3. Consider that two identical firms (firm 1 and firm 2) in a homogenous-product
ID: 1137642 • Letter: 3
Question
3. Consider that two identical firms (firm 1 and firm 2) in a homogenous-product market compete in prices. The capacity of each firm is k. The firms have constant marginal cost equal to 1 up to the capacity constraint. The demand in the market is given by P-10-Q. If the firms set the same price, they split the demand equally. If a firm sets a higher price than his competitor, the demand of the firm becomes residual demand.v a. If each firm's capacity is 9 ( 9), then what would be market equilibrium price?e b. If each firm's capacity is 3 and frim 1 charges his price at the marginal cost, then does firm 2 charge the same price? If not, what would be firm 2's optimal price? c. If each firm's capacity is still 3 and firm 2 charges $4 for his price, then what would be firm 1' optimal price?- d. From result from part b and c, can you say that the Cournot model can be interpreted as a special case of Bertrand model with limited capacity?Explanation / Answer
a. It is given that it is a homogenous product market where firms compete in prices. That means that it is a comepetetive market. Therefore, the equilibrium will be reached when price = marginal cost.
Price = P = 10 - Q where Q is the total quantity and P is the market price
MC is given as 1 for both firms
Therefore equating P = MC
P = 1
Therefore equilibrium market price = 1
Note: We also have to check if at this price, both firms are producing a quantity within their respective capacity. At P = 9, we get quantity from demand equation.
=> P = 10 - Q
=> 1 = 10 - Q
=> Q = 9
This is the market demand. Since firms are identical each will produce 9/2 units. That is each firm will produce 4.5 units < 9. Hence each firm is operating within its capacity and P = 1 is the equalibrium.
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b. Firm one is charging price equal to its marginal cost. Its marginal cost = 1. Therefore it will charge P =1. At this price (and assuming firm 2 supplies nothing), it will want to supply,
=> P = 10 - Q
=> 1 = 10 - Q
=> Q = 9.
Therefore it will want to supply 9 units of good. However its capacity is only 3, therefore firm 1 will supply 3 units of good.
Now residual demand for firm 2 , after 3 units has been supplied by firm 1 will be:
=> P = 10 - (Q-3)
Or in other words, residual demand = intial demand - supply by firm 1
=> P = 10 - Q - 3
Therefore residual demand = P = 7 - Q
In this scenario, now firm 2 wants to maximise its profits given the residual demand. Therefore it wants to maximise profits:
=> profits = Revenue - Cost
Revenue = Price * quantity
Cost = marginal cost * quantity
=> Profits = P*Q - Q as MC =1
=> Profits or R(Q) = P*Q - Q
=> R(Q) = (7-Q)Q - Q
=> R(Q) = 7Q - Q2 - Q
=> R(Q) = 6Q - Q2
In order to find the optimal Q which will maximise this profit function, we differentitate the profit function R(Q) with respect to Q and equate it to zero, to find first order condition.
=> Rule of differentiation = If f(x) = Xn, then d(f(x))/dx = nXn-1
Therefore, d(R(Q))/dQ = 6 - 2Q
First order condtition: d(R(Q))/dQ = 0
=> 6 - 2Q = 0
=> 2Q = 6
=> Q = 3
Therefore putting this Q in residual demand curve,
P = 7 - Q
=> P = 7 - 3
=> P =4
Hence firm 1 should charge $4 and produce 3 units of good (which is also under its capacity)
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c. If firm 2 charges a price = $4, then according to the demand cuve P = 10 - Q, it will want to supply
=> 4 = 10 - Q
=> Q = 6
It will want to supply 6 units of good. However, ots capacity is only 3 therefore, it will supply 3 units of good.
The residual demand for firm 1
=> Residual demand = intial demand - supply by firm 2
=> Residual demand = P = 10 - Q - 3
=> Residual demand for firm 1 = P = 7 -Q
Note that this residual demand is identical to the residual demand we got for firm one in earlier question.
We will do the same exercise as before, that is maximise profits of firm 1. We will maximise the
R*(Q) = (7 -Q)Q - Q (MC = 1)
and this is done by differentiating R*(Q) function with respect to Q and equating it to zero to find first order conditions exactly like in the previous answer.
R*Q = 6Q - Q2
Therefore, d(R*(Q))/dQ = 6 - 2Q
First order condtition: d(R*(Q))/dQ = 0
=> 6 - 2Q = 0
=> 2Q = 6
=> Q = 3
Therefore putting this Q in residual demand curve,
P = 7 - Q
=> P = 7 - 3
=> P =4
Hence we see that firm 1 in this case will want to also charge $4 in order to maximise its profits.
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d. In cournot model, each firm tries to predict the other firm's output choice and then tries to maximise its own profit. In Bertrand competition, each firm selects a quantity to be supplied and lets the market decide the price. However bertrand competiiton works exactly like a perfectly competitive market.
But from the questions b and c, we observe that this is not the case when firms are operating under a limited capacity. From option a, we saw that when the capacity was not a constraint, then both firms competed and P = MC equilibrium was reached.
However in option b, where both firms were constraint by their capacity, firm 2 decided to take advantage of the fact that firm 1 could only supply 3 units. Therefore there was a opportunity for firm 2 to have a residual demand and hence optimise its profits. Therefore, instead of supplying according to the what it did bertrand competition, it maximised its own profits given the other firms output choice.
This is in line with cournot market equilibrium.
In part c, we saw that firm 1 observing the behaviour of firm 2, also decided to optimise it's own profits. Therefore counot equiliborum was reached.
Therefore we could say that cournot equilibrium is a special case of Bertrand model with limited capacity, since each firm is constrained by a capacity limit, there is an opportunity for residual demand which the firms can take advantage of and maximise profits. Had the constraint not been there, then each firm could simply quote a marginally lesser price than the other firm and get the entire market share (and it would continue till bertrand market equilibirum was reached).
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