<p>An arbitrarily large ratio should occur with demand curve </p>

<p>$P=\begin{cases}
\frac{1}{Q} &amp; \text{if } Q&gt;1 \\
2-Q &amp; \text{if } Q\leq 1 \\
\end{cases}$.</p>

<p>The monopolist prices at $P=1$, but the consumers' surplus if $P=0$ is infinite, because the area under the demand curve contains $\int_1^\infty \frac{1}{Q}dQ=\infty$.</p>

CC BY-SA 3.0
2015-04-18T17:28:17.787Z
Thanks! Is there any reference where this issue is discussed? I would expect it to appear in standard textbooks in mircoeconomics, but didn't find it in any book I looked at.
Erel Segal-Halevi
CC BY-SA 3.0
2015-04-18T23:05:01.363Z
I don't know of any references, sorry.
Sander Heinsalu
CC BY-SA 3.0
2015-04-18T05:28:46.373Z
2015-04-18T05:28:46.373Z
Sander Heinsalu
<p>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 $W$.</p>

<p>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, $V$.</p>

<p>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:</p>

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

<hr>

<p>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:</p>

<p><img src="https://i.sstatic.net/uSaVw.png" alt="monopoly deadweight loss"></p>

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

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

<blockquote>
  <p>Let $u(x)=(1-x)^{t-1}$, where $t&gt;1$ is a parameter. Then:</p>
  
  <ul>
  <li>$U(x)=-(1-x)^{t}/t$. </li>
  <li>The first-order condition gives: $x_m=1/t$.</li>
  <li>$W=U(1)-U(0) = 1/t$</li>
  <li>$V=U(x_m)-U(0)=(1-(\frac{t-1}{t})^{t})/t$</li>
  <li>$W/V=1/[1-(\frac{t-1}{t})^{t}]$</li>
  </ul>
  
  <p>When $t\to\infty$,  $W/V \to 1/(1-1/e)\approx 1.58$, so for this family, $W/V$ is bounded.</p>
</blockquote>

<p>But what happens with other families? Here is another example:</p>

<blockquote>
  <p>Let $u(x)=e^{-t x}$, where $t&gt;0$ is a parameter. Then:</p>
  
  <ul>
  <li>$U(x)=-e^{-t x}/t$. </li>
  <li>The first-order condition gives: $x_m=1/t$.</li>
  <li>$W=U(1)-U(0) = (1-e^{-t})/t$</li>
  <li>$V=U(x_m)-U(0)=(1-e^{-1})/t$</li>
  <li>$W/V=(1-e^{-t})/(1-e^{-1})$</li>
  </ul>
  
  <p>When $t\to\infty$, again $W/V \to 1/(1-1/e)\approx 1.58$, so here again $W/V$ is bounded.</p>
</blockquote>

<p>And a third example, which I had to solve numerically:</p>

<blockquote>
  <p>Let $u(x)=\ln(a-x)$, where $a&gt;2$ is a parameter. Then:</p>
  
  <ul>
  <li>$U(x)=-(a-x)log(a-x)-x$. </li>
  <li>The first-order condition gives: $x_m=(a-x_m)\ln(a-x_m)$. Using this <a href="https://www.desmos.com/calculator/bmu1ilr0jp" rel="noreferrer">desmos graph</a>, 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.</li>
  <li>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. </li>
  </ul>
</blockquote>

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

CC BY-SA 3.0
2015-04-16T18:15:00.220Z
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?
Alecos Papadopoulos
CC BY-SA 3.0
2015-04-16T18:18:39.807Z
"Let u(x) be the consumers' utility function (which is also the inverse demand function)." $$. $$Isn´t it the consumers $\texttt{marginal}$ utility function ?
callculus42
CC BY-SA 3.0
2015-04-16T18:38:23.027Z
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.
FooBar
CC BY-SA 3.0
2015-04-16T19:28:43.473Z
@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.
Erel Segal-Halevi
CC BY-SA 3.0
2015-04-16T19:39:55.820Z
@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". https://en.wikipedia.org/wiki/Deadweight_loss
Erel Segal-Halevi
<microeconomics><mathematical-economics><social-welfare><monopoly>
For what demand function is a monopoly most harmful?
Erel Segal-Halevi
# For what demand function is a monopoly most harmful?

(Asked by: username_0 on 2015-04-16)

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:

![monopoly deadweight loss](https://i.sstatic.net/uSaVw.png)

* $W = U(1)-U(0)$, 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](https://www.desmos.com/calculator/bmu1ilr0jp), 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?

**Tags:** microeconomics, mathematical-economics, social-welfare, monopoly

> **Comments:**
> **username_1**: 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?  
> **username_2**: "Let u(x) be the consumers' utility function (which is also the inverse demand function)." $$. $$Isn´t it the consumers $\texttt{marginal}$ utility function ?  
> **username_3**: 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.  
> **username_0**: @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.  
> **username_0**: @username_3 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". https://en.wikipedia.org/wiki/Deadweight_loss  

---

## 1 answers

### Answer by username_4 on 2015-04-18. Score: 4 | Accepted answer

An arbitrarily large ratio should occur with demand curve

$P=\begin{cases}
\frac{1}{Q} & \text{if } Q>1 \\
2-Q & \text{if } Q\leq 1 \\
\end{cases}$.

The monopolist prices at $P=1$, but the consumers' surplus if $P=0$ is infinite, because the area under the demand curve contains $\int\_1^\infty \frac{1}{Q}dQ=\infty$.

> **Comments:**
> **username_0**: Thanks! Is there any reference where this issue is discussed? I would expect it to appear in standard textbooks in mircoeconomics, but didn't find it in any book I looked at.  
> **username_4**: I don't know of any references, sorry.  

---

