Chapter 13, Problem 13.1P

a

To determine

To plot:

Graphical representation of production frontier.

Explanation of Solution

  $P_x$ is the price of product x.

  $P_y$ is the price of product y. There are two products x and y.

The utility function is given by as-

  $P_y=1$

The budget line is given as:

  $M=P_{x}x+P_{y}y$

With the help of Langraingian expression this problem can be explained as:

  $L=x_2^{\frac{1}{2}}{y}_2^{{}^{\frac{1}{2}}}+\lambda \left[{(500-x)}^{\frac{1}{2}}-{(500-y)}^{\frac{1}{2}}-\overline{U_2}\right]$

Taking the differentials w.r.t x,y and $\lambda$

  $\frac{\frac{1}{2}{x}^{-\frac{1}{2}}{y}^{\frac{1}{2}}}{P_x}=\frac{1}{2}\frac{x^{\frac{1}{2}}{y}^{\frac{1}{2}}}{P_y}$

  $\frac{P_y}{P_x}=\frac{x}{y}$

Demand for x is given by:

  $x=y\frac{P_y}{P_x}$

Demand for y is given as:

  $y=x\frac{P_x}{P_y}$

His or her demand for x and y depends on relative prices of these value.

Graph 1

Economics Concept Introduction

Introduction:

Utility is the amount of satisfaction derived for consumption of goods or services. It is usually measured in utils.

Marginal utility is the additional or extra utility that an individual gets by consuming one extra unit of that good or service.

b)

To determine

To find:

Quantity of x and y to be produced.

Explanation of Solution

The total available goods are 1000. From the equilibrium condition is given as-

  $x=y\frac{P_y}{P_x}$

Now put this value of x in the budget equation:

  $M=x{P}_x+y{P}_y$

Now put this value of x in the budget equation:

  $\begin{array}{l}M=x{P}_x+y{P}_y\\ 1000=y\frac{P_y}{P_x}{P}_x+y{P}_y\end{array}$

  $\begin{array}{l}M=x{P}_x+y{P}_y\\ 1000=y{P}_y+y{P}_y\end{array}$

  $\begin{array}{l}1000=2y{P}_y\\ 500=y\end{array}$

Here

  $x=y\frac{P_y}{P_x}$

  $P_y=1$

The total endowment is 1000. So, y = 500 and x = 500

Economics Concept Introduction

Introduction:

Marginal Rate of technical substitution is the rate at which one factor of input can be substituted for another input, output remaining same. It shows efficiency of inputs for productivity.

Indifference curve is a curve which represents combination of goods which gives equal satisfaction.

c)

To determine

To know:

RPT and price ratio of production possibility frontier.

Explanation of Solution

For this $\overline{x}=\overline{y}=500$ , now consider there are two different individuals for the other individual the utility function is:

  $U_2(x_2,y_2)=x_2^{\frac{1}{2}}{y}_2^{{}^{\frac{1}{2}}}$

  $L=x_2^{\frac{1}{2}}{y}_2^{{}^{\frac{1}{2}}}+\lambda \left[{(500-x)}^{\frac{1}{2}}-{(500-y)}^{\frac{1}{2}}-\overline{U_2}\right]$

  $\frac{y}{(500-y)}=\frac{x}{(500-x)}$

The equation shows the pareto optimal allocation. It is given that

y = 500 and x = 500

The price ratio can be calculated as follows:

  $\frac{x}{y}=\frac{P_y}{P_x}$

  $\frac{x}{y}=\frac{P_y}{P_x}=\frac{500}{500}$

  $\frac{P_y}{P_x}=1$

This expression shows pareto efficient condition.

Economics Concept Introduction

Introduction:

Marginal Rate of technical substitution is the rate at which one factor of input can be substituted for another input, output remaining same. It shows efficiency of inputs for productivity.

Indifference curve is a curve which represents combination of goods which gives equal satisfaction.

d)

To determine

To find:

Price ratio and MRS.

Explanation of Solution

Suppose total amount of the product be allocated between x and y with respect to x=600 and y = 400. The given utility function is given as:

  $U(x,y)=x^{\frac{1}{2}}{y}^{\frac{1}{2}}$

So, the MRS is given as:

  $MRS=\frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}}$

  $MRS=\frac{x^{-1/2}{y}^{1/2}}{x^{1/2}{y}^{1/2}}$

  $MRS=\frac{y}{x}$

  $\begin{array}{l}=\frac{400}{600}\\ =0.67\end{array}$

It is known that marginal rate of substitution is equal to price ratio = 0.67

Economics Concept Introduction

Introduction:

Marginal Rate of technical substitution is the rate at which one factor of input can be substituted for another input, output remaining same. It shows efficiency of inputs for productivity.

Indifference curve is a curve which represents combination of goods which gives equal satisfaction.