<p>For example when you are talking about profit maximization starting from a profit function $\pi(q)$, the main condition for a maximum is that: $$\frac{\partial \pi}{\partial q}=0$$ This is the FOC (first order condition).</p>

<p>Though, to be sure that what you have found above is a <em>true</em> maximum you should also check a 'secondary' condition which is: $$\frac{\partial^2 \pi}{\partial q^2}&lt;0$$ This is called the SOC (second order condition).</p>

CC BY-SA 3.0
2015-01-21T22:50:30.563Z
2015-01-21T22:50:30.563Z
ChicagoCubs
<p>The target is to find a local maximum (or minimum) of a function.</p>

<p>If the function is differentiable twice:</p>

<ul>
<li>The <a href="http://mathworld.wolfram.com/FirstDerivativeTest.html" rel="nofollow noreferrer">first derivative test</a> will tell you if it's an local extremum.</li>
<li>The <a href="http://mathworld.wolfram.com/SecondDerivativeTest.html" rel="nofollow noreferrer">second derivative test</a> will tell you if it's a local maximum or a minimum.</li>
</ul>

<p>In case you function is not differentiable, you can do a more general <a href="http://mathworld.wolfram.com/ExtremumTest.html" rel="nofollow noreferrer">extremum test</a>.</p>

<p>Note: it's impossible to construct an algorithm to find a <a href="http://mathworld.wolfram.com/GlobalExtremum.html" rel="nofollow noreferrer">global maximum for an arbitrary function</a>.</p>

<p>Neoclassical economists certainly rename those two mathematical methods to <em>first-order conditions</em> and <em>second-order conditions</em> to look cool or for other historical reasons. Why use a name widely used when you can just make one up?</p>

<p>The term is also used on <a href="https://en.wikipedia.org/wiki/Constrained_optimization" rel="nofollow noreferrer">constrained maximization</a> when they use the <a href="https://en.wikipedia.org/wiki/Lagrange_multiplier" rel="nofollow noreferrer">Lagrange multiplier</a> method and <a href="https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" rel="nofollow noreferrer">Karush–Kuhn–Tucker conditions</a>. Again, I don't thinks the term is used by non-economist.</p>

CC BY-SA 4.0
2018-11-18T20:34:42.187Z
2018-11-18T20:59:01.360Z
2018-11-18T20:59:01.360Z
gagarine
<p>Suppose you have a differentiable function $f(x)$, which you want to optimize by choosing $x$. If $f(x)$ is utility or profit, then you want to choose $x$ (i.e. consumption bundle or quantity produced) to make the value of $f$ as large as possible. If $f(x)$ is a cost function, then you want to choose $x$ to make $f$ as small as possible. FOC and SOC are conditions that determine whether a solution maximizes or minimizes a given function. </p>

<p>At the undergrad level, what is usually the case is that you need to choose $x^*$ such that the derivative of $f$ is equal to zero:
$$f'(x^*)=0.$$
This is the FOC. The intuition for this condition is that a function attains its extremum (either maximum or minimum) when its derivative is equal to zero (see picture below). [You should be aware that there are more subtleties involved: look up terms like "interior vs corner solutions", "global vs local maximum/minimum", and "saddle point" to learn more].</p>

<p><img src="https://i.sstatic.net/AGgAo.png" alt="Example functions where x_star is a maximum and a minimum"></p>

<p>However, as the picture illustrates, simply finding $x^*$ where $f'(x^*)=0$ is not enough to conclude that $x^*$ is the solution that maximizes or minimizes the objective function. In both graphs, the function attains a zero slope at $x^*$, but $x^*$ is a maximizer in the left graph, but a minimizer in the right graph. </p>

<p>To check whether $x^*$ is a maximizer or a minimizer, you need the SOC. The SOC for maximizer is 
$$f''(x^*)&lt;0$$ 
and the SOC for minimizer is
$$f''(x^*)&gt; 0.$$
Intuitively, if $x^*$ maximizes $f$, the slope of $f$ around $x^*$ is decreasing. Take the left graph, where $x^*$ is a maximizer. We see that the slope of $f$ is positive on the left of $x^*$ and negative on the right. Thus, around the neighborhood of $x^*$, as $x$ increases, $f'(x)$ decreases. The intuition for the case of minimizer is similar. </p>

CC BY-SA 4.0
2018-11-10T16:34:55.110Z
But why it's not called "First derivative test" is still a mystery for me.
gagarine
CC BY-SA 3.0
2015-01-21T23:47:00.703Z
2015-01-22T01:30:58.850Z
2015-01-22T01:30:58.850Z
Herr K.
<p>I keep seeing the terms first-order conditions and second-order conditions used in my undergrad economics class on production functions, monopolies, etc but I have no idea what these terms mean. It seems like a completely ambiguous term. What kind of conditions? </p>

<p><strong>Can someone explain what these terms mean? If it is context dependent, provided some of them most elementary meanings you associate with the term.</strong></p>

<microeconomics>
What are FOCs and SOCs?
Stan Shunpike
# What are FOCs and SOCs?

(Asked by: username_0 on 2015-01-21)

I keep seeing the terms first-order conditions and second-order conditions used in my undergrad economics class on production functions, monopolies, etc but I have no idea what these terms mean. It seems like a completely ambiguous term. What kind of conditions?

**Can someone explain what these terms mean? If it is context dependent, provided some of them most elementary meanings you associate with the term.**

**Tags:** microeconomics


---

## 3 answers

### Answer by username_3 on 2015-01-21. Score: 30 | Accepted answer

Suppose you have a differentiable function $f(x)$, which you want to optimize by choosing $x$. If $f(x)$ is utility or profit, then you want to choose $x$ (i.e. consumption bundle or quantity produced) to make the value of $f$ as large as possible. If $f(x)$ is a cost function, then you want to choose $x$ to make $f$ as small as possible. FOC and SOC are conditions that determine whether a solution maximizes or minimizes a given function.

At the undergrad level, what is usually the case is that you need to choose $x^\*$ such that the derivative of $f$ is equal to zero:
$$f'(x^\*)=0.$$
This is the FOC. The intuition for this condition is that a function attains its extremum (either maximum or minimum) when its derivative is equal to zero (see picture below). [You should be aware that there are more subtleties involved: look up terms like "interior vs corner solutions", "global vs local maximum/minimum", and "saddle point" to learn more].

![Example functions where x_star is a maximum and a minimum](https://i.sstatic.net/AGgAo.png)

However, as the picture illustrates, simply finding $x^\*$ where $f'(x^\*)=0$ is not enough to conclude that $x^\*$ is the solution that maximizes or minimizes the objective function. In both graphs, the function attains a zero slope at $x^\*$, but $x^\*$ is a maximizer in the left graph, but a minimizer in the right graph.

To check whether $x^\*$ is a maximizer or a minimizer, you need the SOC. The SOC for maximizer is
$$f''(x^\*)<0$$
and the SOC for minimizer is
$$f''(x^\*)> 0.$$
Intuitively, if $x^\*$ maximizes $f$, the slope of $f$ around $x^\*$ is decreasing. Take the left graph, where $x^\*$ is a maximizer. We see that the slope of $f$ is positive on the left of $x^\*$ and negative on the right. Thus, around the neighborhood of $x^\*$, as $x$ increases, $f'(x)$ decreases. The intuition for the case of minimizer is similar.

> **Comments:**
> **username_2**: But why it's not called "First derivative test" is still a mystery for me.  

---

### Answer by username_1 on 2015-01-21. Score: 6

For example when you are talking about profit maximization starting from a profit function $\pi(q)$, the main condition for a maximum is that: $$\frac{\partial \pi}{\partial q}=0$$ This is the FOC (first order condition).

Though, to be sure that what you have found above is a *true* maximum you should also check a 'secondary' condition which is: $$\frac{\partial^2 \pi}{\partial q^2}<0$$ This is called the SOC (second order condition).


---

### Answer by username_2 on 2018-11-18. Score: 4

The target is to find a local maximum (or minimum) of a function.

If the function is differentiable twice:

* The [first derivative test](http://mathworld.wolfram.com/FirstDerivativeTest.html) will tell you if it's an local extremum.
* The [second derivative test](http://mathworld.wolfram.com/SecondDerivativeTest.html) will tell you if it's a local maximum or a minimum.

In case you function is not differentiable, you can do a more general [extremum test](http://mathworld.wolfram.com/ExtremumTest.html).

Note: it's impossible to construct an algorithm to find a [global maximum for an arbitrary function](http://mathworld.wolfram.com/GlobalExtremum.html).

Neoclassical economists certainly rename those two mathematical methods to *first-order conditions* and *second-order conditions* to look cool or for other historical reasons. Why use a name widely used when you can just make one up?

The term is also used on [constrained maximization](https://en.wikipedia.org/wiki/Constrained_optimization) when they use the [Lagrange multiplier](https://en.wikipedia.org/wiki/Lagrange_multiplier) method and [Karush–Kuhn–Tucker conditions](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions). Again, I don't thinks the term is used by non-economist.


---

