The production function of Langtang Cheese Pvt. Ltd. is Q = 4L + 11L^2 L^3 . Complete the following table. No. of Labor Total Production Average Production Marginal Production : : : : 0 1 2 3 4 5 6…

Solution to the Problem

Step 1: Completing the Table

The production function is given as: $$Q=4L+11L^2-L^3$$

We calculate Total Production (Q), Average Production (AP), and Marginal Production (MP) for each labor input.

• Total Production (Q) is directly obtained by substituting $L$ in the production function.
• Average Production (AP) is calculated as: $$AP=\frac{Q}{L}$$
• Marginal Production (MP) is the change in total production when one additional unit of labor is employed: $$MP=\frac{\Delta Q}{\Delta L}=Q_{L+1}-Q_L$$

The completed table is shown above.

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Step 2: Analysis of the Three Stages of the Law of Variable Proportions

The Law of Variable Proportions states that as more units of a variable input (labor) are added to a fixed input (capital), the marginal product (MP) of the variable input changes in three distinct stages:

1. First Stage (Increasing Returns to Labor)

   • Range: $L=1$ to $L=4$
   • Characteristics:
     • MP is increasing (15 → 23 → 31 → 39).
     • AP is rising (15 → 19 → 23 → 27).
     • Total Production (Q) is increasing at an increasing rate.
   • Reason: Specialization and efficient use of fixed factors lead to higher productivity per additional worker.

2. Second Stage (Diminishing Returns to Labor)

   • Range: $L=5$ to $L=7$
   • Characteristics:
     • MP starts declining (47 → 43 → 29).
     • AP reaches its maximum at $L=6$ (AP = 33) and then falls.
     • Total Production (Q) is still increasing but at a decreasing rate.
   • Reason: Overcrowding of fixed factors (e.g., machinery, workspace) reduces efficiency per additional worker.

3. Third Stage (Negative Returns to Labor)

   • Range: $L=8$ to $L=9$
   • Characteristics:
     • MP becomes negative (5 → -31).
     • AP falls sharply (29 → 22.33).
     • Total Production (Q) starts declining (232 → 201).
   • Reason: Excessive labor leads to congestion, inefficiency, and possible damage to fixed inputs.

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Step 3: Finding the Equilibrium Level of Labor and Output

Given:

• Market price of cheese (P) = Rs. 1000 per kg
• Wage rate (W) = Rs. 34,000 per month

The equilibrium condition for a firm in the short run is: $$MP=\frac{W}{P}$$ $$MP=\frac{34,000}{1,000}=34\text{ kg per additional labor}$$

From the table, MP = 34 occurs between $L=4$ (MP = 39) and $L=5$ (MP = 47). However, since MP must be exactly equal to 34, we interpolate between $L=4$ and $L=5$:

• At $L=4$: $Q=108$, $MP=39$
• At $L=5$: $Q=155$, $MP=47$

Since MP decreases from 39 to 47, the equilibrium occurs before $L=5$. However, since MP is not exactly 34 at any integer value, we consider the second stage (where MP is positive and diminishing) and choose the labor level where MP is closest to 34.

From the table:

• At $L=4$: $MP=39$ (higher than 34)
• At $L=5$: $MP=47$ (even higher)

Since MP is decreasing, the firm should hire labor until MP = W/P (34). However, since MP never equals 34 exactly, we consider the optimal labor level where MP is just above 34 (i.e., $L=4$) because hiring beyond this would lead to diminishing returns.

But let’s verify using profit maximization condition: $$\text{Profit}=TR-TC=P\times Q-W\times L$$ We calculate profit for $L=4$ and $L=5$:

• At $L=4$: $$TR=1,000\times 108=108,000$$ $$TC=34,000\times 4=136,000$$ $$\text{Profit}=108,000-136,000=-28,000\text{ (Loss)}$$

• At $L=5$: $$TR=1,000\times 155=155,000$$ $$TC=34,000\times 5=170,000$$ $$\text{Profit}=155,000-170,000=-15,000\text{ (Loss)}$$

Since profit is higher (less loss) at $L=5$, but MP is still above 34, we check $L=6$:

• At $L=6$: $$TR=1,000\times 198=198,000$$ $$TC=34,000\times 6=204,000$$ $$\text{Profit}=198,000-204,000=-6,000\text{ (Loss)}$$

• At $L=7$: $$TR=1,000\times 227=227,000$$ $$TC=34,000\times 7=238,000$$ $$\text{Profit}=227,000-238,000=-11,000\text{ (Loss)}$$

The maximum profit (least loss) occurs at $L=6$ (Profit = -6,000), but since MP at $L=6$ is 43 (still > 34), we must consider that the firm should stop hiring when MP falls below W/P (34).

However, since MP never equals 34 exactly, the optimal labor level is where MP is closest to 34 but still positive. From the table, MP = 39 at $L=4$ and MP = 47 at $L=5$. Since MP is decreasing, the firm should hire up to $L=5$ (where MP is still above 34) but not beyond, as MP drops below 34 at $L=6$ (MP = 43, still > 34).

But this contradicts the profit maximization approach. Therefore, the correct equilibrium level is where MP = W/P (34). Since MP never equals 34 exactly, we consider the last labor unit where MP ≥ 34, which is $L=5$ (MP = 47).

However, economic theory suggests hiring until MP = W/P. Since MP is always above 34 in the second stage, the firm should hire labor until MP falls below 34, which happens after $L=6$ (MP = 43 at $L=6$, MP = 29 at $L=7$).

But MP at $L=7$ is 29 (< 34), so the firm should stop at $L=6$ (since hiring $L=7$ would reduce profit).

Thus, the optimal labor level is $L=6$ (where MP is still above 34 but hiring more would reduce profit).

Final Equilibrium:

• Optimal Labor (L) = 6
• Optimal Output (Q) = 198 kg

However, let’s re-evaluate using exact calculation:

We need to find $L$ where: $$MP=\frac{dQ}{dL}=4+22L-3L^2=34$$ $$4+22L-3L^2=34$$ $$-3L^2+22L-30=0$$ $$3L^2-22L+30=0$$

Solving the quadratic equation: $$L=\frac{22\pm \sqrt{(-22{)}^2-4\times 3\times 30}}{2\times 3}$$ $$L=\frac{22\pm \sqrt{484-360}}{6}$$ $$L=\frac{22\pm \sqrt{124}}{6}$$ $$L=\frac{22\pm 11.14}{6}$$

Two solutions:

1. $L=\frac{22+11.14}{6}=\frac{33.14}{6}\approx 5.52$
2. $L=\frac{22-11.14}{6}=\frac{10.86}{6}\approx 1.81$

Since we are in the second stage (diminishing returns), we take $L\approx 5.52$. Since labor must be an integer, we check $L=5$ and $L=6$.

• At $L=5$: $MP=47$ (higher than 34)
• At $L=6$: $MP=43$ (still higher than 34)
• At $L=7$: $MP=29$ (lower than 34)

Thus, the optimal labor level is $L=6$ (since hiring $L=7$ would make MP < 34, reducing profit).

Final Answer:

• Equilibrium Labor (L) = 6
• Equilibrium Output (Q) = 198 kg