# For what demand function is a monopoly most harmful?

Consider a firm with zero marginal cost. If it gives the product for free, then all the demand is satisfied and the social welfare increases by the maximum possible amount; call this increase $W$.

But because the firm is a monopoly, it reduces the demand and increases the price in order to optimize its revenue. Now the social welfare increases by a smaller amount, say, $V$.

Define the relative loss of welfare (deadweight loss) as: $W/V$. This ratio depends on the shape of the demand function. So my question is: is this ratio bounded, or can it be arbitrarily large? In particular:

• If $W/V$ is bounded, then for what demand function is it maximized?
• If $W/V$ is unbounded, then for what family of demand functions can it become arbitrarily large?

Here is what I tried so far. Let $u(x)$ be the consumers' marginal utility function (which is also the inverse demand function). Assume that it is finite, smooth, monotonically decreasing, and scaled to the domain $x\in[0,1]$. Let $U(x)$ be its anti-derivative. Then:

• $W = U(1)-U(0)$, the total area under $u$.
• $V = U(x_m)-U(0)$, where $x_m$ is the amount produced by the monopoly. This is the area under $u$ except the "deadweight loss" part.
• $x_m = \arg \max (x \cdot u(x))$ = the quantity which maximizes the producer's revenue (the marked rectangle).
• $x_m$ can usually be calculated using the first-order condition: $u(x_m) = -x_m u'(x_m)$.

To get some feeling of how $W/V$ behaves, I tried some function families.

Let $u(x)=(1-x)^{t-1}$, where $t>1$ is a parameter. Then:

• $U(x)=-(1-x)^{t}/t$.
• The first-order condition gives: $x_m=1/t$.
• $W=U(1)-U(0) = 1/t$
• $V=U(x_m)-U(0)=(1-(\frac{t-1}{t})^{t})/t$
• $W/V=1/[1-(\frac{t-1}{t})^{t}]$

When $t\to\infty$, $W/V \to 1/(1-1/e)\approx 1.58$, so for this family, $W/V$ is bounded.

But what happens with other families? Here is another example:

Let $u(x)=e^{-t x}$, where $t>0$ is a parameter. Then:

• $U(x)=-e^{-t x}/t$.
• The first-order condition gives: $x_m=1/t$.
• $W=U(1)-U(0) = (1-e^{-t})/t$
• $V=U(x_m)-U(0)=(1-e^{-1})/t$
• $W/V=(1-e^{-t})/(1-e^{-1})$

When $t\to\infty$, again $W/V \to 1/(1-1/e)\approx 1.58$, so here again $W/V$ is bounded.

And a third example, which I had to solve numerically:

Let $u(x)=\ln(a-x)$, where $a>2$ is a parameter. Then:

• $U(x)=-(a-x)log(a-x)-x$.
• The first-order condition gives: $x_m=(a-x_m)\ln(a-x_m)$. Using this desmos graph, I found out that $x_m \approx 0.55(a-1)$. Of course this solution is only valid when $0.55(a-1)\leq 1$; otherwise we get $x_m=1$ and there is no deadweight loss.
• Using the same graph, I found out that $W/V$ is decreasing with $a$, so its supremum value is when $a=2$, and it is approximately 1.3.

Is there another family of finite functions for which $W/V$ can grow infinitely?

• Zero marginal cost does not imply zero production cost. Who bears the burden of this cost if the product is given away for free, and in what sense does social welfare is maximized then? Commented Apr 16, 2015 at 18:15
• "Let u(x) be the consumers' utility function (which is also the inverse demand function)." $$.$$Isn´t it the consumers $\texttt{marginal}$ utility function ? Commented Apr 16, 2015 at 18:18
• Without having read most of it, harmful depends on the concept of social welfare, and how we weight those two. If we only look at household surplus, a smaller price-elasticity allows the firms to reap more of the surpluses. Consequently, the demand function D(p) = x, is "worst", if we focus consumer surplus. Commented Apr 16, 2015 at 18:38
• @AlecosPapadopoulos By $W$ I meant increase in social welfare due only to the trade (maybe I should have called it $\Delta W$). In this sense, the production costs are irrelevant. Commented Apr 16, 2015 at 19:28
• @FooBar I referred to the common definition of social welfare, which is the sum of the welfare of the consumers and the producers. The textbooks (or at least those that I read) differentiate between a loss of welfare to a certain sector in the economy and a loss of welfare to the entire economy. The latter is called "deadweight loss". en.wikipedia.org/wiki/Deadweight_loss Commented Apr 16, 2015 at 19:39

$P=\begin{cases} \frac{1}{Q} & \text{if } Q>1 \\ 2-Q & \text{if } Q\leq 1 \\ \end{cases}$.
The monopolist prices at $P=1$, but the consumers' surplus if $P=0$ is infinite, because the area under the demand curve contains $\int_1^\infty \frac{1}{Q}dQ=\infty$.