Question

Suppose the supply function in a market is and the current market price is p* = 4. What is the producer surplus?
b) Demand function is given by
i) Find the interval of prices for which demand is positive.
ii) Express total revenue TR = PQ as a function of price. When is total revenue maximized?
iii) For which price is the own-price elasticity equal to -1?

Answer 1

Solution:

a.). Producer surplus = $\frac{1}{2} \times (B \times H)$

Q_s = P² = 4² = 16

Q_d = 10 – P^0.5 = 10 – 4^0.5 = 10 – 2 = 8

Producer surplus = $\frac{1}{2} \times (8 \times 16)$ = 64

 

b. i.). The interval of prices for which demand is positive are 2 and 4

 

ii.). TR = P $\times$ Q

Price = 4

To get quantity, derive the inverse demand function:

Q = 10 – P^0.5

P = Q² – 20Q + 100

TR = (Q² – 20Q + 100) $\times$ Q = Q³ – 20Q² + 100Q

TR function = Q³ – 20Q² + 100Q

 

Total revenue is maximized when the marginal revenue is equal to zero.

 

iii).  PED = $\frac{\triangle Q}{\triangle P} \times \frac{\ P}{\ Q}$

-1 = $1\times \frac{P}{8}$

P = -8

A price reduction of 8 is equal to the own-price elasticity of demand of -1