Let, cost function TC = 50 + 6Q^2 , Revenue function TR = 100Q 4Q^2 . a. Compute TFC, TVC, TC, TR and profit at output range of 0 to 10 units. [5] b. Using schedules, explain the behavior of TFC,…

a. Computation of TFC, TVC, TC, TR, and Profit

Given:

• Total Cost (TC) = $50+6Q^2$
• Total Revenue (TR) = $100Q-4Q^2$

Step 1: Derive TFC and TVC

• Total Fixed Cost (TFC) is the cost that does not vary with output. From the TC equation, the constant term (50) represents TFC. $$TFC=50$$
• Total Variable Cost (TVC) is the cost that varies with output. It is obtained by subtracting TFC from TC: $$TVC=TC-TFC=(50+6Q^2)-50=6Q^2$$

Step 2: Compute TC, TR, and Profit

• TC is already given as $50+6Q^2$.
• TR is given as $100Q-4Q^2$.
• Profit (π) is calculated as: $$\pi =TR-TC=(100Q-4Q^2)-(50+6Q^2)=100Q-10Q^2-50$$

Step 3: Numerical Computation for Q = 0 to 10

The computed values are tabulated above.

Key Observations:

• TFC remains constant at 50 for all output levels.
• TVC increases quadratically with output due to the $6Q^2$ term.
• TC follows TVC since TFC is constant.
• TR increases initially, reaches a maximum, and then declines due to the $-4Q^2$ term.
• Profit is maximized at Q = 5 (π = 240).

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b. Behavior of TFC, TVC, and TC with Explanation

1. Total Fixed Cost (TFC)

• Definition: Costs that do not change with output (e.g., rent, salaries, insurance).
• Behavior:
  • Constant across all output levels (TFC = 50).
  • Graph: A horizontal line parallel to the x-axis.
• Reason:
  • Fixed costs are incurred regardless of production level. Even if $Q=0$, the firm must pay these costs.

2. Total Variable Cost (TVC)

• Definition: Costs that vary with output (e.g., raw materials, wages of production workers).
• Behavior:
  • Increases at an increasing rate due to the $6Q^2$ term.
  • Non-linear growth: As $Q$ increases, TVC rises faster (e.g., TVC at $Q=10$ is 600, while at $Q=5$ it is 150).
• Reason:
  • Diminishing marginal returns may not apply here since TVC is purely quadratic. However, in real-world scenarios, TVC may rise at a decreasing rate initially (due to specialization) and then at an increasing rate (due to inefficiencies).

3. Total Cost (TC)

• Definition: Sum of TFC and TVC ($TC=TFC+TVC$).
• Behavior:
  • Increases at an increasing rate because TVC dominates.
  • Shape: Similar to TVC but shifted upward by TFC (50 units).
• Reason:
  • Since TVC grows quadratically, TC also grows quadratically, maintaining the same curvature but starting at 50.

Schedule Explanation

Graphical Behavior:

• TFC: Horizontal line at 50.
• TVC: Curved upward (parabola opening upward).
• TC: Parallel to TVC but shifted up by 50.

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c. Graph of TR, TC, and Profit with TR - TC Approach to Firm Equilibrium

1. Graph of TR, TC, and Profit

2. TR - TC Approach to Firm Equilibrium

The TR - TC approach determines the profit-maximizing output by comparing total revenue and total cost.

Key Steps:

1. Find where MR = MC (Alternative Approach):

   • Marginal Revenue (MR) = $\frac{d(TR)}{dQ}=100-8Q$.
   • Marginal Cost (MC) = $\frac{d(TC)}{dQ}=12Q$.
   • Set MR = MC: $$100-8Q=12Q⟹100=20Q⟹Q=5$$
   • This confirms that Q = 5 is the profit-maximizing output.

2. TR - TC Approach:

   • Profit ($\pi$) is maximized where the vertical distance between TR and TC is greatest.
   • From the table, profit peaks at Q = 5 ($\pi =240$).
   • Beyond Q = 5, TR declines faster than TC, reducing profit.

3. Graphical Interpretation:

   • The TR curve is a downward-opening parabola, peaking at $Q=12.5$ (where MR = 0).
   • The TC curve is an upward-opening parabola.
   • The profit curve ($\pi =TR-TC$) is also a downward-opening parabola, peaking at Q = 5.
   • The firm’s equilibrium is at Q = 5, where profit is maximized.

Why TR - TC Approach?

• Firms aim to maximize profit, which is the difference between TR and TC.
• The TR - TC approach directly shows where this difference is largest.
• In this case, Q = 5 is the output where profit is highest before losses set in at higher outputs.

Shutdown Rule (Additional Insight)

• If TR < TC for all $Q>0$, the firm should shut down.
• Here, TR > TC for Q = 1 to 9, so the firm continues production.
• At Q = 10, profit drops to 130, but the firm may still operate if it can cover variable costs (TVC = 600, TR = 780 > TVC).

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Program to Compute TR, TC, and Profit (Python)

import numpy as np
import matplotlib.pyplot as plt

# Define cost and revenue functions
def TC(Q):
    return 50 + 6 * Q**2

def TR(Q):
    return 100 * Q - 4 * Q**2

def Profit(Q):
    return TR(Q) - TC(Q)

# Generate output range
Q = np.arange(0, 11, 1)

# Compute values
TFC = np.full_like(Q, 50)
TVC = TC(Q) - TFC
TC_values = TC(Q)
TR_values = TR(Q)
Profit_values = Profit(Q)

# Print table
print("{:<10} {:<10} {:<10} {:<10} {:<10} {:<10}".format(
    "Output (Q)", "TFC", "TVC", "TC", "TR", "Profit"))
for q, tfc, tvc, tc, tr, pi in zip(Q, TFC, TVC, TC_values, TR_values, Profit_values):
    print("{:<10} {:<10.2f} {:<10.2f} {:<10.2f} {:<10.2f} {:<10.2f}".format(
        q, tfc, tvc, tc, tr, pi))

# Plot
plt.figure(figsize=(10, 6))
plt.plot(Q, TR_values, label="Total Revenue (TR)", color="green")
plt.plot(Q, TC_values, label="Total Cost (TC)", color="red")
plt.plot(Q, Profit_values, label="Profit (π = TR - TC)", color="blue")
plt.axvline(x=5, color="black", linestyle="--", label="Profit-Maximizing Q")
plt.xlabel("Output (Q)")
plt.ylabel("Value (NPR)")
plt.title("TR, TC, and Profit Curves")
plt.legend()
plt.grid()
plt.show()

How it works:

• Computes TC, TR, and profit for Q = 0 to 10.
• Plots the curves and marks the profit-maximizing output (Q = 5).