Question

Consider a two-goods quasilinear model where the number of consumers and firms are both 1 (ie, I=1 and J=1 ). The initial endowment of denominated goods is ωm>0, and the initial endowment of commodity ∣ is 0 . Let the consumer's quasilinear utility function be φ(x)+m, where φ(x)=α+βlnx holds for some (α,β) ≫0. Also, let the firm's cost function be c(q)=σq for some real number σ>0. Suppose the consumer gets all the profits of the business. Businesses and consumers are price takers. Normalize the price of commodity m to 1 , and denote the price of commodity I as p. a) Derive the competitive equilibrium price and output of good I. How do these quantities vary with α,β,σ ?

Answer 1

To derive the competitive equilibrium price and output of good I in the given quasilinear model, we need to find the point where the consumer's utility is maximized and the firm's profit is maximized simultaneously.

Consumer's Problem:

The consumer aims to maximize their utility, which is given by φ(x) + m. The utility function is φ(x) = α + β ln(x), where x represents the quantity of good I consumed.

The consumer's problem can be stated as follows:

Maximize α + β ln(x) + m, subject to the budget constraint p*x + m = ωm, where p is the price of good I, and ωm is the consumer's initial endowment of denominated goods.

To solve the consumer's problem, we need to find the optimal value of x that maximizes the utility function subject to the budget constraint.

Differentiating the utility function with respect to x:

d(α + β ln(x))/dx = β/x

Setting the derivative equal to zero to find the maximum:

β/x = 0

This implies that x = ∞, which is not feasible. However, since the consumer has only a finite endowment ωm, the maximum value of x is ωm/p.

Substituting this value back into the budget constraint:

p * (ωm/p) + m = ωm

ωm + m = ωm

m = 0

Therefore, in the competitive equilibrium, the consumer does not consume any of good I (x = 0) and the consumer's initial endowment of commodity ∣ is 0.

Firm's Problem:

The firm aims to maximize its profit, which is given by the revenue from selling the quantity q of good I, minus the cost of production. The cost function is c(q) = σq, where σ is a positive real number.

The firm's problem can be stated as follows:

Maximize pq - c(q), where p is the price of good I.

Substituting the cost function into the profit function:

Maximize pq - σq

Differentiating the profit function with respect to q:

d(pq - σq)/dq = p - σ

Setting the derivative equal to zero to find the maximum:

p - σ = 0

This implies that p = σ.

Competitive Equilibrium:

In a competitive equilibrium, the price of good I, denoted as p, is equal to the marginal cost of production, which is σ according to the firm's problem. Therefore, p = σ.

From the consumer's problem, we know that x = 0, which means there is no consumption of good I.

Hence, in the competitive equilibrium:

Price of good I (p) = σ

Output of good I (x) = 0

The quantities of price and output in the competitive equilibrium do not vary with α and β since they only affect the consumer's utility function. However, they do vary with σ since it determines the cost function and thus the firm's behavior. A higher σ would lead to a higher price and possibly a lower output of good I in the competitive equilibrium.