Question

Regulating a Monopoly

Say you have a monopoly that provides a transportation service to a city.

Total investment in infrastructure and busses amount to $65,000,000

The company sells tickets to riders for a fixed price- i.e. one monthly price for any length journey

Monthly demand for tickets is

P = 20.25 - .00005Q

Where P = price of a ticket and Q the number of tickets

TC = $4,625,000 + 30Q + .0003Q2 MC =

30 + .0006Q

A)

Without regulation, how much would a monopoly charge per ticket and how many

tickets would it make available?

B)

What would be its ROI?

C)

Assume that you want to regulate the monopoly so that its return on investment does

not exceed 8%. How many units must it produce and how much would it charge for

each unit?

Answer 1

GIVEN THAT

• The company sells tickets to riders for a monthly fixed price for any length journey.
• Monthly demand for tickets is P = 20.25 - .00005Q
• Where P is the price, Q is the quantity multiplied by the demand to price coefficient, i.e. 0.00005

1. Now without regulation, to maximize profit monopoly would produce where the marginal cost of producing each unit is equal to the marginal revenue earned from selling each unit, or MR= MC

Now to derive MR we need to calculate total revenue generated i.e.

= Price*Quantity

And MR is Total revenue of q^th unit less the total revenue from selling q+1^th unit.

Or, MR= dTR/DQ

Or, MR= d(20.25-0.00005Q)Q/dQ

Or MR= 20.25Q-0.00005Q²/dQ

Or, MR= 20.25- 0.0001Q

On the other hand, similarly, Marginal cost can be calculated using total cost,

Where, MC is given as,

MC= 30+0.0006Q is the cost of producing each unit.

Now at equilibrium without regulation,

MR=MC

Implies,

20.25- 0.0001Q = 30+0.0006Q

Or, 20.25-30 = (0.0006+0.0001)Q

Or, -9.75 = 0.0007Q

Or, Q= -13,928.57

Due to the specification the quantity in figures calculates to be dubious.

It can be seen that the equilibrium cannot be in negative due to common sense, hence we infer that the total production for would be 13,928 units for MR=MC.

Using the Q we can calculate per unit price that is,

P= 20.25-0.00005Q

Supplanting Q gives

P= 20.25- 0.6964

Or, P= 19.55

2. Return on investment is calculated from

That is, RoI = (Current Value of Investment - Cost of Investment) / Cost of Investment

Now current value of the investment would be

Total quantity*per unit price

Or, Q*P= 13,928 *19.55

Or, current value = 272292.4

Therefore, RoI= (272292.4-65000000)/ 65000000

Or, RoI= -0.9958

Or, taking modulus, to remove negative,

Return on investment is 0.99% per annum

Or, alternatively,

Using the TC as the Cost of investment,

TC= $4,625,000 + 30(13928) +0 .0003(13928)2 = 5101037

Implies,

RoI = (Current Value of Investment - Cost of Investment) / Cost of Investment

RoI= (272292.4-5101037)/5101037

And the return on investment is= -0.94662,

Again, removing negative, RoI is 0.94%

3. Lets Assume that the total value of the output for the RoI to be 8% be x, that is, P*Q= x

0.8= (x-65000000)/ 65000000

Or, 0.8*65000000= x -65000000

Or, 52,000,000+65000000=x

Or, x = 117000000 is the total revenue that can be generated,

Thus, If it keeps its price fixed, it can sell 5,984,655 (approximate)

Or if it wants to sell 13,928 units, it can charge a price (117000000/13928)= 8400 or $84 per unit.

Because the monopoly cannot decide both price and quantity together.