# Does marginal revenue actually refer to marginal revenue per time unit?

When I was learning about why marginal revenue is lower than demand, I referred to this derivation:

$$\frac{d(TR)}{dQ} = \frac{d(PQ)}{dQ} = \frac{PdQ + QdP}{dQ} = P + Q\frac{dP}{dQ}$$

Wikipedia explains that marginal revenue is the "change in revenue for some change in quantity sold, to that change in quantity sold" (https://en.wikipedia.org/wiki/Marginal_revenue) and uses TR in the derivation, as in total revenue.

However, this is confusing for me, because the function that describes the total revenue over the history of the firm is not equal to the function used above, PQ. Is marginal revenue instead the change in daily revenue for its corresponding change in daily quantity sold? (Or maybe hourly, or weekly, depending on the rate at which that good is actually sold?)

This makes more sense to me, since total revenue is an accumulation over time of the daily marginal revenue.

• Marginal revenue (MR) is not "lower than demand", it is lower than inverse demand. (MR is measured in monetary units, demand in quantity units.) Apr 16, 2022 at 11:03

Total revenue in economics is defined as $$TR=PQ$$ (if competition is not perfect $$P=P(Q)$$). You can see this definition of total revenue in basically any economic textbook. For example, have a look at Mankiw Principles of Economics pp 267.

In a static model there is no time dimension, total revenue is function that tells you at a point in time what revenue is given $$P$$ and $$Q$$.

Note that marginal revenue is a simplifying theoretical construct.

The way you describe it above, it is essential that the inverse demand function defines the same price for all units sold. This usually means that marginal revenue is calculated for a single time period in a single market.

• If price is constant during this time period, would marginal revenue not be equal to the inverse demand function? Or do we assume the price change is very small? Apr 15, 2022 at 18:29
• I don't understand why that would be the case. As to your second question: yes, marginal revenue is a marginal measure. Apr 15, 2022 at 20:17
• Ok, thanks. For my first question: since P is constant, dP/dQ = 0, and dR/dQ = P, which is also the inverse demand. But I think one can assume P to be constant while dP/dQ is nonzero, as long as dP is small. Apr 15, 2022 at 20:33
• I didn't write that "$P$ is constant", because it is not constant w.r.t. $Q$. Perhaps you mean that the price is same for all the buyers? This is true, unless there is price discrimination. In the case of first degree price discrimination marginal revenue is equal to the inverse demand. (Again, this is for a single time period in a single market.) Apr 16, 2022 at 5:54
• My main confusion is that when the price P is decreased to (P-dP) to allow for a higher quantity (Q+dQ) sold, it is not as if the seller started afresh and sold Q+dQ units; the seller only sold an additional dQ units. However, the formula above calculates marginal revenue as [(P-dP)(Q+dQ) - PQ] / dQ = P + P dQ/dP. This formula implies that the seller actually sold another Q+dQ units at a later time, then calculated the difference in revenue over dQ. Apr 16, 2022 at 16:15

Yes, marginal revenue refers to marginal revenue per time unit.

These basic models are static, i.e. variables such as quantity, revenue, profit, demand, supply, etc., are measured per time unit. Therefore, your claim that total revenue is an accumulation over time of the daily marginal revenue does not hold for these models. The total revenue function TR is defined over quantities, $$TR=TR(Q)=P(Q)Q$$, not over time.