<p><span class="math-container">$F(K, L) = K^{\frac{1}{4}}L^\frac{3}{4}+K^{\frac{3}{4}}L^\frac{1}{4}$</span></p>

CC BY-SA 4.0
2022-06-23T14:32:34.100Z
How do we know this is not just another form of $$\sqrt{5} \cdot K^{1/3} \cdot L^{2/3} \ \ ?? $$
Giskard
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2022-06-23T14:35:59.430Z
It cannot be of this form because your expression does not satisfy $F(K, L) = F(L, K)$ whereas mine does.
Amit
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2022-06-23T14:43:41.140Z
You're totally right, i overseen this clear fact that the sum of two Cobb-Douglas functions (with CRS) again fulfill the in the question mentioned conditions. Based on this do you know if all functions that are neoclassical in the sense of the question are given as an infinite sum or finite sum of Cobb-Douglas functions (with CRS)?
I am strongly interested in how the family of functions has to look in order to fulfill the conditions to be a neoclassical production function, i.e., does there exist an "explicit" presentation of function family.
maximilian43
CC BY-SA 4.0
2022-06-23T14:44:56.337Z
@Giskard Likewise, we can rule out all functions such that $\alpha \neq \frac{1}{2}$. So, there is only one possibility left which is $G(K, L) = 2K^{\frac{1}{2}}L^{\frac{1}{2}}$. This can be ruled out because $G(16, 1) \neq F(16, 1)$.
Amit
CC BY-SA 4.0
2022-06-23T17:29:14.350Z
@maximilian43 I don't think that this is true either. I think it is possible to construct functions that satisfy diminishing returns to factors, are differentiable in the interior, satisfy inada conditions and CRS, and are not Cobb Douglas or sums of Cobb Douglas.
Amit
CC BY-SA 4.0
2022-06-23T17:43:34.447Z
@maximilian43 For example, if we don't assume differentiability, and just want a function that satisfy CRS, diminishing returns, inada conditions then the following function also works: $F(K, L) = \min(K^{\frac{1}{4}}L^{\frac{3}{4}}, K^{\frac{3}{4}}L^{\frac{1}{4}})$
Amit
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2022-06-23T17:44:32.483Z
But inada conditions are conditions on the derivative of the function, so we need to assume differentiability?
maximilian43
CC BY-SA 4.0
2022-06-23T17:48:37.260Z
This function is differentiable everywhere in the interior except at $K = L$, and inada conditions are on the limit of the derivatives as $L$ goes to $0$ or $\infty$ (holding $K$ fixed), so inada conditions will be satisfied.
Amit
CC BY-SA 4.0
2022-06-23T18:09:21.863Z
That's true, i'm still curious about if differentiability leads to my aforementioned conjecture that the functions are a, possible infinite , sum of Cobb-Douglas functions. In any case, all your suggestions are of high quality, thank you!
maximilian43
CC BY-SA 4.0
2022-06-27T09:35:55.760Z
@maximilian43 Consider $F(K, L) = \begin{cases} K^\frac{1}{4}L^\frac{3}{4} & \text{if } K < \frac{1}{3}L \\  K^\frac{3}{4}L^\frac{1}{4} & \text{if } K > 3L \\ \frac{3^{\frac{3}{4}}}{4}(K+L) & \text{otherwise} \end{cases} $
Amit
CC BY-SA 4.0
2022-06-27T09:52:26.233Z
@maximilian43 $F$ defined above is monotonically increasing,  concave, differentiable, and satisfy both CRS and Inada conditions.
Amit
CC BY-SA 4.0
2022-06-27T10:11:32.217Z
@Amit But this function is clearly not $C^2$ since the function defined for "otherwise" has $0$ as second derivative in $K$ and $L$
maximilian43
CC BY-SA 4.0
2022-06-27T10:20:21.537Z
@maximilian43 It is not strictly concave that I agree.
Amit
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2022-06-27T10:22:56.687Z
@maximilian43 And yes in the region $\frac{1}{3}L < K < 3L$, $\frac{\partial^2 F}{\partial L^2} = 0$, $\frac{\partial^2 F}{\partial K^2} = 0$.
Amit
CC BY-SA 4.0
2022-06-27T10:29:51.483Z
But this $F$ satisfy everything you wrote in your question: diminishing returns to factor + Inada conditions + constant returns + differentiability in the interior + monotonicity. Only diminishing returns to factor is replaced by diminishing (or constant ) returns to factor.
Amit
CC BY-SA 4.0
2022-06-27T10:37:57.070Z
But it is not twice continuous differentiable since $\partial^2 F/ \partial L^2 = 0$ in $1/3 L < K < 3L$. So the continuity is lost.
maximilian43
CC BY-SA 4.0
2022-06-27T10:40:35.227Z
It is linear and hence twice continuously differentiable in the region $\frac{1}{3}L < K < 3L$.
Amit
CC BY-SA 4.0
2022-06-27T10:46:48.650Z
But the whole function $F$ is not twice continuous differentiable, piecewise yes.
maximilian43
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2022-06-27T10:59:19.830Z
You're now adding more requirements. Anyways it is also possible to construct a twice differentiable function also.
Amit
CC BY-SA 4.0
2022-06-27T11:02:22.620Z
This was the original requirement from the base questions. Yes every piece of the function $F$ is even $C^\infty$, but $F$ itself not twice continuous differentiable.
maximilian43
CC BY-SA 4.0
2022-06-23T13:39:21.483Z
2022-06-23T13:39:21.483Z
Amit
<p>Assume the neoclassical production function <span class="math-container">$$F(K,L)\colon [0,\infty) \times [0,\infty) \to [0,\infty)$$</span> twice continuously differentiable, i.e., F is montone increasing and concave, i.e.,
<span class="math-container">$$
\partial F(\bar K,\bar L) /\partial K &gt; 0, \quad \partial F(\bar K,\bar L) /\partial L &gt; 0,\\
\partial^2 F(\bar K,\bar L) /\partial K^2 &lt; 0, \quad \partial^2 F(\bar K,\bar L) /\partial L^2 &lt; 0,
$$</span>holds, i.e., <span class="math-container">$F$</span> has diminishing returns, for all <span class="math-container">$\bar K, \bar L \in (0,\infty)$</span>. Moreover, the Inada conditions hold, i.e.,
<span class="math-container">$$\lim_{K\to 0} \partial F(\bar K,\bar L)/\partial K = \infty, \quad \lim_{K\to \infty} \partial F(\bar K,\bar L)/\partial K = 0,\\
\lim_{L\to 0} \partial F(\bar K,\bar L)/\partial L = \infty, \quad \lim_{L\to \infty} \partial F(\bar K,\bar L)/\partial L = 0$$</span>
for all <span class="math-container">$\bar K, \bar L \in (0,\infty)$</span>.
Finally <span class="math-container">$F$</span> yields constant returns to scale, i.e., <span class="math-container">$F$</span> is positive homogeneous of degree <span class="math-container">$1$</span>, i.e., <span class="math-container">$F(\lambda K, \lambda L) = \lambda F(K,L)$</span> for all <span class="math-container">$\lambda \in (0,\infty)$</span> and <span class="math-container">$K,L \in [0,\infty)$</span>.</p>
<p>I am aware of the paper <a href="https://www.sciencedirect.com/science/article/abs/pii/S0165176503002180" rel="nofollow noreferrer">&quot;Inada conditions imply that production function must be asymptotically Cobb–Douglas&quot;
</a>, by Barelli and Pessôa.</p>
<p>So the Inada conditions with the monotonicity and concavity imply asymptotically Cobb-Douglas behaviour of every <span class="math-container">$F$</span> fulfilling those conditions. However, this argument is without imposing a constant returns to scale assumption.</p>
<p><strong>The question is:</strong> Does diminishing returns + Inada conditions + constant returns to scale uniquely determine <span class="math-container">$F$</span> as given by
<span class="math-container">$$F(K,L) = c K^\alpha L^{1-\alpha}$$</span>
with <span class="math-container">$c \in (0,\infty)$</span> and <span class="math-container">$\alpha \in (0,1)$</span>?</p>
<p>I know without constant returns to scale in 1D, i.e. utility functions twice continously differentiable that are monotone increasing and concave and fulfill Inada conditions, we can choose two functions, e.g., <span class="math-container">$\tilde u(c)$</span> and <span class="math-container">$\bar u(c)$</span>, with <span class="math-container">$\tilde u \neq \bar u$</span>, where <span class="math-container">$\tilde u(c_0) = \bar u(c_0)$</span>, <span class="math-container">$\tilde u'(c_0) = \bar u'(c_0)$</span>, and <span class="math-container">$\tilde u''(c_0) = \bar u''(c_0)$</span> holds.
Then is the piecewise defined function <span class="math-container">$$u(c) = \begin{cases}\tilde u(c), &amp; c \le c_0 \\ \bar u(c) &amp; c &gt; c_0\end{cases}$$</span> also monotone increasing and concave and fulfills the Inada conditions (and clearly twice continously differentiable).</p>
<p>Somehow i feel that the constant returns to scale enforces that this piecewise &quot;trick&quot; is not working. But i couldn't manage to find information about if these neoclassical conditions on the production function imply Cobb-Douglas type.</p>

CC BY-SA 4.0
2022-06-23T16:26:30.733Z
The result on the production function being asymptotically Cobb-Douglas [is not correct](https://doi.org/10.1016/j.econlet.2007.09.035).
Michael Greinecker
CC BY-SA 4.0
2022-06-23T16:56:14.370Z
To clarify the comment for others, reading the content without following the links, the result that the function being asymptotically Cobb-Douglas is still right but the arguments of the paper cited in the question are wrong, this is what you want to say?
maximilian43
CC BY-SA 4.0
2022-06-23T18:36:25.900Z
No, the function need not be Cobb-Douglas. It just needs to have an elasticity of substitution asymptotically to one. The paper gives an example of that that is not CD.
Michael Greinecker
CC BY-SA 4.0
2022-06-23T18:42:10.047Z
I am not sure about this fact, are you aware of [this](https://mpra.ub.uni-muenchen.de/82304/1/MPRA_paper_82304.pdf) paper?
maximilian43
CC BY-SA 4.0
2022-06-23T18:45:37.793Z
I am not; I will take a look at it.
Michael Greinecker
CC BY-SA 4.0
2022-06-23T22:21:23.957Z
There is a [newer published paper](https://doi.org/10.1016/j.econlet.2021.109786) that suggests the CD-result is actually correct, though the original proof was not.
Michael Greinecker
CC BY-SA 4.0
2022-06-23T23:02:07.303Z
My institute does not subscribe to that site, therefore i sticked and verified for myself the public available result from the aforementioned link. Thank you for sharing the more recent paper, especially to add value to this discussion if someone finds it later.
maximilian43
<utility><economic-growth><production-function><ramsey>
Does a neoclassical production with constant returns to scale implies type of Cobb-Douglas
maximilian43
# Does a neoclassical production with constant returns to scale implies type of Cobb-Douglas

(Asked by: username_0 on 2022-06-23)

Assume the neoclassical production function $$F(K,L)\colon [0,\infty) \times [0,\infty) \to [0,\infty)$$ twice continuously differentiable, i.e., F is montone increasing and concave, i.e.,
$$
\partial F(\bar K,\bar L) /\partial K > 0, \quad \partial F(\bar K,\bar L) /\partial L > 0,\\
\partial^2 F(\bar K,\bar L) /\partial K^2 < 0, \quad \partial^2 F(\bar K,\bar L) /\partial L^2 < 0,
$$holds, i.e., $F$ has diminishing returns, for all $\bar K, \bar L \in (0,\infty)$. Moreover, the Inada conditions hold, i.e.,
$$\lim\_{K\to 0} \partial F(\bar K,\bar L)/\partial K = \infty, \quad \lim\_{K\to \infty} \partial F(\bar K,\bar L)/\partial K = 0,\\
\lim\_{L\to 0} \partial F(\bar K,\bar L)/\partial L = \infty, \quad \lim\_{L\to \infty} \partial F(\bar K,\bar L)/\partial L = 0$$
for all $\bar K, \bar L \in (0,\infty)$.
Finally $F$ yields constant returns to scale, i.e., $F$ is positive homogeneous of degree $1$, i.e., $F(\lambda K, \lambda L) = \lambda F(K,L)$ for all $\lambda \in (0,\infty)$ and $K,L \in [0,\infty)$.

I am aware of the paper ["Inada conditions imply that production function must be asymptotically Cobb–Douglas"](https://www.sciencedirect.com/science/article/abs/pii/S0165176503002180), by Barelli and Pessôa.

So the Inada conditions with the monotonicity and concavity imply asymptotically Cobb-Douglas behaviour of every $F$ fulfilling those conditions. However, this argument is without imposing a constant returns to scale assumption.

**The question is:** Does diminishing returns + Inada conditions + constant returns to scale uniquely determine $F$ as given by
$$F(K,L) = c K^\alpha L^{1-\alpha}$$
with $c \in (0,\infty)$ and $\alpha \in (0,1)$?

I know without constant returns to scale in 1D, i.e. utility functions twice continously differentiable that are monotone increasing and concave and fulfill Inada conditions, we can choose two functions, e.g., $\tilde u(c)$ and $\bar u(c)$, with $\tilde u \neq \bar u$, where $\tilde u(c\_0) = \bar u(c\_0)$, $\tilde u'(c\_0) = \bar u'(c\_0)$, and $\tilde u''(c\_0) = \bar u''(c\_0)$ holds.
Then is the piecewise defined function $$u(c) = \begin{cases}\tilde u(c), & c \le c\_0 \\ \bar u(c) & c > c\_0\end{cases}$$ also monotone increasing and concave and fulfills the Inada conditions (and clearly twice continously differentiable).

Somehow i feel that the constant returns to scale enforces that this piecewise "trick" is not working. But i couldn't manage to find information about if these neoclassical conditions on the production function imply Cobb-Douglas type.

**Tags:** utility, economic-growth, production-function, ramsey

> **Comments:**
> **username_1**: The result on the production function being asymptotically Cobb-Douglas [is not correct](https://doi.org/10.1016/j.econlet.2007.09.035).  
> **username_0**: To clarify the comment for others, reading the content without following the links, the result that the function being asymptotically Cobb-Douglas is still right but the arguments of the paper cited in the question are wrong, this is what you want to say?  
> **username_1**: No, the function need not be Cobb-Douglas. It just needs to have an elasticity of substitution asymptotically to one. The paper gives an example of that that is not CD.  
> **username_0**: I am not sure about this fact, are you aware of [this](https://mpra.ub.uni-muenchen.de/82304/1/MPRA_paper_82304.pdf) paper?  
> **username_1**: I am not; I will take a look at it.  
> **username_1**: There is a [newer published paper](https://doi.org/10.1016/j.econlet.2021.109786) that suggests the CD-result is actually correct, though the original proof was not.  
> **username_0**: My institute does not subscribe to that site, therefore i sticked and verified for myself the public available result from the aforementioned link. Thank you for sharing the more recent paper, especially to add value to this discussion if someone finds it later.  

---

## 1 answers

### Answer by username_2 on 2022-06-23. Score: 4 | Accepted answer

$F(K, L) = K^{\frac{1}{4}}L^\frac{3}{4}+K^{\frac{3}{4}}L^\frac{1}{4}$

> **Comments:**
> **username_3**: How do we know this is not just another form of $$\sqrt{5} \cdot K^{1/3} \cdot L^{2/3} \ \ ?? $$  
> **username_2**: It cannot be of this form because your expression does not satisfy $F(K, L) = F(L, K)$ whereas mine does.  
> **username_0**: You're totally right, i overseen this clear fact that the sum of two Cobb-Douglas functions (with CRS) again fulfill the in the question mentioned conditions. Based on this do you know if all functions that are neoclassical in the sense of the question are given as an infinite sum or finite sum of Cobb-Douglas functions (with CRS)? I am strongly interested in how the family of functions has to look in order to fulfill the conditions to be a neoclassical production function, i.e., does there exist an "explicit" presentation of function family.  
> **username_2**: @username_3 Likewise, we can rule out all functions such that $\alpha \neq \frac{1}{2}$. So, there is only one possibility left which is $G(K, L) = 2K^{\frac{1}{2}}L^{\frac{1}{2}}$. This can be ruled out because $G(16, 1) \neq F(16, 1)$.  
> **username_2**: @username_0 I don't think that this is true either. I think it is possible to construct functions that satisfy diminishing returns to factors, are differentiable in the interior, satisfy inada conditions and CRS, and are not Cobb Douglas or sums of Cobb Douglas.  
> **username_2**: @username_0 For example, if we don't assume differentiability, and just want a function that satisfy CRS, diminishing returns, inada conditions then the following function also works: $F(K, L) = \min(K^{\frac{1}{4}}L^{\frac{3}{4}}, K^{\frac{3}{4}}L^{\frac{1}{4}})$  
> **username_0**: But inada conditions are conditions on the derivative of the function, so we need to assume differentiability?  
> **username_2**: This function is differentiable everywhere in the interior except at $K = L$, and inada conditions are on the limit of the derivatives as $L$ goes to $0$ or $\infty$ (holding $K$ fixed), so inada conditions will be satisfied.  
> **username_0**: That's true, i'm still curious about if differentiability leads to my aforementioned conjecture that the functions are a, possible infinite , sum of Cobb-Douglas functions. In any case, all your suggestions are of high quality, thank you!  
> **username_2**: @username_0 Consider $F(K, L) = \begin{cases} K^\frac{1}{4}L^\frac{3}{4} & \text{if } K < \frac{1}{3}L \\  K^\frac{3}{4}L^\frac{1}{4} & \text{if } K > 3L \\ \frac{3^{\frac{3}{4}}}{4}(K+L) & \text{otherwise} \end{cases} $  
> **username_2**: @username_0 $F$ defined above is monotonically increasing,  concave, differentiable, and satisfy both CRS and Inada conditions.  
> **username_0**: @username_2 But this function is clearly not $C^2$ since the function defined for "otherwise" has $0$ as second derivative in $K$ and $L$  
> **username_2**: @username_0 It is not strictly concave that I agree.  
> **username_2**: @username_0 And yes in the region $\frac{1}{3}L < K < 3L$, $\frac{\partial^2 F}{\partial L^2} = 0$, $\frac{\partial^2 F}{\partial K^2} = 0$.  
> **username_2**: But this $F$ satisfy everything you wrote in your question: diminishing returns to factor + Inada conditions + constant returns + differentiability in the interior + monotonicity. Only diminishing returns to factor is replaced by diminishing (or constant ) returns to factor.  
> **username_0**: But it is not twice continuous differentiable since $\partial^2 F/ \partial L^2 = 0$ in $1/3 L < K < 3L$. So the continuity is lost.  
> **username_2**: It is linear and hence twice continuously differentiable in the region $\frac{1}{3}L < K < 3L$.  
> **username_0**: But the whole function $F$ is not twice continuous differentiable, piecewise yes.  
> **username_2**: You're now adding more requirements. Anyways it is also possible to construct a twice differentiable function also.  
> **username_0**: This was the original requirement from the base questions. Yes every piece of the function $F$ is even $C^\infty$, but $F$ itself not twice continuous differentiable.  

---

