Revenue Calculations from Demand Data

The Core Concepts of Revenue

Before we begin calculating, let's establish the definitions of our key terms. The provided text from a calculus course offers clear and concise definitions.

As the reading highlights, the key revenue concepts are:

• Total Revenue (TR): The total income from selling a given quantity of a good. It is calculated as Price multiplied by Quantity: $TR=P\times Q$.
• Average Revenue (AR): The revenue per unit sold. It is Total Revenue divided by Quantity: $AR=TR/Q$. As you can see, this simplifies to $(P\times Q)/Q=P$. Therefore, the average revenue curve is identical to the firm's demand curve.
• Marginal Revenue (MR): The additional revenue generated from selling one more unit of output. Mathematically, it is the rate of change of total revenue with respect to quantity. For discrete changes, it's $\Delta TR/\Delta Q$. For continuous functions, which will be our main focus, it is the first derivative of the total revenue function: $MR=d(TR)/dQ$.

Revenue in Imperfect Competition: A Numerical and Graphical View

Let's see how these concepts play out for a firm that has some market power, meaning it faces a downward-sloping demand curve. To sell more, it must lower its price. This has important consequences for its revenue.

The following video provides an excellent walkthrough using a numerical example to calculate TR, AR, and MR, and then plots the resulting curves.

Let's consolidate the key relationships shown in the video.

From the table and the graphs, we can observe several critical points:

1. As price decreases, the quantity demanded increases, which is the law of demand.
2. The Average Revenue (AR) column is identical to the Price (P) column, confirming that the AR curve is the demand curve.
3. The Marginal Revenue (MR) is initially positive but decreases as quantity increases. It is always less than the price (except for the very first unit).
4. The MR curve lies below the AR curve and is steeper.
5. The Total Revenue (TR) curve is an inverted U-shape. It increases as long as MR is positive, reaches its maximum point exactly where MR equals zero, and then decreases when MR becomes negative.

Why is MR less than price? When a firm with a downward-sloping demand curve lowers its price to sell one more unit, it doesn't just get a lower price for that extra unit. It must also accept that lower price for all the previous units it could have sold at a higher price. This loss on the previous units is why the marginal (additional) revenue is less than the price of the new unit sold.

Deriving Revenue Functions with Calculus

While tables are useful, your analytical background makes you well-equipped for the more precise and generalizable method: using calculus on demand functions. This allows us to derive exact functions for TR, AR, and MR.

The process is straightforward:

1. Start with the inverse demand function, which expresses price as a function of quantity: $P(Q)$.
2. Find the Total Revenue function by multiplying the inverse demand function by quantity: $TR(Q)=P(Q)\times Q$.
3. Find the Marginal Revenue function by taking the first derivative of the Total Revenue function with respect to quantity: $MR(Q)=\frac{d(TR)}{dQ}$.

The following video demonstrates this process perfectly.

Let's work through an example to solidify this. Consider the inverse demand function from the EconGraphs resource:

1. Find Total Revenue (TR):

   $$TR(Q)=P(Q)\times Q=(20-Q)\times Q=20Q-Q^2$$

2. Find Average Revenue (AR):

   $$AR(Q)=\frac{TR(Q)}{Q}=\frac{20Q-Q^2}{Q}=20-Q$$

   As expected, $AR(Q)$ is identical to the inverse demand function $P(Q)$.

3. Find Marginal Revenue (MR):

   $$MR(Q)=\frac{d(TR)}{dQ}=\frac{d}{dQ}(20Q-Q^2)=20-2Q$$

Notice the pattern that the video called a "trick." Our demand curve $P=20-1Q$ has a vertical intercept of 20 and a slope of -1. The resulting marginal revenue curve $MR=20-2Q$ has the same vertical intercept (20) but twice the slope (-2).

This is a general rule for any linear inverse demand curve of the form $P=a-bQ$: the corresponding marginal revenue curve will be $MR=a-2bQ$. This is a powerful shortcut you can use in calculations.

Conclusion

In this lesson, we have shifted our focus to the firm's revenue, defining and calculating the three key measures: total, average, and marginal revenue. We established the fundamental identity between the average revenue curve and the demand curve. Most importantly, we applied calculus to formalize the concept of marginal revenue as the derivative of the total revenue function, a direct parallel to our earlier work on marginal cost.

Key Takeaways:

• Total Revenue (TR) is $P\times Q$.
• Average Revenue (AR) is $TR/Q$, which is always equal to the price $P$. The AR curve is the demand curve.
• Marginal Revenue (MR) is the derivative of total revenue, $d(TR)/dQ$. It represents the change in revenue from selling one additional unit.
• For a firm facing a downward-sloping demand curve, MR is less than price ($MR<P$).
• For a linear inverse demand curve $P=a-bQ$, the marginal revenue curve is $MR=a-2bQ$.
• Total revenue is maximized when marginal revenue is zero.

We have now independently analyzed both the cost side and the revenue side of the firm's operations. The next logical step is to bring them together. In our upcoming lesson, we will combine marginal cost (MC) and marginal revenue (MR) to find a firm's profit-maximizing level of output, using the fundamental condition $MC=MR$.