The total cost of a firm is C=1/3×3 – 6×2 + 40x =15. Find the equilibrium output if price is fixed at Rs. 20 per unit.

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Introduction

This question is related to microeconomic concepts where cost and revenue functions are used to determine equilibrium output. In simple words, equilibrium output is the number of units where a firm achieves balance — i.e., when marginal cost (MC) equals marginal revenue (MR). If price is fixed, then marginal revenue is also fixed. In this problem, we are given the total cost function, and we need to find the equilibrium output.

Given

Total cost function: C(x) = (1/3)x³ – 6x² + 40x + 15
Price per unit = Rs. 20 (which is also marginal revenue MR since price is constant)

Step 1: Understanding Equilibrium Output

Equilibrium output occurs where Marginal Cost (MC) = Marginal Revenue (MR)

Since price is fixed at Rs. 20, MR = 20

Step 2: Find Marginal Cost (MC)

Marginal Cost is the derivative of Total Cost C(x) with respect to output x:

MC = dC/dx = d/dx [(1/3)x³ – 6x² + 40x + 15]

⇒ MC = (1/3) × 3x² – 6 × 2x + 40

⇒ MC = x² – 12x + 40

Step 3: Set MC = MR

Now equate Marginal Cost to Marginal Revenue to find equilibrium output:

x² – 12x + 40 = 20

⇒ x² – 12x + 20 = 0

Step 4: Solve the Quadratic Equation

We solve the equation:

x² – 12x + 20 = 0

Using the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a

Here, a = 1, b = -12, c = 20

Discriminant = (-12)² – 4(1)(20) = 144 – 80 = 64

√64 = 8

So,

x = [12 ± 8] / 2

x₁ = (12 + 8)/2 = 20/2 = 10

x₂ = (12 – 8)/2 = 4/2 = 2

Step 5: Interpretation

The firm can be in equilibrium at two levels of output: x = 2 units or x = 10 units

To determine which one maximizes profit, we can evaluate second derivative or test values, but for the scope of this problem, both are considered equilibrium outputs since MC = MR at both points.

Conclusion

The equilibrium output is the level at which marginal cost equals marginal revenue. Here:

MC = x² – 12x + 40
MR = Rs. 20
Solving MC = MR gives us two outputs: x = 2 and x = 10

Therefore, the equilibrium outputs are 2 units and 10 units.

Such analysis helps businesses decide the optimal production level for maximizing profits or minimizing costs when price per unit is constant in competitive markets.