Exercise about Monopoly price charging

Question

In a two-period lived economy, one consumer wishes to buy a TV set in period 1. The consumer lives for two periods, and is willing to pay a maximum price of $100 per period of TV usage. In period 2, two consumers (who live in period 2 only) are born. Each of the newly born consumers is willing to pay a maximum of fifty dollars for using a TV in period 2. Suppose that in this market there is only one firm producing TV sets, that TV sets are durable, and that production is costless.

(a) Calculate the prices the monopoly charges for TV sets in periods 1 and 2.

(b) Answer the previous question assuming that in the first period, a consumer who lives two periods is willing to pay no more than twenty dollars per period for TV usage.

My attempt

For this I consider cases whether person 1 buys tv in period one or not. If he not buys then monopolist sell tv to other person for 50 yielding profit of 100 but also he can charge 100 for person 1 so total profit is 200. So my question is whether he charge 50 or 100 from person 1 as there is not given about price discrimination. Also if he buys tv in period 1 then 100 for second period person is same but then how to calculate it for period 1 price is it 150 or 100 as person 1 not want to give more than 100 for period.

Answer 1

I do not understand your attempt very well.

Assuming that the monopolist cannot directly price discriminate (i.e. charge different prices to old and new consumers in the same period), then in the second period it is optimal to offer a price of 50 and sell to everyone, regardless of whether the old guy bought or not.

Knowing this, the old guy would buy in the first period iff the price gives him a total surplus of 50 (which he can get by waiting). Hence, the initial price should be 150.

If, on the other hand, the old guy has a value of only 20, then in the second period the monopoly price would still be 50. Hence, the old guy has no surplus from waiting and the monopolist should charge 40 to him in the first period.