Difference between representation of LM function in different textbooks

Question

In Blanchard, the LM curve equation is represented in the following manner:

$$ \frac{M}{P} = L(y,r) $$ Whereas in Branson it is given as

$$ \frac{M}{P} = L(r) + k(y) $$

L(r) is speculative demand for money and k(y) is transactions demand.

How are both the representations appropriate? Can we do this for any two(or more) variable functions?

Answer 1

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

1. $L(y,r)\equiv y+r$;
2. $L(y,r)\equiv y + \cos(y)+\sin(r)$;
3. $L(y,r)\equiv y \ln (r)$;
4. $L(y,r)\equiv yr$;
5. etc.

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$L(r)+k(y)$ 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.

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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 $f(x_1,\ldots,x_n)$.