Question

Propose a second degree cost function and explain it (crossing points, F, Cv, concavity, etc.)

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1. **Identify the Cost Function**:

Given the general form of a quadratic cost function:

C(x) = ax^2 + bx + c

2. **Intercept with y-axis (Fixed Costs)**:

The termc represents the fixed costs (F ), where the cost function intersects the y-axis (i.e., when x = 0 ).

C(0) = c

3. **Variable Costs**:

The termbx corresponds to the variable costs (Cv ), where the cost increases linearly with the increase in production x .

C_{v}(x) = bx

4. **Quadratic Term and Concavity**:

The termax^2 shows the rate of change in cost with respect to production. The sign of a determines the concavity of the function:

- Ifa > 0 , the function is concave up (U-shaped), indicating increasing costs at increasing rates as production rises.

- Ifa < 0 , the function is concave down (∩-shaped), indicating increasing costs at decreasing rates as production rises.

5. **Break-even Points**:

Solving for the points where the cost function intersects the x-axis (break-even points) by settingC(x) = 0 and solving the quadratic equation:

ax^2 + bx + c = 0

Using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

**Answer**:

C(x) = ax^2 + bx + c

Given the general form of a quadratic cost function:

2. **Intercept with y-axis (Fixed Costs)**:

The term

3. **Variable Costs**:

The term

4. **Quadratic Term and Concavity**:

The term

- If

- If

5. **Break-even Points**:

Solving for the points where the cost function intersects the x-axis (break-even points) by setting

Using the quadratic formula:

**Answer**:

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