Consider the model of a circular city of length one where consumers are located
ID: 1124881 • Letter: C
Question
Consider the model of a circular city of length one where consumers are located with density one. Firm 1 is located at location 0 and Firm 2 at location a. Consumers assign utility to consuming the good, which is that large that every consumer always buys. Both firms have marginal production cost being equal to c. Transportation costs are equal to tr2 with z being the distance. Both firms maximize profits. a. What is the location of the two indifferent consumers? (8 points) b. Determine the optimal output price of firms 1 and 2. (8 points) c. Determine the optimal location a. (8 points)Explanation / Answer
Consumer is indifferent to buy from any of the firm when,
P1+tx^2=P2+t(1/2-x)^2
P1-P2=t((1/4-x+x^2)-x^2)=t((1/4-x))
P1-P2-t/4= -tx
(t/4-P1+P2)/2t =x
As x is the demand for good by the consumers who are indifferent to buy from any firm
Now let's solve for firms to maximise the profit , Profit is given as below
Profit = ((P1-c)(P2-P1+t/4)/2t) to maximise we check FOC
(P1*P2-P1^2+tP1/4-c*P2+cP1-ct/4)/2t then FOC with P1 will be
P2-2P1+t/4+c =0
P2+t/4+c =2P1
Let P1=P2=P
t/4+c=P
Putting this value into profit function
(P-c)*(t/4) then (t/4+c-c)*t/4=t^2/16
Hence profit among two buyers will be shared equally t^2/32
Optimal location will be the mid point as any other location would give that firm to increase the profit hence other than mid point is not stable
Hence optimal point is diametrically opposite