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In a perfectly competitive market, the short-run supply curve of an individual firm is crucial to understanding its behavior and responses to market prices. This curve depicts the quantity of goods a firm is willing to supply at different price levels in the short run.

To find a firm's supply curve from the cost function, we first derive the marginal cost (MC) by taking the derivative of the total cost function. For our exercise, the total cost function is given by:

• \(C(q) = \frac{1}{300} q^{3} + 0.2 q^{2} + 4 q + 10\).

After deriving, the MC becomes:

• \(MC(q) = \frac{3}{300} q^{2} + 0.4 q + 4\).

In perfect competition, firms equate their marginal cost to the market price (P), solving for output quantity (q) gives the firm's supply curve. Rearranging the equation \(P = MC(q)\), we get the supply function:

• \(q(P) = q's \, expression \, in \, terms \, of \, P\).

This equation now represents the firm's responsive supply curve in the short run based on external market prices.

The industry supply curve is an aggregation of individual firms' supply curves and is essential for determining the overall market supply. It tells us the total quantity of goods supplied by all firms at varying price levels in the short run.

Given 100 identical firms in our problem, we multiply each firm's supply function by 100 to form the industry supply curve. Suppose each firm follows the supply curve:

• \(q(P) = \text{individual \, supply \, equation \, in \, terms \, of \, P}\).

The short-run industry supply curve then is:

• \(Q(P) = 100 \, \cdot \, q(P)\),

where \(Q(P)\) represents the total industry output. This mathematical relationship shows how sensitive the industry is to price changes and ensures a connection between individual supply action and industry-wide outcomes. The approach assumes no interaction effects among firms, meaning their actions don’t affect each other, which keeps calculations straightforward in perfect competition.

Understanding the equilibrium price and quantity in a competitive market is fundamental to predicting how supply and demand interactions balance out. Equilibrium occurs where the quantity demanded equals the quantity supplied, aligning market interest.

In this exercise, we are given the market demand function:

• \(Q = -200P + 8000\).

To find equilibrium, we set the industry supply function \(Q(P)\) equal to the demand function. Solving this equation provides the equilibrium price:

• \(P = \frac{7500}{10200} = \frac{25}{4}\).

Once price \(P\) is known, substitute back into either the supply or demand equation to find the equilibrium quantity \(Q\). For instance, using the demand equation yields:

• \(Q = -200 \times \frac{25}{4} + 8000 = 6875\).

Thus, the equilibrium in this competitive market sits at a price of \(\frac{25}{4}\) dollars and a quantity of 6875 units. This point reflects an optimal balance where no excess supply or unmet demand exists at the given price.