Elasticity of demand for entire market?

Question

Market of books with 4 buyers: A, B, C, D

individual demand curve for each:

P= 50 - 5qA

P= 50 - 5qB

P= 100 - 20qC

P= 75 - 5qD

what is market demand? i got that q total: was qA +qB +qC+qD = 40 - 13/20P

(by rearranging the above eqns to solve for q and adding together)

***but how do i find the elasticity of demand for the entire market?

dQ/dP = -13/20

i know that Elasticity = P/Q *dQ/dP but i don't have a P or Q value given here, so i can't find what the elasticity is?

what do i do?

Answer 1

Market demand =P(A)+P(B)+P(C)+P(D)=

=(50-5q)+(50-5q)+(100-20q) +(75-5q)=

=275-35q

Elasticity=PδQ/QδP

δP/δQ=(275-35q)'=-35

δQ/δP= -1/35

Elasticity= -P/35Q = 35Q-275/35Q =1-(55/7Q)

Now you need some extra data to find elasticity - Price or quantity. If you will have quantity - then just put in equation, but if you have price then calculate quantity from market demand equation: P=275-35Q or 35Q=275-P or Q=(275-P)/35

P.S. It doesn't' matter from what to find derivative

Q=(275-P)/35

δQ/δP=(275/35 - P/35)'= -1/35

P=275-35Q

δP/δQ= (275-35Q)' = -35

Because of math rule in calculus (X)'=1/(Y)'

Answer 2

The first thing to do is to add the demand of all the 4 buyers keeping Price at the same level each time.

How do we do this. We assume P to be fixed and add the q's.

qA= 10 - P/5  ( got from P= 50 - 5qA)., Similarly, we get,

qB= 10 -P/5,

qC=  5 - P/20,

qD=  15/ -P/5.

Now by adding, Q= qA+qB+qC+qD= 30 - 13P/20

Now we get from this dQ/dP = - 13/20.

And, Q/P= 30/P - 13/20

0r, P/Q =1/ [30/P - 13/20]

So, Elasticity = P/Q *dQ/dP ={1/ [30/P - 13/20]}* {-13/20} ..(Z)

This is the formula for elasticity and the value of elasticty varies depending on the value of P. For each level of P, you will get one particular value of P from thus equation.

Your answer will be correct if you deduce upto (Z) above. In addition you can assume teo alternative values of P and calculate the elasticity. There is no need to know Q here.

For example, if P= 30, elasticity equals

{1/ [30/P - 13/20]}* {-13/20} = {1/[30/30 - 13/20]}*{-13/20}

= {1/ [7/20]}*{-13/20}=  (20/7)*(-13/20)= -13/7

I am glad that you did not add up price equations to get Q.