Consider a perfectly competitive industry with n firms. Assume the demand curve is linear, i.e. P=a-bQ, where P is the price and Q is the aggregate quantity. Assume each firm has a quadratic cost function, i.e.,C= cq^2, where q is the quantity produced by an individual firm. Each firm produces an output level where price equals marginal cost, i.e.,P=2cq , and aggregate output is simply the sum of all the individual outputs, i.e., Q=nq. Assume that in the short run the number of firms is fixed. (a) Write this as a system of 3 linear equations in 3 unknowns. What are the unknowns? What are the exogenous parameters here? (b) Write this linear system in matrix form and find the determinate of the A matrix. Under what condition will the system of equations have a unique solution. (c) Use Cramers Rule to find the equilibrium price (P*) and quantity per firm (q*). (d) How will the equilibrium price change if the number of firms increases? (e) This system can be reduced to a system of 2 linear equations in 2 unknowns by eliminating Q. Write this new system in matrix form and use Cramers Rule to solve for P* and q* again. Verify that you get the same results you got in (c). (f) Now suppose the government imposes a per unit tax of t on producers, which increases their marginal cost to 2cq+t. Write the conditions for equilibrium under the tax as a system as 2 linear equations in 2 unknowns and use Cramers Rule to solve for the equilibrium price and output per firm. (g) Suppose that the tax is imposed on consumers instead. This increases the price consumers pay, so that the demand curve becomes P+t=a-bQ. Write the conditions for equilibrium under this tax as a system of 2 linear equations in 2 unknowns and use Cramers Rule to solve for the equilibrium price and output per firm. How do the equilibrium quantities compare under the two taxes? How do the equilibrium prices compare? (h) Try to explain the result you got in (g), i.e., how the outcomes under the two taxes compare, using a simple supply-demand graph.