Question

Ace Washtub Company is currently the sole producer of washtubs. Its cost function is C(q)=49+2q, and the market demand function is D(P)=100-P. There is a large pool of potential entrants, each of which has the same cost function as Ace. Assume the Bain-Sylos postulate. Let the incumbent firm’s output be denoted qI.

Suppose we now assume active firms expect to achieve a Cournot solution. Does entry depend on qI? Will there be entry?

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Answer 1

1)The residual inverse function for the new entrant (q^E) is the demand left unsatisfied by the incumbent firm and it is represented by the below equation:

P = 100 - q^1 - q^E,

2)q^E = (98 - q^1)/2 is the optimal quantity made the entrant firm,

3) The limit price is 9 dollars.

4) No, entry does not rely on the quantity made by the existing firms. Yes, the firms will enter as long as there are profits.

Explanation:

1)

The inverse demand function is:

D(P) = 100 -P, it can be written as:

q = 100 - P(Where 'q' is the quantity demanded by the entire market) . It, the residual demand function, is derived as follows:

q^(E) = q - q^1,

q^(E) = 100 - P - q^1, it can written in terms of q^1 and q^E as:

P = 100 - q^1 - q^E---------(1)

2)

Please note that the Bain-Sylos postulate states that quantity supplied by the incumbent firm remains unchanged in both pre-entry and post-entry period. Using this and the equation (1) to maximize the profits for the entrant firm as below:

Profit = TR - C(q^(E)),

Profit = (100 - q^1 - q^E)*q^E - (49 + 2*q^E), simplifying it:

Profit = 100*q^E * q^1*q^E - (q^E)^2 - 49 - 2*q^E, differentiate with respect to q^E (keeping q^1 as constant:

dProfit / dq^E = 100 - q^1 - 2*q^E - 2, putting it equal to zero:

dProfit / dq^E = 0 ,

100 - q^1 - 2*q^E - 2 = 0,

98 - 2*q^E = q^1 ------(1a)

Solving for q^E:

q^E = (98 - q^1)/2 ----(2)

The equation (2) is the optimal quantity made by the new firm given the quantity made by the incumbent one.

3)

The limit price signifies the price chosen by the incumbent firm with the objective to rule out the entry of the entrant firm. In simple words, it is price where the incumbent firm can earn only normal profits. Thus, we will equate the average cost curve faced by the entrant firm with the market price to find this limit price as follows:

Equation of AC:

TC = 49 + 2*q,

AC = 49 /q^E + 2----(3)

Now, based on the optimal quantity of entrant and incumbent firms, the market price would be:

P = 100 - q,

P = 100 - (q^1 + (98 - q^1)/2 )

Setting P = AC:

100 - (q^1 + q^E ) = 49 /q^E + 2, Using the (1a) equation:

100 - (98 - 2*q^E + q^E ) = 49 / q^E + 2

2 +q^E = 49 /q^E + 2, solving for q^E:

q^E = 7 units and plugging it in (1a):

98 - 2*7 = q^1

q^1 = 84 units.

So, the market price at which the entrant firm would not find profitable (that is limit price) is:

P = 100 - 84 - 7,

P = 9 dollars

4)

In case of Cournot solution, the entry of any entrant firms does not rely on the quantity already made by the existing firms. Rather, new firms continue to enter the industry until all profits from business are wiped out. So, we need to evaluate whether the existing firm is earning profits or not for the entry purpose.

Using the condition of MR= MC:

MR = MC:

100 - 2*q = 2,

q = 49 units and plugging in the demand function:

P = 100 - 49,

P = 51 dollars.

Now, the amount of profits are:

= TR - TC,

= 51*49 - (49 + 2*49),

=2352 dollars > 0, which means that the entry will happen.