# Compute variables in perfect competition, in monopoly and duopoly

There are a given demand and MC value. How could I calculate quantity, price, and profit from these values in case of perfect competition, in monopoly and duopoly(Cournot and Stackelberg)?

Under perfect competition, $p=MC$. So you may substitute this price into the demand equation to obtain the quantity.

A monopolist chooses output in a way such that $MR=MC$. So you have $P‘(Q)*Q+P=MC(Q)$, where $P(Q)$ is the inverse demand equation.

For Cournot, it is also simple. Note that the profit function of firm i is given by $$\pi(i)=P(q_{i}+q_{-i})*q_{i}-MC(q_{i})*q_{i}$$. You just use the first order condition to solve for the price and quantitiy respectively. $q_{-i}$ denotes the output of all players except for $i$.

Finally, for Stackelberg I would like to know how many players/firms you have. Then, I would be able to explain it to you in a suitable way. I hope this helps.

• Thanks for the answer, but I am not sure, how to use this equation Q=1000-10*P and MC=50 value in these formulas. Could you please explain it with exact numbers? I have two firms. – plaidshirt May 13 '18 at 8:55
• under perfect competition, price would be $p=MC=50$. Substitute this into the demand equation to get $Q=1000-10*50=500$. – saguru May 13 '18 at 21:16
• for a monopolist: the inverse demand is $P(Q)=100-Q/10$. Hence $-1/10*Q+100-Q/10=50$. Hence, $Q=250$ and $P=75$. For the other models use the same logic, i.e. just substitute in your equations in what I told you above. – saguru May 13 '18 at 21:22
• What is the meaning of q−i in your equation? – plaidshirt May 14 '18 at 7:38
• In your case it would be the quantity produced by the other firm. So if you’re computing the quantity for firm 1, $i$ would be 1, $q_{-i}$ is the notation for the quantity produced by all other firms, which would in this case boil down to $q_{2}$. – saguru May 14 '18 at 8:02