Chapter 3, Problem 3.21P

To determine

(A)

To explain:

The given assumption of more is better is sufficient for both goods.

Answer to Problem 3.21P

The given assumption more is better is not sufficient for both goods.

Explanation of Solution

Given that −

  $U=A{x}^{\alpha }{y}^{\beta }$

  $M{U}_x=\alpha A{x}^{\alpha -1}{y}^{\beta }$

  $M{U}_y=\beta A{x}^{\alpha }{y}^{\beta -1}$

And,

  $M{U}_x=\frac{du}{dx}=\alpha A\left(\alpha -1\right)x^{\alpha }{y}^{\beta }>0$

  $M{U}_y=\frac{du}{dy}=\beta A\left(\beta -1\right)x^{\alpha }{y}^{\beta }>0$

In the above equation both MU_x and MU_y are greater than zero. Therefore, the given assumption more is better is not sufficient for both goods.

Economics Concept Introduction

Introduction:

Marginal utility shows the additional utility that the consumer is deriving from consuming additional units of that product or service.

To determine

(B)

To explain:

Whether the marginal utility of x decrease, or remain constant or increase with the consumer buying more of x, if the preferences of consumer are represented by the utility function of Cobb Douglas, $U=A{x}^{\alpha }{y}^{\beta }$, where A, α and β being positive constants, and the marginal utilities are represented by $M{U}_x=\alpha A{x}^{\alpha -1}{y}^{\beta }$, and $M{U}_y=\beta A{x}^{\alpha }{y}^{\beta -1}$.

Answer to Problem 3.21P

The marginal utility of x will diminish will not diminish but increases with an increase in the consumption of x by the individual.

Explanation of Solution

The given marginal utility function of x will not diminish but increases, as the individual consumes more of good x.

Economics Concept Introduction

Introduction:

Marginal utility shows the additional utility that the consumer is deriving from consuming additional units of that product or service.

To determine

(C)

Tofind:

The MRS_xy.

Answer to Problem 3.21P

After necessary derivations, the MRS_xy is given by $\frac{\alpha y}{\beta x}$.

Explanation of Solution

Marginal utility shows the additional utility that the consumer is deriving from consuming additional units of that product or service.

Through dividing the marginal utility of x with the marginal utility of y, the marginal rate of substitution can be arrived at.

Implies,

  $\begin{array}{l}MR{S}_{x,y}=\frac{MUx}{M{U}_y}\\ =\frac{(\alpha A{x}^{\alpha -1}{y}^{\beta })}{\beta A{x}^{\alpha }{y}^{\beta -1}}\\ =\frac{\alpha y}{\beta x}\end{array}$

Thus, the MRS of x for y will be $\frac{\alpha y}{\beta x}$.

Now, it can be concluded that the MRS_xy will decrease when x increases, and MRS_xy also will decrease when y decreases. Since, in both these cases MRS_xy decreases, it stays decreasing.

To determine

(D)

To evaluate:

Whether the MRS_xy will decrease or increase or remain constant, if the individual decides to substitute x for y along an indifference curve.

Answer to Problem 3.21P

After necessary observations, it can be said that the MRS_xy will be diminishing.

Explanation of Solution

Marginal utility shows the additional utility that the consumer is deriving from consuming additional units of that product or service.

MRS_xy, denoting the marginal rate of substitution of x for y, equaling to $\frac{\alpha y}{\beta x}$ will diminish as the individual substitutes x for y, along an indifference curve.

To determine

(E)

To plot:

A graph with x denoted by horizontal axis and y denoted by vertical axis, an indifference curve, and to label the curve U₁.

Answer to Problem 3.21P

As can be seen in the graph plotted in the next section, the indifference curve is bowed towards the origin, owing to the diminishing MRS.

Explanation of Solution

Marginal utility shows the additional utility that the consumer is deriving from consuming additional units of that product or service.

  

In the graph above, as explained in the requirements, it can be seen that with the diminishing MRS, the indifference curve is bowed towards the origin.

To determine

(F)

Toplot:

On the same graph, with x denoted by horizontal axis and y denoted by vertical axis, another indifference curve, U₂, as U₂> U₁.

Answer to Problem 3.21P

As can be seen in the graph plotted in the next section, the indifference curves, U₁ and U₂, as U₂> U₁.

Explanation of Solution

Marginal utility shows the additional utility that the consumer is deriving from consuming additional units of that product or service.

  

In the graph above, as explained in the requirements, it can be seen that with the diminishing MRS, the indifference curves are bowed towards the origin, satisfying the rule that U₂> U₁.