Section 7.1 Market power and marginal revenue
We now return to look at firms with market power. Let’s have a quick refresher of Sections 4.1 and 4.2. Market power can manifest itself in many ways, which we categorize using three characterizations of market power:
- MP1: A seller with market power can influence the price for which it sells in the market.
- MP2: A firm with market power faces a downward-sloping residual demand curve. A firm with NO market power faces a horizontal residual demand curve.
- MP3: A firm with market power can charge a price above its marginal cost. A firm with NO market power charges a price equal to its marginal cost.
Firms with market power can influence their price, which allows firms to charge a price above their marginal cost. Market power manifests itself as a downward-sloping demand curve faced by the seller, since at different prices, consumers will demand different quantities of their product.
We have seen that market power can have several origins. If there are barriers to entry in a given market, this could reduce the number of sellers, and give existing sellers market power. Market power can also arise if a firm produces a differentiated product: even if there are many sellers in a market, consumers will be responsive to price changes for a unique product, and the firm can maintain its influence over the price.
It is important to remember that market power forces the firm to make a tradeoff. Since a firm with market power faces a downward-sloping demand curve, if the firm tries to increase its quantity, it must lower its price; this tradeoff (price must \(\downarrow\) to get quantity \(\uparrow\)) undercuts the firm’s revenue, and gives the firm a weak incentive to produce high levels of output - or conversely, a strong incentive to limit output.
Marginal analysis shows (Section 3.4) that any firm will maximize its profit by choosing a level of output \(q^*\) where its marginal revenue (\(MR\)) equals its marginal cost (\(MC\)). That is, at \(q^*\text{,}\)
\begin{equation*}
MR = \frac{\Delta TR}{\Delta q} = \frac{\Delta TC}{\Delta q} = MC
\end{equation*}
We can observe this with our current toolkit by simultaneously graphing the firm’s total revenue and total cost:
- Any firm will have an increasing and convex total cost function under assumption P2;
- A firm with market power will have an upside-down U-shaped total revenue function (see the end of Chapter 1).
Since marginal revenue is the slope of total revenue curve, and marginal cost is the slope of the total cost curve, the profit-maximizing quantity of output occurs where the two curves have the same slope. This corresponds to where the distance between \(TR\) and \(TC\) is the greatest, since, by definition, profit is this difference: \(\pi = TR - TC\text{.}\)
Just as we saw in perfect competition 1 , it is very useful to be able to visualize optimal output on a graph showing marginal revenue and marginal cost. This will take a little bit of work, though.
Unlike in perfect competition, where the marginal revenue is equal to the price, marginal revenue is not equal to the price for a firm with market power. Remember why this happens. For a competitive firm who cannot influence the price, each additional unit sold will generate exactly \(P\) dollars, so each additional unit sold must generate exactly \(P\) dollars of revenue. Therefore, \(MR = P\) must be true in perfect competition.
For a firm with market power,
\begin{equation*}
MR \neq P
\end{equation*}
For each additional unit sold by a firm with market power, the price decreases. This is the tradeoff, the double-edged nature of market power we just discussed! But notice: when the price decreases, the firm collects a lower price for every unit sold! This significantly undercuts the added revenue from the additional unit, and weakens marginal revenue. As a result, it must be true 2 that
\begin{equation*}
MR \lt P
\end{equation*}
Numerically, for example, if the price of a good is \(P = 10\text{,}\) then an additional unit sold by a firm with market power will not generate 10 dollars of added revenue. This is because the price will need to decrease (below 10) to sell the additional unit, and the firm will collect less than dollars from all units now sold. Even if the price drops to 9.90, collecting 9.90 instead of 10 dollars from each unit sold may lead to the firm only collecting, say, 6 dollars of added revenue. As a result, the nature of market power keeps marginal revenue below the price.
This syncs up with incentives for firms with market power. We know that a firm with market power has a weak incentive to increases its output, since price will drop. A lower marginal revenue reinforces the weakness of this incentive: since additional units of output are not generating as much revenue, the firm with market power will not find increasing output as appealing.
Therefore, marginal revenue must be less than the price for a firm with market power. To see exactly how marginal revenue and price compare, therefore, we turn back to the downward-sloping demand faced by the firm (MP2). First, price comes from the demand faced by the firm: by definition, since the demand curve gives the relationship between price and quantity, it will give us the highest price the firm can charge to sell any desired quantity of output.
What about marginal revenue? By definition, marginal revenue is the change in total revenue given a change in output, and can be seen as the slope of total revenue. But total revenue is an upside-down U-shaped curve for a firm with market power! This means 1) for low levels of output, MR is positive; 2) for high levels of output, MR is negative; 3) at some level of output in between, MR is zero.
This conclusion is consistent with how market power affects marginal revenue. We know that increases in output will force the firm to lower its price, eroding the added revenue generated by the output increase. At low level of output, the added unit of output will increase TR, and MR will be positive:
\begin{equation*}
MR = \frac{\Delta TR}{\Delta q} = \frac{+}{+} = +\text{ (left side) }
\end{equation*}
As the firm produces more and more output, however, the price gets lower and lower. This amplifies the price tradeoff the firm must make when it produces more output. If adding another unit of output lowers the price which can be charged on a high number of units, the tradeoff is more damaging to TR. In particular, if the firm produces a unit on the right side of TR (beyond the peak of the curve), that added unit actually decreases total revenue! The price decrease entirely offsets any revenue gains from selling another unit.
\begin{equation*}
MR = \frac{\Delta TR}{\Delta q} = \frac{-}{+} = -\text{ (right side) }
\end{equation*}
This has an important implication for market power profit maximization. A firm with market power will never want to choose a quantity on the right side of the peak! If producing an additional unit decreases revenue (negative \(MR\)) and increases cost (assumption P1), this unit cannot ever be optimal. Why produce a unit that will add to your cost and lose you revenue?
So marginal revenue starts off positive, hits zero, then becomes negative. This suggests that marginal revenue is decreasing, which makes sense: the slope of TR decreases as \(q\) increases. 3 We can then draw marginal revenue as a decreasing function which cuts through the x-axis and goes negative for high values of output. 4
But we have also shown that marginal revenue must be less than the price. Therefore, while demand and marginal revenue are both decreasing functions, marginal revenue must lie below demand.
In a numerical example we saw earlier, a firm with market power faces demand curve \(Q_D = 200 - 10P\text{.}\) This corresponds to an inverse demand curve of \(P = 20 - \frac{1}{10}Q\text{.}\) We know the firm’s marginal revenue curve must be below the demand curve, and in this example, the firm’s marginal revenue is \(MR = 20 - \frac{1}{5}Q\text{.}\) To interpret, if the firm would like to sell \(Q = 80\) units of output, then
- from the inverse demand, we know the firm will be able to charge a price of \(P = 20 - (\frac{1}{10})(80) = 12\text{;}\)
- from the marginal revenue, the added revenue of the last unit of output for the firm is \(MR = 20 - (\frac{1}{5})(80) = 4\text{.}\)
These calculations are consistent with the \(MR \lt P\) result from earlier. Even though the \(80^{th}\) unit of output sells for a price of 12, it will only generate 4 dollars of additional revenue for the firm, as the price decrease to 12 erodes revenue gains from the other 79 units.
One last point: the equation for marginal revenue here does not appear from thin air. In fact, there is a simple rule for determining the marginal revenue equation in cases where inverse demand is linear:
- When inverse demand is linear, the firm’s marginal revenue curve will have (1) the same y-intercept as inverse demand, and (2) twice the slope of the inverse demand function.
With inverse demand \(P = 20 - \frac{1}{10}Q\text{,}\) we then know the marginal revenue curve will also have y-intercept of 20, and a slope twice as steep as \(\frac{1}{10}\text{,}\) which is \(\frac{1}{5}\text{.}\) This gives the full equation \(MR = 20 - \frac{1}{5}Q\text{.}\) The “twice as steep” rule guarantees that marginal revenue lies below the demand curve. 5
