Help Me With This Econ Question?

Question

Three silver mines near Goldenstate, are part of a national triopoly that consist of three main firms a, b and c.
P = 1000 – 1000x, the demand function,
p = price per ounce of gold
x = number of ounces of gold produced per day.
Marginal costs, although constant, vary at each firm.

Firm A is $100 per ounce
Firm B is $ 350 per ounce
Firm C is $ 400 per ounce

Determine the Cournot triopoly equilibrium, including

(i) The number of ounces of gold produced per day?
(ii) The price of gold?
(iii) The social surplus?
(iv) The M – Firm concentration ratio of this gold triopoly?
(v) The Herfindahl – Hirschman Index applicable

I have no idea how to do it. please help.

Answer 1

Is this a trick problem? silver mines producing gold?

If not, it is interesting. A bit difficult, but with constant marginal costs, still feasible.

Look at the example in:
http://en.wikipedia.org/wiki/Cournot_duopoly

It is for a duopoly, but the assumptions are otherwise the same:

- Everywhere you see (q1 + q2) substitute (qa + qb + qc)

- In the equation for q1, substitute (qb + qc) for q2 to get the equation for qa.

- Similarly for the equations for q2, and q3, getting equations for qb and qc.

That gives you three equations in three unknowns.

Solve these to get qa, qb, and qc.

The total amount produced is their sum, Qs = qa + qb + qc

The price is determined by the total amount sold, Qs, and the demand function.

To compute the social surplus, check out:
http://en.wikipedia.org/wiki/Economic_surplus

In this case the demand function is linear so the consumer surplus is the area of a triangle.

As for the producer surplus, it is total income minus total costs.

The M-firm concentration ratios are straightforward:

1-firm ratio = qa/Qs
etc.
http://en.wikipedia.org/wiki/Concentration_ratio

And the Herfindahl – Hirschman Index would be

(qa/Qs)^2 + (qb/Qs)^2 + (qc/Qs)^2

http://en.wikipedia.org/wiki/Herfindahl_index

In case you haven't gotten the message, Wikipedia is your friend. :-)