Home | | Business Maths 11th std | Demand, supply, cost, revenue and profit functions

# Demand, supply, cost, revenue and profit functions

In a market, the quantity of a commodity demanded by the consumer depends on its price.

Demand, supply, cost, revenue and profit functions

## Demand function

In a market, the quantity of a commodity demanded by the consumer depends on its price. If the price of the commodity increases, then the demand decreases and if the price of the commodity decreases, then the demand increases.

The relationship between the quantity and the unit price of a commodity demanded by consumer is called as demand function and is defined as x = f ( p) or p = f (x) , where x>0 and p>0 .

Graph of the demand function, x = f(p) Observations

(i) Price and quantity of the demand function are in inverse variation.

(ii) The graph of the demand function lies only in first quadrant.

(iii) Angle made by any tangent to the demand curve with respect to the positive direction of x ŌĆō axis is always an obtuse angle.

(iv) Slope of the demand curve is negative( ŌĆōve).

## Supply function

In a market, the quantity of a commodity supplied by producer depends on its price. If the price of the commodity increases, then quantity of supply increases and if the price of the commodity decreases, then quantity of supply decreases.

The relationship between the quantity and the unit price of a commodity supplied by producer is called as supply function and is defined as x = g(p) or p=g(x) where x > 0 and p > 0

## The graph of the supply function, x= g(p) Observations

(i) Price and quantity of the supply function are in direct variation.

(ii) The graph of supply function lies only in first quadrant.

(iii) Angle made by any tangent to the supply curve with respect to positive direction of x ŌĆō axis is always an acute angle.

(iv) Slope of the supply curve is positive (+ve).

## Equilibrium Price

The price at which the demand for a commodity is equal to its supply is called as Equilibrium Price and is denoted by pE.

Equilibrium Quantity

The quantity at which the demand for a commodity is equal to its supply is called as Equilibrium Quantity and is denoted by xE.

NOTE

Usually the demand and supply functions are expressed as x in terms of p, so the equilibrium quantity is obtained either from the demand function (or) from the supply function by substituting the equilibrium price.

### Equilibrium Point

The point of intersection of the demand and supply function (pE, xE) is called as equilibrium point.

## Diagrammatical explanation of equilibrium price, equilibrium quantity and equilibrium point ## Average and Marginal concepts

Usually, the variation in the dependent quantity ŌĆśyŌĆÖ with respect to the independent quantity ŌĆśxŌĆÖ can be described in terms of two concepts namely

(i) Average concept  and

(ii) Marginal concept.

### (i) Average concept

The average concept expressed as the variation of y over a whole range of x and is denoted by y/x .

### (ii) Marginal concept

The marginal concept expressed as the instantaneous rate of change of y with respect to x and is denoted by dy/dx .

### Remark:

If Ōłåx be the small change in x and Ōłåy be the corresponding change in y of the function y=f(x), then Instantaneous rate of change of y with respect to x is defined as the limiting case of ratio of the change in y to the change in x.  ## Cost function

The amount spent for the production of a commodity is called its cost function.

Normally, total cost function [TC] consists of two parts.

(i) Variable cost

(ii) Fixed cost

### Variable cost

Variable cost is the cost which varies almost in direct proportion to the volume of production.

### Fixed cost

Fixed cost is the cost which does not vary directly with the volume of production.

If f(x) be the variable cost and k be the fixed cost for production of x units, then total cost is C(x) = f(x) + k, x>0.

NOTE

(i) Variable cost f(x) is a single valued function.

(ii) Fixed cost k is independent of the level of output.

(iii) f(x) does not contain constant term.

## Some standard results

If C(x) = f(x) + k be the total cost function, then (vi) Total cost [TC]=Average cost ├Ś output

(vii)  Average cost [AC] is minimum, when MC = AC

### Remark:

The marginal cost [MC] is approximately equal to the additional production cost of (x+1)th unit, when the production level is x units.

## Diagrammatical explanation of marginal cost [MC]

Marginal cost is the change in aggregate cost when the volume of production is increased or decreased by one unit. ## Revenue function

Revenue is the amount realised on a commodity when it is produced and sold. If x is the number of units produced and sold and p is its unit price, then the total revenue function R(x) is defined as R(x) =px, where x and p are positive.

### Some standard results

If R(x) =px be the revenue function, then ### Remarks:

(i) Average revenue [AR] and price [p] are the same. [i.e. AR=p]

(ii) The marginal revenue [MR] is approximately equal to the additional revenue made on selling of (x+1)th unit, whenx the sales level is x units.

## Diagrammatical explanation of Marginal Revenue [MR]

Marginal revenue is the change in aggregate revenue when the volume of selling unit is increased by one unit. ## Profit function

The excess of total revenue over the total cost of production is called the profit. If R(x) is the total revenue and C(x) is the total cost, then profit function P(x) is defined as P(x) = R(x) ŌĆō C(x)

## Some standard results

If P(x) = R(x) ŌĆō C(x) be the profit function, then (iv) Profit [P(x)] is maximum when MR = MC

Tags : Applications of differentiation in business and economics , 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation : Demand, supply, cost, revenue and profit functions | Applications of differentiation in business and economics