a monopolist supplies twwwo markets, one at home, the other one abroad. The dema

Question

a monopolist supplies twwwo markets, one at home, the other one abroad. The demand function are

q1 = 10 - P1, q2 = 5 - 0.5 P2

Where q1 denotes home sales and q2 foreign sales. The firm's total cost-function is

C = 0.5(q1 + q2)2

No arbitrage between the market is possible.

1. Write down the optimization problem this monopolist is facing.

2. Write down the first order conditions for this optimization problem.

3. Solve the system of first-order condition to find the optimal home and foreign sales.

4. Write down the second-order condition and check whether they are satisfied.

5. Suppose now that price regulation is imposed in the home market, in the form of a maximum price $b. Investigate the possibility of a corner solution in which the optimal home price is $b. What would it be the optimal quantities and prices?

Explanation / Answer

1) Assuming that the monopolist can discriminate prices, the optimization condition requires it to produce a profit maximizing quantity which is achieved when its marginal cost in both markets equals the respective marginal revenues.

Marginal cost is the first derivative of total cost with respect to the output. Marginal revenue is also the first derivative of total revenue with respect to the output. Thus the optimization condition requires that MR1 = MC and MR2 = MC

2) As mentioned, compute the marginal revenues and marginal cost in separate markets. First transform the demand functions to get the inverse demand functions as: p1 = 10 – q1 and p2 = 10 – 2q2

Home

MR = dTR/dQ

= d(p1*q1)/dq1

= d(10q1-q12)/dq1

=10 – 2q1

Marginal cost MC1 = dC/dq1

= d(0.5(q1 + q2)2)/dq1

= q1 + q2

Foreign

MR = dTR/dQ

= d(p2*q2)/dq2

= d(10q2-2q22)/dq2

=10 – 4q2

Marginal cost MC2 = dC/dq2

= d(0.5(q1 + q2)2)/dq2

= q1 + q2

3) Solve the equations using the fact that marginal revenues and marginal cost in respective markets should be equal

MR1 = MC1                                         and                              MR2 = MC2

10 – 2q1 = q1 + q2                                                                  10 – 4q2 = q1 + q2

10 = 3q1 + q2                                                                          10 = q1 + 5q2

Solve them to get q1 = 2q2. Optimal sales can be found by using this relation:

10 – 4q2 = 2q2 + q2 or 10 = 7q2. This gives q2 = 10/7 or 1.42. Similarly, 10 – 2q1 = q1 + (q1)/2 or 10 = 3.5q1. This gives q1 = 10/3.5 or 2.85.

This implies that the price at home is p1 = 10 – 2.85 or p1 = 7.15 and price in foreign is 10 – 2*1.42 or p1 = 7.15. Hence the monopolist is not discriminating the price

d) Second order conditions require that the Marginal revenue function is increasing and marginal cost fucntion is decreasing. Note that the derivatives of marginal revenue functions are negative while that of marginal cost is positive

dMR1/dq1 = - 2 and dMR2/dq2 = - 4

Similarly, dMC2/dq1 = 1 and dMC2/dq2 = 1

Hence the second-order condition are satisfied.

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