The term *fiscal policy* describes the spending and taxation decisions taken by government and is used to differentiate these policies from other economic policies of government such as the setting of interest rates (*monetary policy*). The impact of the circular flow of money on incomes is complicated by government spending and taxation and the effect is encapsulated in a concept called *the fiscal multiplier* which is the focus of this post.

To start with, let's call government spending $G$. When the government spends it is hiring folk to do productive work or buying things from private businesses. The spending therefore constitutes an income of $G$ for the "private" or "non-governmental" sector. We can state that:

$$Y_0 = G$$that is, aggregate income (so far) is simply equal to government spending. The subscript $_0$ tells us that this is the *initial* income and the reason for using it will become clear soon. Next, we'll say that the government imposes a tax on income at a rate $\theta$, which is a number between 0 and 1 corresponding to a percentage. For example, if the tax rate is 20%, $\theta = 0.2$. So on the initial income, $Y_0$, we can say that the initial amount of tax paid is $\theta Y_0$. A simple but useful corollary to this is that the amount of income left over after tax is paid is $(1 - \theta) Y_0$. This amount can be described as "disposable income" and the cruicial consideration is what happens with it next.

Assume that the government initially spends 100 pounds and sets an income tax rate of 20%. The private sector gains an income of 100 pounds but pays 20 pounds back to the government immediately as tax. The holders of the remaining disposable income now have two choices: they can spend it or not spend it, and *not spending* is simply what we refer to as *saving*. For now we'll assume that the 80 pounds of disposable income is entirely spent on consumption goods and services - i.e. no saving (yet). This additional consumption spending becomes an additional 80 pounds of income for the economy of which 20% - 16 pounds - is paid to the government as tax. Total income so far is therefore 180 pounds: 100 from the initial government spend, and 80 pounds from the subsequent consumption spending. The total amount of tax paid to the government is 36 pounds (20 plus 16). And the private sector *now* holds 64 pounds as disposable income (80 pounds income - 16 pounds tax = 64 pounds) which also entirely spent and a similar pattern of further income and tax receipts ensues.

round | spending | income | tax | disposable income |
---|---|---|---|---|

0 | 100 | 100 | 20 | 80 |

1 | 80 | 80 | 16 | 64 |

2 | 64 | 64 | 13 | 51 |

3 | 51 | 51 | 10 | 41 |

4 | 41 | 41 | 8 | 33 |

The table describes 5 such rounds of spending as the money injected by the initial government spend of 100 pounds circulates. There are a few things to note:

- income in each round is exactly equal to spending. This is simply down to the fact that spending and income are the two sides of any transaction
- tax is levied on the income of each round at 20%
- the disposable income from any round is what is available for spending in the next round (so the first column of each row is identical to last column of the row above it).
- On each successive round of spending, more new income is generated and more new tax is paid, though each is smaller than in the previous round.

Now, we can discover exactly where this is heading with a bit of fairly simple maths. Remember, the initial income, $Y_0$, is equal to the initial government spend, $G$. The tax paid on this income is $\theta Y_0$ and the income left over for subsequent spending is $(1-\theta) Y_0$. If this disposable income is spent in its entirety then the next round of income, $Y_1$, has the same value:

$$Y_1 = (1-\theta)Y_0$$The amount of tax paid on *this* round of income is $\theta Y_1$ and the amount remaining is $(1 - \theta) Y_1$. Therefore, as before, we can describe the *next* round of income:

In general, we can see that:

$$ Y_n = (1-\theta)Y_{n-1}$$i.e. the income earned in the *n*th spending round is equal to the income earned on the previous round *less the tax paid* on that income.

Now, here's where we can use a bit of ingenuity. We know that $Y_0$ is equal to $G$. So substituing $G$ for $Y_0$ in the expression for $Y_1$ we get:

$$Y_1 = (1-\theta)G$$But now we can substitute this expression for $Y_1$ into the expression for $Y_2$:

$$Y_2 = (1-\theta)(1-\theta)G$$or, more succinctly:

$$Y_2 = (1-\theta)^2G$$We've managed to express the first 3 rounds of income ($Y_0, Y_1, Y_2$) entirely in terms of only the initial government spend and the tax rate. We can keep going with this, or just generalise to the *n*th round of spending:

So now that we can express the income arising from any single round of spending simply in terms of the "fiscal policy" parameters only: the government spending and tax rate choices, $G$ and $\theta$.

If we know the income for each individual round of spending then the total amount of income over all spending rounds is simply the sum total of the individual spending rounds:

$$\sum_{n=0}^{\infty} Y_{n} = Y_0 + Y_1 + Y_2 + Y_3 + ... $$or

$$\sum_{n=0}^{\infty} Y_{n} = G + (1-\theta)G + (1-\theta)^2G + (1-\theta)^3G + ...$$Note that $\theta$ is a number between $0$ and $1$ (in our example $0.2$, i.e. 20%), and so the term $1-\theta$ is also always between $0$ and $1$. As this term is raised to increasing powers through successive terms in the sum above ($^{0,1,2,3...}$), we can see that it becomes smaller, and what is being added are therefore successively smaller fractions of the initial quantity $G$. The first term in the sum is just $G$ itself, the initial goverment spend which represents the initial flow of income. This term is equivalent to $(1 - \theta)^0 G$ although we don't bother writing it out in full. The next term, with $\theta = 0.2$ evaluates to $0.8 G$, and the subsequent terms to $0.64 G$, $0.51 G$, $0.41 G$ and so on.

We can fairly easily add together many more terms because the last expression is known in mathematics as a geometric series - a sum in which the successive terms share a *common ratio*. If we want to know the total income arising from a certain number, say $m$, of spending rounds, we can use this standard solution:

where $a$ is the common ratio, which in our case is $(1-\theta)$, giving:

$$\sum_{n=0}^{m-1} Y_{n} = \frac {1-(1-\theta)^m}{\theta}G \hspace{1cm} (2)$$As an illustration, in the plot below the blue bars represent income associated with each individual spending round (based on equation 1) and the red bars show cumulative income across spending rounds (equation 2). Graphically, the size of the red bars is equal to all of the blue bars up to that point stacked on top of one another.

```
In [2]:
```import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline
G = 100 # government spending
theta = 0.2 # tax rate
n_rounds = 30 # number of rounds we'll consider
# create an array of numbers from 0-30, one for each spending round
r = np.arange(0,n_rounds)
# solve equation 1 for each individual round
y = G*(1-theta)**r
# solve equation 2 for each individual round
sum_y = G*(1-(1-theta)**(r+1))/(1-(1-theta))
# plot
plt.bar(r,sum_y, color='r',label='cumulative income')
plt.bar(r,y, color='b', label='spending round income')
plt.grid()
plt.legend(loc='center right')
plt.xlabel('Spending round, n')
plt.ylabel('Income')
plt.tight_layout()
plt.show()

```
```

If we consider a very large number of spending rounds (i.e. $n \to \infty $) equation 2 converges to:

$$\sum_{n=0}^{\infty} Y_{n} = \frac {G}{\theta} \hspace{1cm} (3)$$which represents the *total* income from *all* concievable spending rounds. With $G = 100$ and $\theta = 0.2$ equation 3 evaluates to $\frac {100}{0.2} = 500$ and so we conclude that total amount of income achieved by the successive spending of 100 pounds with a 20% withdrawal rate due to taxation is 500 pounds. And the plot above corroborates this: the income converges to 500 pounds after ~20 rounds of spending.

So the affect of government spending is to cause a cascade of repeated spending which ultimately generates incomes which are some multiple of the initial government spend. We can summarise this for our case above by omitting the subscripts and simply writing:

$$Y = \frac {G}{\theta} \hspace{1cm} (4)$$Now $Y$ simply represents total income from all spending. Equation 4 is basically an analytical solution for our simple economy, describing the income for any given pair of fiscal policy settings. It is clear that in this simple economy, incomes can be raised by either increasing the government spending ($G$) or decreasing the tax rate ($\theta$), though these conclusions may not necessarily hold for more complicated scenarios.

Rearranging equation 4 we can get:

$$ \frac {Y}{G} = \frac {1}{\theta}$$This expression represents the total income earned *per pound spent by the government*. The right-hand side $\frac {1}{\theta}$ is called the *fiscal multiplier* and can be considered to describe the stimulatory effect of government spending on aggregate incomes in the economy. In our case the multiplier is $\frac {1}{0.2} = 5$ - the aggregate income attained is 5 times the size of the goverment spend. Another way of conceptualizing this is that each pound circulates 5 times on average before being collected in tax.

One last thing worth looking at is the total amount of tax that has been paid. Let's call tax revenue $T$. We know that $T$ is paid as a fraction, $\theta$, of income. In other words (well, symbols):

$$T = \theta Y$$Now, we can substitute our expression for $Y$ (equation 4):

$$T = \theta \frac {G}{\theta}$$which simplifies to $T = G$. Tax revenue for the government is exactly equal to the initial government outlay. The government has recouped all of its spending - it has a *balanced budget*. This explains why the cycle of spending and incomes slows down and ultimately stops: the money circulating is being withdrawn and is eventually entirely withdrawn. So tax acts as a "leakage" of money from the cycle of spending and this is seen in the decreasing size of the terms in the geometric sum. Essentially, the total amount of income that can be generated by the spending and re-spending of the government money is limited because the money is gradually recouped by the government through taxation. A simple but quite profound consequence of this model is that if the private sector reflexively spend all of their income, and the government levies an *income tax*, then it is an inevitability that *all* the money will end up back at government. There are only two uses of money - spending and paying tax - and spending simply results in income which is subject to the same two choices and therefore merely postpones the taxation. Eventually every pound is taxed and the tax rate determines how fast it occurs and therefore how much income can be generated in the interim.

In summary, we've assumed a pretty simple scenario wherein the government spends an amount of money and taxes income at a certain rate, and where the private sector spend all of their income, i.e. no saving occurs. The upshot of this is that the private sector directly receives an income from the government spending and this income drives further cycles of spending and therefore more income. The total aggregate income generated by all spending is a multiple, $\frac {1}{\theta}$, of the initial government spending. And the government money is gradually withdrawn via income taxation meaning that the government eventually recoups *all* of its spending via taxation - that is, it has a balanced budget

A number of questions emerge from this. What if the private sector *do* want to save some of their income? How many spending rounds do we need to consider? And what stops all the money being drained out of the economy by taxation? These will have to wait for another time.