Suppose a company plans to produce a new product that incurs some costs (fixed and variable) and let the company plan to sell the product at a fixed price. Prepare a mathematical model to examine the profitability.

![Cost and revenue lines meeting at the break-even point](/figur

Question

Suppose a company plans to produce a new product that incurs some costs (fixed and variable) and let the company plan to sell the product at a fixed price. Prepare a mathematical model to examine the profitability.

![Cost and revenue lines meeting at the break-even point](/figures/application-of-derivatives/cost-revenue-break-even-point-fig-a2-6.svg)

Accepted Answer

> [!TLDR]
> With $x$ units produced and sold, fixed cost $a$, variable cost $b$ per unit and selling price $s$ per unit, the cost is $C=a+bx$, the income is $I=sx$ and the profit is $P=(s-b)x-a$. The firm breaks even at $x=\dfrac{a}{s-b}$; above it there is profit, and $\dfrac{dP}{dx}=s-b$ is the profit earned per extra unit.

**Step 1 — Identify.** Build a relationship that tells the company, before production, whether and when the new product turns a profit.

**Step 2 — Formulate the assumptions.** Costs are of two kinds: **fixed** costs (rent, rates) that do not change with output, and **variable** costs that grow with output. Assume the variable cost is directly proportional to the number of units produced, and that every unit produced is sold immediately at a fixed price.

**Step 3 — The mathematical model.** Let
$$x = 	ext{units produced and sold},\quad C = 	ext{total cost},\quad I = 	ext{income from sales},\quad P = 	ext{profit}.$$
With fixed cost $a$ (rupees) and variable cost $b$ (rupees per unit),
$$C = a + bx. \quad(1)$$
With selling price $s$ (rupees per unit),
$$I = sx. \quad(2)$$
Profit is income minus cost:
$$P = I - C = sx - (a + bx) = (s - b)x - a. \quad(3)$$
Here $x$ is the independent variable, $C, I, P$ are dependent variables, and $a, b, s$ are parameters the manufacturer knows.

**Step 4 — Solve for the break-even point.** The break-even point is where the company makes neither profit nor loss, i.e. $P = 0$. From $(3)$,
$$(s-b)x - a = 0 \ \Rightarrow\ x = \frac{a}{s - b}\ 	ext{units} \qquad (s > b).$$ …

Q.Suppose a company plans to produce a new product that incurs some costs (fixed and variable) and let the company plan to sell the product at a fixed price. Prepare a mathematical model to examine the profitability.

Concept understanding — Mathematical Modelling

Method: The Five-Step Mathematical Modelling Process

Steps

Common Mistakes

Q.Suppose a company plans to produce a new product that incurs some costs (fixed and variable) and let the company plan to sell the product at a fixed price. Prepare a mathematical model to examine the profitability.

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