How to show the production function is concave in K and L but not strictly so? 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.
So we want the Hessian to be NSD, so we need the PMs to alternate weakly. 

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

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

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

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

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.