<p>If you consider the production function <span class="math-container">$f(K, L) = F(K, L, A = 1)= K^{\frac{1}{2}}L^{\frac{1}{2}}$</span>, this function meets all the conditions specified in assumption 1 within the interior, but not on the axes due to its lack of differentiability on the axes. Typically, this detail is not explicitly mentioned. Check that <span class="math-container">$f(K, L)= K^{\frac{1}{2}}L^{\frac{1}{2}}$</span> is concave and not strictly concave. To show that this is not strictly concave, consider <span class="math-container">$(K_1, L_1)=(0,0)$</span>, <span class="math-container">$(K_2, L_2)=(2,2)$</span> and <span class="math-container">$\lambda = \frac{1}{2}$</span>,</p>
<p><span class="math-container">$f(\lambda (K_1, L_1) +(1-\lambda) (K_2, L_2))  = f(1,1) =1 =\frac{1}{2}(0) + \frac{1}{2}(2) = \lambda f(K_1, L_1) + (1-\lambda) f(K_2, L_2) $</span></p>

CC BY-SA 4.0
2023-04-02T10:46:12.730Z
I understand the definition of concave function as you wrote above, and makes sense. But I also know that if the second (partial) derivatives are strictly negative, then $F$ is strictly concave. Why this is not the case?
John M.
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2023-04-02T11:06:20.997Z
Thanks. I see. Ok, my statement was simply wrong
John M.
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2023-04-02T10:02:59.873Z
2023-04-02T10:17:06.477Z
2023-04-02T10:17:06.477Z
Amit
<p>I'm studying the Solow growth model from the Acemoglu's book.</p>
<p>Consider the following standard assumptions:</p>
<p><a href="https://i.sstatic.net/1jkIt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1jkIt.png" alt="enter image description here" /></a></p>
<p>The fact that <span class="math-container">$F()$</span> exhibits constant returns to scale means that F() is linear homogenous. Why the book says that <span class="math-container">$F()$</span> is concave? Why it is not strictly concave?</p>

CC BY-SA 4.0
2023-04-01T00:49:21.153Z
I'm not certain that I've understood your question fully. If the book assumes that $F$ is concave, this is a weaker condition than assuming it is strictly concave. Therefore, the analysis conducted in the book, which holds for concave functions, will also hold for strictly concave functions.
Amit
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2023-04-01T13:35:48.967Z
Thanks for your answer. The book says : "Prove that Assumption 1 implies that F(A,K, L) is concave in K and L but not strictly so."
John M.
CC BY-SA 4.0
2023-04-01T18:45:29.823Z
@JohnM. Are you asking if there any concave functions that are not strictly concave, or what exactly is your question here?
Giskard
CC BY-SA 4.0
2023-04-01T21:52:09.543Z
@JohnM. Let me ask you a follow-up question. Can you think of any production function $F$ that satisfy assumption $1$?
Amit
CC BY-SA 4.0
2023-04-02T08:38:33.537Z
Thanks for your replies. Assumption 1, in my understanding implies that $F$ is strictly concave. Strict concavity implies concavity. Why the book says that $F$ is concave only (and not strictly concave as well)?
John M.
<macroeconomics><economic-growth><solow>
Solow model - concavity vs strict concavity
John M.
# Solow model - concavity vs strict concavity

(Asked by: username_0 on 2023-03-31)

I'm studying the Solow growth model from the Acemoglu's book.

Consider the following standard assumptions:

[![enter image description here](https://i.sstatic.net/1jkIt.png)](https://i.sstatic.net/1jkIt.png)

The fact that $F()$ exhibits constant returns to scale means that F() is linear homogenous. Why the book says that $F()$ is concave? Why it is not strictly concave?

**Tags:** macroeconomics, economic-growth, solow

> **Comments:**
> **username_1**: I'm not certain that I've understood your question fully. If the book assumes that $F$ is concave, this is a weaker condition than assuming it is strictly concave. Therefore, the analysis conducted in the book, which holds for concave functions, will also hold for strictly concave functions.  
> **username_0**: Thanks for your answer. The book says : "Prove that Assumption 1 implies that F(A,K, L) is concave in K and L but not strictly so."  
> **username_2**: @JohnM. Are you asking if there any concave functions that are not strictly concave, or what exactly is your question here?  
> **username_1**: @JohnM. Let me ask you a follow-up question. Can you think of any production function $F$ that satisfy assumption $1$?  
> **username_0**: Thanks for your replies. Assumption 1, in my understanding implies that $F$ is strictly concave. Strict concavity implies concavity. Why the book says that $F$ is concave only (and not strictly concave as well)?  

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## 1 answers

### Answer by username_1 on 2023-04-02. Score: 5 | Accepted answer

If you consider the production function $f(K, L) = F(K, L, A = 1)= K^{\frac{1}{2}}L^{\frac{1}{2}}$, this function meets all the conditions specified in assumption 1 within the interior, but not on the axes due to its lack of differentiability on the axes. Typically, this detail is not explicitly mentioned. Check that $f(K, L)= K^{\frac{1}{2}}L^{\frac{1}{2}}$ is concave and not strictly concave. To show that this is not strictly concave, consider $(K\_1, L\_1)=(0,0)$, $(K\_2, L\_2)=(2,2)$ and $\lambda = \frac{1}{2}$,

$f(\lambda (K\_1, L\_1) +(1-\lambda) (K\_2, L\_2)) = f(1,1) =1 =\frac{1}{2}(0) + \frac{1}{2}(2) = \lambda f(K\_1, L\_1) + (1-\lambda) f(K\_2, L\_2) $

> **Comments:**
> **username_0**: I understand the definition of concave function as you wrote above, and makes sense. But I also know that if the second (partial) derivatives are strictly negative, then $F$ is strictly concave. Why this is not the case?  
> **username_0**: Thanks. I see. Ok, my statement was simply wrong  

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