Suppose that the inverse market demand for pumpkins is given by P = 10-0.05Q. Pu

Question

Suppose that the inverse market demand for pumpkins is given by P = 10-0.05Q. Pumpkins can be grown by anybody at a constant marginal cost of a $1. If there are lots of pumpkin growers in town so that the pumpkin industry is competitive, how many pumpkins will be sold, and what price will they sell for? Suppose that a freak weather event wipes out the pumpkins of all but two producers, Linus and Lucy. Both Linus and Lucy have produced bumper crops, and have more than enough pumpkins available to satisfy the demand at even a zero price. If Linus and Lucy collude to generate monopoly profits, how many pumpkins will they sell, and what price will they sell for? Suppose that the predominant form of competition in the pumpkin industry is price competition. In other words, Units and Lucy are Bertrand competitors. What will be the final price of pumpkins in this market - i.e., what is the Bertrand price? At the Bertrand equilibrium price, what will be the final quantity of pumpkins sold by both Linus and Lucy individually, and for the industry as a whole? How profitable will Linus and Lucy be? Would the results you found in part (c) and (d) be likely to hold if Linus let it be known that his pumpkins were the most orange in town, and Lucy let it be known that hers were the tastiest? Explain.

Explanation / Answer

Inverse Market Demand for pumpkins is given by: P = 10 - 0.05Q, where Q is the total quantity demanded and P is the price of pumpkins. Marginal Cost is given to be $1.

(a)

If the market is competitive in nature then each producer will sell pumpkins at the marginal cost. This is beacause if any other producer will try to increase the price of the pumpkins above the marginal cost its market share will be reduced to zero as other producers are selling the same commodity at a lower price and so there is no incentive to increase the price. Also, selling pumpkins at a price below the marginal cost will force the producers to incur losses and thus producers won't deviate from the marginal cost pricing.

Therefore, P = MC = $1

Considering the market demand function:

P = 10 - 0.05Q

1 = 10 - 0.05Q

0.05Q = 9

Q = 9/0.05 = 180

So, In competitive market 180 units of pumpkins would be sold at a price of $1.

(b)

If Linus and Lucy will collude to generate monopoly profits then the market will face a single entity that is selling pumpkins and thus it will be price maker.

Now, in this case Total Revenue(TR) can be calculated as follows:

TR = P*Q = (10 - 0.05Q)*Q

To calculate the Marginal Revenue(MR), we can simply differentiate TR w.r.t. Q

MR = dTR/dQ = 10 - 0.1Q

We know that under Monoply market the profit maximizing condition of the producer is:

MR = MC

with MC = $1

MR = 10 - 0.1Q = 1 = MC

0.1Q = 9

Q = 9/0.1 = 90

The Price can now be calculated through inverse demand function:

P = 10 - 0.05(90) = 5.5

So, If Linus and Lucy collude to generate monoploy profits then 90 units of pumpkins would be sold at a price of $5.5.

(c)

If Linus and Lucy are Bertrand competitors then the competition is price competition i.e. both the producers compete with each other by changing the price of their commodity. In this scenario, since both the producers are identical in nature in terms of their commodity, marginal cost and capacity of production the Nash Equilibrium in this case will come out to be Price = Marginal Cost.

The reason of this equilibrium is that none of the two producers have an incentive to deviate from this point. For explaining this let us consider the case of Lucy,let us suppose initially both, Linus and Lucy, are selling their goods at a price equal to marginal cost, now if Lucy tries to sell at a price above the marginal cost then because the same commodity is being sold in the market by Linus at a lower rate and Linus have the capacity to supply for the whole market demand, Lucy's market share will be reduced to zero and thus there would be no revenue generated by Lucy. So, there is no incentive to move the price upwards. Also, if Lucy tries to sell pumpkins at a price below the marginal cost then she would be incurring losses as revenue would be less than the cost and thus no incentivve to move price downwards. Similarly we can say for Linus. Thus the nash eqilibrium in this case will be:

Price = Marginal Cost = $1 (Competitive Outcome)

So, Berttrand Price is the marginal cost which is equal to $1.

(d)

Total Industry's demand function is given to be:

P = 10 - 0.05Q

Here, Q = Q1 + Q2

where Q1 is the amount sold by Linus and Q2 is the amount sold by Lucy. Since both the producers have identical pumpkins and their capacity and marginal cost are same the Bertrand Nash Equilibrium will be Price = Marginal Cost and so both the producers would be selling same amount of commodity in the market i.e. Q1 = Q2.

So, Total Industry's output would be:

P = MC = 10 - 0.05Q

1 = 10 - 0.05Q

Q = 180.

Q = Q1 + Q2 = 2(Q1) = 180

so, Q1 = Q2 = 90

Since Price = Marginal Cost, Profit of the two producers will be equal to zero. They are not earning any supernormal profit.

(e)

If Linus' pumpkins are the most orange in twon and Lucy's pumpkins are the tastiest then in this case the goods can be differentiated from one another and thus this won't result in Bertrand Nash Equilibrium of Price = Marginal Cost. This is beacuse of the fact that now the market is monopolistic in nature and with differentiated goods both the producers have an opportunity to alter the prices and earn supernormal profits.

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