Suppose that the monopolist can produce with a constant marginal cost of $20 per

Question

Suppose that the monopolist can produce with a constant marginal cost of $20 per unit and there is no       fixed cost. Assume also that he monopolist sells its goods in two different markets separated by some distance. The demand curves in the first market and the second market are given by Q1=200-P1 and Q2 =100-P2.

a)If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? Why are the two prices different? Verify the Lerner Index for each market.

b)How would your answer to part (a) change if it only cost demanders $10 to transport goods between the two markets? What would be the monopolist’s new profit level in this situation?

c)How would your answer to part (a) change if it only cost demanders $60 to transport goods between the two markets? What would be the monopolist’s new profit level in this situation?

d)How would your answer to part (a) change if transportation costs were zero and firm was forced to follow a single-price policy?

e)Compare the profit levels in parts (a), (b) and (d). Explain why this difference obtains.

Explanation / Answer

It is given to us that the monopolist operates in two markets 1 and 2. His demand functions for both markets are given by Q1 = 200 - P1 and Q2 = 100 - P2. It is also given to us that marginal cost to the monpolist is $20 in both markets.

a) If monopolist can maintain separation between each market, the quantities and prices in each market can be calculated with the help of the inverse demand function.

Market 1

The demand function for market 1 is given as Q1 = 200 - P1 . The inverse demand function here would be P1 = 200 - Q1. The total revenue is TR = P x Q. So, the total revenue is TR = (200 - Q1)Q1 = 200Q1 - Q12.

Marginal revenue is the first derivative of total revenue so, MR = dTR/dQ1 = 200 - 2Q1

Now, a profit-maximising firm will always produce at a point where MR = MC. So, on equating the two we get :

MR = MC

200 - 2Q1 = 20

- 2Q1 = 20 - 200

2Q1 = 180

Q1 = 180/2 = 90

The price P1 in market 1 from the inverse demand function is then,

P1 = 200 - Q1

P1 = 200 - 90 = 110

So, price in market 1 is 110 and quantity sold in market 1 is 90.

Similarly, in market 2

The demand function for market 1 is given as Q2 = 100 - P2 . The inverse demand function here would be P2= 100 - Q2. The total revenue is TR = P x Q. So, the total revenue is TR = (100 - Q2)Q2 = 200Q2 - Q22.

Marginal revenue is the first derivative of total revenue so, MR = dTR/dQ2 = 100 - 2Q2

Now, a profit-maximising firm will always produce at a point where MR = MC. So, on equating the two we get :

MR = MC

100 - 2Q2 = 20

- 2Q2 = 20 - 100

2Q2 = 80

Q2 = 80/2 = 40

The price P1 in market 1 from the inverse demand function is then,

P2 = 100 - Q2

P2 = 100 - 40 = 60

So, price in market 1 is 60 and quantity sold in market 1 is 40.

The difference in prices in both markets is because market 1 has a higher market power. Market power represents capacity of a firm to raise its market price and also raise profits. This can be measured using the Lerner Index. The Lerner Index measures the market power of every market and ranges from 0 to 1 with higher numbers depicting higher market power. The formula for Lerner Index is given as follows :

L = (P - MC) / P

For market 1, the Lerner Index is :

L1 = (P1 - MC) / P1

= (110 - 20) / 110

= 90/110

= 0.81

The Lerner Index for market 2 is

L2 = (P2 - MC) / P2

= (60 - 20) / 60

= 40/60

= 0.67

Since, the Lerner Index for market 1 is higher at 0.81, it shows that market 1 has a higher market power than market 2 and hence demands a higher market price of 110.

The total revenue for market 1 in this situation is TR1 = P1 x Q1 = 110 x 90 = 9,900

Since there is no fixed cost, the total cost for firm is 20 x Q1. Total cost = 20 x 90 = 1800

The total profit from market 1 is Profit = TR1 - TC1 = 9,900 - 1,800 = $8,100.

The total revenue for market 2 in this situation is TR = P2 x Q2 = 60 x 40 = 2,400

Since there is no fixed cost, the total cost for firm is 20 x Q2. Total cost = 20 x 40 = 800

The total profit from market 2 is Profit = TR2 - TC2 = 2,400 - 800 = $1,600

So, the total profit for the firm in the earlier situation is profit from market 1 + profit from market 2 = 8,100 + 1,600 = $9,700

b) Now, if it costs the demanders only $10 to transport goods between two markets, then they would transport all the goods from market 1 into market 2 and pay only $70 per unit (60 + 10) instead of $110 per unit in market 1. So, now the demanders will buy all quantities Q = Q1 + Q2 = 90 + 40 = 130 from market 2 at price $70. So, now the total revenue for the firm is TR = P x Q = 70 x 130 = $9,100 and the total cost is TC = 20 x 130 = 2,600. The profit now is Profit = TR - TC = 9,100 - 2,600 = $6,500

In the new situation, when demanders can transports goods for $10, the profit of the firm reduces from $9,700 to $6,500.

c) Now, if it costs the demanders only $60 to transport goods between two markets, then they would have to pay $120 per unit (60 + 60) if they choose to tranport goods from market 1 to 2 instead of $110 per unit in market 1 and $60 in market 2 separately. Since, $120 is the most expensive price available to the demanders they will choose not to transport the goods from one market to another. In this case the profit for the firm remains the same as in part a)

d) If there are no transportation costs, then all demanders transport the goods from market 1 to market 2 because market 2 offers a cheaper price of $60 per unit. in this case, The firm will be forced to follow a single-price policy and charge only $60 per unit. The demanders will now buy all quantities Q = Q1 + Q2 = 90 + 40 = 130 from market 2 at price $60. So, now the total revenue for the firm is TR = P x Q = 60 x 130 = $7,800 and the total cost is TC = 20 x 130 = 2,600. The profit now is Profit = TR - TC = 7,800 - 2,600 = $5,200

The profit to the firm in this case is the lowest at $5,200.

e) The profit levels in part a), b) and d) stand at $9,700, $6,500 and $5,200, respectively. This difference is prevalent because in the first case, the monopolist is able to separate both the markets 1 and 2 and also maintain a considerable price difference between the two. This changes in part b), when the demanders have the option of transporting the goods at a cheaper rate and hence can obtain the goods at a lower rate in market 2 than in market 1. This reduces the market power of market 1 and hence lowers the profit for the monopolist firm. In the fourth situation in part d), there are no transportation costs so all demanders transport the goods from market 1 to 2 at no cost. This lowers the profit of the monopolist further as he does not have the privilege of price-differentiation anymore.

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