Question

A local supplier of widgets serves two types of clients: restaurants and gas stations. The demand for widgets by a restaurant is Qr = 24-4P, whereas the gas station’s demand is Qg = 15 –3P. The number of restaurants and gas stations is the same. The marginal cost of a widget is $3.

(a) What price will the supplier set for a widget if she cannot discriminate between the two groups? What will be her profit in this case?

(b) Suppose now that the supplier can discriminate and decides to issue membership cards to gas stations and restaurants. To purchase widgets the clients must become members: a gas station must pay the supplier a membership fee Fg , while a restaurant must pay the fee Fr . Once a member, a buyer can order supplies at a price per widget – gas stations at a (unit) price Pg and restaurants at a (unit) price Pr . What are the membership fees charged? And what are the unit prices charged to each group of clients?

Answer 1

a) When the supplier can't discriminate between two groups----
Equilibrium ( profit maximising) price for widget between two groups=$6.91.,profit=$27.37

There will be a single equilibrium price and combined quantity which will be determined where -------------

combined MR=MC

Calculating combined MR------

# Demand function of widgets at restaurants---- Qr=24-4p

or P(r)=6-1/4Q

TR= PQ= 6Q-1/4Q2

MRr= 6-1/2Q

# Demand function for widgets at gas stations---- Qg=15-3p

or P(Q)= 5-1/3Q

TR=5Q-1/3Q2

MRg= 5-2/3Q

# Combined MR= MRr+MRg

6-1/2Q+5-2/3Q=11-7/6Q

At equilibrium------

MR=MC

as MC= 3

so, 11-7/6Q=3

Q=6.85 units or 7 units

putting this value in combined demand function----

6-1/4Q+5-1/3Q= 11-7/12Q

= 11-7/12(7)= 83/12=$6.91

# calculation of profit= Q(P-AC)

​​​​​​​7(6.91-3) ( assuming MC=AC)

=$27.37

b) Price under discrimination policy------

Pr=$4.5

Pg=$4

​​​​​​​Calculations---------

For calculating discriminatory prices,we will apply MR-MC rule for both clients---- -----: when MRr=3,& when MRg=3

# when MRr=3

6-1/2Q=3

Q=6

putting this value in demand equation-----

p= 6-1/4(6)=$4.5

# when MRg=3

5-2/3Q=3

Q=3

putting this value in demand equation----

p=5-1/3(3)=$4