The profit maximizing or loss minimizing equilibrium level of output (Q*) for a firm operating in a perfect competition market is determined by the intersection of its total revenue (TR) and total cost (TC) functions. In this instance, the TR and TC equations are: TR = 6Q and TC = Q3 – 2Q2 + 50Q + 25.

## Solving for the Equilibrium Quantity

The profit maximizing output (Q*) is determined by setting TR equal to TC and solving for Q. When doing so, the equation becomes: 6Q = Q3 – 2Q2 + 50Q + 25, which can be solved for Q using the quadratic equation. Doing so yields a solution of Q* = 3.

## Calculating Profit or Loss at Equilibrium

The level of profit or loss at the equilibrium quantity can be calculated by substituting the equilibrium quantity (Q*) into either the TR or TC equations. For this example, profit or loss (π) is equal to TR – TC, so π = 6Q – (Q3 – 2Q2 + 50Q + 25). When substituting in Q*, the equation simplifies to π = 6(3) – (33 – 2(3)2 + 50(3) + 25) = -27. This indicates that the firm is operating at a loss.

## Deriving the Supply Function

The supply function of the firm can be derived mathematically and graphically. Mathematically, the supply function of the firm is determined by setting the marginal cost (MC) equal to the market price (P). The MC equation is the derivative of the TC equation, which in this case is MC = 3Q2 – 4Q + 50. Setting this equation equal to P and solving for Q yields the supply equation of Q = (4 + P)/3.

Graphically, the supply curve is the curve formed by the points on the TR and TC graphs where they intersect. This point of intersection is the profit maximizing output (Q*), which in this case is 3. Thus, the supply function of the firm is the line connecting the origin to the point (3, 18).

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