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

**Elasticity**

Elasticity â€˜Î·â€™ 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.*

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 *h** _{d}* =

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 *h*_{s}

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

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

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

(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.

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 + 4*x* + 3âˆš*x* . Find
the marginal cost of the product at 9 units of output.

*Solution:*

*MC* is â‚¹ 4.50 , 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 âˆ’ *p*) and
Supply : *x* = 2*p*â€“3

*Solution:*

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.

Find the elasticity of supply for
the supply function *x* = 2*p*^{2} - 5 *p* + 1

*Solution:*

**Example 6.7**

A demand function is given by *xp ^{n}* =

*Solution:*

**Example 6.8**

For the given demand function *p* = 40â€“*x*, find the value of the output when Î·* _{d}* =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:*

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

*Solution:*

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

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

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

**Example 6.12**

**Example 6.13**

is the manufacturerâ€™s average cost function. What is the marginal cost when 50 units are produced and interpret your result.

*Solution:*

If the production level is
increased by one unit from *x* = 50,
then the cost of additional unit is approximately equal to â‚¹ 391.

For the function *y* *x*^{3}+19,
find the values of *x* when its
marginal value is equal to 27.

*Solution:*

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.

Rate of change of demand with
respect to the price at *p*
= â‚¹ 4 is âˆ’ 0.25

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

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

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

Revenue: *R *=* px*

The cost function of a firm is *C* = 1/3 *x*^{3} âˆ’ 3*x*^{2} + 9*x* . Find the level of output (*x*>0)
when average cost is minimum.

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

Tags : Applications of differentiation in business and economics , 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation

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

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