<p>Sorry I don't have enough reputation yet in the Economics community to comment. But the answer provided by @Alecos Papadopoulos can be illustrated further. Please feel free to move my "answer" to comments of his answer. </p>

<p>With the assumption of only $F' &gt; 0$ and $F &gt; 0$ and $h' &gt; 0$, then there is a simple case where the minimization problem has no solution. Consider when $F(q) = \sqrt{q}$ and $h(x) = x$. Then the objective function becomes $1 / \sqrt{q} - \sqrt{q} - c$, and if you're considering for any $q &gt; 0$, then there is no global minimizer for this (i.e. globally decreasing for all $q &gt; 0$ and hence the minimizer is $q \to \infty$. Thus, the FOC you'd derived is not true. Indeed, the condition you'd provided would render as $4 \le 1$ --- clearly impossible. </p>

<p>In general, an increasing function $F$ over a linear function (i.e. say $F(x) / x$ in your case) will <em>not</em> necessarily be convex. And furthermore, when you subtract off the form $h \circ F$, the resulting $-h \circ F$ could again depend on the form of $F$ (which may not have any particular geometric form you could exploit). I think what would fix this model is a form such that $F(q) / q$ is convex in $q$ and $h \circ F$ is concave, so that $-h \circ F$ is convex, then the whole problem will be convex. </p>

CC BY-SA 3.0
2015-08-26T18:28:50.303Z
2015-08-26T18:40:46.487Z
2015-08-26T18:40:46.487Z
user32416
<p>This is perhaps a bit more complex than you might suspect. Let $F(q), F'&gt;0$ the fixed (setup) cost, which is a positive function of the productive capacity, which here is represented by actual production level (we assume that the factory will operate at full capacity). Also, variable cost per unit produced is</p>

<p>$$V(q, F) = [c-h(F)]\cdot q,\;\; h&lt;c , h'&gt; 0$$</p>

<p>This incorporates the assumption that there are cost-reducing economies of scale if the factory is larger.</p>

<p>Before even factoring in also "time to build", consider an average cost minimization problem</p>

<p>$$\min_q AC = \Big[\frac {1}{q}F(q) + \frac {1}{q} [c-h(F)]\cdot q \Big]$$</p>

<p>which simplifies to </p>

<p>$$\min_q AC =\Big[\frac {1}{q}F(q) + c-h(F(q))\Big]$$</p>

<p>The first-order condition for minimization is </p>

<p>$$\frac{\partial AC}{q} = \frac{F'q-F(q)}{q^2}-h'\cdot F' =0$$</p>

<p>$$\implies F'q-F(q) =q^2h'\cdot F' \implies (h'\cdot F')q^2-F'q+F(q) = 0$$</p>

<p>This is an (implicit) quadratic in $q$, and in order to have a real and strictly positive $q^*$ as a candidate minimizer, one can deduce that the following condition must be imposed (otherwise the AC will always be increasing):</p>

<p>$$4h'(F/F') \leq 1$$</p>

<p>My point is that all these must acquire specific functional forms that behave realistically for a wide range of values... and then you say you want to bring in the intertemporal aspect, going for optimal control/dynamic programming.... hmm.</p>

CC BY-SA 3.0
2015-08-26T17:30:49.770Z
2015-08-26T17:30:49.770Z
Alecos Papadopoulos
<p>I know there has got to be a standard set of theories or formulas to chart this but I don't know how to search for it. Other than a great place like this. :)</p>

<p>I'm a software developer and not an economist but I need to analyze data of factories that have simple defined expense to set up, each has a specified output and a length of time to build. </p>

<p>While a larger factory might have the lowest cost per unit of production and therefore be the best bang the buck (assuming the setup cost was zero), the actual setup cost might be so large that it would take a long time to actually build the factory. It is actually more economical in the long run to build less costly and lower performing factories because in the long run you will have produced more and aquire the cost of the larger factory faster. </p>

<p>What I am attempting to build is a formula/program that will advise on the next most profitable move and the required time to get there. </p>

<p>If you haven't guessed this is actually to be tested an ran on a game scenario. However, I have many other applications and thought this would be a good start. </p>

<p>I hope that makes sense. The cost of production, the output and such are all intentionally simplistic. There's no need to account for outside/variable factors in this particular scenario. </p>

<p>If you can lead me to a theory/concept, perhaps a link or two or examples of this type of analysis given as set of inputs that would be great!</p>

<p>Thanks!</p>

<reference-request><investment><producer-theory><profit-maximization>
What is this economic theory? Cost of investment vs production accounting for time?
Matt Penner
# What is this economic theory? Cost of investment vs production accounting for time?

(Asked by: username_0 on 2015-08-24)

I know there has got to be a standard set of theories or formulas to chart this but I don't know how to search for it. Other than a great place like this. :)

I'm a software developer and not an economist but I need to analyze data of factories that have simple defined expense to set up, each has a specified output and a length of time to build.

While a larger factory might have the lowest cost per unit of production and therefore be the best bang the buck (assuming the setup cost was zero), the actual setup cost might be so large that it would take a long time to actually build the factory. It is actually more economical in the long run to build less costly and lower performing factories because in the long run you will have produced more and aquire the cost of the larger factory faster.

What I am attempting to build is a formula/program that will advise on the next most profitable move and the required time to get there.

If you haven't guessed this is actually to be tested an ran on a game scenario. However, I have many other applications and thought this would be a good start.

I hope that makes sense. The cost of production, the output and such are all intentionally simplistic. There's no need to account for outside/variable factors in this particular scenario.

If you can lead me to a theory/concept, perhaps a link or two or examples of this type of analysis given as set of inputs that would be great!

Thanks!

**Tags:** reference-request, investment, producer-theory, profit-maximization


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## 2 answers

### Answer by username_1 on 2015-08-26. Score: 2

Sorry I don't have enough reputation yet in the Economics community to comment. But the answer provided by @username_2 can be illustrated further. Please feel free to move my "answer" to comments of his answer.

With the assumption of only $F' > 0$ and $F > 0$ and $h' > 0$, then there is a simple case where the minimization problem has no solution. Consider when $F(q) = \sqrt{q}$ and $h(x) = x$. Then the objective function becomes $1 / \sqrt{q} - \sqrt{q} - c$, and if you're considering for any $q > 0$, then there is no global minimizer for this (i.e. globally decreasing for all $q > 0$ and hence the minimizer is $q \to \infty$. Thus, the FOC you'd derived is not true. Indeed, the condition you'd provided would render as $4 \le 1$ --- clearly impossible.

In general, an increasing function $F$ over a linear function (i.e. say $F(x) / x$ in your case) will *not* necessarily be convex. And furthermore, when you subtract off the form $h \circ F$, the resulting $-h \circ F$ could again depend on the form of $F$ (which may not have any particular geometric form you could exploit). I think what would fix this model is a form such that $F(q) / q$ is convex in $q$ and $h \circ F$ is concave, so that $-h \circ F$ is convex, then the whole problem will be convex.


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### Answer by username_2 on 2015-08-26. Score: 1

This is perhaps a bit more complex than you might suspect. Let $F(q), F'>0$ the fixed (setup) cost, which is a positive function of the productive capacity, which here is represented by actual production level (we assume that the factory will operate at full capacity). Also, variable cost per unit produced is

$$V(q, F) = [c-h(F)]\cdot q,\;\; h 0$$

This incorporates the assumption that there are cost-reducing economies of scale if the factory is larger.

Before even factoring in also "time to build", consider an average cost minimization problem

$$\min\_q AC = \Big[\frac {1}{q}F(q) + \frac {1}{q} [c-h(F)]\cdot q \Big]$$

which simplifies to

$$\min\_q AC =\Big[\frac {1}{q}F(q) + c-h(F(q))\Big]$$

The first-order condition for minimization is

$$\frac{\partial AC}{q} = \frac{F'q-F(q)}{q^2}-h'\cdot F' =0$$

$$\implies F'q-F(q) =q^2h'\cdot F' \implies (h'\cdot F')q^2-F'q+F(q) = 0$$

This is an (implicit) quadratic in $q$, and in order to have a real and strictly positive $q^\*$ as a candidate minimizer, one can deduce that the following condition must be imposed (otherwise the AC will always be increasing):

$$4h'(F/F') \leq 1$$

My point is that all these must acquire specific functional forms that behave realistically for a wide range of values... and then you say you want to bring in the intertemporal aspect, going for optimal control/dynamic programming.... hmm.


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