Chapter 17, Problem 17.1P

a)

To determine

To find:

MRS = 1+r

Explanation of Solution

Given utility function:

$U(c_1,c_2)$

Budget constraint :

$W=c_1+\frac{c_2}{1+r}$

The above equations are put in Langrange equation:

$L=U(c_1,c_2)+\lambda \left(W-c_1+\frac{c_2}{1+r}\right)$

Taking the fisrt order derivative and equating it to 0.

$\frac{\partial L}{\partial c_1}=\frac{\partial U}{\partial c_1}-\lambda =0$

$\frac{\partial L}{\partial c_2}=\frac{\partial c_1}{\partial r}<0-\lambda \frac{1}{1+r}=0$

Divide the above two equations, we get:

$MRS=\frac{1}{1+r}$

Economics Concept Introduction

Introduction:

b)

To determine

To know:

Price eslaticity of demand of c₂.

Explanation of Solution

Substitution effect is always negative which implies that any increase in value of $\frac{1}{1+r}$ leads to fall in $c_2$ .

While income effect is positive in case of normal good, so it implies $c_2$ and r are positively related. Numerically,

$\frac{\partial c_2}{\partial r}>0$

However, $\frac{\partial c_1}{\partial r}$ has no specific sign as substitution and income effect has both opposite result.

It is assumed that substitution effect has more impact than income effect, that is, $\frac{\partial c_1}{\partial r}<0$

Economics Concept Introduction

Introduction: Envelop theorem states that changes in exogeneous variables must be considered for profit maximizing equations, ignoring the change in endogeneous variable.

c)

To determine

To ascertain:

Changes in part b due to change in budget constraint.

Explanation of Solution

Budget constraint is given as: $y_1-c_1+\frac{y_2-c_2}{1+r}=0$

Rearranging the terms:

$y_1+\frac{y_2}{1+r}=c_1+\frac{c_2}{1+r}$

The above equation is a slope of budget line.

When $c_1>y_1$ , consumption is greater than income, it implies the consumer borrows and repay in period 2 and vice-versa.

Economics Concept Introduction

Introduction: