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Let's go through each part of the question step-by-step:
### Part (a): Calculate the Marginal Product of Labor (MP_L)
To calculate the marginal product of labor, we need to find the change in total product when an additional unit of labor is employed. The marginal product of labor (MP_L) for each level of labor can be calculated as follows:
[tex]\[ MP_L(i) = \Delta \text{Total Product} = \text{Total Product}(i) - \text{Total Product}(i-1) \][/tex]
Using this formula, we find:
- [tex]\( MP_L(1) = 100 - 0 = 100 \)[/tex]
- [tex]\( MP_L(2) = 180 - 100 = 80 \)[/tex]
- [tex]\( MP_L(3) = 240 - 180 = 60 \)[/tex]
- [tex]\( MP_L(4) = 280 - 240 = 40 \)[/tex]
- [tex]\( MP_L(5) = 300 - 280 = 20 \)[/tex]
The marginal product of labor is therefore:
[tex]\[ MP_L = [None, 100, 80, 60, 40, 20] \][/tex]
### Part (b): Calculate the Value of Marginal Product of Labor (VMP_L)
The value of the marginal product of labor is calculated by multiplying the marginal product of labor by the price of the product. Given that the price of the product is Rs. 600, we calculate [tex]\(VMP_L\)[/tex] as follows:
[tex]\[ VMP_L(i) = MP_L(i) \times \text{price of product} \][/tex]
Using this formula, we find:
- [tex]\( VMP_L(1) = 100 \times 600 = 60000 \)[/tex]
- [tex]\( VMP_L(2) = 80 \times 600 = 48000 \)[/tex]
- [tex]\( VMP_L(3) = 60 \times 600 = 36000 \)[/tex]
- [tex]\( VMP_L(4) = 40 \times 600 = 24000 \)[/tex]
- [tex]\( VMP_L(5) = 20 \times 600 = 12000 \)[/tex]
The value of marginal product of labor is therefore:
[tex]\[ VMP_L = [None, 60000, 48000, 36000, 24000, 12000] \][/tex]
### Part (c): Determine the Profit-Maximizing Number of Labor with a Wage Rate of Rs. 36000
To maximize profit, a firm will employ labor up to the point where the value of the marginal product of labor equals the wage rate. Given a wage rate of Rs. 36000, we identify the point where [tex]\(VMP_L\)[/tex] is just above the wage rate and the next [tex]\(VMP_L\)[/tex] is below the wage rate.
From the calculated [tex]\(VMP_L\)[/tex] values:
[tex]\[ \begin{align*} VMP_L(1) &= 60000 \\ VMP_L(2) &= 48000 \\ VMP_L(3) &= 36000 \\ VMP_L(4) &= 24000 \\ \end{align*} \][/tex]
At [tex]\( L = 3 \)[/tex], [tex]\( VMP_L = 36000 \)[/tex], which is equal to the wage rate. Beyond this, [tex]\( VMP_L \)[/tex] falls below 36000. Therefore, the profit-maximizing level of labor is 3.
Summary in table:
\begin{tabular}{|c|c|c|}
\hline
Labor & VMP_L & Wage Rate = 36000 \\
\hline
0 & & \\
1 & 60000 & \\
2 & 48000 & \\
3 & 36000 & Equilibrium \\
4 & 24000 & \\
5 & 12000 & \\
\hline
\end{tabular}
### Part (d): Determine the Profit-Maximizing Number of Labor with a Wage Rate of Rs. 48000
Similarly, with a wage rate of Rs. 48000, we find:
[tex]\[ \begin{align*} VMP_L(1) &= 60000 \\ VMP_L(2) &= 48000 \\ \end{align*} \][/tex]
At [tex]\( L = 2 \)[/tex], [tex]\( VMP_L = 48000 \)[/tex], which is equal to the wage rate. Beyond this, [tex]\( VMP_L \)[/tex] falls below 48000. Therefore, the profit-maximizing level of labor is 2.
Summary in table:
\begin{tabular}{|c|c|c|}
\hline
Labor & VMP_L & Wage Rate = 48000 \\
\hline
0 & & \\
1 & 60000 & \\
2 & 48000 & Equilibrium \\
3 & 36000 & \\
4 & 24000 & \\
5 & 12000 & \\
\hline
\end{tabular}
### Graphical Representation
For both scenarios, plot the [tex]\( VMP_L \)[/tex] curve against labor. Indicate the wage rates as horizontal lines and mark the equilibrium points where the wage rate intersects the [tex]\( VMP_L \)[/tex] curve. This will visually show the profit-maximizing levels of labor at different wage rates.
Graph notes:
- Wage rate of Rs. 36000 intersects [tex]\( VMP_L \)[/tex] curve at [tex]\( L = 3 \)[/tex].
- Wage rate of Rs. 48000 intersects [tex]\( VMP_L \)[/tex] curve at [tex]\( L = 2 \)[/tex].
By following these steps, you get a comprehensive view of how the profit-maximizing number of labor changes with varying wage rates.
### Part (a): Calculate the Marginal Product of Labor (MP_L)
To calculate the marginal product of labor, we need to find the change in total product when an additional unit of labor is employed. The marginal product of labor (MP_L) for each level of labor can be calculated as follows:
[tex]\[ MP_L(i) = \Delta \text{Total Product} = \text{Total Product}(i) - \text{Total Product}(i-1) \][/tex]
Using this formula, we find:
- [tex]\( MP_L(1) = 100 - 0 = 100 \)[/tex]
- [tex]\( MP_L(2) = 180 - 100 = 80 \)[/tex]
- [tex]\( MP_L(3) = 240 - 180 = 60 \)[/tex]
- [tex]\( MP_L(4) = 280 - 240 = 40 \)[/tex]
- [tex]\( MP_L(5) = 300 - 280 = 20 \)[/tex]
The marginal product of labor is therefore:
[tex]\[ MP_L = [None, 100, 80, 60, 40, 20] \][/tex]
### Part (b): Calculate the Value of Marginal Product of Labor (VMP_L)
The value of the marginal product of labor is calculated by multiplying the marginal product of labor by the price of the product. Given that the price of the product is Rs. 600, we calculate [tex]\(VMP_L\)[/tex] as follows:
[tex]\[ VMP_L(i) = MP_L(i) \times \text{price of product} \][/tex]
Using this formula, we find:
- [tex]\( VMP_L(1) = 100 \times 600 = 60000 \)[/tex]
- [tex]\( VMP_L(2) = 80 \times 600 = 48000 \)[/tex]
- [tex]\( VMP_L(3) = 60 \times 600 = 36000 \)[/tex]
- [tex]\( VMP_L(4) = 40 \times 600 = 24000 \)[/tex]
- [tex]\( VMP_L(5) = 20 \times 600 = 12000 \)[/tex]
The value of marginal product of labor is therefore:
[tex]\[ VMP_L = [None, 60000, 48000, 36000, 24000, 12000] \][/tex]
### Part (c): Determine the Profit-Maximizing Number of Labor with a Wage Rate of Rs. 36000
To maximize profit, a firm will employ labor up to the point where the value of the marginal product of labor equals the wage rate. Given a wage rate of Rs. 36000, we identify the point where [tex]\(VMP_L\)[/tex] is just above the wage rate and the next [tex]\(VMP_L\)[/tex] is below the wage rate.
From the calculated [tex]\(VMP_L\)[/tex] values:
[tex]\[ \begin{align*} VMP_L(1) &= 60000 \\ VMP_L(2) &= 48000 \\ VMP_L(3) &= 36000 \\ VMP_L(4) &= 24000 \\ \end{align*} \][/tex]
At [tex]\( L = 3 \)[/tex], [tex]\( VMP_L = 36000 \)[/tex], which is equal to the wage rate. Beyond this, [tex]\( VMP_L \)[/tex] falls below 36000. Therefore, the profit-maximizing level of labor is 3.
Summary in table:
\begin{tabular}{|c|c|c|}
\hline
Labor & VMP_L & Wage Rate = 36000 \\
\hline
0 & & \\
1 & 60000 & \\
2 & 48000 & \\
3 & 36000 & Equilibrium \\
4 & 24000 & \\
5 & 12000 & \\
\hline
\end{tabular}
### Part (d): Determine the Profit-Maximizing Number of Labor with a Wage Rate of Rs. 48000
Similarly, with a wage rate of Rs. 48000, we find:
[tex]\[ \begin{align*} VMP_L(1) &= 60000 \\ VMP_L(2) &= 48000 \\ \end{align*} \][/tex]
At [tex]\( L = 2 \)[/tex], [tex]\( VMP_L = 48000 \)[/tex], which is equal to the wage rate. Beyond this, [tex]\( VMP_L \)[/tex] falls below 48000. Therefore, the profit-maximizing level of labor is 2.
Summary in table:
\begin{tabular}{|c|c|c|}
\hline
Labor & VMP_L & Wage Rate = 48000 \\
\hline
0 & & \\
1 & 60000 & \\
2 & 48000 & Equilibrium \\
3 & 36000 & \\
4 & 24000 & \\
5 & 12000 & \\
\hline
\end{tabular}
### Graphical Representation
For both scenarios, plot the [tex]\( VMP_L \)[/tex] curve against labor. Indicate the wage rates as horizontal lines and mark the equilibrium points where the wage rate intersects the [tex]\( VMP_L \)[/tex] curve. This will visually show the profit-maximizing levels of labor at different wage rates.
Graph notes:
- Wage rate of Rs. 36000 intersects [tex]\( VMP_L \)[/tex] curve at [tex]\( L = 3 \)[/tex].
- Wage rate of Rs. 48000 intersects [tex]\( VMP_L \)[/tex] curve at [tex]\( L = 2 \)[/tex].
By following these steps, you get a comprehensive view of how the profit-maximizing number of labor changes with varying wage rates.
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