<p>The function $L(y,r)$ is a placeholder for <strong>any</strong> rule that says</p>

<blockquote>
  <p>"give me a $y$ and and an $r$, and I will spit-out a value for $L$."</p>
</blockquote>

<p>For example, we could have </p>

<ol>
<li>$L(y,r)\equiv y+r$;</li>
<li>$L(y,r)\equiv y + \cos(y)+\sin(r)$;</li>
<li>$L(y,r)\equiv y \ln (r)$;</li>
<li>$L(y,r)\equiv yr$;</li>
<li>etc.</li>
</ol>

<hr>

<p>$L(r)+k(y)$ 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".</p>

<p>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).</p>

<p>This additional restriction means the Blanchard function is a <em>more general</em> formulation than that in Branson.</p>

<hr>

<p>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 $f(\cdot)$).</p>

<p>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 $f(x_1,\ldots,x_n)$.</p>

CC BY-SA 3.0
2017-02-11T13:07:07.357Z
2017-02-11T13:18:51.920Z
2017-02-11T13:18:51.920Z
Ubiquitous
<p>In Blanchard, the LM curve equation is represented in the following manner:</p>

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

<p>$$ \frac{M}{P} = L(r) + k(y) $$</p>

<p>L(r) is speculative demand for money and k(y) is transactions demand.</p>

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

<macroeconomics>
Difference between representation of LM function in different textbooks
SDM
# Difference between representation of LM function in different textbooks

(Asked by: username_0 on 2017-02-11)

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?

**Tags:** macroeconomics

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## 1 answers

### Answer by username_1 on 2017-02-11. Score: 3

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.

---

$L(r)+k(y)$ 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 $f(x\_1,\ldots,x\_n)$.


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