The function $L(y,r)$ is a placeholder for any rule that says
"give me a $y$ and and an $r$, and I will spit-out a value for $L$."
For example, we could have
- $L(y,r)\equiv y+r$;
- $L(y,r)\equiv y + \cos(y)+\sin(r)$;
- $L(y,r)\equiv y \ln (r)$;
- $L(y,r)\equiv yr$;
$L(r)+k(y)$ imposes an additional restriction. We con no longer pick any rule. Now we are only allowed to use rules where any terms involving $r$ are separated from those involving $y$ by a $+$. we call such function "additively separable".
Thus, in the Branson formulation encompasses rules such as 1 and 2 above, but not 3 and 4 (because the latter two are not additively separable).
This additional restriction means the Blanchard function is a more general formulation than that in Branson.
If we have a result that we know to be true for any function $f(x_1,\ldots,x_n)$ then we automatically know it must be true for additively separable functions of the form $f_1(x_1)+\ldots+f_n(x_n)$ (because the additively separable form is just a less general subcase of the more general form $f(\cdot)$).
But the same is not true in reverse. A result that we know holds for additively separable functions $f_1(x_1)+\ldots+f_n(x_n)$ need not hold for more general functions $f(x_1,\ldots,x_n)$.