estion 2: Springfield has only two pastry shops, which were established around t
ID: 1126790 • Letter: E
Question
estion 2: Springfield has only two pastry shops, which were established around the same time and have a similar market share. Consumers in Springfield appear to regard both pastry shops' cakes to be equally good they cannot taste the difference. Each pastry shop bakes its cakes in the premises, and neither can easily expand output given its limited oven capacity. The manager of one of the pastry shops has estimated the market demand for cakes (Y) in this all town to be Yd(p) 280- 10p, where p is the price of a cake and Ythe total number of cakes sold in this markct per weck. This pastry shop's weckly cost function is C (1/4)y, 2 + 428.75, which implies the marginal cost function MGO) = (1/2)., where y, is the number of cakes sold by this shop per week (similarly, ½ is the number of cakes sold by the rival shop, and obviously yty2 -Y). The manager expects its rival to be using the same technology and access the same input markets, and therefore have the same cost function. Based on the description of the industry above, what is the most appropriate model to use to represent this market? Why? a) b) Given your choice of model in (a), how many cakes should this pastry shop produce per weck and at what pricc should it sell them?Explanation / Answer
a). We can see that, here there 2 cake maker shop both produces totally identical good and the market price depend on the total market production. It does not matter which shop is producing how much it only depends on the “total production. Finally both the shop will compete each other to maximize profit by choosing different quantity.
So, the above model is a cournot duopoly model
b). Here the market demand curve be, “Y = 280 - 10*P, => 10*P = 280 – Y,
=> P = 28 – (1/10)Y, => P = 28 – 0.1*Y and the MCi = (1/2)*Yi for ith firm.
So, the 1st shops profit function is given by,
p1 = [ 28 0.1*(Y1 + Y2) ] ´ Y1 C1 .
Now, to find the best response function for shop-1 we differentiate the payoff function with respect to Y1 , set equal to zero.
p1/Y1 = 28 (0.1*Y2 + 2*0.1Y1) MC1 = 0, where MC1 = (1/2)*Y1.
=> 28 0.1*Y2 - 2*0.1Y1 – 0.5*Y1 = 0, => 28 0.1*Y2 – 0.7*Y1 = 0,
=> 28 0.1*Y2 = 0.7*Y1, => Y1 = (28/0.7) – (0.1/0.7)*Y2, => Y1 = 40 – (0.1428)*Y2.
=> Y1 = 40 – (0.1428)*Y2 ……………………….(1).
Now, we do the same for firm 2:
p2/Y2 = 28 (0.1*Y1 + 2*0.1*Y2) – 0.5*Y2 = 0,
=> 28 – 0.1*Y1 – 2*0.1*Y2 – 0.5*Y2 = 0, => 28 – 0.1*Y1 – 0.7Y2 = 0.
Solve this for q2:
=> Y2 = 40 – (0.1428)*Y1 ……………………….(2).
Solve (1) and (2) for Y*1 and Y*2, we will get the COURNOT-NASH equilibrium quantities.
By substituting (2) into (1).
=> Y1 = 40 – (0.1428)*Y2, => Y1 = 40 – (0.1428)*[40 – (0.1428)*Y1].
=> Y1 = 40 – 5.712 + 0.0204*Y1, => 0.9796*Y1 = 34.3, => Y1 = 34.3 / 0.9796.
=> Y1 = 35.01 = 35.
Now, substituting “Y1=35” in to “2” we will get “Y2=35”.
So, Y = Y1+Y2 = 70.
So, the market price is, “P=28 – 0.1*Y, => P = 28 – 0.1*70 = 21.
So, both the firms will produce same amount of cakes and will sale at P=21.
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