Some Pedantic Things (that probably nobody cares)

I never imagined that some basic concepts could confuse so many people, until I recently noticed certain mistakes appearing repeatedly. Perhaps this is because some classic references propagate these errors, and others simply adopt them without critically examining the logic.

Constant returns to scale and strict concavity are incompatible!!!

A common error occurs in macroeconomic notes, where constant returns to scale and strict concavity are simultaneously assumed for $F(K,L)$. However, this is incorrect.

Take $x \neq 0$ (where $x$ and $0$ can be vectors if you like), so $x \neq 2x$. Then take $\lambda = 1/2$.

Then $(1/2)x + (1/2)2x = (3/2)x$. If $h$ were strictly concave, it must follow that $h((3/2)x) > (1/2)h(x) + (1/2)h(2x)$.

However, by constant returns to scale, $h((3/2)x) = (3/2)h(x)$,

while
$(1/2)h(x) + (1/2)h(2x)$, using constant returns to scale, equals $(1/2)h(x) + h(x)$, which is $(3/2)h(x)$ again, leading to a contradiction (Since only the weak inequality, rather than the strict one, can be guarenteed).

Actually, you can find exercises explicitly asking for a proof of this incompatibility in textbooks on mathematical economics, such as Sydsæter, K., Seierstad, A., & Strøm, A. (2008)., which I learned from a lot.

It is okay to state that $F(K,L)$ is constant returns to scale and concave (but not strictly concave), or that $F(K,L)$ is constant returns to scale and $f(k) = \frac{F(K,L)}{L}$ is strictly concave.

However, the assumption of constant returns to scale and strict concavity for $F(K,L)$ is fundamentally wrong.