Chapter 3, Problem 3.1P

a

To determine

To prove:

Graphical representation of utility function whether IC curve is convex or not.

Explanation of Solution

The utility function is linear. This means that the goods x and y are perfect substitutes. For simplicity, the value of utility is taken to be constant at 60. Then the equation for the indifference curve becomes:

  $60=3x+y$

To graph the indifference curve:

Graph 1

  $MRS=-\frac{dy}{dx}=\frac{U_x}{U_y}$

  $\begin{array}{l}y=60-3x\\ -\frac{dy}{dx}=3\end{array}$

The MRS is constant at 3. Hence, the IC curve is not convex.

Economics Concept Introduction

Introduction:

Rate of substitution is the ratio of two goods at which consumer gives away a quantity of good 1 in order to get good 2

b)

To determine

To prove:

Graphical representation of utility function whether IC curve is convex or not.

Explanation of Solution

Let utility function be set equal to 10.

Then the equation will be:

  $10=\sqrt{xy}$

  $100=xy$

The IC for utility equal to 10 can be obtained by graphing the following equation:

  $y=\frac{100}{x}$

Graph 2

  $MRS=-\frac{dy}{dx}=\frac{1}{2\sqrt{x}}$

For equation, $y=\frac{100}{x}$

  $-\frac{dy}{dx}=\frac{100}{x^2}$

As x is in the denominator, the MRS decreases when x increases. Hence indifference curve is convex.

Economics Concept Introduction

Introduction:

Rate of substitution is the ratio of two goods at which consumer gives away a quantity of good 1 in order to get good 2

c)

To determine

To prove:

Graphical representation of utility function whether IC curve is convex or not.

Explanation of Solution

To simplify the function, arbitrarily set the utility at 8. So, the utility function becomes

  $8=\sqrt{x}+y$

Graphical representation:

Graph 3

  $MRS=-\frac{dy}{dx}=\frac{U_x}{U_y}$

  $-\frac{dy}{dx}=\frac{1}{2\sqrt{x}}$

As x is in the denominator, the MRS decreases when x increases. Hence indifference curve is convex.

Economics Concept Introduction

Introduction:

Rate of substitution is the ratio of two goods at which consumer gives away a quantity of good 1 in order to get good 2

d)

To determine

To prove:

Graphical representation of utility function whether IC curve is convex or not.

Explanation of Solution

Utility is set equal to 4. Then the equation for equation is:

  $4=\sqrt{x^2-y^2}$

  $16=x^2-y^2$

The indifference curve for utility equal to 4:

The graph is shown below:

Graph 4

  $MRS=-\frac{dy}{dx}=-\frac{x}{y}$

In the above equation, x is numerator. Though there is a negative sign in front of the fraction. This means that as x increases, the marginal rate of substitution increases.

Hence, the indifference curves are not convex.

Economics Concept Introduction

Introduction:

Rate of substitution is the ratio of two goods at which consumer gives away a quantity of good 1 in order to get good 2

e)

To determine

To prove:

Graphical representation of utility function whether IC curve is convex or not.

Explanation of Solution

To simplify the problem, we arbitrarily take utility equal to 1.

  $1=\frac{xy}{x+y}$

To graph the indifference curve,

  $\begin{array}{l}{U}_x=2x\\ U_y=-2y\\ y=\frac{x}{x-1}\end{array}$

Graph 5

MRS = $-\frac{dy}{dz}=\frac{\left(\left(x-1\right)\times 1\right)-\left(x-1\right)}{{\left(x-1\right)}^2}$

In the above equation, x is in numerator. Hence, as x increases the marginal rate of substitution decreases. Hence, the indifference curves are convex.

Economics Concept Introduction

Introduction:

Rate of substitution is the ratio of two goods at which consumer gives away a quantity of good 1 in order to get good 2