Consumption modeling

Until now we have only spoken about consumption generally, however more detail will help us to better understand the relationship between output, consumption, taxes and government spending. Following the Great Depression, John Maynard Keynes developed his theories that would revolutionize macroeconomics. Importantly, in his view aggregate demand was more important than aggregate supply and the largest component of $AD$ in developed economies is consumption.

Consumption can be viewed as having two pieces. The first piece is autonomous consumption, the amount we spend even if we have zero income. This can be viewed as basic needs such as food, shelter, and clothing. The second piece depends on our income, but in particular disposable income, our net income after taxes.

Here we will treat taxes as lump sum taxes, a fixed amount that is the same regardless of the income of a household. In reality taxes are proportional to income with a progressive structure, so the tax rate on additional dollars goes up as income rises. Simplifying taxes allows us to get at the core relationships and concepts without spending time on algebra that does not significantly change the final conclusions.

These two pieces of consumption lead to a preliminary version of the consumption function,

$$\begin{array}{r}C=a+b\times (Y-T),\end{array}$$

where $a$ represents autonomous consumption and $b$ is the fraction of disposable income that is used for consumption.

The consumption-savings decision

The last piece we need is dig deeper into $b$. This requires discussion of how households decide the split between consumption and savings.

For macroeconomics we focus on aggregate behavior over the personal finance considerations of individual households. That allows us to make a simple assumption, that households on average consume and save fixed percentages of their income.

Definitions

Marginal propensity to consume (MPC)

The fraction of every new dollar of disposable (after-tax) income that is used for consumption. May also be viewed as a percentage expressed as a decimal.

Marginal propensity to save (MPS)

The fraction of every new dollar of disposable (after-tax) income that is used for savings. May also be viewed as a percentage expressed as a decimal.

Consumption and saving are the only options available to households, so it is always true that $MPC+MPS=1$. This highlights a tricky piece of terminology to keep in mind. In AP Macroeconomics, for households savings refers to any use other than current consumption. This will be reinforced when we get to the loanable funds market.

Adding these concepts gives us the final Keynesian consumption function,

$$\begin{array}{r}C=a+MPC\times (Y-T),\end{array}$$

where $a$ is autonomous consumption and $(Y-T)$ is disposable income.

$MPC$ gives the marginal change in consumption as income increases. For every new dollar of disposable income, whether it comes from an increase in $Y$ or a decrease in $T$, consumption increases by $MPC$.

Detailed consumption shifts

Now we can tie in our consumption shifts and be a little more specific about how they impact consumption and aggregate demand. Nothing will contradict relationships that we have already discussed, we will just provide a more detailed story.

For nearly all AP exam material, knowing the first and last columns is what is most important for graphs, while the middle can matter when it comes to explaining what happened.

Factor Increasing	Change	Impact on $C$ and $AD$
Household wealth	$a\uparrow$	$\uparrow$
Current income	$Y\uparrow$	$\uparrow$
Expectations of future income	$a\uparrow$	$\uparrow$
Consumer confidence (animal spirits)	$a\uparrow$ or $MPC\uparrow$	$\uparrow$
Interest rates	$MPC\downarrow$ and $MPS\uparrow$	$\downarrow$
Taxes	$-(b\times T)\uparrow$	$\downarrow$
Prices	$(Y-T)\downarrow$	$C\downarrow$, see below

Prices require a special comment. The $AD$ curve is graphed as how $Y$ changes with $PL$. An increase in the price level makes real income fall, since nominal income does not immediately change. Consumers can afford to buy fewer goods and services than they were able to before the change in prices. Because of this relationships, changes in the price level move the economy along a single $AD$ curve.

Fiscal multipliers

The Keynesian consumption function and in particular the $MPC$ gives us new perspective on how fiscal policy impacts the $AD$ curve.

The key insight is that following any fiscal policy change there is new spending, and that becomes income for someone else. That income in turn leads to more spending, which is income for another seller. This process continues on and on. Because of this, $1 of new spending leads to more than a $1 change in real GDP.

Assuming that the Keynesian consumption function represents all households in our economy, every household has the same behavior. Any new income has the same percentage split between spending, $MPC$, and saving, $(1-MPC)=MPS$.

We start with the spending multiplier, before moving on to the taxation multiplier.

The spending multiplier

First consider a change in $G$, which shifts the $AD$ curve. Recall that the Keynesian consumption function we reviewed was for a closed economy, so all of the new spending becomes domestic income.

The circulation of initial spend leads to the following equation for the spending multiplier,

$$\begin{array}{r}\text{Spending multiplier }=\frac{1}{(1-MPC)}=\frac{1}{MPS}.\end{array}$$

The spending multiplier gives the total change in GDP for a one dollar change in $G$.

Although it can be an awkward story, multiplier effects also occur in reverse. If spending is cut, that reduces income for another household which leads to additional reductions in spending.

If $G$ shifts by a larger amount than one dollar, the total change in GDP is just the shift amount times the multiplier.

$$\begin{array}{rl}\text{Change in }Y& =\text{Spending Multiplier}\times \text{change in }G\\ & =\frac{1}{(1-MPC)}\times \text{ change in}G\end{array}$$

The spending multiplier also applies to any non-tax change in a piece of the $AD$ curve that is unrelated to prices. This includes shifts in autonomous consumption ($a$), $I$ and $NX$ in addition to changes in $G$. In each case a one dollar increase initially makes $Y$ increase by a dollar, which becomes new income and leads to an expansion of $C$.

The formulas are the same, regardless of which piece of $AD$ changes. However, $G$ is the only component directly affected by fiscal policy. Except for changes in interest rates, the other changes may be summarized as demand shocks.

The tax multiplier

With the previous stories, the first dollar of activity is directly spent. For taxes and transfers, the first dollar of activity is an increase in income for a household. So part of that first dollar is saved. This makes the tax multiplier smaller than the spending multiplier. In fact, it is the spending multiplier minus one, since the first dollar is not spent.

$$\begin{array}{rl}\text{Tax multiplier }& =\text{Spending multiplier}-1\\ & =-\frac{MPC}{(1-MPC)}=-\frac{MPC}{MPS}\end{array}.$$

If taxes rise disposable income $(Y-T)$ falls and so does consumption. From this we can see that taxes have the opposite effect of spending on GDP.

For transfers, the multiplier would be the negative of the tax multiplier, since transfers increase income directly. A result of this is that a one dollar tax cut and a one dollar transfer have the same effect on disposable income and therefore on consumption.

A complication to this is that households may not treat transfers the same as other income, not subject to the same $MPC$ because it is not a permanent income change. This is some evidence for this in recent history, with data showing a higher percentage saved than expected. Overall there is mixed evidence on the effectiveness of transfers as a stimulus,

Regardless of the specific story, we can say that the total change in real GDP is nothing more than the first change times the correct multiplier.

$$\begin{array}{rl}\text{Change in }Y& =\text{Multiplier}\times \text{initial change}\end{array}$$

Application with calculations

For the AP exam some smaller calculations do show up, things similar to the following examples.

A value of $MPC$ around 0.75 to 0.9 is realistic for most advanced economies. This translates to a multiplier between $\frac{1}{(1-0.75)}=\frac{1}{0.25}=4$ and $\frac{1}{(1-0.9)}=\frac{1}{0.1}=10$.

These $MPC$ values translate to a tax multiplier between $-\frac{0.75}{(1-0.75)}=-\frac{0.75}{0.25}=-3$ and $-\frac{0.9}{(1-0.9)}=-\frac{0.9}{0.1}=-9$.

Compared to the spending multiplier values of $4$ and $10$, we can see that indeed the tax multiplier is smaller and has the opposite sign of the spending multiplier.

Finally, we can find the total impact on $Y$ following standard fiscal policy options, changes in $G$ or $T$. Given $MPC=0.5$ and a $1000 policy change we find that,

• A $1000 increase in $G$ will result in a $2\times 1000=2000$ increase in $Y$

• A $1000 tax cut will result in a $-1\times -1000=1000$ increase in $Y$.

So to achieve an equal change in $Y$, a tax cut must be larger than a spending increase.

If we include transfers in this analysis, the surface level take is that a transfer must be the same size as a tax cut, just with the opposite sign. So we would expect a $1000 transfer and a $1000 tax cut to have the same total effect on real GDP.