Derivation of Expenditure Multiplier

Marginal Propensity to Consume

Proportionate change in consumption due to change in income will be same (Positively)

$$\begin{array}{rl}MPC& =\text{Marginal Propensity to Consume}\\ & =\frac{\Delta C(\text{Consumer Spending})}{\Delta Y(\text{Disposible Income})}=\frac{\Delta C}{\Delta Y}\end{array}$$

Also, Marginal Propensity to Save

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Change in planned investment or change in government planned expenditure changes the equilibrium level of income in a positive manner.

The change in income due to initial change in investment or government expenditure depends on the slopes of the consumption function or savings function which is $(s+t)$ schedule.

Ratio of $\frac{dy}{di}$ (or $\frac{dy}{dg}$), gives us the change in the equilibrium level of income per unit change in income (or government expenditure), is referred to as the multiplier effect of investment expenditure or government expenditure.

Lump Sum Tax, $\overline{t}$

Tax revenue is assumed to be a fixed sum, $\overline{t}$

$$c(y-\overline{t})+\overline{i}+g=y=c(y-\overline{t})+s(y-\overline{t})+\overline{t}\text{\hspace{1em}(13)}$$

Subtract consumption, $c$ from each part

$$\overline{i}+g=y=s(y-\overline{t})+\overline{t}\text{\hspace{1em}(14)}$$

To find change in the equilibrium income following change in planned investment, we can differentiate the L.H.S of the equation $(13)$, and treating $g$ and $\overline{t}$ as constants.

$$c^{\prime }dy+d\overline{i}=dyanddy(1-c^{\prime })=d\overline{i}$$

Change in the equilibrium level of income $dy$ relative to change in investment is

$$\frac{dy}{d\overline{i}}=\frac{1}{1-c^{\prime }}\text{\hspace{1em}(15)}$$

Example

1. $c’=0.80:\frac{1}{1-c^{\prime }}=\frac{1}{0.2}=5$
2. $c’=0.50:\frac{1}{1-c^{\prime }}=\frac{1}{0.5}=2$
3. $c’=0.90:\frac{1}{1-c^{\prime }}=\frac{1}{0.1}=10$

If change in investments is $\Delta i=100,000$ then

1. $\Delta y=5\times 100,000=500,000$ (base MPC)
2. $\Delta y=2\times 100,000=200,000$ (lower MPC)
3. $\Delta y=10\times 100,000=1000,000$ (higher MPC)

So, higher the MPC, higher is the change in income level.

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Change in $\overline{i}$, planned investment leads to change in $i_0+g\to i_1+g$ which leads to the increase in $y$ (income) from $y_0\to y_1$.

• where a small change in $\overline{i}$ leads to a bigger change in $y$, due to the multiplier effect.
• Higher the MPC, higher the change or multiplier effect.

General Equilibrium Multiplier

$$y=c(y-\overline{t})+\overline{i}+g$$

Changes in $y$ as a function of change in $\overline{t}$, $\overline{i}$ and $g$ by differentiation equations.

$$c(y-\overline{t})+i+g=y=c(y-\overline{t})+s(y-\overline{t})+\overline{t}\text{\hspace{1em}(13)}$$ $$\begin{array}{rl}dy& =c^{\prime }(dy-d\overline{t})+d\overline{i}+dg\\ dy(1-c^{\prime })& =-c^{\prime }d\overline{t}+d\overline{i}+dg\end{array}$$ $$\therefore \boxed{dy=\frac{-c^{\prime }d\overline{t}+d\overline{i}+dg}{1-c^{\prime }}}\text{\hspace{1em}(16)}$$

Equation $(16)$ is the General Multiplier equation.

To obtain a specific multiplier we can set the other components as zero and differentiate only with respect to the multiplier specifically we are looking at:

e.g. To obtain investment multiplier we set $d\overline{t}$ and $dg$ as zero and divide it by $d\overline{i}$.

Balanced Budget Multiplier

Government expenditure, $g$ increases by $1$ unit if this is financed through increase in taxation by $1$ unit

What happens to $y$ if we rise the government expenditure and tax revenue by the same amount by holding investment as constant.

Substituting $dg=d\overline{t}$ in equation $(16)$ holding $d\overline{i}$ as constant we get

$$\begin{array}{rl}dy=\frac{-c^{\prime }dg+dg}{1-c^{\prime }}& =dg\frac{1-c^{\prime }}{1-c^{\prime }}\\ \frac{dy}{dg}& =\frac{1-c^{\prime }}{1-c^{\prime }}=1\end{array}\text{\hspace{1em}(17)}$$

Example

1. If consumer spending increases from $$200\to$ $$265$ and disposable income increases from $$425\to$ $$550$, what is the MPC?

$$\begin{array}{rl}\Delta \text{Consumer Spending}& =\$65\\ \Delta \text{Disposable Income}& =\$125\\ & \\ \therefore MPC& =\frac{65}{125}=\boxed{0.52}\\ \text{and }MSC& =1-0.52=\boxed{0.48}\end{array}$$

2. If $MPC=0.4$, what is expenditure multiplier?

$$\begin{array}{rl}\text{Expenditure multiplier}& =\frac{1}{1-MPC}\text{ or }\frac{1}{1-c^{\prime }}\\ & =\frac{1}{1-0.4}=\frac{1}{0.6}=1.667\end{array}$$

Taxes as a function of income

Now, lets assume that $t=t(y)$ which is a more realistic scenario. the equilibrium condition is

$$c(y-t(y))+\overline{i}+g=y=c(y-t(y))+s(y-t(y))+t(y)\text{\hspace{1em}(18)}$$

Subtract $c(y-t(y))$ from each side of equation $(18)$

$$\overline{i}+g=y=s(y-t(y))+t(y)\text{\hspace{1em}(19)}$$

General multiplier is definition of equation $(19)$

$$\begin{array}{rl}dy& =c^{\prime }(dy-t^{\prime }dy)+d\overline{i}+dg\text{and}\\ dy& =c^{\prime }(1-t^{\prime })dy+d\overline{i}+dg\end{array}$$

So, the multiplier is

$$\boxed{dy=\frac{d\overline{i}+dg}{1-c^{\prime }(1-t^{\prime })}}$$

Introducing tax as a function of income has reduced the multiplier.

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With taxes fixed at $\overline{t}$, increase in investments from $i_0\to i_1$ will increase the equilibrium level of income from $y_0\to y_1$

In an alternative scenario, where we assume taxes as a function of income, the same increase in investment from $i_0\to i_1$ would increase the equilibrium level of income from $y_0\to y_2$.

Thus, the presence of tax functions $t(y)$ reduce the increase in the disposable income relative to those in the total income at each stage of expansion ($y\uparrow$). Then reducing eventually the increase in level of income from $(y_1-y_0)\to (y_2-y_0)$.

Tax rate multiplier

Tax multiplier is the most relevant for stabilization of economy. $\boxed{t(y)=Ty}$ where $T$ = percentage tax rate. and thus, $Ty$ = Tax revenue.

With equilibrium,

$$y=c(y-Ty)+\overline{i}+g$$

Since, $d(Ty)$ is approximately equal to $Tdy+ydT$¹, we differentiate the above equation to get

$$\begin{array}{rl}dy& =(dy-d(Ty))c^{\prime }+d\overline{i}+dg\\ dy& =(dy-Tdy-ydT)c^{\prime }+d\overline{i}+dg\\ dy& =(1-T)c^{\prime }-c^{\prime }ydT+d\overline{i}+dg\end{array}$$ $$\boxed{dy=\frac{d\overline{i}+dg-c^{\prime }ydT}{1-c^{\prime }(1-T)}}$$

Thus, the tax rate multiplier just translates a tax rate change into a direct impact on consumer expenditure² and the multiplier in a usual form.

Example

1. $c’=0.80$ (Lump Sum) Multiplier $=\frac{1}{1-c’}=\frac{1}{0.2}=\boxed{5}$
2. $c’=0.80;t^{\prime }=10\%$ (as a function) Multiplier $=\frac{1}{1-c’(1-t’)}=\frac{1}{0.28}=\boxed{3.591}$
3. $c’=0.80;T=20\%$ (tax rate, $T$) Multiplier $=\frac{1}{1-c’(1-T)}=\frac{1}{0.36}=\boxed{2.778}$

We see the difference, the multiplier effect reduces $$ \textrm{Lump Sum} > \textrm{Function} > \textrm{Tax Rate}

Footnotes

1. Total Differential ↩

2. Look how $T$ deflates the value of $c’$ in the expression $c’(1-T)$. This increases the multiplier value. ↩