Breakeven

Total cost (CT) is the sum of fixed costs (CF) and variable costs (CV):

CT = CF + CV

Variable cost is a function of demand 

CV  = cv× D

Where, cv = variable cost per unit

Therefore, 

CT = CF + cv× D

Equation 2-6

Demand is a Function of Price

The total cost equation is a straight line with a positive slope, indicating that total cost increases as production volume or demand increases, with the fixed cost representing the y-intercept of the equation.

The total revenue function is a concave-down parabola, indicating that the total cost function may intersect it at two points – representing breakeven points where no profit or loss occurs. Producing below or above the demand levels at these intersections results in a loss, while operating between them yields a positive profit.

Total Profit Function

Total profit is calculated by subtracting total cost (in Equation 2 6) from total revenue (in Equation 2 2).

Profit (loss)

= Total Revenue - Total Costs 

= (aD - bD² )- (CF  + cv D)

= -bD²+ (a - cv )D - CF 

for 0 ≤ D ≤  a/b  and a > 0,b > 0.

Equation 2-7

To make any profit, the following two conditions must be made:

1. (a - cv )>0: The maximum price (intercept) that results in no demand must be greater than the variable cost per unit, ensuring non-negative demand.

2. Total Revenue (TR) > Total Cost (CT): Revenue must exceed cost.

If both conditions hold, the optimal demand (where maximum profit occurs) can be found by:

• Taking the first derivative of the profit equation with respect to demand, D,

• Setting it equal to zero and solving.

d(profit)/dD =0

Therefore, 

a - cv - 2bD = 0.

Profit is maximized when D equals: 

D*  = (a - cv)/2b

Equation 2-8

To confirm that profit is maximized, the second derivative of the profit function must be negative, which is true since d²(profit)/dD²=-2b

The maximum profit can be obtained by substituting the optimum demand into the profit(loss) equation as given below. 

Profit (loss) 

= -bD² + (a - cv)D - CF 

by replacing the value for optimum demand, D* for maximizing profit,

= -b * ((a - cv) / (2b))² + (a - cᵥ) * ((a - cv) / (2b)) - CF  

= -(1 / (4b)) * (a - cv)² + (1 / (2b)) * (a - cv)² - CF  

Maximum Profit = (1 / (4b)) * (a - cvᵥ) - CF

Equation 2-9

Breakeven

The breakeven point occurs when total revenue equals total cost (same as setting the profit in Equation 2-7 to zero.):

aD-bD²=CF+cᵥ D

-bD²  + (a - cᵥ)D - CF = 0 

The two breakeven demand values, where neither profit nor loss occurs, can be found by solving:

D'=(-(a - cᵥ) ± √((a - cᵥ)²  - 4(-b)(-CF)))/(2(-b))

Equation 2-10

With the profit conditions met, the discrimination (the quantity in the brackets of the numerator) in Equation 2-10 is positive, ensuring two distinct, real, and positive breakeven demand values D₁'  and D₂'.