The U.S. market for hand sanitizer is controlled by a monopoly (firm I, for incu

Question

The U.S. market for hand sanitizer is controlled by a monopoly (firm I, for incumbent) that has a total cost given by TC(qi) = 0.025qi^2. The market demand for hand sanitizer is given by P = 50 – 0.1Q. Under monopoly, Q = Qi. Now let there be a foreign firm (firm E, for entrant) that is considering entry into the market. Because the entrant must ship hand sanitizer all the way across the ocean, its costs are higher. Specifically, the entrant’s costs are given by TC(qe) = 10qe + 0.025qe^2.

Question: Show that the monopolist would need to commit to produce 400 units in order to deter entry of the foreign firm. (Hint: figure out the monopolist’s output level q* such that the entrant loses money if it exports anything other than zero.) What are the incumbent’s profits if it commits to this output level and deters entry?

Explanation / Answer

Firm I is a monopoly in the U.S. market for hand sanitizer, i.e. the domestic market for hand sanitizer is currently being serviced by the Firm I.

As given the Total Cost as TC(Qi) = 0.025Qi2 …………………… (1)

And the market demand as P = 50 – 0.1Q, ……………………… (2)

First of all we will find the price and the profit maximizing quantity for this firm.

The monopolist’s profit maximizing level of output/quantity is found by equating its Marginal Revenue with its Marginal Cost. Because if the firm produces at an output level where MR>MC or MR<MC will yield lower profits. Since the goal of the monopolist is to maximize the profits, i.e. difference between TR and TC OR maximizes profits when MR = MC, where MR(Marginal Revenue) is the change in the total revenue associated with the change in quantity and MC (Marginal Cost) is the change in the total cost associated with the change in quantity.

From (1), TC(Qi) = 0.025Qi2

MC(Qi) = (0.025*2) Qi = 0.05Qi

OR MC = 0.05Qi

Also, from (2):

TR = P*Q = [(50 – 0.1Qi)*Qi] = 50Qi - 0.1Qi2

Therefore, MR = TR/Q = 50 – (0.1*2)Qi

OR MR = 50 – 0.2Qi

As per the profit maximizing condition, MR = MC

50 – 0.2 Qi = 0.05 Qi

50 = 0.25 Qi

Qi = 200 units

Thus, from (2), P = 50 – 0.1 Qi

P = 50 – 0.1(200) = 50 – 20 = $ 30

The firm will have profits equal to

i = TR - TC = 30*200 – (0.025)(200)2 = 6000 – 1000 = 5000

Total profits = $ 5000

………………………………………………………………………………………………

Now, if the new entrant enters into US to export the hand sanitizer, thus the industry demand curve can be written as

P = 50 – 0.1Q = 50 – 0.1(Qi + Qe) = 50 – 0.1Qi – 0.1Qe

Therefore, P = 50 – 0.1(200) – 0.1Qe

P = 30 – 0.1Qe

Also, TR for new entrant is TR = P*Q = [(30 – 0.1Qe)*Qe] = 30Qe – 0.1Qe2

Thus, Marginal Revenue for the new entrant will be:

MRe = 30 - (0.1*2)Qe = 30 -0.2Qe

Also, TC(Qe) = 10Qe + 0.025Qe2

MCe = 10 + (0.025*2)Qe

MCe = 10 + 0.05Qe

Thus, for the new entrant, the profit maximizing level of output would occur at

MR = MC

30 – 0.2Qe = 10 + 0.05Qe

20 = 0.25Qe

Qe = 80 units

Therefore, P = 30 – 0.1Qe

P = 30 – 0.1(80) = $ 22

The new entrant will export 80 units to the US market and price will fall from $ 30 to $ 22.

The total quantity will also rise from 200 to 280. (200 is when only Firm I was trading, and 280 is the combined output level for both the firms)

Profits for both the firms would be

e = (22)(80) – (10)(80) – (0.025)(80)2 = 1760 – 800 – 160 = $ 800

i = (22)(200) – (0.025)(200)2 = 4400 – 1000 = $ 3400

Finally, we need to find the monopolist’s output level q* such that the entrant loses money if it exports anything other than zero.

Writing the residual demand curve:

P = 50 – 0.1Q

P = 50 – 0.1Qi – 0.1Qe

Total Revenue for new entrant would be TR = P*Q = 50Qe – 0.1QiQe – 0.1Qe2

Marginal Revenue for the entrant firm will be

MRe = 50 – 0.1Qi – 0.2Qe

Setting MR = MC we obtain

MRe = 50 – 0.1Qi – 0.2Qe = 10 + 0.05Qe = MCe

0.25Qe = 40 – 0.1Qi

Qe =160 – 0.4 Qi

If the incumbent then chooses qI such that the optimal qE = 0, then the entrant will not enter. This implies

Qe = 160 – 0.4Qi = 0

i.e. 160 = 0.4Qi

or Qi = 400 units or Q = 400 units, the level of Qi = Q such that the best response of the entrant is to produce zero output.

With this level of output price and profits for the two firms are

P = 50 – 0.1Qi – 0.1Qe

P = 50 – 0.1(400) – 0.1(0)

P = 50 – 40

P = $ 10

i = (10)(400) – (0.025)(400)2 = 4000 – 4000 = 0

e = (10)(0) – (10)(0) – (0.025)(0)2 = 0

So, in this case, the incumbent would accommodate the enter.

iaccommodate = 3400 > 0 = ideter

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