Can a monopoly supply negative units to one of its two markets?

Question

A question I have seen in my microeconomics textbook is as follows:

Consider a monopoly supply to both a domestic and foreign market, where the market demands are:

Yd(pd) = 20 - 2pd, hence pd(yd) = 10 - Yd/2

Yf(pf) = 20 - pf, hence pf(Yf) = 20 - Yf

Total cost = .5Y^2 +20

Total output, Y = Yd + Yf

We are assuming the monopolist cannot price discriminate and therefore pf = pd = p

We are asked what price should the monopolist now set for its output? What quantity will it sell to each market?

To solve this question i first calculated the new total market demand, or aggregate demand by summing both the foreign and domestic markets.

D(p) = 20 - 3p + 20 - p = 40 -3p, hence p = 40/3 - Y/3

I then calculated total revenue and marginal revenue.

TR = (40/3)Y - (Y^2)/3 , hence MR = 40/3 - 2Y/3

I then calculated marginal cost.

MC = Y

I then solved for Y where MR = MC and substituted the value into the market price function.

Y = 40/3 - 2Y/3, hence Y = 8, p(8) = 32/3

My problem arises when i substitute this value for P into the domestic market's output function to calculate how many units will now be supplied to the domestic market.

Yd(8) = 20 - 2(32/3) = -4/3

My interpretation of this is that the monopoly is purchasing 4/3 units from the domestic market to sell into the international market, however this concept is not familiar to me. Having a negative answer for units does not sit well with me. Any guidance as to something i did wrong in calculating the question or an economic interpretation of the solution would be much appreciated! I am very confused.

Answer 1

Marginal cost is only guaranteed to be equal to marginal revenue in optima where $Y > 0$ (and when the profit function is concave, but in your case it is). If $$ MC(0) > MR(0) $$ the producing the first unit results in a loss, and a profit maximizing monopolist would not produce at all (if the profit function is concave).

An example: I am willing to buy golf ball sized diamonds from you. I am willing to pay 5 dollars for the first, 4 dollars for the second, 3 dollars for the third and so on, ending in 0 dollars for the sixth and any further units. How many golf ball sized diamonds will you sell me?