Why do firms produce where marginal revenue equals marginal cost

Profit is total revenue minus total cost. A firm chooses the quantity that makes this difference as large as possible. The calculus condition for a maximum is that the derivative of profit with respect to output, which is marginal revenue (MR) minus marginal cost (MC), equals zero. Setting MR = MC therefore identifies the profit‑maximizing output.

Deriving the MR = MC rule

Let (pi(Q)=R(Q)-C(Q)) denote profit as a function of quantity (Q). Differentiating gives ($\frac{dpi}{dQ}$=R'(Q)-C'(Q)=MR(Q)-MC(Q)). The first‑order condition for a maximum requires ($\frac{dpi}{dQ}$=0), which simplifies to (MR(Q)=MC(Q)). The second‑order condition demands that marginal cost rise faster than marginal revenue beyond the optimum, ensuring the point is a peak rather than a trough.

Key implications of the rule

• If MR exceeds MC, profit can be increased by expanding output.
• If MC exceeds MR, profit can be increased by reducing output.
• The rule applies to any market structure, but the shape of MR differs for competitive firms versus monopolists.

How to find the profit‑maximizing output for a simple linear demand

1. 1Write the inverse demand function, for example (P=100-2Q).
2. 2Calculate total revenue (R(Q)=P\times Q=(100-2Q)Q).
3. 3Differentiate to obtain marginal revenue (MR=100-4Q).
4. 4Set (MR) equal to marginal cost (assume constant (MC=20)) and solve for (Q).

Worked example with linear demand and constant marginal cost

In the example, setting (MR=MC) gives (100-4Q=20), so (Q=20). Substituting back, price is (P=100-2·20=60) and profit equals ((60-20)·20=800). Any quantity lower than 20 leaves unused profit potential, while any higher quantity incurs a cost larger than the extra revenue, reducing profit. This illustrates why the MR=MC condition is the decisive rule for profit maximization.