Definitions [10]
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,
\[MR=p+x\frac{dp}{dx}\]
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.
The marginal cost, denoted by MC, is defined as the rate of change of the total cost with respect to output
\[\mathrm{MC}=\frac{dC}{dx}\]
Formulae [2]
\[\frac{d}{dx}(AC)=\frac{1}{x}(MC-AC)\]
\[\mathrm{AC}=\frac{\mathrm{Total~Cost}}{\text{Quantity of output}}=\frac{TC}{x}\]
Since the total cost is the sum of total fixed cost and total variable cost, the average cost is given by
\[\mathrm{AC}=\frac{TFC}{Q}+\frac{TVC}{Q}=AFC+AVC\]
Key Points
-
MC < AC → AC falls
-
MC = AC → AC is minimum
-
MC > AC → AC rises
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:
\[\frac{dP}{dx}=0\Rightarrow MR-MC=0\Rightarrow MR=MC\]
| Feature | Perfect Competition | Monopoly |
|---|---|---|
| Sellers | Many | One |
| Price | Constant | Depends on output |
| AR curve | Horizontal | Downward sloping |
| MR curve | Coincides with AR | Lies below AR |
| TR curve | Straight line | Curved (∩ shape) |
| MR = 0 | Not applicable | TR is maximum |
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:
\[R=px\quad\Rightarrow\quad p=\frac{R}{x}\]
\[\text{Total revenue for }a\text{ units sold }=\int_0^aMRdx\]
Average Cost (AC) Formula:
\[AC=\frac{C}{x}\]
Condition for Minimum Average Cost:
\[\frac{d(AC)}{dx}=0\quad\mathrm{and}\quad\frac{d^2(AC)}{dx^2}>0\]
