Chapter 2, Problem 2.10P

A

To determine

State the reason behind the utility function.

Answer to Problem 2.10P

The decimal exponents resulting in $\alpha =\beta =\frac{1}{2}$ , makes the functions a special case.

Explanation of Solution

The utility function used is under the Cobb Douglas form:

  $\begin{array}{l}U=\sqrt{A.B}\\ U=A^{\frac{1}{2}}.B^{\frac{1}{2}}\end{array}$

From the above function, it can be derived that

  $\alpha =\beta =\frac{1}{2}$ , thus, it can be seen that $\alpha +\beta =1$ .

Similarly, in the other given problem also;

  $\begin{array}{l}U=\sqrt{C.D}\\ U=C^{\frac{1}{2}}.D^{\frac{1}{2}}\end{array}$

  $\alpha =\beta =\frac{1}{2}$ , thus, see that $\alpha +\beta =1$ .

This result makes this function a special case.

B

To determine

Prove that the individual will spend a fraction of the income on commodity X and a fraction of income on commodity Y.

Answer to Problem 2.10P

While the fraction of income spent on commodity X is $\frac{P_{X}X}{I}=\alpha$ , and the fraction of income spent on commodity Y is $\frac{P_{Y}Y}{I}=\beta$ .

Explanation of Solution

The Marginal Rate of Substitution for the utility function will be as follows:

  $\begin{array}{l}MRS=\frac{M{U}_X}{M{U}_Y}\\ =\frac{\alpha Y}{\beta X}\end{array}$

Further, $\alpha +\beta =1$ .

The condition according to the utility maximization states that MRS equals to the ratio between the price of X and price of Y.

  $\frac{\alpha Y}{\beta X}=\frac{P_X}{P_Y}$

Now, rearrange it in terms of P_XX,

  $P_{X}X=\frac{\alpha P_{Y}Y}{\beta }$

Now, the budget constraint will be as follows:

  $P_{X}X+P_{Y}Y=I$

Now, substituting the value of P_XX in the above equation;

  $\begin{array}{l}\frac{\alpha P_{Y}Y}{\beta }+P_{Y}Y=I\\ P_{Y}Y\left(\frac{\alpha }{\beta }+1\right)=I\\ \frac{P_{Y}Y}{I}=\frac{\beta }{\alpha +\beta }=\beta \end{array}$

Now, substituting the value of P_YY in the equation, derive $\frac{P_{X}X}{I}=\alpha$ .

Thus, now the fraction of income spent on commodity X is $\frac{P_{X}X}{I}=\alpha$ , and the fraction of income spent on commodity Y is $\frac{P_{Y}Y}{I}=\beta$ .

Economics Concept Introduction

Introduction: The consumer waiving off a commodity in exchange for another good through maintaining the same level of utility is called the marginal rate of substitution.

C

To determine

Prove that the determined expenditure on commodity X does not alter.

Answer to Problem 2.10P

After careful observations, the fraction of income spent on commodity X does not alter and remains a.

Explanation of Solution

First, let us assume that the price of commodity X has changed to P`_x.

With this value, the utility maximizing condition will be as follows:

  $\frac{\alpha Y}{\beta X}=\frac{P^{\iota }{}_x}{P_y}$

Now, rearrange this condition in terms of P`_xX;

  $P{`}_{x}X=\frac{\alpha P_{Y}Y}{\beta }$

The budget constraint function of the individual consuming commodities X and Y will be as follows:

  $P^{1}X+P_{Y}Y=I$

Now, substitute this value of P`_xX in the above equation;

  $\begin{array}{l}\frac{\alpha P_{Y}Y}{\beta }+P_{Y}Y=I\\ P_{Y}Y(\frac{\alpha }{\beta }+1)=I\\ \frac{P_{Y}Y}{I}=\frac{\beta }{\alpha +\beta }=\beta \end{array}$

Now, substitute this value of P`_YY in the equation;

  $\begin{array}{l}{P}_{x}X=\frac{\alpha (\beta I)}{\beta }\\ \frac{P_{x}X}{I}=\alpha \end{array}$

Now, the fraction of income spent on commodity X remains a, this means, $\frac{P_{x}X}{I}=\frac{P_{Y}Y}{I}=\alpha$ .

Economics Concept Introduction

Introduction: Utility in Economics refers to the total satisfaction received by the individual from consuming goods and services in consideration. And hence, the utility level implies a direct influence on the demand and price of those goods and services in consideration.

D

To determine

Prove that the altering of price of Y does not have any impact on the quantity purchased of X.

Answer to Problem 2.10P

Since the fraction of income spent on commodity X remains a, the quantity purchased of X also does not change.

Explanation of Solution

Now, supposing that the price of commodity Y changes to P_Y¹, the condition of utility maximization will be as follows:

  $P_{X}X=\frac{\alpha P^{`}{}_{Y}Y}{\beta }$

As usual the budget constraint function;

  $P_{X}X+P^{`}{}_{Y}Y=I$

Now, substitute the value of $P_{X}X=\frac{\alpha P^{`}{}_{Y}Y}{\beta }$ in the above equation;

  $\begin{array}{l}\frac{\alpha P^{`}{}_{Y}Y}{\beta }+P^{`}{}_{Y}Y=I\\ P^{`}{}_{Y}Y(\frac{\alpha }{\beta }+1)=I\\ \frac{P^{`}{}_{Y}Y}{I}=\frac{\beta }{\alpha +\beta }=\beta \end{array}$

Now, substitute this value of $P^{`}{}_{Y}Y=\beta I$ in the given equation;

  $\begin{array}{l}{P}_{X}X=\frac{\alpha (\beta I)}{\beta }\\ \frac{P_{X}X}{I}=\alpha \end{array}$

Thus, since the fraction of income spent on commodity X remains a, the quantity purchased of commodity X does not change.

Economics Concept Introduction

Introduction: Utility in Economics refers to the total satisfaction received by the individual from consuming goods and services in consideration. And hence, the utility level implies a direct influence on the demand and price of those goods and services in consideration.

E

To determine

Using this utility function prove that the income when doubled with no price changes of the commodities will lead to the purchases of these commodities being doubled.

Answer to Problem 2.10P

By observing the doubled fraction of income spent on commodities, the purchases of both X and Y being doubled.

Explanation of Solution

As usual the budget constraint function, with income being doubled;

  $P_{X}X+P^{`}{}_{Y}Y=2I$

Now, substitute the value of $P_{X}X=\frac{\alpha P^{`}{}_{Y}Y}{\beta }$ in the above equation;

  $\begin{array}{l}\frac{\alpha P^{`}{}_{Y}Y}{\beta }+P^{`}{}_{Y}Y=2I\\ P^{`}{}_{Y}Y(\frac{\alpha }{\beta }+1)=2I\\ \frac{P^{`}{}_{Y}Y}{I}=\frac{\beta }{\alpha +\beta }=2\beta \end{array}$

Now, substitute this value of $P^{`}{}_{Y}Y=2\beta I$ in the given equation;

  $\begin{array}{l}{P}_{X}X=\frac{\alpha (2\beta I)}{\beta }\\ \frac{P_{X}X}{I}=2\alpha \end{array}$

Thus, since the fraction of income spent on commodities X and Y are doubled, the quantity purchased of commodities X and Y are doubled.

Economics Concept Introduction

Introduction: Utility in Economics refers to the total satisfaction received by the individual from consuming goods and services in consideration. And hence, the utility level implies a direct influence on the demand and price of those goods and services in consideration.