Consider the perfectly competitive market for one good. That is, both consumers and producers are price takers. The demand function is given by QD = −aP +b, where a > 0 and b > 0 are real numbers an

Consider the perfectly competitive market for one good. That is, both consumers and producers are price takers. The demand function is given by QD = −aP +b, where a > 0 and b > 0 are real numbers and P is the market price. The supply is given by QS = cP, where c > 0 is a real number. Assume the government introduces a per-unit tax,τ , that is paid by the producers.

1. (2 points) Suppose that a = c = 2 and b = 10. Derive an expression for the tax revenue as a function of the per-unit tax τ .

2. (3 points) Sketch the graph of the tax revenue. That is, plot the tax-revenue on the vertical axis ("the y-axis") against the per-unit tax, τ , on the horizontal axis ("the x−axis"). Recall that 'sketching a graph' involves: find the intercept with the y-axis, the intercepts with the x-axis and the shape of the graph.

3. (6 points) Repeat part (1.) and (2.) without assuming specific values for a,b and c. Recall that 'sketching a graph' involves: find the intercept with the y-axis, the intercepts with the x-axis and the shape of the graph. Some of your answers might dependent on the parameters a, b and c.

Consider a monopolist firm, facing the demand QD = −aP +b, where a > 0 and b > 0 are real numbers. The monopolist's total production costs are linear and take the following form: C(Q) = cQ + d, where c > 0 and d > 0 are real numbers and the domain of C is [0,∞).

1. (1 points) Derive the monopolist's revenue function, R(Q).

2. (1 points) Derive the monopolist's profit function, Π(Q).

3. (1 points) What is the shape of the profit function, Π(Q)? Explain your answer.

4. Assume now that costs are quadratic and have the form C(Q) = gQ2 + d, where d > 0 and g are some real number. The domain of C is [0, ∞).

(a) (2 points) For which choice of the parameters, g and d > 0, are the profits inverse u-shaped? Your answer might depend on a or b.

(b) (2 points) For which choice of the parameters, g and d > 0, are the profits u-shaped? Your answer might depend on a or b.

(c) (2 points) Assume a = 1. Choose particular values for the parameters, g and d > 0, that are consistent with your answer from part (b) and sketch the cost function.

(d) (2 points) Assume a = 1. Choose particular values for the parameters, g and d > 0, that are consistent with your answer from part (a) and sketch the cost function.

(e) (1 points) Based on your sketches of the cost function, is there any case you find reasonable? Explain your answer.