Chapter 9, Problem 9.1P

a

To determine

To plot:

Graphical representation of isoquant and quantity of K and L.

Explanation of Solution

Since q = 40,000 has been produced with $5\times 2=10units$ of capital and $5\times 1=5unit$ of labor. However, substitutability is not allowed by the production technology. Thus, the production function took the shape of a flexographic curve as shown below:

In the above diagram, x-axis measures labor while y-axis measures capital, q represents isoquant of Large Mowers as depicted above.

Economics Concept Introduction

Introduction:

Isoquant curve is a graphical representation showing the same level of output production with different combinations of inputs.

b)

To determine

To plot:

Graphical function of isoquant for small mower.

Explanation of Solution

This diagram shows the plotting of q = 40,000, K = 8 and L = 8

In the above diagram, x-axis measures labor while the y-axis measures capital. Q represents isoquant of Small Mowers as depicted above.

Economics Concept Introduction

Introduction:

Expansion path is a combination of optimal output when input changes due to prices.

c)

To determine

To know:

Quantities of K and L for different lawn divisions.

Explanation of Solution

Production functions are given below:

$\begin{array}{l}q=5,000\times k\\ =5,000\times l\\ q=8,000\times 2l\\ =8,000\times k\end{array}$

When lawn is cut by 1st method:

$\begin{array}{l}k=l=\frac{20,000}{5,000}\\ =4\end{array}$

When lawn is cut by 2nd method:

$\begin{array}{l}l=\frac{20,000}{8,000}\\ =2.5\\ k=2l\\ =2\times 2.5\\ =5\end{array}$

Total capital requirement = 9

Total labor requirement = 6.5

Production functions are given below:

$\begin{array}{l}q=\text{}5,000\times k\\ =5,000\times l\end{array}$

$\begin{array}{l}q=\text{}8,000\times 2l\\ =8,000\times k\end{array}$

When lawn is cut by 1st method:

$\begin{array}{l}k=l=\frac{10,000}{5,000}\\ =2\end{array}$

When lawn is cut by 2nd method:

$\begin{array}{l}l=\frac{30,000}{8,000}\\ =3.75\end{array}$

$\begin{array}{l}k=2l\\ =2\times 3.75\\ =7.50\end{array}$

Total capital requirement = 9.5

Total labor requirement = 5.75

The fraction of K and L means the amount of capital input used per unit of labor.

This table shows the resources uses in the Large Mowers and Small Mowers.

40,000 sq.ft.	LM		SM
LM/SM mix	L	K	L	K
0/100$%	0	0	8	8
25%/75%	1.25	2.5	6	6
50%/50%	2.5	5	4	4
75%/25%	3.75	7.5	2	2
100%/0%	5	10	0	0

Economics Concept Introduction

Introduction:

Isoquant curve is a graphical representation showing the same level of output production with different combinations of inputs.

d)

To determine

To plot:

Graphical representation of isoquant.

Explanation of Solution

Calculation of labor and capital when 40000 square foot of lawn is cut by 1st method with p fraction.

$\begin{array}{l}k=l\\ =\frac{40,000p}{5,000}\\ =8p\end{array}$

Calculation of labor and capital when (1-p) fraction of 40,000 square foot lawn is cut by 2nd method.

$\begin{array}{l}l=\frac{40,000(1-p)}{8,000}\\ =5(1-p)\end{array}$

$\begin{array}{l}k=2l\\ =2\times (1-p)\\ =10(1-p)\end{array}$

Calculation of labor requirement:

$\begin{array}{l}l=8p+5-5p\\ =5+3p\end{array}$

Calculation of total capital requirement:

$\begin{array}{l}k=8p+10-10p\\ =10-2p\end{array}$

Two equations are given as follows:

$\begin{array}{l}l=5+3p\\ k=10-2p\end{array}$

Solving first equation for p and putting the value of p in second as shown below:

$\begin{array}{l}l=5+3p\\ l-5=3p\\ p=\frac{l-5}{3}\\ k=10-2\left(\frac{l-5}{3}\right)\\ =\frac{30-2l+10}{3}\\ =\frac{40-2l}{3}\end{array}$

The Table shows the different calculations of Labor (L) and capital (K) in Small Mowers as shown below:

40,000 sq.ft.	LM
LM/SM mix	L	K
0/100$%	8	8
25%/75%	7.25	8.5
50%/50%	6.5	9
75%/25%	5.75	9.5
100%/0%	5	10

Following graph is plotted from the given table shows the isoquant for the combined production function as shown below:

In the above diagram, x-axis measures labor while y-axis measures capital, q represents isoquant for the combined production function as depicted above.

Economics Concept Introduction

Introduction:

Expansion path is a combination of optimal output when input changes due to prices.