There are two firms 1 and 2 in an industry, each producing output Q1 and Q2 respectively and facing the industry demand given by P=50-2Q, where P is the market price and Q represents the total industry output, that is Q= Q1 + Q2. Assume that the cost function is C = 10 + 2q. Solve for the Cournot equilibrium in such an industry

In a Cournot duopoly, each firm determines its output level taking the output of the other firm as given.

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The profit-maximizing output for each firm is where marginal cost equals marginal revenue.

Given the demand function (P = 50 – 2Q), where (Q = Q1 + Q2), and the cost function (C = 10 + 2q), where (q = Q1 + Q2), let’s find the equilibrium output for each firm.

1. Determine the Total Output (Q):
   [Q = Q1 + Q2]
2. Derive the Marginal Revenue (MR) for Each Firm:
   [MR = \frac{d(TR)}{dQ} = \frac{d(50Q – Q^2)}{dQ}]
3. Equate Marginal Revenue to Marginal Cost (MC) for Each Firm:
   [MR = MC]
   [10 – 2Q1 = 2]
   [10 – 2Q2 = 2]
4. Solve for the Output of Each Firm (Q1 and Q2) in Equilibrium:
   Solve the system of equations to find (Q1) and (Q2). [Q1 = \frac{1}{2}(10 – 2Q2)]
   [Q2 = \frac{1}{2}(10 – 2Q1)]
5. Calculate the Market Price (P) at Equilibrium:
   Substitute the equilibrium (Q) into the demand function to find the market price. [P = 50 – 2Q]

The values obtained for (Q1), (Q2), and (P) represent the Cournot equilibrium in the industry. Keep in mind that the solution involves solving simultaneous equations, and the exact steps can vary based on the specific equations given.