Due to a lack of competition, a monopoly increases its profits by reducing its output and raising the price. Consumers suffer a social cost because there is less of the good available. An example is Microsoft Word. In a perfectly competitive market, the price of Microsoft Word would be equal to its marginal cost (MC), which may be only $20. But instead Microsoft, by virtue of its near-monopoly, can sell it for perhaps $200.
Quantifying the Social Loss
It is a challenge to quantify the social cost of a monopoly. But it can be done as follows.
In a perfectly competitive market, each unit would have sold at its marginal cost (MC). But a monopoly can increase its profits by reducing its output and selling its good at a price greater than MC.
Consider the price P at the precise point when a particular unit goes from being sold to being unsold. When consumers were still willing to buy that unit, the overall value of that particular unit to society is P, because someone is willing to buy it for that price. Based on our assumption about this price, when P for this unit is increased slightly, then no one buys it.
In a perfectly competitive market, this particular unit would sell at price P=MC. In that transaction society would enjoy a windfall benefit of (P-MC), because it was willing to pay an extra amount of (P-MC) for that same unit.
When the good is withdrawn from the market, the net loss to society is its total benefit from that unit minus its overall cost. Its total benefit is what society was willing to pay for that unit: the price P. Its total cost is what society paid to make that unit: MC. Thus the net loss to society from a withdrawal of that unit is (P-MC).
To find the overall loss to society imposed by a monopoly, we do the same calculation for every unit that goes unsold due to the monopoly's higher price. The overall loss to society imposed by a monopoly is thus the sum of all the individual (P-MC) for each unit withdrawn from the market. Because P is a function of output, the P is different for each unit withdrawn from the market as the monopoly raises its price. The summation must take into account the different prices at which the units go unsold.
The social loss can also be described in graphical form as the area under the demand curve between the points P and MC, and above a price line equal to MC. The summation can be performed by doing an integral in calculus.
Here is an example:
Question: Suppose Anthony owns a company having marginal costs of $5 for all his units. If he sells only one, then he reaps $11; selling two fetches a price of $10 piece; selling 3 attains a price of $9; selling four reaps $8; Q=5 would have P=$7; Q=6 has P=$6, Q=7 has P=$5, etc. A competitive firm would have the same cost and demand numbers. What does Anthony sell at, and what is the social cost of his monopoly?
Answer: If Anthony's company has monopoly and a marginal cost of $5 per widget, then using the described demand curve, his company should sell three widgets at $9 apiece or 4 widgets at $8 apiece. Either approach will give Anthony's company a maximum profit of $12.
But while Anthony has two different price points that maximize his profits, the social loss is the not same for those different prices. We would expect the social loss to be greater when the output is less, because society is deprived of more units of the good.
If Anthony sells only three widgets at $9, then that is four less than what a competitive market would sell. The social cost is the sum of (P-MC) over each of the withheld units, noting that the social cost for each withheld unit is different because the unit goes unsold at a different P.
We assume that society would have purchased the unit as its price increased until just below the price at which the unit goes unsold and is withdrawn from the market. Thus when the price went from $5 to $6, we assume society would have purchased the unit at a price of $5.999. But at $6, one unit went unsold and the loss to society was very nearly (P-MC=$6-$5=$1). Likewise, another unit went unsold at $7 (P-MC=$2), another unit went unsold at $8 and another unit went unsold at $9. That total social cost is very nearly $1 + $2 + $3 + $4 = $10.
If Anthony instead maximized his profits by selling more output, then the social loss is less. If Anthony sold four units at $8, then the social cost is very nearly $1 + $2 + $3 = $6. The "very nearly" is so close to the number that we drop the "very nearly" and simply provide the number as the estimated social cost.