Overview of Application of Calculus in Commerce and Economics

If x denotes the quantity produced of a commodity at total cost C, then the cost function is expressed as C = C(x)​

Thus, the cost function represents the functional relationship between the cost of production and the level of output.

The demand function expresses the functional relationship between the quantity demanded of a commodity and its price, all other factors being constant.

If p denotes the price per unit and xxx the quantity demanded, then x = f(p)​

Revenue is the amount of money received from the sale of goods.

If x units of a commodity are sold at price p per unit, then the total revenue is R = px​

Thus, the revenue function is R = R(x)

Profit is defined as the excess of total revenue over total cost.

If R(x) is the revenue function and C(x) the cost function, then the profit function is

P(x) = R(x) − C(x)

The break-even point is that level of output at which total revenue equals total cost.
At this point, there is neither profit nor loss.

Mathematically,

\[P(x)=0\mathrm{~or~}R(x)=C(x)\]

Average Fixed Cost (AFC) is the fixed cost per unit of production.
It is obtained by dividing the total fixed cost by the corresponding level of output

\[\mathrm{AFC}=\frac{\mathrm{TFC}}{Q}\]

where

• TFC = Total Fixed Cost

• Q = Level of output

Average Variable Cost (AVC) is the variable cost per unit of production.
It is obtained by dividing the total variable cost by the corresponding level of output.

\[\mathrm{AVC}=\frac{\mathrm{TVC}}{Q}\]

where

• TVC = Total Variable Cost

• Q = Level of output

If C = C(x) is the total cost of producing and marketing x units of a commodity, then the average cost (AC) or average total cost (ATC) is the total cost per unit of output.

\[\mathrm{AC}=\frac{\mathrm{Total~Cost}}{\text{Quantity of output}}=\frac{TC}{x}\]

Since the total cost is the sum of total fixed cost and total variable cost, the average cost is given by

\[\mathrm{AC}=\frac{TFC}{Q}+\frac{TVC}{Q}=AFC+AVC\]

The marginal cost, denoted by MC, is defined as the rate of change of the total cost with respect to output

\[\mathrm{MC}=\frac{dC}{dx}\]

\[\frac{d}{dx}(AC)=\frac{1}{x}(MC-AC)\]

• MC < AC → AC falls

• MC = AC → AC is minimum

• MC > AC → AC rises

Average Revenue:

Average revenue (AR) is the revenue received per unit.

\[AR=\frac{TR}{x}\]

Since TR = px, AR = p

Marginal Revenue:

Marginal revenue (MR) is defined as the rate of change of total revenue with respect to quantity sold.

\[MR=\frac{dR}{dx}\]

Since R = px,

\[MR=p+x\frac{dp}{dx}\]

Total Revenue: 

R = px = R(x)

Condition for Maximum Total Revenue:

\[\frac{dR}{dx}=0\quad\mathrm{and}\quad\frac{d^2R}{dx^2}<0\]

Profit Function:

P(x) = R(x) − C(x)

 Condition for Maximum Profit:

\[\frac{dP}{dx}=0\Rightarrow MR-MC=0\Rightarrow MR=MC\]

Average Cost (AC) Formula:

\[AC=\frac{C}{x}\]

Condition for Minimum Average Cost:

\[\frac{d(AC)}{dx}=0\quad\mathrm{and}\quad\frac{d^2(AC)}{dx^2}>0\]

Core Formula:

\[MC=\frac{dC}{dx}\]

Inverse Relation:

\[C=\int MCdx+k\]

Average Cost Formula:

\[AC=\frac{C}{x}\]

Alternate Cost Formula:

\[\text{Total cost of producing }a\mathrm{~units}=\int_0^aMC\mathrm{~}dx\]

Marginal revenue is defined as:

\[MR=\frac{dR}{dx}\]

Finding Total Revenue from MR:

\[R=\int MRdx+k\]

Determination of Demand Function:

\[R=px\quad\Rightarrow\quad p=\frac{R}{x}\]

\[\text{Total revenue for }a\text{ units sold }=\int_0^aMRdx\]