partial \text{MRS}}{\partial \text{x}}=-\text{0}\text{.667}\left(\frac{\text{y}}{{\text{x}}^2}\right)\hfill \end{array}$

As, $\frac{\partial \text{MRS}}{\partial \text{x}}<0.$

Thus, it can be concluded that marginal rate of substitution of the two goods, x and y diminishes, as the consumer substitutes x for y along an indifference curve.

To determine

(e)

A typical indifference curve of the given utility function. And whether the curve intersects X-axis or Y-axis or not.

Explanation of Solution

The figure (1) below depicts a typical indifference curve U_1. The declining slope of the curve represents diminishing MRS_{x, y.}

  

The indifference curve U₁ does not intersect either axis.

To determine

(f)

The second indifference curve U_2, with U₂> U₁.

Explanation of Solution

The figure (2) below represents second indifference curve U_2. Here, U₂ >U_1.

  

Since U₂ lies above U₁,U₂ represents higher level of satisfaction. Thus, U₂ >U_1. A consumer always prefers higher indifference curves.