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Applications of differentiation in business and economics - Elasticity | 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation

Chapter: 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation

Elasticity

Price elasticity of demand is the degree of responsiveness of quantity demanded to a change in price.

Elasticity

Elasticity ‘η’ of the function y = f (x ) at a point x is defined as the limiting case of ratio of the relative change in y to the relative change in x.


(i) Price elasticity of demand

Price elasticity of demand is the degree of responsiveness of quantity demanded to a change in price.

If x is demand and p is unit price of the demand function x = f(p), then the elasticity of demand with respect to the price is defined as hd


(ii)  Price elasticity of supply :

Price elasticity of supply is the degree of responsiveness of quantity supplied to a change in price.

If x is supply and p is unit price of the supply function x = g(p), then the elasticity of supply with respect to the price is defined as hs 


Some important results on price elasticity

(i) If | η |>1, then the quantity demand or supply is said to be elastic.

(ii) If | η |=1, then the quantity demand or supply is said to be unit elastic.

(iii) If | η |<1, then the quantity demand or supply is said to be inelastic.

Remarks:

(i) Elastic : A quantity demand or supply is elastic when its quantity responds greatly to changes in its price.

Example: Consumption of onion and its price.

(ii) Inelastic : A quantity demand or supply is inelastic when its quantity responds very little to changes in its price.

Example: Consumption of rice and its price.

(iii) Unit elastic : A quantity demand or supply is unit elastic when its quantity responds as the same ratio as changes in its price.

Relationship among Marginal revenue [MR], Average revenue [AR] and Elasticity of demand[ηd].

We know that R(x)=px


 

Example 6.1

The total cost function for the production of x units of an item is given by


(i) Average cost

(ii) Average variable cost

(iii) Average fixed cost

(iv) Marginal cost and

(v) Marginal Average cost

Solution:


 

Example 6.2

The total cost C in Rupees of making x units of product is C (x ) = 50 + 4x + 3x . Find the marginal cost of the product at 9 units of output.

Solution:


MC is4.50 , when the level of output is 9 units.

 

Example 6.3

Find the equilibrium price and equilibrium quantity for the following demand and supply functions.

Demand : x = 1/2 (5 p) and Supply : x = 2p–3

Solution:


 

Example 6.4

For the demand function x=20/P+1, find the elasticity of demand with respect to price at a point p = 3. Examine whether the demand is elastic at p = 3.

Solution:


 

Example 6.5

Find the elasticity of supply for the supply function x = 2p2 - 5 p + 1

Solution:


 

Example 6.6


 

Example 6.7

A demand function is given by xpn = k where n and k are constants. Prove that elasticity of demand is always constant.

Solution:


 

Example 6.8

For the given demand function p = 40–x, find the value of the output when ηd =1

Solution:


 

Example 6.9

Find the elasticity of demand in terms of x for the demand law p = (a-bx)1/2 . Also find the values of x when elasticity of demand is unity.

Solution:


 

Example 6.10

Verify the relationship of elasticity of demand, average revenue and marginal revenue for the demand law p = 50 - 3x .

Solution:


 

Example 6.11

Find the elasticity of supply for the supply law x = p / p+5  when p = 20 and interpret your result.



Interpretation:

·        z If the price increases by 1% from p = 20, then the quantity of supply increases by 0.2% approximately.

·        z If the price decreases by 1% from p = 20, then the quantity of supply decreases by 0.2% approximately.

 

Example 6.12


 

Example 6.13

 is the manufacturer’s average cost function. What is the marginal cost when 50 units are produced and interpret your result.

Solution:


 

Interpretation:

If the production level is increased by one unit from x = 50, then the cost of additional unit is approximately equal to 391.

 

Example 6.14

For the function y x3+19, find the values of x when its marginal value is equal to 27.

Solution:


 

Example 6.15

The demand function for a commodity is p = 4/x,where p is unit price. Find the instantaneous rate of change of demand with respect to price at p=4. Also interpret your result.

Solution:


Rate of change of demand with respect to the price at p =4 is0.25

 

Interpretation:

When the price increases by 1% from the level of p = 4, the demand decreases (falls) by 0.25%

 

Example 6.16

The demand and the cost function of a firm are p = 497-0.2x and C = 25x+10000 respectively. Find the output level and price at which the profit is maximum.

Solution:

We know that profit is maximum when marginal revenue [MR] = marginal cost [MC].

Revenue:     R = px


 

Example 6.17

The cost function of a firm is C = 1/3 x3 3x2 + 9x . Find the level of output (x>0) when average cost is minimum.

Solution:

We know that average cost [AC] is minimum when average cost [AC] = marginal cost [MC].





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11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation


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