Question: Suppose a monopolist produces a good in two plants, with quantity q₁ produced in Plant 1 and q₂ produced in Plant 2. The demand and cost functions function of the good is given by: P = a – b(q₁ + q₂), \({{C}_{1}}={{a}_{1}}{{q}_{1}}+{{b}_{1}}q_{1}^{2}\), \({{C}_{2}}={{a}_{2}}{{q}_{2}}+{{b}_{2}}q_{2}^{2}\)

a) Derive the FOC and SOC conditions of profit maximization for this firm. Show that SOC is satisfied (impose necessary conditions).

b) Suppose that there is a shift in demand (change in a). Show that output will increase in both plants as a result.

c) Which plant will have a greater increase in output? Please explain why.