Chapter 6, Problem 6.9P

a.

To determine

To draw: Isoquants at output of 1000 boxes of puffed rice per hour.

Explanation of Solution

The given production function is given by $q=100\sqrt{KL}$

Where K is the number of puffing guns used and L is the number of workers hired each hour.

When q is 1000, the value of $\sqrt{KL}$ must be 10, therefore the product of K and L must be 100.

The isoquant shall be made by joining the points which provide answer to such function, this shall, random numbers can be taken for K, to calculate the value of L.

Value of K	Calculation	Value of L
10	$L=\frac{100}{10}$	10
25	$L=\frac{100}{25}$	4
4	$L=\frac{100}{4}$	25

  

Economics Concept Introduction

Introduction: Isoquants are curves that represent same level of output at various combination of inputs and are used in determining efficient input mix for lower operational costs.

b.

To determine

To calculate: Value of L when K is 10 and average labor productivity.

Explanation of Solution

The given production function is given by $q=100\sqrt{KL}$

Where K is the number of puffing guns used and L is the number of workers hired each hour.

When q is 1000, the value of $\sqrt{KL}$ must be 10, therefore the product of K and L must be 100.

Therefore, when the value of K is 10, the value of L must be $\frac{100}{10}$ , that is, 10 units.

In other words, when 10 puffing guns are employed, 10 workers must work each hour to produce 1000 boxes of puffed rice.

  ${\text{Average Productivity}}_{K=10}=\frac{\text{Total Output}}{\text{Total Labour Input}}=\frac{1000}{10}=100\text{ boxes per labour per hour}\text{.}$

Economics Concept Introduction

Introduction: For any given value of labor (L) or capital (K), one can be calculated from the other by inputting the available value in the production function.

Average productivity refers to the arithmetic mean of total production with total labor input employed.

c.

To determine

To state: Impact of technical progress on sub-part (a) and (b).

Explanation of Solution

The new production function is given by $q=200\sqrt{KL}$

Where K is the number of puffing guns used and L is the number of workers hired each hour.

When q is 1000, the value of $\sqrt{KL}$ must be 5, therefore the product of K and L must be 25.

The isoquant shall be made by joining the points which provide answer to such function, this shall, random numbers can be taken for K, to calculate the value of L.

Value of K	Calculation	Value of L
2	$L=\frac{25}{2}$	12.5
5	$L=\frac{25}{5}$	5
10	$L=\frac{25}{10}$	2.5
12.5	$L=\frac{25}{12.5}$	2

  

As stated in sub part (b), value of one variable may be calculated by inputting the other in production function, therefore, when the value of K is 10, L shall be $\frac{25}{10}$ , that is, 2.5.

  ${\text{Average Productivity}}_{K=10}=\frac{\text{Total Output}}{\text{Total Labour Input}}=\frac{1000}{2.5}=400\text{ boxes per labour per hour}\text{.}$

Economics Concept Introduction

Introduction: Isoquants are curves that represent same level of output at various combination of inputs and are used in determining efficient input mix for lower operational costs.

Average productivity refers to the arithmetic mean of total production with total labor input employed.

d.

To determine

To state: Impact of technical progress on sub-part (a) and (b).

Explanation of Solution

The new production function is given by $q={(1+0.05)}^{t}100\sqrt{KL}$

Where K is the number of puffing guns used and L is the number of workers hired each hour.

The value of ${(1+0.05)}^t$ denotes the factor of increase productivity year on year due to progress in resource utilization due to technical development, ‘t’ denotes time in years.

Separate isoquants will be observed every year since the input required to produce the desired output of 1000 boxes would undergo shift.

When q is 1000, the value of $\sqrt{KL}$ must be $\frac{1000}{100{(1+0.05)}^t}$ , therefore the product of K and L must be ${(\frac{1000}{100{(1+0.05)}^t})}^2$ .

In order to plot the isoquants, arbitrary value of one variable can be taken to find the other variable.

Year	Required K x L
1	${(\frac{1000}{100{(1+0.05)}^1})}^2=90.70$
2	${(\frac{1000}{100{(1+0.05)}^2})}^2=82.27$
3	${(\frac{1000}{100{(1+0.05)}^3})}^2=74.62$

The isoquant shall be made by joining the points which provide answer to such function, this shall, random numbers can be taken for K, to calculate the value of L.

Using ‘1’, ‘2’ and ‘3’ to denote the years values for K and L at various levels shall be as follow:

K1	L 1	K2	L 2	K3	L 3
5.00	18.14	5.00	16.45	5.00	14.92
10.00	9.07	10.00	8.23	10.00	7.46
15.00	6.05	15.00	5.48	15.00	4.97

Isoquants can be drawn based on the calculated data as:

  

As stated in sub part (b), value of one variable may be calculated by inputting the other in production function, therefore, when the value of K is 10, L shall be $\frac{90.70}{10}$ , that is, 9.7 in year 1, $\frac{82.3}{10}$ that is, 8.23 in year 2, $\frac{74.6}{10}$ that is, 7.46 in year 3 and so on.

  ${\text{Average Productivity}}_{K=10}=\frac{\text{Total Output}}{\text{Total Labour Input}}=\frac{1000}{9.07}=110.25\text{ boxes per labour per hour in year 1}\text{.}$

  ${\text{Average Productivity}}_{K=10}=\frac{\text{Total Output}}{\text{Total Labour Input}}=\frac{1000}{8.23}=121.5\text{ boxes per labour per hour in year 2}\text{.}$

  ${\text{Average Productivity}}_{K=10}=\frac{\text{Total Output}}{\text{Total Labour Input}}=\frac{1000}{7.46}=134.05\text{ boxes per labour per hour in year 3}\text{.}$

Since technical progress is constant year on year, average productivity per hour will increase year on year as well.

Economics Concept Introduction

Introduction: Isoquants are curves that represent same level of output at various combination of inputs and are used in determining efficient input mix for lower operational costs.

Average productivity refers to the arithmetic mean of total production with total labor input employed.

Technical progress refers to increase the productivity factor, thereby increasing the production with same available inputs.