Profit Maximization: MC = MR

The Mathematics of Profit Maximization

A firm's profit, denoted by $\pi$, is the difference between its total revenue (TR) and its total cost (TC), both of which are functions of the quantity produced, $Q$.

The firm's goal is to choose the quantity $Q$ that maximizes this profit function. From calculus, we know that to find the maximum (or minimum) of a function, we must find the point where its first derivative with respect to the variable of interest is equal to zero. In this case, we differentiate the profit function with respect to $Q$:

To find the maximum profit, we set this derivative to zero:

Recalling our definitions from previous lessons, we know that $\frac{d(TR)}{dQ}$ is the marginal revenue (MR) and $\frac{d(TC)}{dQ}$ is the marginal cost (MC). Substituting these into the equation gives us the cornerstone of microeconomic analysis:

This elegant result provides a universal rule: a firm maximizes its profit by producing at the level of output where marginal revenue equals marginal cost.

The logic is intuitive:

• If MR > MC, producing and selling one more unit adds more to revenue than it adds to cost, so profit increases. The firm should expand production.
• If MR < MC, the last unit produced added more to cost than it added to revenue, so profit fell. The firm should reduce production.
• If MR = MC, the firm has exhausted all opportunities to increase profit by changing its output level. Profit is at its maximum.

For a more detailed mathematical and graphical explanation of this rule, the following resource provides an excellent supplement.

Applying the Rule: A Worked Example

Let's now see this rule in action. We'll watch a video that works through two scenarios, reinforcing the theory and demonstrating the calculation process. The second example, in particular, combines concepts from our previous lessons on cost and revenue functions.

As the video demonstrates, the procedure is methodical:

1. Derive the Marginal Revenue (MR) function from the firm's demand (or total revenue) function.
2. Derive the Marginal Cost (MC) function from the firm's total cost function.
3. Set $MR=MC$ and solve for the quantity, $Q^{\ast }$. This is the profit-maximizing level of output.

The Graphical Representation

Visually, the profit-maximizing quantity is found where the marginal cost and marginal revenue curves intersect.

An important point to remember is that this graph only gives you the optimal quantity. To find the price the firm will charge, you must take this quantity $Q^{\ast }$ and find the corresponding price on the demand curve (which is also the Average Revenue curve), not the MR curve.

A Step-by-Step Problem

Let's now apply this method to a typical problem you might encounter. The following resource presents a clear, step-by-step solution.

Let's summarize the key steps from that example:

• Demand: $q=100-\frac{1}{2}p$
• Inverse Demand: $p=200-2q$
• Marginal Revenue: $mr=200-4q$ (using the double-the-slope rule)
• Marginal Cost: $mc=4$ (given as constant)

1. Find Q:*
Set $mr=mc$:

2. Find P:*
Substitute $q^{\ast }=49$ into the inverse demand function:

3. Find Profit π:*
Total Revenue: $T{R}^{\ast }=p^{\ast }\times q^{\ast }=102\times 49=4998$
Total Cost: $T{C}^{\ast }=MC\times q^{\ast }=4\times 49=196$ (since fixed costs are zero)
Profit: ${\pi }^{\ast }=T{R}^{\ast }-T{C}^{\ast }=4998-196=4802$

A Final Note on Fixed Costs

You might wonder how fixed costs (FC) fit into this picture. Fixed costs are crucial for determining the total profit, but they do not affect the profit-maximizing quantity.

Remember that marginal cost is the derivative of total cost: $MC=d(TC)/dQ$. Since $TC=VC+FC$, where VC is variable cost, the derivative is:

Because fixed cost (FC) is a constant, its derivative is zero. Therefore, $MC=d(VC)/dQ$. Marginal cost is solely determined by the change in variable costs. Since the $MR=MC$ decision rule only involves marginal values, fixed costs do not enter the calculation for the optimal quantity.

Conclusion

In this lesson, we unified our analyses of cost and revenue to derive and apply the fundamental condition for profit maximization: $MR=MC$. This powerful rule provides a clear-cut method for determining a firm's optimal output level. The process involves deriving the MR and MC functions, equating them to solve for the optimal quantity $Q^{\ast }$, and then using the demand curve to find the corresponding price $P^{\ast }$.

Key Takeaways:

• The profit function is $\pi (Q)=TR(Q)-TC(Q)$.
• Profit is maximized when its derivative with respect to quantity is zero, which leads to the condition $MR=MC$.
• Fixed costs affect the level of profit but not the profit-maximizing quantity, as they do not influence marginal cost.
• The process for finding the profit-maximizing outcome is:
  1. Find $MR(Q)$ and $MC(Q)$.
  2. Solve for $Q^{\ast }$ in the equation $MR(Q)=MC(Q)$.
  3. Find $P^{\ast }$ by substituting $Q^{\ast }$ into the inverse demand function $P(Q)$.

We have now completed our toolkit for analyzing the firm. In the upcoming module, we will apply this $MR=MC$ framework to understand how firms behave in different market structures, starting with the two polar opposites: perfect competition and monopoly.