Finding the best response for Beta is as follows;
The demand function for Beta whines is
Fin the function of the price in the above equation
The price for Beta’s product can be equated as follows;
Find the marginal revenue of the firm;
The marginal revenue (MRA) also can be equated as;
From the information, the marginal cost for Beta Winery is = $10 per bottle.
The marginal cost is uniform to the marginal revenue as follows
QB=50* (-10+90+2/5PA) =4000+20PA
Equating the quantity function to the demand function of Beta Winery to find the best response function.
The best response function in this case is
At equilibrium the two functions of the two firms can equate as follows;
Equating PA=1/2PB+110 to PB=1/5PA+50, gives PB=80
Equating the PB=80 into the first equation, you get that PA=150
If PA=$150 and PB=$80, then the equation QA=200-150+80=130 bottles and QB=9000-i.e. 100*80+40*150=7000bottles.
We need to find the profit of the two firms.
To determine the profit of:
Alpha: 130 bottles *$150per bottle -$6000(fixed costs) =$13500
Beta: 7000bottles *$80per bottle-$10000(fixed costs) = $550000.
From this calculations we can inference that at equilibrium, Alpha and Beta charge a bottle of wine in $150 per bottle and $80 per bottle. In conclusion, Alpha and Beta can sell 130 bottles and 7000bottles respectively. Alpha and Beta’s profits are $13500 and $550000 respectively.
Using the Lerner index (L= (P-MC)/P) to measure market power.
P is for the price of the product and MC is a marginal cost of the product. In this case, P is the equilibrium price, (P=$150) for Alpha and (P=$80) for Beta. Marginal cost is $20 per bottle for Alpha and $10 per bottle for Beta. So substituting the values gives;
L Alpha= (150-20)/150=0.867
L Beta= (80-10)/80=0.875.
From the above L Beta has lower Lerner index than L-Alpha. From the above calculation, we see that Beta has greater market power as it has lower marginal cost.
An increase in Beta’s Winery’s fixed cost to $100000 the equilibrium price won’t change. This is because the equilibrium function is related to the amount of other product. But not to the fixed costs. (Varian, 2014).