Math Challenge: Zero Economic Profit in Electricity Production

Key Concepts

• Zero Economic Profit: Price should equal average cost.
• Average Cost (AC): Total cost divided by quantity produced.
• Total Cost (TC): Sum of variable costs and fixed costs.
• Variable Cost (VC): Cost per unit (constant marginal cost) multiplied by quantity.

Calculations

1. Variable Cost:

   • Each unit of electricity costs $0.05 to produce.
   • VC = $0.05 * Q (where Q is the quantity produced).

2. Fixed Cost:

   • Fixed cost is $1,000,000.
   • Total Cost (TC) = VC + Fixed Cost.

3. Average Cost (AC):

   • AC = (VC + Fixed Cost) / Q.
   • Simplification leads to formula: AC = 0.05 + (1,000,000 / Q).*

Demand and Price Relationship

• To find where demand crosses with this price, need another equation involving price and quantity.
• Demand Equation: Quantity demanded = 15,000,000 - 0.5 * Price.*

Solving the Equation

1. Equation Setup:

   • Substituting the price equation into the demand equation.
   • Rearrange to form a quadratic: Q^2 = 15,000,000 - 0.025Q - 500,000.

2. Quadratic Form:

   • Q^2 - 0.025Q + 15,000,000 - 500,000 = 0.
   • Coefficients: a = 1, b = -0.025, c = 500,000.

3. Quadratic Solution:

   • Use quadratic formula to solve for Q.
   • Solutions found using a calculator or software (Wolfram Alpha).
   • Largest value for Q = 14,999,999.4 units.

4. Finding Price (P):

   • Substitute Q into AC equation: P = 0.116666.
   • Price per unit = over 11 cents.

Conclusion

• Set price to where average cost equals zero economic profit.
• Largest viable production Q = 14,999,999.4 units.
• Tools and technology like calculator or Wolfram Alpha are recommended for solving quadratic equations.

Study Tips

• Understand the relationship between cost, demand, and price.
• Practice solving quadratic equations, utilizing tools for assistance if needed.