Reference no: EM132408231
ECON4044/L1/01 Microeconomics: Consumer and Firm
Behaviour
Assignment
1. Derive the profit function Π(w, p) and supply correspondence y(w, p) for the single-output technologies whose production functions f(z) are given by
(a) f(z) = √(z1 + z2)
(b) f(z) = √ (min{z1, z2})
(c) f(z) = (zρ1 + zρ2)1/ρ , for ρ ≤ 1
2. Derive the cost function c(w, q) and conditional factor demand correspondence z(w, q) for each of the single-output constant return technologies whose production functions f(z) are given by
(a) f(z) = z1 + z2
(b) f(z) = min{z1, z2}
(c) f(z) = (zρ1 + zρ2)1/ρ , for ρ ≤ 1
3. The Cobb-Douglas production function with two inputs is given by f(z1, z2) = zα1zβ2, where
z1 and z2 are the firm's inputs and α, β ≥ 0 are constants.
(a) Derive the conditional factor demand equations and the cost function for the firm.
(b) Derive the supply function and profit equation for the firm, assuming that the firm operates in a perfectly competitive market.
(c) Using second order conditions show that it is impossible to maximize profit when α+β > 1. Explain why this is the case.
4. Consider a market for an homogenous good consisting of n firms. Each firm produces at zero marginal cost and there are no fixed costs of production and no capacity constraints. Market demand is given by D(p), where D'(p) < 0.
Firms compete by setting price. When firms choose the same price p they earn a per period profit of ∏(p) = pαD(p)/n. When firm i deviates and charges a price pi < p, then it earns a profit ∏(p) = piαD(pi) and all other firms get zero profit. The parameter α represents the state of demand. It is assumed that in the current period α = 1, but for each of the periods following on from the current period demand is characterized by α = θ. Firms have a common discount factor, which is denoted by δ.
(a) Assume θ > 1 and consider the following trigger strategies: Each firm sets the monopoly price pm and continues to charge this price until a profit equal to zero is observed (i.e. one of the firms undercuts the cartel price). When this happens each firm drops their price to zero forever.
Under what conditions is this strategy an equilibrium (i.e. how do the size of θ and n impact on the equilibrium? Give some economic intuition for this result).
(b) Can other prices be sustained at equilibrium under strategies similar to the ones above? Describe the conditions that are required
(c) Assume θ < 1. Find the conditions under which the trigger strategy explained above represents an equilibrium.
5. Consider the following model of Cournot oligopoly. There are n firms in an industry. Let qi denote the quantity produced by firm i and let Q = q1 + · · · + qn denotes the total output of the industry. For each firm i, denote
Q-i = Σj≠i qj
as the sum of output produced by all firms other than i. The market price of the output, P, is determined by the following inverse demand function:
Each firm has a constant marginal cost of production c < a. Firms choose their output levels simultaneously and independently. Each firm seeks to maximize its own profit.
(a) Write down firm i's profit as a function of its own output, qi, and the sum of all other firms' output, Q-i.
(b) Find the best response of firm i against Q-i. That is, find the qi that maximizes firm i's profit given Q-i.
(c) A profile of output levels (q1, . . . , qn) is a symmetric pure strategy Nash equilibrium of this game if all firms are producing the same level of output q (i.e., qi = q for all i) and that each firm is best responding to the other firms' output level. Using your answer to (ii), solve for the level of output q each firm produces in a symmetric pure strategy Nash equilibrium.
(d) Show what happens when N → ∞. Explain this answer.