Let the inverse demand function be
P = a - xQ
Where, P is price
Q = quantity
a = constant
X = slope of demand curve
When P =£ 40, Q = 10
40 = a - 10x ....eq 1
When P = £ 15, Q = 60
15 = a - 60x ....eq 2
Subtracting equation 2 from 1 we get
40 - 15 = (a-10x)-(a-60x)
25 = 50x
X = 0.5
Plug in x = 0.5 in equation 1 we get
40 = a - 10*0.5
40 = a - 5
a = 45
The required inverse demand function is
P = 45 - 0.5Q (solution of part A)
B. Elasticity of demand at P =£ 40
P = 45 - 0.5Q
0.5Q = 45 - P
Q = 90 - 2P
Differentiate Q wrt P we get
P = £ 40, Q = 10
Elasticity,
Elasticity (@£40) = -8
C.
i. Relationship between price and MR
TR = PQ
Differentiate TR with respect to q we get marginal revenue
The above equation is termed as Lerner index and it represents the relationship between MR, P and elasticity of demand.
ii. Same equation is used to determine the monopoly power.
As we know under Monopoly the monopolist maximizes profit by equating MR with MC. Therefore the monopolist power can be determined using the following formula
iii.
Determine the total revenue, TR = PQ
Differentiate total revenue with respect to q we get marginal revenue
MR = 45 - Q
Now equate it to MC =£15
MR = MC
45 - Q = 15
Q* = 30
P* = 45 -0.5*30 = 45 -15 = £ 30 per unit
Profit maximizing price =£ 30 per unit
Profit maximizing quantity = 30 units
Monopoly power
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