5.1 Monopolist's Costs and Revenues

In the previous two chapters, we examined the behavior of firms in a perfectly competitive market, where in the long run each firm produces at minimum average cost, and the equilibrium price is established at this level. In this chapter, we will focus on the analysis of monopoly, a market structure in which a single firm controls the entire supply of goods.

A market can become monopolistic for various reasons. In some cases, a firm may acquire exclusive control over essential resources or technologies for production, thereby preventing other firms from entering. In other cases, barriers to entry such as high initial costs or patents may exist. Still, in other cases, a monopoly can be created by state policies that grant a single firm the exclusive right to produce and sell a particular good or service.

To allow for a direct comparison between perfect competition and monopoly, we assume that the industry’s technological structure is the same — in each production unit, employing $L$ units of labor yields $Q=A\sqrt{L}$ units of output. Here, too, maintaining a production unit costs $FC$, and the price of labor is $W$. The crucial difference between perfect competition and monopoly lies in the industry’s ownership structure. While a competitive firm owns (and is therefore constrained to operate with) a single production unit, the monopolist owns all possible production units and can choose how many to employ.

This difference has an important consequence on the cost structure: in the case of monopoly, both marginal cost and average cost are constant and equal to $AC_\text{min}$ , as the monopolist can exploit all production units at the minimum possible cost. In other words, the monopolist, having access to all production units, can produce at the minimum cost for any quantity produced. This differs from perfect competition, where each firm produces at the minimum average cost but cannot expand production beyond its single production unit.

The second crucial difference from perfect competition is that the monopolist, unlike the competitive firm, is a price-maker and not a price-taker. In other words, while firms in perfect competition take the market price as given (being small relative to the market), the monopolist has the power to influence the price through its decision on how much to produce.

Thus, the monopolist differs from a competitive firm in two ways. The first concerns costs, as discussed earlier. The second concerns revenues, as the monopolist faces a negatively sloped demand curve and must reduce the price to sell additional quantities.

We will now begin addressing the monopolist’s decision problem. The problem may seem much more complicated than that of a competitive firm. The latter only needs to decide how much to produce. In contrast, to identify its optimal choice, the monopolist must answer three questions:

• how much to produce

• how many production units to use

• how to allocate production across the production units

However, as we will see, given our assumptions, the problem will turn out to be much simpler than it appears. Once we analyze the monopolist’s costs, only the first question will remain to be answered.

Monopolist's Costs

In the previous chapter, while discussing the efficiency of competitive markets, we observed that there was no alternative allocation of production between firms—other than each of them producing the same quantity—that would allow for lower costs. The same reasoning applies to a monopolist. Regardless of the quantity the monopolist intends to produce, and regardless of the number of production units they choose to operate, to minimize costs, the monopolist will find it optimal to allocate production equally across production units. The following figure, in which we assume the monopolist uses two production units, helps to reinforce this point.

In the figure, we have assumed that the monopolist uses two production units, but the same reasoning applies to any other number of units. It is always optimal for the monopolist to allocate production equally across the production units. Not only does this make economic sense, but it also simplifies the monopolist’s problem significantly. If the monopolist wants to use $n$ production units to produce $Q$ units of output, they will always choose to produce $Q/n$ units in each of their $n$ units. The corresponding cost is therefore

See Section 3.2 on how to compute the variable cost of a productive unit.

\(\begin{gathered} C_n (Q) = n \times \big[ FC + \underbrace{W \times (Q/n)^2 / A^2}_{\substack{\text{variable cost of each}\\ \text{production unit}}} \big] \end{gathered}\)

Having answered the third question posed earlier, “how to allocate production across the production units,” we are now ready to answer the second: how many production units to use? In other words, given any amount of output $Q$ that the monopolist might want to produce, which value of $n$ minimizes the cost $C_n(Q)$?

Rather than delving into calculations, we proceed intuitively. The number of production units that minimizes the cost of producing $Q$ units must be the one that minimizes the average cost of producing $Q$ units. Thus, we have answered the second question as well. Assuming, for simplicity, that the monopolist wants to produce an amount of output equal to $n$ times $Q^\text{eff}$, the least costly way to do this is to use $n$ production units and produce $Q^\text{eff}$ units of output in each of them.

FIGURE 5.2

At this point, the monopolist’s cost structure should be clear. The monopolist’s cost function is a straight line with slope equal to $AC_\text{min}$, which is therefore the marginal cost (equal to average cost) of the monopolist.

Monopolist's Revenues

The revenue function of a competitive firm is simple: each unit of output is sold at the market price, which is determined by forces that are virtually external to the firm — the production decisions of the individual firm have a negligible impact on the price. The revenue function of a competitive firm is therefore a straight line, with a slope equal to the market price.

In the case of monopoly, the situation changes radically. The monopolist has the power to influence the market price through its own production choices. Unlike the competitive firm, the monopolist does not sell its output at a pre-determined price: it is the monopolist who chooses the price through its quantity decision. The monopolist faces a downward-sloping demand curve, i.e. knows that selling more units requires lowering the price. This leads to a revenue function that is more complex than that of a competitive firm.

Assuming the market demand curve has the linear form $P = a - bQ$, the monopolist’s revenue function is

\(\begin{gathered} R = PQ = aQ - bQ^2 \end{gathered}\)

As illustrated in the figure below, this function has the shape of an inverted U: as quantity increases, revenue first rises and then falls. This should not be surprising: since the monopolist is the only seller in the market, its revenue is nothing but the consumers’ total expenditure! As we saw in Section 4.3, when quantity is low and price is high, we are on the elastic portion of the demand curve: total expenditure increases as price falls and quantity rises. The opposite occurs when quantity is high and price is low — in that case, we are on the inelastic portion of demand.

To better understand what is going on, let us compute the monopolist’s marginal revenue function. Suppose the monopolist wants to increase production by a small amount: $\Delta Q$ units of output. In order to sell these additional units, the price With linear demand $P=a-bQ$, selling $\Delta Q$ additional units of output requires selling each unit for $b\times\Delta Q$ euros less. In other words, $\Delta P=-b\times\Delta Q$. must decrease: $\Delta P < 0$. The variation in revenue is therefore \(\Delta R = (P+\Delta P)(Q+\Delta Q)- PQ\) which we can write as the sum of two terms: \(\underbrace{(P+\Delta P)\times\Delta Q}_{ \text{quantity effect } (> 0) }\;\; + \underbrace{\Delta P \times Q}_{ \text{price effect } (< 0) }\) The first term, which is positive, is the additional revenue from selling more units. The second, negative term is the lost revenue from having to sell the units already produced at a lower price. Dividing by $\Delta Q$ and taking the limit as $\Delta Q$ approaches zero, we obtain the rate at which revenue changes with quantity — the marginal revenue: \(MR=\frac{dR}{dQ} = P + \frac{dP}{dQ}\times Q\) From this decomposition, and recalling the formula for the price elasticity of demand, we The formula on the left seems different from that of perfect competition, $MR = P$, but it’s not. In fact, a competitive firm can be viewed as a “monopolist” facing a perfectly elastic demand: operating in a market with many perfect substitutes, it cannot influence the price. When the elasticity tends to infinity, the term $1/E^D$ vanishes, and we obtain the condition $MR = P$. obtain:

\(\begin{gathered} MR = P \times \Big(1 + \frac{1}{E^D}\Big) \end{gathered}\)

When demand is elastic ($E^D < -1$), we have $MR>0$: revenue increases as output increases. When demand is inelastic ($-1 < E^D < 0$), we have $MR<0$: revenue decreases as output increases.

In the case of a linear demand curve $P = a - bQ$, we have:

\(\begin{gathered} MR = a - 2bQ \end{gathered}\)

The marginal revenue function therefore has the same intercept as the demand curve (i.e., $a$), and like the demand curve, it is decreasing: if the monopolist wants to produce more, it must lower the price — each additional unit yields less revenue than the previous ones. However, the marginal revenue curve lies below the demand curve, because it has twice the slope ($-2b$ instead of $-b$): to sell more, the monopolist must lower the price for all units sold, not just the additional ones.

The following figure illustrates these conclusions in the specific case where $a = 5$ and $b = 0.001$: