Question

5. There are 100 patients who could benefit from a new drug, Tipilor, manufactured by Zifer. Patient i has willingness to pay i, i=1,…,100. Ignore fixed costs and assume marginal cost, c, is constant.

a) What is the profit maximising price and resulting profit?

b) Now assume all patients have insurance which means they only pay 10% of the price as a co-payment. What is the profit maximising price and resulting profit?

c) Zifer sets up a charity to cover the co-payment for poor people. Patients 1, 2, …, 50 are considered poor. What is the profit maximising price and resulting profit?

d) For c = 10, calculate optimal profit in a), b) and c) and conclude whether setting up the charity is profitable.

Answer 1

Solution :-

(a) :-

There are 100 patients who could benefit from a new drug, Tipilor, manufactured by Zifer.

Patient i has willingness to pay i, i=1,…,100. Ignore fixed costs and assume marginal cost, c, is constant.

Value of each player = i

If a firm charges vi, ( 100 - vi + 1) will buy

Revenue = vi * ( 100 - vi + 1)

Marginal revenue = d(revenue)/dvi

Marginal cost ( MC) = c

Now, equating

MR = MC

100 - 2vi + 1 = c

101 - 2vi = c

101 - c = 2vi

(101 - c )/2 = vi

So, this will be the profit maximizing price , therefore whose are equal to or more than Vi will buy.

So,

100 - vi + 1

100 - (101 - c )/2 + 1 will buy

100 - 50.5 + c/2 + 1 = 50.5 + c/2 will buy

If 50.5 + c/2 is not an integer , then the next possible integer is [ 50.5 + c/2]

Where [ ] - refers to least integer function

Therefore

Profit = vi * ( 50.5 + c/2) - c * ( 50.5 + c/2)

Where vi = (101 - c)/2

Profit = vi * ( 50.5 + c/2) - c * ( 50.5 + c/2)

= ( vi - c) [ 50.5 - c/2]

= [( 101 - c)/2 - c] [ 50.5 + c/2]

= (101 - c - 2c)/2 [ 50.5 + c/2]

= ( 101 - 3c)/2 [ 50.5 + c/2]

(b) :-

Now assume all patients have insurance which means they only pay 10% of the price as a co-payment

If firm charges vi then ( 100 - vi + 1) will buy in addition to that ( vi - 0.1vi)

Let's say firm2 changes to 20

Then,

( 100 - 20 + 1) will buy in which the one with the valuation 20 is also included.

Now since patients have to buy 10% only so all those with valuation 2, 3 ....... ,19 will also buy since now they only have to buy 10% of their valuation

Therefore ( vi - 0.1vi) need to be added

So,

Revenue = vi ( 100 - vi + 1 + vi - 0.1vi )

= vi ( 101 - 0.1vi)

= 101vi - 0.1vi^2

Marginal revenue = d( revenue)/dvi

= 101 - 0.2vi

Now ,

MR = MC

101 - 0.2vi = c

101 - c = 0.2vi

( 101 - c )/0.2 = vi......... optimal price

[ Note 0 vi 100 ]

( 101 - 0.1vi) no.of will buy [ ] refers to least integer function,

Now,

Profit = vi * ( 101 - 0.1vi ) - c * ( 101 - 0.1vi)

= ( vi - c ) [ 101 - 0.1]

= [( 101 - c)/0.2 - c] [ 101 - 0.1Vi]

= ( 101 - c - 0.2c)/0.2 [ 101 - 0.1Vi]

= ( 101 - 1.2c)/0.2 [ 101 - 0.1 ( 101 - c)/0.2 ]

= ( 101 - 1.2c)/0.2 [ 101 - ( 101 - c) /2]

= ( 101 - 1.2c)/0.2 [ 202 - 101 + c]/2

= ( 101 - 1.2c)/0.2 [ 101 + c]/2

(c) :-

Zifer sets up a charity to cover the co-payment for poor people. Patients 1, 2, …, 50 are considered poor.

Patient 51 + 100 are lot poor

Revenue = vi * [ 50 + ( 50 - 0.1vi +1]

Whatever be the price changed by firm all those who are not poor( 51 to 100) will buy since they only have to pay 10% of the price , hence 50 is added.

So, ( 50 - 0.1vi +1) because charity will take care of those people's payment

Marginal Revenue = 100 - 0.2vi + 1

= 101 - 0.2vi

Now,

MR = MC

101 - 0.2vi = c

101 - c = 0.2vi

(101 - c)/2 = vi...... optimal price

[ Note 0 vi 100 ]

( no of people who will buy)

= [ 101 - 0.1vi ]

= [ 101 - 0.1 ( 101 - c)/0.2]

= [ 101 - ( 101 - c)/2 ]

= [ 202 - 101 + c]/2

= [101 + c ]/2

Take [ 101 + c ] /2 if (101 + c)/2 is not an integer

Profit in this case will be same as that in part(b) price and no of people who brought the drugs are the same, therefore

Profit =  ( 101 - 1.2c)/0.2 [ 101 + c]/2

(d) :-

For c = 10

Profit in a =  ( 101 - 3c)/2 [ 50.5 + c/2]

= ( 101 - 3 * 10)/2 [ 50.5 + 10/2]

= ( 101 - 30)/2 [ 50.5 + 5 ]

= 35.5 * 56

= 1988

Profit in c = profit in b = ( 101 - 1.2c)/0.2 [ 101 + c]/2

= ( 101 - 1.2 * 10)/0.2[ 101 + 10]/2

= ( 101 - 12)/0.2 [ 111/2]

= 89/0.2 * 56

= 445 * 56

= 24920

Profit b = 24920

Profit c = 24920

Since profit in case b and case c are same therefore we can conclude that setting up the charity is at least profitable when campare with case a.