Question

A dominant or price setting firm and several smaller price takers serve a market where total market demand is Qd = 560 – 2P and the combined supply from all the smaller firms is Qs = - 60 + 2P.

1. State the demand (Q_df=) and inverse demand (P_df=) function for the dominant firm (df).

2. If the dominant firm decides to produce 220 units, determine the price (P) it will set for the market and the (combined) quantity supplied by all the small firms (Qs).

3. Does this market meet the assumptions that the dominant firm has at least a market share of 50%? (you must motivate your answer numerically)

Answer 1

Qs=-60+2P

We can say that quantity supplied by smaller firms is zero for price less than $30.

First we estimate the demand function for dominant firms.

Quantity demanded for dominant firm, Qdf, is

Qdf=Market demand-combined quantity supplied by smaller firms

Qdf=560-2P-(-60+2P)=620-4Pdf (for price, Pdf30)

Qdf=620-4Pdf (Demand curve for dominant curve curve for Pdf30)

Qdf=Qd=560-2Pdf (Demand curve for dominant curve curve for Pdf<30)

Let us find inverse demand function for P30

Qdf=620-4Pdf

4Pdf=620-Qdf

Pdf=155-0.25*Qdf (Inverse Demand curve for dominant curve curve for Pdf30)

Let us find inverse demand function for Pdf<30

Qdf=560-2Pdf

2Pdf=560-Qdf

Pdf=280-0.5*Qdf (Inverse Demand curve for dominant curve curve for Pdf<30)

Now Set Qdf=220

Let us assume Pdf<30

Pdf=280-220*0.5=170

It is not true. So, ignore

Let us assume Pdf30

Pdf=155-0.25*220=$100

Qs=-60+2P=-60+2*100=140

So,

Price=Pdf=$100

Combined quantity supplied by smaller firms=140 units

Total market output=Qt=140+220=360

Share of dominant firm=Qdf/Qt=220/360=61.11%

Yes, we can see that the share of dominant firm is higher than 50%