problem firm 1 produces cars and the total cost

Question

Problem: Firm 1 produces cars and the total cost of producing q cars is given as C(q) = 2q² + 5q.

a) Assuming the ?rm operates in a perfectly competitive market. Write down the pro?t function of the ?rm and solve for the ?rst order condition. What is the supply curve of the ?rm?

b) Suppose ?rm 2 also produces cars at a cost of C(q) = 3q² . What is the supply curve of this ?rm?

c) Solve for the aggregate supply curve of cars. Illustrate this in a diagram.

d) Is it e?cient if both ?rms produce 10 cars? Why or why not?

e) Suppose there are 2 consumers. Both consumers have individual demand functions of p = 20 - 2q_i for cars. What is the aggregate demand curve? Illustrate in a diagram.

f) What is the market equilibrium (where price equals demand). What is the equilibrium quantity and price? How much does each ?rm produce and how much does each consumer consume at the equilibrium.

g) What is the marginal cost of each ?rm at the equilibrium? What is the marginal willingness to pay of each consumer at the equilibrium? Is the equilibrium Pareto optimal?