Chapter 13, Problem 15P

(a)

To determine

Mathematical expression of marginal revenue product of labor.

Explanation of Solution

To derive the mathematical expression of marginal revenue product of labor, it is necessary to derive the intercept and slope of the MRP_L.

The general formula for calculating slope of the MRP_L curve is given as follows:

$\text{Slope}=\frac{{\text{Present value}}_{\text{y variable}}-{\text{Previous value}}_{\text{yvariable}}}{{\text{Present value}}_{\text{x variable}}-{\text{Previous value}}_{\text{x variable}}}$ (1)

Substitute the respective values in Equation (1) to calculate the slope of MRP_L.

$\begin{array}{c}\text{Slope}=\frac{\left(42-14\right)}{(0-56)}\\ =-\frac{28}{56}\\ =-\frac{1}{2}\end{array}$

Slope is$-\frac{1}{2}$ .

Substitute the slope in the general formula of equation to derive the expression of MRP_L.

$\text{y}=\text{mx}+\text{b}$, where “m” is slope and “b” is the intercept.

$\text{y}=-\left(\frac{1}{2}\right)\text{x}+42$

Substitute “MRP_L” for “y” and “L’’ for “x”.

${\text{MRP}}_{\text{L}}=42-0.5\text{L}$ (2)

Mathematical expression of marginal revenue product of labor is${\text{MRP}}_{\text{L}}=42-0.5\text{L}$.

Economics Concept Introduction

Marginal revenue product of labor: Marginal revenue product of labor is the change in the firm’s revenue as a result of employing an extra unit of labor.

(b)

To determine

Mathematical expression of supply of labor in the lemongrass industry.

Explanation of Solution

Substitute the respective values in Equation (1) to calculate the slope of supply curve.

$\begin{array}{c}\text{Slope}=\frac{\left(14-0\right)}{(56-0)}\\ =\frac{14}{56}\\ =0.25\end{array}$

Slope is 0.25.

Substitute the slope in the general formula of equation to derive the expression of supply curve.

$\text{y}=\text{mx}+\text{b}$, where “m” is slope and “b” is the intercept. The intercept is zero. Thus, the supply equation can be written as follows:

$\text{y}=0.25\text{x}$

Substitute “W” for “y” and “L’’ for “x”.

$\text{W}=0.25\text{L}$ (3)

Mathematical expression of supply of labor is$\text{W}=0.25\text{L}$.

(c)

To determine

Number workers hired and wage rate paid.

Explanation of Solution

The competitive outcome is characterized by equating the wage rate and the marginal revenue product of labor. Symbolically, it is shown below:

$\text{W}={\text{MRP}}_{\text{L}}$

The value of “W” and “MRP_L” is already calculated in part (a) and in part (b), respectively.

$\begin{array}{c}0.25\text{L}=42-0.5\text{L}\\ \text{0}\text{.75L}=42\\ \text{L}=\frac{42}{\text{0}\text{.75}}\\ \text{L}=56\end{array}$

The number of laborers hired is 56 units.

Substitute the quantity of laborers hired in the supply curve of labor (Equation (3)).

$\begin{array}{l}\text{W}=0.25\text{L}\\ \text{W}=0.25\left(56\right)\\ \text{W}=14\end{array}$

Wage rate paid is $14.

(d)

To determine

Mathematical expression of marginal expenditure curve and draw the graph of the marginal expenditure curve.

Explanation of Solution

As per the rule of marginal expenditure, that is, marginal expenditure curve has twice the slope of the labor supply curve. The labor supply curve is $\text{W}=0.25\text{L}$. Thus, marginal expenditure curve function is shown below:

$\begin{array}{c}\text{W}=0.25\text{L}\\ \text{ME}=2\left(0.25\right)\text{L}\\ \text{ME}=0.5\text{L}\end{array}$

Marginal expenditure curve function is$\text{ME}=0.5\text{L}$.

Graphical illustration of marginal expenditure curve is depicted in the Figure below:

In Figure 1, the vertical axis measures wage rate, and horizontal axis measures quantity of labor. The downward sloping curve is the marginal revenue product of labor. The upward sloping curve S_L is the supply curve of labor, and ME is the marginal expenditure curve of labor. The slope of marginal expenditure curve is twice the slope of supply curve of labor.

(e)

To determine

Units of labor will a monopsolistic lemongrass producer hire and wage rate paid.

Explanation of Solution

The monopsolistic outcome is characterized by equating the marginal revenue product of labor and marginal expenditure. Symbolically, it is shown below:

${\text{MRP}}_{\text{L}}=\text{ME}$

The values of “MRP_L” and “ME” are already calculated in part (a) and part (c), respectively.

$\begin{array}{c}42-0.5\text{L}=0.5\text{L}\\ \text{L}=42\end{array}$

Units of labor will a monopsolistic lemongrass producer hire (Monopsony quantity) is 42 units.

Substitute the quantity of labor a monopsolistic producer hires in the supply curve of labor (Equation (3)).

$\begin{array}{l}\text{W}=0.25\text{L}\\ \text{W}=0.25\left(42\right)\\ \text{W}=10.5\end{array}$

Wage rate (Monopsony wage) paid is $10.5.

(f)

To determine

Fall in employment.

Explanation of Solution

The labor market goes from competitive on the buyer’s side to monopsonistic side, that result in decrease the employment. That is, at competitive market, the unit of laborers hired is 56, but in monopsonistic market, the unit of laborers hired is 42. Thus, the quantity of labor hired is reduced by 14 units $\left(56-42\right)$.

(g)

To determine

Deadweight loss.

Explanation of Solution

Deadweight loss can be calculated as follows:

$\begin{array}{c}\text{Deadweight loss}=\frac{1}{2}\left(\begin{array}{l}{\text{Wage}}_{\text{Equilibrium}}\\ -{\text{Wage}}_{\text{Monoposony}}\end{array}\right)\left(\begin{array}{l}{\text{Quantity}}_{\text{Equilibrium}}\\ -{\text{Quantity}}_{\text{Monoposony}}\end{array}\right)\\ =\frac{1}{2}\left(\text{21}-\text{10}\text{.50}\right)\left(56-42\right)\\ =73.50\end{array}$

Thus, deadweight loss is $73.50.

The graphical illustration of deadweight loss is depicted below in Figure 2.

In Figure 2, the vertical axis measures wage rate, and horizontal axis measures quantity of labor. The downward sloping curve is the marginal revenue product of labor. The upward sloping curve S_L is the supply curve of labor, and ME is the marginal expenditure curve of labor. The shaded area is the deadweight loss created in the society.