<p>It simply depends on the shape of the marginal external damages. If the <strong>total</strong> damages are linear in output, then the <strong>marginal</strong> damages are constant, and the gradient of the marginal social cost is the same as that of the marginal private cost. </p>

<p>If the <strong>total</strong> damages are quadratic (or some other form with increasing <em>marginal</em> damages), the marginal social cost has a steeper slope than the marginal private cost.</p>

<p>In a simple example: if the marginal <em>private</em> costs are <span class="math-container">$MPC(Q)=70+2Q$</span>, where <span class="math-container">$Q$</span> is quantity produced and <em>total</em> damages are <span class="math-container">$D(Q)=50Q$</span>, then marginal external damages are <span class="math-container">$MED=50$</span> and the marginal social costs of production would be:</p>

<p><span class="math-container">$MSC=MPC+MED=70+2Q+50=120+2Q$</span></p>

<p>If in contrast the damage function would be <span class="math-container">$D(Q)=50Q+2Q^2$</span>, then marginal external damages would be <span class="math-container">$MED=50+4Q$</span> and the marginal social costs of production:</p>

<p><span class="math-container">$MSC=MPC+MED=120+6Q$</span></p>

<p>As you can see the latter has a steeper slope than the original MPC, whereas the former does not.</p>

CC BY-SA 4.0
2019-05-28T08:29:27.583Z
2019-05-28T08:29:27.583Z
Maarten Punt
<p><a href="https://i.sstatic.net/GdVbZ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GdVbZ.jpg" alt="Market Failure (negative externalities diagram"></a></p>

<p>Why does the social cost supply curve in a negative externalities diagram have a steeper gradient than the private cost to firms?</p>

<p>I assumed it should only be shifted upwards as the negative social costs are directly proportional to the production quantity? </p>

<p>Any explanation is greatly appreciated.  </p>

<supply-and-demand><market-failure>
Understanding the gradient of the social cost curve in a market failure/ negative externalities diagram
Luke Wanless
# Understanding the gradient of the social cost curve in a market failure/ negative externalities diagram

(Asked by: username_0 on 2019-05-27)

[![Market Failure (negative externalities diagram](https://i.sstatic.net/GdVbZ.jpg)](https://i.sstatic.net/GdVbZ.jpg)

Why does the social cost supply curve in a negative externalities diagram have a steeper gradient than the private cost to firms?

I assumed it should only be shifted upwards as the negative social costs are directly proportional to the production quantity?

Any explanation is greatly appreciated.

**Tags:** supply-and-demand, market-failure


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## 1 answers

### Answer by username_1 on 2019-05-28. Score: 3 | Accepted answer

It simply depends on the shape of the marginal external damages. If the **total** damages are linear in output, then the **marginal** damages are constant, and the gradient of the marginal social cost is the same as that of the marginal private cost.

If the **total** damages are quadratic (or some other form with increasing *marginal* damages), the marginal social cost has a steeper slope than the marginal private cost.

In a simple example: if the marginal *private* costs are $MPC(Q)=70+2Q$, where $Q$ is quantity produced and *total* damages are $D(Q)=50Q$, then marginal external damages are $MED=50$ and the marginal social costs of production would be:

$MSC=MPC+MED=70+2Q+50=120+2Q$

If in contrast the damage function would be $D(Q)=50Q+2Q^2$, then marginal external damages would be $MED=50+4Q$ and the marginal social costs of production:

$MSC=MPC+MED=120+6Q$

As you can see the latter has a steeper slope than the original MPC, whereas the former does not.


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