In the market for a product, there are 100 identical...

In the market for a product, there are 100 identical competitive firms, each firm having the cost function $c(q)=200+0.5 q^2$, where $q$ is the quantity of output in tons produced by each firm. The market demand curve is given by $Q^d=3200-100 p$. (a) Find the market equilibrium price $p^*$ and quantity produced by each firm, $q^*$. (b) What will be the market equilibrium price in the long run, $p_{\text {lr }}$, after entry or exit? How many firms will there be after entry or exit?

In the market for a product, there are 100 identical competitive firms, each firm having the cost function $c(q)=200+0.5 q^2$, where $q$ is the quantity of output in tons produced by each firm. The market demand curve is given by $Q^d=3200-100 p$.
(a) Find the market equilibrium price $p^*$ and quantity produced by each firm, $q^*$.
(b) What will be the market equilibrium price in the long run, $p_{\text {lr }}$, after entry or exit? How many firms will there be after entry or exit?

Key Concepts

Long-Run Equilibrium

In the long run, all inputs are variable, and firms can enter or exit the market. This adjustment continues until firms earn zero economic profit. The long-run equilibrium is characterized by firms operating at the point where average costs are minimized, and the market price equals both marginal cost and average cost.

Perfect Competition

In a perfectly competitive market, many identical firms sell homogeneous products. Each firm is a price taker, meaning that no single firm can influence the market price. The competitive environment ensures that decisions by firms are based on market conditions, and the equilibrium is achieved where supply equals demand.

Market Equilibrium

Market equilibrium occurs when the quantity supplied equals the quantity demanded at a specific price. In both the short run and long run, the intersection of the market supply and demand curves determines the equilibrium price and quantity, ensuring that the market clears, with no excess supply or demand.

Cost Function Analysis

A cost function describes how total production cost varies with the quantity of output. It is used to derive marginal and average costs, which are essential for making production decisions. Understanding the cost function helps determine at what output level a firm minimizes its costs and maximizes its profit.

Derivation of the Supply Curve

In a competitive market, individual firms determine their supply decisions based on marginal cost, which is the additional cost of producing one more unit of output. By equating marginal cost to the market price, the firm's supply function is derived, and summing these individual supplies leads to the market supply curve.

Short-Run Equilibrium

Short-run equilibrium refers to the period when some factors, such as the number of firms, are fixed. Firms choose their optimal output level by equating marginal cost to the prevailing market price to maximize profits or minimize losses, establishing an immediate market equilibrium that may yield positive or negative economic profits.

Zero Economic Profit Condition

The zero economic profit condition ensures that, in the long run, firms in a competitive market just cover all their costs, including opportunity costs. This condition prevents further entry or exit of firms, leading to a stable market structure where firms operate at minimum efficient scale.

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