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Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

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FUNCTIONS IN ECONOMICS AND COMMERCE 

Demand Function
Let q be the demand (quantity) of a commodity and p the price of that commodity. The demand function is defined as q = f(p) where p and q are positive. Generally, p and q are inversely related.
Observe the graph of the demand function q = f(p)

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Following y can be made from the graph (Fig 3.1) (i) only the first quadrant portion of the graph of the demand function is shown since p and q are positive. (ii) slope of the demand curve is negative.

 Supply Function
Let x denotes amount of a particular commodity that sellers offer in the market at various price p, then the supply function is given by x = f(p) where x and p are positive.
Generally x and p are directly related Observe the graph of the supply function, x = f (p)

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Following observations can be made from the graph (Fig 3.2)
(i) only the first quadrant portion of the graph of the supply function is shown since the function has meaning only for nonnegative values of q and p.
(ii) slope of the supply function is positive.

Cost Function
Normally total cost consists of two parts.
(i) Variable cost and
(ii) fixed cost. Variable cost is a single - valued function of output, but fixed cost is independent of the level of output.

Let f (x ) be the variable cost and k be the fixed cost when the output is x units. The total cost function is defined as C(x) = f(x) + k, where x is positive.

Note that f(x) does not contain constant term.
We define Average Cost (AC), Average Variable Cost (AVC), Average Fixed Cost (AFC), Marginal Cost (MC), and Marginal Average Cost (MAC) as follows.

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Note d (AC) dx If C(x) is the total cost of producing x units of some product then its derivative C'(x) isthemarginal cost which isthe approximate cost of producing 1 more unit when the production level is x units. The graphical representation is shown here (Fig 3.3).

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Revenue Function

Let x units be sold at Rs. p per unit. Then the total revenue R(x) is defined as R(x) = px , where p and x are positive.

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Note If R(x) be the total revenue gained from selling x units of some product, then its derivative, R'(x) is the marginal revenue, which is approximate revenue gained from selling 1 more unit when the sales level is x units. The graphical representaion is shown here (Fig 3.4)

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Profit Function
The profit function P(x) is defined as the difference between the total revenue and the total cost. i.e. P(x) = R(x)  C (x).

Elasticity
The elasticity of a function y = f(x), with respect to x, is defined as

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Thus the elasticity of y with respect to x is the limit of the ratio of the relative increment in y to the relative increment in x, as the increment in x tends to zero. The elasticity is a pure number, independent of the units in x and y.

Elasticity of Demand
Let q = f(p) be the demand function, where q is the demand and p is the price. Then the elasticity of demand isApplications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Since the slope of the demand curve is negative and elasticity is a positive quantity the elasticity of demand is given by
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Elasticity of Supply
Let x = f(p) be the supply function, where x is the supply and p is the price. The elasticity of supply is defined as
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Equilibrium Price
The price at which quantity demanded is equal to quantity supplied is called equilibrium price.

Equilibrium Quantity
The quantity obtained by substituting the value of equilibrium price in any one of the given demand or supply functions is called equilibrium quantity.
Relation between Marginal Revenue and Elasticity of Demand Let q units be demanded at unit price p so that p = f(q) where f is differentiable.
The revenue is given by
R(q) = qp
R(q) = q f(q)                           [ p = f(q)]

Marginal revenue is obtained by differentiating R(q) with respect to q

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 1
A firm produces x tonnes of output at a total cost C(x) = 1/10 x3 - 4x 2 + 20x + 5 10
Find
(i) Average cost
(ii) Average Variable Cost
(iii) Average Fixed Cost
(iv) Marginal Cost and
(v) Marginal Average Cost.


Solution : 
C(x) = 1/10 x3 - 4x 2 + 20x + 5

; Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 2
The total cost C of making x units of product is C = 0.00005x3 - 0.06x2 + 10x + 20,000. Find the marginal cost at 1000 units of output.


Solution :
C = 0.00005x3 - 0.06x2 + 10x + 20,000

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

At x = 1000 units, Marginal Cost is Rs. 40

Example 3 Find the elasticity of demand for the function x = 100 - p - p 2 when p = 5.
Solution :

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 4 
Find the elasticity of supply for the supply function x = 2p2+8p+10

Solution : x = 2p2+8p+10

dx/dy = 4p+8
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 5  For the function y = 4x-8 find the elasticity and also obtain the value when x = 6.

Solution : y = 4x- 8

dx/dy = 4

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

 

Example 6 If y = Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com find Ey/Ex . Obtain the values of ψ when x = 0 and x = 2.

Solution :

We have y = Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Differntiating with respect to x, we get

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

when x = 0, η = 0
when x = 2, η = 7/12

Example 7
A demand function is given by xpn = k, where n and k are constants. Calculate price elasticity of demand.

Solution : Given x pn = k
⇒x = k p-n

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 8
Show that the elasticity of demand at all points on the curve xy2 = c (c is constant), where y represents price will be numerically equal to 2.

Solution : We have

We have xy2 = c 
x = c/y2

Differentiating with respect to y,

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 9
The demand curve for a monopolist is given by x = 100-4p (i) Find the total revenue, average revenue and marginal revenue. (ii) At what value of x,the marginal revenue is equalto zero?

Solution : We have

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 10
If AR and MR denote the average and marginal revenue at any output level, show that elasticity of demand is equal to Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com . Verify this for the linear demand law p = a + bx. AR – MR where p is price and x is the quantity.

Solution :

  Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Now,

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Given p = a + bx

Differentiating with respect to x

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Example 11  Find the equilibrium price and equilibrium quantity for the following demand and supply functions, Qd = 4-0.06p and Qs = 0.6+0.11p

Solution :
At the equilibrium price
Qd = Qs

⇒ 4-0.06p = 0.6 + 0.11p

⇒ 0.17p = 3.4

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

when p = 20, Qd = 4 -(0.06)(20)
             = 4-1.2 = 2.8

∴ Equilibrium price = 20 and Equilibrium quantity = 2.8

Example 12 The demand for a given commodity is given byq = p/p - 5 (p>5), where p is the unit price. Find the elasticity of demand when p =7. Interpret the result.

Solution :

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
This means that if the price increases by 1% when p = 7, the quantity demanded will decrease by approximately 2.5%. Also if the price decreases by 1% when p = 7, the quantity demanded will increase by approximately2.5%.

Example 13 The demand for a given commodity is q = - 60p + 480, (0 < p < 7) where p is the price. Find the elasticity of demand and marginal revenue when p = 6. Solution : Demand function q = -60p + 480 Differentiating with respect to p, we get

Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com
Applications - Differentiation, Business Mathematics & Statistics Notes | Study Business Mathematics and Statistics - B Com

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