Question

A firm can sell 10 gizmos when the market price is £40 and can sell 60 gizmos when the market price is £15. The firm faces constant marginal cost of £15, with no other costs of production a) Assuming demand is linear, find the market demand function for gizmos and the associated inverse demand function. b) Determine the price elasticity of demand when the market price is £40. c) Suppose the firm is a monopolist in this market. i.) Explain the connection between marginal revenue and price elasticity of L) Explain the connection between price elasticity of demand and the firm's monopoly power Calculate the firm's profit maximising output. Quantify the firm's monopoly power. iii.) d) Now suppose there are two identical firms competing in this market, characterised by the above demand and costs. Solve for the Cournot equilibrium outputs, market price and profit levels.

Answer 1

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

$\large Elasticity, e = \frac{dQ}{dP}*\frac{P}{Q}$

P = 45 - 0.5Q

0.5Q = 45 - P

Q = 90 - 2P

Differentiate Q wrt P we get

$\large \frac{dQ}{dP} = \frac{d(90-2P)}{dP}$

$\large \frac{dQ}{dP} = -2$

P = £ 40, Q = 10

Elasticity,

$\large e = \frac{dQ}{dP}*\frac{P}{Q}= -2*\frac{40}{10}$

$\large e = -8$

Elasticity (@£40) = -8

C.

i. Relationship between price and MR

TR = PQ

Differentiate TR with respect to q we get marginal revenue

$\large \frac{dTR }{dQ}=\frac{d(PQ)}{dQ}$

$\large MR =P\frac{dQ}{dQ} + Q\frac{dP}{dQ}$

$\large MR =P+ Q\frac{dP}{dQ}$

$\large MR =P[1+ \frac{Q}{P}*\frac{dP}{dQ}]$

$\large MR =P[1+ \frac{1}{e}]$

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.

$\large MR =P[1+ \frac{1}{e}]$

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

$\large \frac{1}{e} = \frac{P - MC}{P}$

iii. $\large P = 45 - 0.5Q$

Determine the total revenue, TR = PQ

$\large TR = PQ= (45 - 0.5Q)Q = 45Q - 0.5Q^2$

Differentiate total revenue with respect to q we get marginal revenue

$\large \frac{dTR }{dQ}=\frac{d( 45Q - 0.5Q^2)}{dQ}$

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

$\large MR =P[1+ \frac{1}{e}]$

$\large \frac{1}{e} = \frac{P - MC}{P}$

$\large L = \frac{1}{e} = \frac{30 - 15}{30}$

$\large L = 0.5$

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