Economics question part 4; help!?

Question

I'm having problems with this question and was hoping someone could walk me through it. I appreciate this a LOT!!

The demand function for drangles is q(p)= (p + 1) `-2

(Note: The "-2" is meant to be to the power of 2)

A. what is the price elasticity of demand at price p?

B. At what price is the price elasticity of demand for drangles equal to -1?

C. Write an expression for total revenue from the sale of drangles as a function of their price. Use calculas to find the revenue maximizing price. Don't forget to check the second order condition.

D. Suppose that the demand function for drangles takes the more general form q(p)= (p + a) `-b where a > 0 and b > 1. Calculate an expression for the price elasticity of demand at price p.

(Note: the "-b" is an increase to the power of b)

E. From (D); At what price is the price elasticity of demand equal to -1?

Answer 1

A:

E=P*δQ/Q*δP

δQ/δP=((p + 1)^-2)' = -2*(1+P)^(-3)

E=(-2P*(1+P)^(-3))/Q

B:

(-2P*(1+P)^(-3))/Q=-1

-2P*(1+P)^(-3)=-Q

2P*(1+P)^(-3)=Q

2P*(1+P)^(-3)=(P+1)^(-2)

2P/(1+P)=1

2P=1+P

1P=1

P=1

C:

TR=Q*P = P/(P+1)^2

MR=0

MR=(TR)'

(P/(P+1)^2)' = 0

((1+P)^(-2))-2P/(1+P)^3=0

P=1

Second-order:

(((1+P)^(-2))-2P/(1+P)^3) '=

=(1+P)^(-3)*(6P/(1+P)-4)

P=1

6/16-4/8=3/8-1/2=-1/8

Second-order derivative is negative at this point - conclusion TR is at maximum point with price P=1 Q=0.25 TR(Max)=0.25

D:

E=(δQ/δP)*P/Q

δQ/δP=((p + a) ^-b)'= -b/(a+P)^(1+b)

E=-b*P*(a+P)^(-1-b+b)=-bP/(a+P)

-bP/(a+P)= -1

bP=a+P

bP-P=a

P(b-1)=a

P=a/(b-1)