<p>So we want the Hessian to be NSD, so we need the PMs to alternate weakly. </p>

<p>$H=\begin{bmatrix} F_{kk} &amp; F_{kl} \\
F_{kl} &amp; F_{ll}
\end{bmatrix}~~NSD \iff~~~F_{kk},F_{ll}\leq0~~~\&amp;~~F_{kk}F_{ll}-F_{kl}^2\geq0$</p>

<p>We are given that $F_{kk},F_{ll}&lt;0$ so we need to figure out the cross partials.</p>

<p>Constant returns to scale implies that we have a homogeneous of degree 1 function: </p>

<p>$F(K,L)=KF_k+LF_l \implies  F_l=KF_{kl}+LF_{ll}+F_l~~ \&amp;~~  F_k=LF_{kl}+KF_{kk}+F_k$</p>

<p>Now we can substitute these equations into the expression for the 2nd PM:</p>

<p>$\big(-\frac{L}{K}F_{kl}\big)\big(-\frac{K}{L}F_{kl}\big)-F_{kl}^2=0~~~~~$
 and we're done. </p>

CC BY-SA 3.0
2017-01-15T19:30:55.817Z
I would like to point a mistake... $F^2_{kl}$ is wrong, unless we are talking when $F_{kl}=F_{lk}$. However, this is not always true
Yorgos
CC BY-SA 3.0
2017-01-18T05:00:33.193Z
@Yorgos $F_{KL}=F_{LK}$ by Young's theorem?
VCG
CC BY-SA 3.0
2017-01-18T08:13:50.403Z
Consider a quadaratic productin function $f(k,l)=3kl^{2}+4lk^{2}$, then $f_{kl}\neq f_{lk}$. My point is that when you have a Cobb-Douglas production function then $f_{kl}=f_{lk}$, but it might be cases where young's theorem is not valid
Yorgos
CC BY-SA 3.0
2017-01-18T16:28:21.897Z
@Yorgos  I believe you are mistaken. $f_{kl}=6l+8k$ and $f_{lk}=6l+8k$. Cross derivatives are always identical (given regularity conditions). That is a general mathematical result: https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives
VCG
CC BY-SA 3.0
2017-01-18T21:52:16.880Z
from the link you provided there is a statement "In most "real-life" circumstances the Hessian matrix is symmetric, although there are a great number of functions that do not have this property.". My point was not to say that you were wrong, but rather it would be better to keep is as $f_{kl}f_{lk}$
Yorgos
CC BY-SA 3.0
2016-09-10T14:14:24.733Z
2016-09-10T14:22:29.323Z
2016-09-10T14:22:29.323Z
VCG
<p>Suppose we have a production function with constant returns to scale. Let us denote it by $F(A,K,L)$ where $A$ is the technology, $K$ the capital and $L$ Labor. Further assume the first partial derivative of $L,K$ are both positive and the second partial negative. How to show the production function is concave in $K$ and $L$ but not strictly so?</p>

<p>This problem is from Acemoglu's Intro to Modern Economic Growth. I don't quite understand what it means to say the function is concave in $K$ and $L$. Do I need to show that $F(K,L)$ (treating A as constant) is concave? or do I show $F(K)$ and $F(L)$ are concave (treating L,A or K,A as constants)?</p>

<p>If it is the latter, it seems the problem seems trivial by the second derivative test.</p>

CC BY-SA 3.0
2016-09-10T02:10:27.873Z
With multivariate functions you need to use the Hessian - the matrix of second orders.The univariate 2nd derivative test doesn't work.
VCG
CC BY-SA 3.0
2016-09-10T04:15:29.587Z
@VCG so what do you think the question is asking for? Can you please give a mathematical definition please?
Kun
CC BY-SA 3.0
2016-09-10T13:08:50.487Z
The question is straightforward. It is asking you to find the Hessian and show that the Hessian has the properties of a weak concave function (NSD). The links that Alvaro provided give you the definitions. I can write up an answer if you are still struggling with the definitions.
VCG
CC BY-SA 3.0
2016-09-10T13:10:33.067Z
@VCG yes, I looked at the links. But since I am not given partial F over partial L partial K, how would I even derive the hessian matrix? Oh, I am also given constant return to scale.Thank you.
Kun
<macroeconomics><production-function>
How to show the production function is concave in K and L but not strictly so?
Kun
# How to show the production function is concave in K and L but not strictly so?

(Asked by: username_0 on 2016-09-10)

Suppose we have a production function with constant returns to scale. Let us denote it by $F(A,K,L)$ where $A$ is the technology, $K$ the capital and $L$ Labor. Further assume the first partial derivative of $L,K$ are both positive and the second partial negative. How to show the production function is concave in $K$ and $L$ but not strictly so?

This problem is from Acemoglu's Intro to Modern Economic Growth. I don't quite understand what it means to say the function is concave in $K$ and $L$. Do I need to show that $F(K,L)$ (treating A as constant) is concave? or do I show $F(K)$ and $F(L)$ are concave (treating L,A or K,A as constants)?

If it is the latter, it seems the problem seems trivial by the second derivative test.

**Tags:** macroeconomics, production-function

> **Comments:**
> **username_1**: With multivariate functions you need to use the Hessian - the matrix of second orders.The univariate 2nd derivative test doesn't work.  
> **username_0**: @username_1 so what do you think the question is asking for? Can you please give a mathematical definition please?  
> **username_1**: The question is straightforward. It is asking you to find the Hessian and show that the Hessian has the properties of a weak concave function (NSD). The links that Alvaro provided give you the definitions. I can write up an answer if you are still struggling with the definitions.  
> **username_0**: @username_1 yes, I looked at the links. But since I am not given partial F over partial L partial K, how would I even derive the hessian matrix? Oh, I am also given constant return to scale.Thank you.  

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## 1 answers

### Answer by username_1 on 2016-09-10. Score: 6 | Accepted answer

So we want the Hessian to be NSD, so we need the PMs to alternate weakly.

$H=\begin{bmatrix} F\_{kk} & F\_{kl} \\
F\_{kl} & F\_{ll}
\end{bmatrix}~~NSD \iff~~~F\_{kk},F\_{ll}\leq0~~~\&~~F\_{kk}F\_{ll}-F\_{kl}^2\geq0$

We are given that $F\_{kk},F\_{ll}<0$ so we need to figure out the cross partials.

Constant returns to scale implies that we have a homogeneous of degree 1 function:

$F(K,L)=KF\_k+LF\_l \implies F\_l=KF\_{kl}+LF\_{ll}+F\_l~~ \&~~ F\_k=LF\_{kl}+KF\_{kk}+F\_k$

Now we can substitute these equations into the expression for the 2nd PM:

$\big(-\frac{L}{K}F\_{kl}\big)\big(-\frac{K}{L}F\_{kl}\big)-F\_{kl}^2=0~~~~~$
and we're done.

> **Comments:**
> **username_2**: I would like to point a mistake... $F^2_{kl}$ is wrong, unless we are talking when $F_{kl}=F_{lk}$. However, this is not always true  
> **username_1**: @username_2 $F_{KL}=F_{LK}$ by Young's theorem?  
> **username_2**: Consider a quadaratic productin function $f(k,l)=3kl^{2}+4lk^{2}$, then $f_{kl}\neq f_{lk}$. My point is that when you have a Cobb-Douglas production function then $f_{kl}=f_{lk}$, but it might be cases where young's theorem is not valid  
> **username_1**: @username_2  I believe you are mistaken. $f_{kl}=6l+8k$ and $f_{lk}=6l+8k$. Cross derivatives are always identical (given regularity conditions). That is a general mathematical result: https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives  
> **username_2**: from the link you provided there is a statement "In most "real-life" circumstances the Hessian matrix is symmetric, although there are a great number of functions that do not have this property.". My point was not to say that you were wrong, but rather it would be better to keep is as $f_{kl}f_{lk}$  

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