1. Find analytically the level of aggregate production and prices for N = 5. Ass

Question

1. Find analytically the level of aggregate production and prices for N = 5. Assume that the equilibrium is symmetric which means that in equilibrium all firms produce the same amount of output.

2. Assume N = 2 (duopoly). Assume for this task that the firms choose prices and not quantities. The price decisions are made simultaneously. Since the minivans are identical, the consumers choose the cheapest minivan. If both firms choose same prices, the consumers are indifferent which minivan to buy. In this case, each consumer randomizes and buys a minivan from one of the firms with probability 1/2.

Show carefully that it is a Nash equilibrium for both firms to choose p = MC.

cale documentupere are N (identical) minivan manufacturers in France. Denote by yi the production of the manufacturer i = 1,.., N. The cost function of each minivan manufacturer is C(y) = 10yi, The inverse demand for the minivans is p(y) = 1000 - y where y is the aggregate supply. Assume that only manufacturers located in France supply the minivans for the local market and all production decisions are made simultaneously.

Explanation / Answer

1).

Suppose there are 5 firms are producing identical products, the demand curve is given by, “P=1000-Y”.

“A1” be the profit function of the 1st firm, then it is given by, “A1=P*Y1-10*Y1”

=> A1=P*Y1-10*Y1,

=> A1 = 1000*Y1 – Y1^2 – Y2*Y1 – Y3*Y1 – Y4*Y1 – Y5*Y1 – 10*Y1.

So, the FOC requires dA/dYi = 0.

=> 1000 – 2*Y1 – Y2 – Y3 – Y4 – Y5 – 10 = 0.

=> 990 – Y2 – Y3 – Y4 – Y5 = 2*Y1,

=> Y1 = 495 – (1/2)*(Y2+Y3+Y4+Y5), be the “Reaction Function” of “F1”.

So, similarly we can also derive the same for each Firms.

=> Y2 = 495 – (1/2)*(Y1+Y3+Y4+Y5), be the “Reaction Function” of “F2”.

=> Y3 = 495 – (1/2)*(Y2+Y2+Y4+Y5), be the “Reaction Function” of “F3”.

=> Y4 = 495 – (1/2)*(Y1+Y2+Y3+Y5), be the “Reaction Function” of “F4”.

=> Y5 = 495 – (1/2)*(Y1+Y2+Y3+Y4), be the “Reaction Function” of “F5”.

Now, as we know that all these firms will produce the same level of output,

=> “Y1=Y2=Y3=Y4=Y5, Y=2*Y1.

So, now from the “Reaction Function” of F1, we have,

=> Y1 = 495 – (1/2)*(Y2+Y3+Y4+Y5), => Y1 = 495 – (1/2)*4*Y1.

=> Y1 = 495 – 2*Y1, => 3*Y1=495, => Y1=165.

So, at the equilibrium all the firms will produce, “165”.

So, if all will produce, Yi=165, Y=5*165=825, P=1000-825=175.

2).

Now, consider the “price competition” duopoly case, here all the firms have identical cost structure, so here since both the firms are producing identical product, => if both the firm charge same price => all the consumer will randomize between F1’s product and F2’s product.

So, here given the situation if they want to increase the profit, => they have to reduce the price to capture the entire market share, because 2 goods are identical so if a particular firm will charge lower price compared to other one then that firm will able to capture the entire market and also will be able to maximize profit.

So, here to maximize the total profit they start reducing price, => finally both of them will end the price competition and the P=MC, since they cannot reduce price further, since if they does leads to loses.

So, at the equilibrium P=MC.

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