The fundamental national income accounting identity is: $Y = C + I + G + X - M$. The multiplier effect tells us that an autonomous injection into the economy will result in a multiplied increase in the real national income due to rounds of induced consumption. My question is, if that is indeed the case, then wouldn't the national income identity need to be modified to include the multiplier? That is to say, wouldn't it become: $Y = k_c C + k_i I + k_g G + k_x X - k_m M$?


You can't apply it to the national income identity directly because its components, namely consumption, investment and imports are functions of national income (output) and hence, they are not autonomous. In fact, in simple undergraduate level model, following Blanchard et al:

$$ Y = C(Y-T) + I(Y,i) + G + X(Y^*, \epsilon - M(Y, \epsilon)/ \epsilon $$

Where in addition to variables already described in your post $T$ are taxes, $i$ is interest rate $Y^*$ foreign output and $\epsilon$ is the exchange rate.

So you cant simply apply the multiplayer just to the variables itself as you would apply them to output multiple times. Rather you first need to solve the equation above for output. To do that I will assume two simplifications - $I$ will be fixed (that is $I = \bar{I}$ and consequently $I$ is no longer function of $Y$, and autraky to get rid of $X$ and $M$ - this is to simplify the math as now only $C$ is a function of $Y$ making solution easier (not doing so would only cause multiplier to be more complex and derivations longer). Moreover, assuming that $C$ is given by $C = c_0 + c_1(Y-T)$, the solution to the goods market equilibrium will be given by:

$$Y = \frac{1}{1-c_1} \left( c_0 + \bar{I} + G - c_1T\right)$$

where the term $\frac{1}{1-c_1}$ is the multiplier (which you denote by $k$). Applying multiplier makes sense only at this step because only now Y was eliminated from the right hand side of the equation. Previously, you were applying the same multiplier to Y over and over again without noticing it.

Also this is completely correct:

multiplier effect tells us that an autonomous injection into the economy will result in a multiplied increase in the real national income due to rounds of induced consumption.

But no part of the original accounting formula contains autonomous spending aside form $G$. In fact the solution to the goods market equilibrium given by the equation where we eliminate $Y$ from right hand side gives you effect of autonomous injections as in that solution $c_0$ is the autonomous consumption, $I$ was assumed away to be autonomous by assuming its fixed (admittedly that was bit sloppy from me but it simplifies math and shortens the explanations) and $G$ and $T$ are arguably autonomous by themselves.

  • $\begingroup$ Another 'autonomous' revenge (-1) ? :) $\endgroup$ – 1muflon1 May 25 '20 at 1:43
  • $\begingroup$ Yes, I understand that there is a need to distinguish between "autonomous" and "income-induced". In that case, we can split up the components on the left-hand side into "autonomous" and "income-induced" and then only apply the multiplier to the "autonomous" part? $\endgroup$ – Tan Yong Boon May 25 '20 at 2:47
  • $\begingroup$ Are you saying that there is already the multiplier encompassed within each term? That means to say, if we write out the full expression for each component splitting them up into the "autonomous" and "income-induced" components, then we would see the multiplier appearing? $\endgroup$ – Tan Yong Boon May 25 '20 at 2:48
  • $\begingroup$ @TanYongBoon 1. It’s actually not correct to even think of applying the multiplier - you actually calculate the multiplier by separating out all of the autonomous spending by solving for Y. Using the goods equilibrium equation I showed above is in fact how you prove there is a multiplier there to begin with, but also it does not make sense to apply the multiplier to the same term Y multiple times as multiplier should already gives us the effect of 1e circulating in the economy infinite number of times $\endgroup$ – 1muflon1 May 25 '20 at 10:31
  • $\begingroup$ @TanYongBoon 2. encompassing is not the right term. I think your confusion stems from misunderstanding the idea of the multiplier - the point of the multiplier is the fact that the accounting identity is endogenous- that’s what creates multiplier in the first place consumption increases income but increased income increases a consumption but that must again increase income ad infinitum - the multiplier tries to capture what the effect of this infinite loop is and mathematically to see that you first have to solve for Y and then examine how it changes wrt independent (‘autonomous’) variables $\endgroup$ – 1muflon1 May 25 '20 at 10:49

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