<p>Your notation isn't the cleanest, and my notation is a little shaky, but here is the idea. It's just algebra resulting from plugging <span class="math-container">$$\partial C_W / \partial Y_W = 1-1/k \implies \mathbf{D} C_W = (1-1/k) \mathbf{D}Y_W$$</span> into the identity <span class="math-container">$$\mathbf{D}Y_W = \mathbf{D} C_W+\mathbf{D} I_W.$$</span>
After simplifying, you will see that the two equations are equivalent.</p>

CC BY-SA 4.0
2023-08-01T13:54:38.930Z
2023-08-01T13:54:38.930Z
A. Miller
<p>Since</p>
<p><span class="math-container">$$\Delta Y_w= \Delta C_w+ \Delta I_w$$</span></p>
<p>we have</p>
<p><span class="math-container">$$1=\frac{\Delta C_w}{\Delta Y_w}+\frac{\Delta I_w}{\Delta Y_w}$$</span></p>
<p>or</p>
<p><span class="math-container">$$\frac{\Delta I_w}{\Delta Y_w}=1-\frac{\Delta C_w}{\Delta Y_w}$$</span></p>
<p>Setting <span class="math-container">$1-1/k=\frac{\Delta C_w}{\Delta Y_w}$</span> (the marginal propensity to consume), we have</p>
<p><span class="math-container">$$\frac{\Delta I_w}{\Delta Y_w}=\frac{1}{k}$$</span></p>
<p>so that</p>
<p><span class="math-container">$$\Delta Y_w=k\Delta I_w.$$</span></p>
<hr />
<p>Alternatively, substituting <span class="math-container">$\Delta C_w=(1-1/k)\Delta Y_w$</span> into</p>
<p><span class="math-container">$$\Delta Y_w= \Delta C_w+ \Delta I_w$$</span></p>
<p>gives:</p>
<p><span class="math-container">$$\Delta Y_w= \left(1-\frac{1}{k}\right)\Delta Y_w +\Delta I_w$$</span></p>
<p>or</p>
<p><span class="math-container">$$0= -\frac{1}{k}\Delta Y_w +\Delta I_w$$</span></p>
<p>so that</p>
<p><span class="math-container">$$\Delta Y_w=k\Delta I_w.$$</span></p>

CC BY-SA 4.0
2023-08-01T13:56:00.243Z
2023-08-01T13:56:00.243Z
smcc
<p>In &quot;The General Theory of Employment, Interest and Money&quot;, Chapter 10: <a href="http://gutenberg.net.au/ebooks03/0300071h/chap10.html" rel="nofollow noreferrer">The marginal propensity to consume and the multiplier</a>, John Maynard Keynes says:</p>
<blockquote>
<p>Our normal psychological law that, when the real income of the community increases or decreases, its consumption will increase or decrease but not so fast, can, therefore, be translated¾not, indeed, with absolute accuracy but subject to qualifications which are obvious and can easily be stated in a formally complete fashion into the propositions that <span class="math-container">$\Delta C_w$</span> and <span class="math-container">$\Delta Y_w$</span> have the same sign, but <span class="math-container">$\Delta Y_w &gt;\Delta C_w $</span>, where <span class="math-container">$C_w$</span> is the consumption in terms of wage-units. This is merely a repetition of the proposition already established in Chapter 3 above. Let us define, then, <span class="math-container">$dC_w/dY_w$</span> as the <em>marginal propensity to consume</em>.</p>
<p>This quantity is of considerable importance, because it tells us how the next increment of output will have to be divided between consumption and investment. For <span class="math-container">$\Delta Y_w  =  \Delta C_w + \Delta I_w$</span>, where <span class="math-container">$C_w$</span> and <span class="math-container">$I_w$</span> are the increments of consumption and investment; so that we can write <span class="math-container">$\Delta Y_w   =  k\Delta I_w$</span>, where <span class="math-container">$1-1/k$</span> is equal to the marginal propensity to consume.</p>
<p>Let us call <span class="math-container">$k$</span> the <em>investment multiplier</em>. It tells us that, when there is an increment of aggregate investment, income will increase by an amount which is k times the increment of investment.</p>
</blockquote>
<p>I don't understood how Keynes make this step: &quot;For <span class="math-container">$\Delta Y_w  =  \Delta C_w + \Delta I_w$</span>, where <span class="math-container">$C_w$</span> and <span class="math-container">$I_w$</span> are the increments of consumption and investment; so that we can write <span class="math-container">$\Delta Y_w   =  k\Delta I_w$</span>, where <span class="math-container">$1-1/k$</span> is equal to the marginal propensity to consume.&quot;</p>
<p>How is &quot;<span class="math-container">$\Delta Y_w  = k\Delta I_w$</span>&quot; inferred from &quot;<span class="math-container">$\Delta Y_w  =  \Delta C_w + \Delta I_w$</span>&quot;?</p>

<classical-economics><keynes><multiplier>
The marginal propensity to consume and the multiplier according to Keynes
santos82h
# The marginal propensity to consume and the multiplier according to Keynes

(Asked by: username_0 on 2023-08-01)

In "The General Theory of Employment, Interest and Money", Chapter 10: [The marginal propensity to consume and the multiplier](http://gutenberg.net.au/ebooks03/0300071h/chap10.html), John Maynard Keynes says:

> Our normal psychological law that, when the real income of the community increases or decreases, its consumption will increase or decrease but not so fast, can, therefore, be translated¾not, indeed, with absolute accuracy but subject to qualifications which are obvious and can easily be stated in a formally complete fashion into the propositions that $\Delta C\_w$ and $\Delta Y\_w$ have the same sign, but $\Delta Y\_w >\Delta C\_w $, where $C\_w$ is the consumption in terms of wage-units. This is merely a repetition of the proposition already established in Chapter 3 above. Let us define, then, $dC\_w/dY\_w$ as the *marginal propensity to consume*.
>
> This quantity is of considerable importance, because it tells us how the next increment of output will have to be divided between consumption and investment. For $\Delta Y\_w = \Delta C\_w + \Delta I\_w$, where $C\_w$ and $I\_w$ are the increments of consumption and investment; so that we can write $\Delta Y\_w = k\Delta I\_w$, where $1-1/k$ is equal to the marginal propensity to consume.
>
> Let us call $k$ the *investment multiplier*. It tells us that, when there is an increment of aggregate investment, income will increase by an amount which is k times the increment of investment.

I don't understood how Keynes make this step: "For $\Delta Y\_w = \Delta C\_w + \Delta I\_w$, where $C\_w$ and $I\_w$ are the increments of consumption and investment; so that we can write $\Delta Y\_w = k\Delta I\_w$, where $1-1/k$ is equal to the marginal propensity to consume."

How is "$\Delta Y\_w = k\Delta I\_w$" inferred from "$\Delta Y\_w = \Delta C\_w + \Delta I\_w$"?

**Tags:** classical-economics, keynes, multiplier

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## 2 answers

### Answer by username_2 on 2023-08-01. Score: 2 | Accepted answer

Since

$$\Delta Y\_w= \Delta C\_w+ \Delta I\_w$$

we have

$$1=\frac{\Delta C\_w}{\Delta Y\_w}+\frac{\Delta I\_w}{\Delta Y\_w}$$

or

$$\frac{\Delta I\_w}{\Delta Y\_w}=1-\frac{\Delta C\_w}{\Delta Y\_w}$$

Setting $1-1/k=\frac{\Delta C\_w}{\Delta Y\_w}$ (the marginal propensity to consume), we have

$$\frac{\Delta I\_w}{\Delta Y\_w}=\frac{1}{k}$$

so that

$$\Delta Y\_w=k\Delta I\_w.$$

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Alternatively, substituting $\Delta C\_w=(1-1/k)\Delta Y\_w$ into

$$\Delta Y\_w= \Delta C\_w+ \Delta I\_w$$

gives:

$$\Delta Y\_w= \left(1-\frac{1}{k}\right)\Delta Y\_w +\Delta I\_w$$

or

$$0= -\frac{1}{k}\Delta Y\_w +\Delta I\_w$$

so that

$$\Delta Y\_w=k\Delta I\_w.$$


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### Answer by username_1 on 2023-08-01. Score: 2

Your notation isn't the cleanest, and my notation is a little shaky, but here is the idea. It's just algebra resulting from plugging $$\partial C\_W / \partial Y\_W = 1-1/k \implies \mathbf{D} C\_W = (1-1/k) \mathbf{D}Y\_W$$ into the identity $$\mathbf{D}Y\_W = \mathbf{D} C\_W+\mathbf{D} I\_W.$$
After simplifying, you will see that the two equations are equivalent.


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