suppose that there are two periods and in the following question Alcoa is a mono

Question

suppose that there are two periods and in the following question Alcoa is a monopolist maker of new aluminum with marginal costs of making aluminum of zero.

Part one

Suppose that it not posible to recycle/resell aluminim (e.g. it isn't durable). Demand in period 1 is Q1=100-Psub1 and demand in period 2 is Q2=120-Psub2. Show that Alcoa will set a price of 50 in period 1. What price does it set in period 2? What are Alcoa's total profits? (Hint: this is just a simple static monopoly in each period)

Explanation / Answer

1.

Demand in period 1

Q1=100-Psub1

Psub1 = 100 – Q1 ----------------(1)

Multiplying by Q1 to both sides of eq. 1

Q1*Psub1 = Revenue = 100Q1 – Q12 -----------------(2)

Differentiation of eq. (2) w.r.t. Q1 and setting them to zero.

d(Q1*Psub1)/dQ1 = 100 – 2Q1 = 0

Q1 = 100/2 = 50,                putting the value of Q1 in eq. 1

Price=100-50 = 50

Thus, Alcoa will set a price of 50 in period 1.

2.

Demand in period 2

Q2=120-Psub2

Psub2 = 120 – Q2 ----------------(3)

Multiplying by Q2 to both sides of eq. 3

Q2*Psub2 = Revenue = 120Q2 – Q22 -----------------(4)

Differentiation of eq. (4) w.r.t. Q2 and setting them to zero.

d(Q2*Psub2)/dQ2 = 120 – 2Q2 = 0

Q2 = 120/2 =60,                 putting the value of Q2 in eq. 1

Price=120-60 = 60

Thus, Alcoa will set a price of 60 in period2.

3.

In period1, total profit = P1*Q1 = 50*50 = 2500

In period2, total Profit = P2*Q2 = 60*60 = 3600

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