Chapter 10, Problem 10.6P

a)

To determine

To ascertain:The production possibility curve for region A and B.

Answer to Problem 10.6P

The production possibility curve for region A

The production possibility curve for region B

Explanation of Solution

To calculate the production possibility curves for region A, consider the following production

function for region A: $\begin{array}{l}XA=\sqrt{Ix}\\ YA=\sqrt{Iy}\end{array}$

Here, I_x, is the labor allocated in the production of good x and I, is the labor allocated in the

production of good y

The total amount of labor available to region A is 100, that is, $1x+1y=100$

From the production function, compute the value of I_xand I_y, as follows:

  $\begin{array}{l}xB=\frac{1}{2}\sqrt{Ix}\\ yB=\frac{1}{2}\sqrt{Iy}\\ \end{array}$

Now, substitute the value of I_x, and I_y, in the equation of total available labor and compute the

production function as follows:

  $\begin{array}{l}I\text{x }+Iy=100\\ x^{2}A+y^{2}A=100\end{array}$

Hence, the production for region Ais

The PPF curve for region A is shown below: $x^{2}A+y^{2}A=100$

In this graph the vertical axis shows production of good y and horizontal axis shows production of good x.

To calculate the production possibility curves for region B, consider the following production

function for region B:

  $\begin{array}{l}xB=\frac{1}{2}\sqrt{Ix}\\ \\ yB=\frac{1}{2}\sqrt{Iy}\\ \end{array}$

Here, I_xis the labor allocated in the production of good x and I_yis the labor allocated in the

production of good y .

The total amount of labor available to region A is 100, that is, $I\text{x }+Iy=100$

  $\begin{array}{l}xB=\frac{1}{2}\sqrt{Ix}\\ Ix={\left(2xB\right)}^2\\ yB=\frac{1}{2}\sqrt{Iy}\\ Iy={\left(2yB\right)}^2\end{array}$

From the production function, compute the value of I_x and I_yas follows:

Now, substitute the value of I_x and I_y and in the equation of total available labor and compute theproduction function as follows:

  $\begin{array}{l}I\text{x }+Iy=100\\ {\left(2xA\right)}^2+{\left(\text{2}yA\right)}^2=100\end{array}$

Hence, the production of region B is ${\left(2xA\right)}^2+{\left(\text{2}yA\right)}^2=100$

The PPF curve for region B is shown below:

In the above graph, the vertical axis shows the production of good y and the horizontal axisshows the production of good x.

Economics Concept Introduction

Introduction: Production possibilities of frontier is referring to specify the highest production combination of two products or services by the fixed amount of investment.

b)

To determine

To ascertain:conditions behind work allotted in region A and B.

Answer to Problem 10.6P

The condition must hold for production allocation in country R between regions is

  $\frac{xA}{yA}=\frac{xB}{yB}$

Explanation of Solution

To allocate the production efficiently in a region, the rate of production transformation (RPT) mustbe equal to the marginal rate of substitution (MRS).

Mathematically, the following condition musthold:

  $RPT=-\frac{\partial yA}{\partial xA}$

To allocate the production of two goods efficiently in the region A, the following condition musthold:

  $\begin{array}{l}{\left(xA\right)}^2+{\left(yA\right)}^2=100\\ {\left(yA\right)}^2=100-{\left(xA\right)}^2\\ 2yA\frac{\partial yA}{\partial xA}=-2xA\\ \frac{\partial yA}{\partial xA}=\frac{xA}{yA}\end{array}$

Hence, the rate of production transformation (RPT) in region A is $\frac{xA}{yA}$

To allocate the production of two goods efficiently in region B, the following condition must hold:

  $\begin{array}{l}{\left(2xB\right)}^2+{\left(2yB\right)}^2=100\\ 4{\left(xB\right)}^2=100-4{\left(xB\right)}^2\\ 2yB\frac{\partial yB}{\partial xB}=-2xB\\ \frac{\partial yB}{\partial xB}=\frac{xB}{yB}\end{array}$

Hence, the rate of production transformation (RPT) in region B is $\frac{xB}{yB}$ YA

To achieve the production allocative efficiency between regions, the RPT of two regions must beequal.

Hence, the condition must hold for production allocation in country R between regions is

  $\frac{xA}{yA}=\frac{xB}{yB}$

Economics Concept Introduction

Introduction: The marginal rate of production transformation is can be described as number of quantity of product that inevitable in order to attain a unit of another product.

c)

To determine

To ascertain:production possibility curve for a country.

Answer to Problem 10.6P

The PPF for a country R is shown below:

Explanation of Solution

To determine the combined PPF for a country R,

first, calculate the combine production function asfollows:

Region A

  $\begin{array}{l}Ix=x^{2}A\\ Iy=y^{2}A\end{array}$

Region B

  $\begin{array}{l}Ix={\left(2xB\right)}^2\\ Iy={\left(2yB\right)}^2\end{array}$

Equate the Ix , and Iy , as follows:

  $\begin{array}{l}{x}^{2}A={\left(2xB\right)}^2\\ x^{2}A=2xB\\ y^{2}A={\left(2yB\right)}^2\\ y^{2}A=2yB\end{array}$

Combine production function is shown below:

  $\begin{array}{l}x{c}^2+y^{2}c=225\\ {\left(12\right)}^2+y^{2}c=225\\ y^{2}c=225-144\\ y^{2}c=81\\ yc=9\end{array}$

  $\begin{array}{l}x{c}^2+y^{2}c\\ =9\left(x{B}^2+y^{2}B\right)\\ =9\left(\frac{100}{4}\right)\\ =225\end{array}$

Thus, the combined PPF is $x{c}^2+y^{2}c=225$

The total output of x is 12. Substitute the value of x in the above combined PPF and calculate the

output of y as follows:

  $\begin{array}{l}x{c}^2+y^{2}c=225\\ {\left(12\right)}^2+y^{2}c=225\\ y^{2}c=225-144\\ y^{2}c=81\\ yc=9\end{array}$

Hence, the total output of y produced in a country is 9

The PPF for a country is shown below:

Economics Concept Introduction

Introduction: Production possibilities of frontier is referring to specify the highest production combination of two products or services by the fixed amount of investment.

d)

To determine

To ascertain:production possibilities frontier for whole country.

Answer to Problem 10.6P

If the labor is mobile within the country the production can be increased by allocating the resources in the favor of the more productive region.

Explanation of Solution

The PPF for the whole country can be developed for the whole country in the same way the PPF has been calculated for country R with two regions

If the labor is mobile within the country the production can be increased by allocating the resources in the favor of the more productive region.

Economics Concept Introduction

Introduction: Production possibilities of frontier is referring to specify the highest production combination of two products or services by the fixed amount of investment.