Consider the horizontal quality model on the unit interval from 0 to 1. There ar
ID: 1206083 • Letter: C
Question
Consider the horizontal quality model on the unit interval from 0 to 1. There are Nconsumers located uniformly along the interval. There are two firms, with zero marginal costs, initially located at 0 to 1. Consumers will buy one unit of the good from the lowest cost retailer as long as the effective price is below V. They have transportation costs of t getting from their location to the store and back.
1. If the firms sell to the whole market, derive the function governing demand to each firm as a function of the two prices.
2. Solve for the equilibrium price if both firms sell to the whole market.
3. What condition do we have to check to ensure that the firms sell to the whole market?
4. Solve for the equilibrium price if both firms only sell to a part of the market.
5. Suppose that firm 1 is located at a and firm 2 is located a b (without loss of generality, let 0 a < b 1). Show the profit function for each firm as a function of their prices and the location of both firms.
6. What are the two first order conditions for firm 1 with respect to a and its price? What does the FOC with respect to a imply?
Explanation / Answer
According to the situation of the question I ensure that it follows "Bertrands Duopoly Model" because here the firm don't have the clear knowledge about the demand of the market so and this is the main assumtion of this model and rest assumtion follows accordingly.like 1) homogeneous product produces by 2 firm and 2) there is 2 seller in the market etc.So now we can solve the problem according to Bertrands Model:
Answer of question no1:Bertrand demand function
1)Firms set prices rather than quantities i.e Pi=X-YQi where P=price and Q=quantity demanded i=1,2
2)Customers buy from the firm with the cheapest price.
3)The market is split evenly if firms offer the same price.
So Firm demand depends on the relationship between P1 and P2 and the Demand fuction is:
Qi=0 if Pi>Pj (i,j=1,2)
Qi=X-Pi/Y if Pi<Pj
Qi=X-P/2Y if Pi=Pj=P
Firm 1 should choose C1 P1 P2 (if possible)
Firm 2 should choose C2 P2 P1 (if possible
Here C1=C2=0(Marginal Cost of both firm)
Answer of question no2:Bertrand Equilibrium Price
1)For both firms to sell positive quantities profitably C1 P1 P2 and C2 P2 P1
2)Suppose C=C1=C2
P=C Q1=Q2=(X-C)/2Y here C=0 so P=0 and Qi=X/2Y
Answer of question3:
To check it we have to check that Price should not be less than unit cost of production.The equilibrium is achive when market price is equal to the avearage cost of production and the combined equilibrium output of the two Duopolists is equal to the compititive output.
Answer for profit function
Firm 1ís profit function:
(P1)=(P1- C1) Q1 As here Marginal Cost is Zero so we only consider the transportation cost.
To ensure Q1>0 (recall: P=X-YQ and Q=(X-P)/Y) P1X
To ensure nonnegative profits P1 C1 Firm 1 should choose C1 P1 X
Similarly, firm 2 should choose C2 P2 X
Sorry I certainly realise that this way of answering is wrong I have to change my total approch so now I dont have time to do that if I get time I can reasnswering the question.Thank you.