Chapter 3, Problem 3.19P

To determine

(a)

Whether the assumption that 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-

  $\text{U(x,y)}\text{= }\sqrt{\text{x}}\text{+2}\sqrt{\text{y}}$

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

  $\begin{array}{l}{\text{MU}}_{\text{x}}\text{= }\frac{\text{1}}{\text{2}\sqrt{\text{x}}}\left(\text{1}\right)\hfill \\ {\text{MU}}_{\text{y}}\text{= }\frac{\text{1}}{\sqrt{\text{y}}}\left(\text{2}\right)\hfill \end{array}$

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 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.

  $\begin{array}{c}\frac{{\text{dMU}}_{\text{x}}}{\text{dx}}\text{= }\text{-}\left(\frac{\text{1}}{\text{2}}\right)\left(\frac{\text{1}}{\text{2}}\right){\text{x}}^{\left(\text{ - }\frac{\text{1}}{\text{2}}\text{-1}\right)}\\ \text{= }\text{-}\frac{\text{1}}{\text{4}}{\text{x}}^{\left(\text{ - }\frac{\text{3}}{\text{2}}\right)}\\ \text{= -}\frac{\text{1}}{{\text{4x}}^{\text{3}/\text{2}}}\end{array}$

As, $\frac{{\text{dMU}}_{\text{x}}}{\text{dx}}<0.$

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 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.

  $\begin{array}{c}{\text{MRS}}_{\text{x,y}}\text{=}\frac{{\text{MU}}_{\text{x}}}{{\text{MU}}_{\text{y}}}\\ \text{=}\frac{\frac{\text{1}}{\text{2}\sqrt{\text{x}}}}{\frac{\text{1}}{\sqrt{\text{y}}}}\\ \text{=}\frac{\sqrt{\text{y}}}{\text{2}\sqrt{\text{x}}}\left(3\right)\end{array}$

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 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.

  $\begin{array}{l}\frac{\partial \text{MRS}}{\partial \text{x}}=-\left(\frac{1}{2}\right)\left(\frac{\sqrt{y}}{2}\right)x^{-3/2}\hfill \\ =-\frac{1}{4}\left(\frac{\sqrt{y}}{x^{3/2}}\right)\hfill \end{array}$

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 both the 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.