## Deficit identities

Wed 12 October 2016

It's my birthday today and as a treat, I've turned off my phone and am ignoring emails and social media for a short time so I can write up some stuff on macroeconomic identities. Yes, I know it sounds a bit sad, but it's what makes me happy.

In this post I want to derive an identity that relates two deficits. An identity is an equation that has to be true by definition. As it makes no assumptions and doesn't build on any theory, it provides a solid starting point for understanding an economy.

Let's start with the macroeconomic identity for GDP expressed by types of expenditure:

\begin{equation} Y = C + G + I + X - M \label{income-exp} \end{equation}

It's worth taking a moment to be clear on the definitions of the expenditures involved:

• $$Y$$ is GDP — the total expenditure on all new products created, where products can be goods or services.
• $$C$$ is consumption — spending by all households.
• $$G$$ is government spending — excludes welfare payments for pensions and benefits.
• $$I$$ is investment — investment is spending by businesses excluding that on wages plus the value of all stock waiting to be sold (Note: macroeconomic investment has a meaning separate from the everyday one associated with buying shares).
• $$X - M$$ is net trade — money received from exports less money spent on imports — the trade deficit is the negative of this.

What this identity tells us is that GDP is the sum of all these types of spending into the economy. But, since one person's spending has to be another's income, GDP $$Y$$ can also be understood as the total of everyone's income. Also, it's important to remember that all these quantities are totals over a given time period, usually a year.

We can write another identity that expresses GDP as a total of different kinds of income:

\begin{equation} Y = W + P + V \label{income-use} \end{equation}

where $$W$$ represents monies paid to employees for their time spent working (i.e. wages, also known as compensation of employees in national accounts), $$P$$ represents gross profits (referred to as Gross Operating Surplus) and $$V$$ represents taxes on production and products less any subsidies from government (this is mainly VAT, but includes a number of less well-known taxes too).

You can think of this identity as the sum of employee income, business income and government income. Notice that employees don't get all of $$W$$ as income because some goes to paying tax, mainly income tax and national insurance. Likewise, business pays some of $$P$$ to the government as corporation tax. As such, $$V$$ is not total government income, just that from taxes on products and production.

Let's now focus on the income of the private sector.

$$W$$ and $$P$$ represent incomes to the private sector from supplying new goods and services in the economy, but there is another source of income that we'll call transfers that are not associated directly with any productive activity.

The bulk of transfers come from the government in the form of welfare payments such as pensions and various kinds of benefits, but they also include monies from interest payments from the public sector, notably government bonds. We'll denote transfers from the public sector by $$U$$.

There will also be income from returns on assets held overseas and returns to foreign-owned assets going in the other direction. Let's denote the net income from this (which may be negative) by $$Z$$.

We can now write an identity that expresses the balance of the private sector:

\begin{equation} W + P + U + Z = C + S + T \label{private-sector} \end{equation}

The left hand side is the total of all income to the private sector and the right hand side represents what the private sector can do with that income. Some of the income must go to the government as tax $$T$$, mainly income tax, national insurance and corporation tax and the remainder is either spent as consumption $$C$$ or kept as "savings" $$S$$. The colloquial notion of savings can easily mislead at this point, so I prefer to think of $$S$$ as just the income that is not-consumed.

Using equations (\ref{income-use}) and (\ref{private-sector}) we can express GDP as

\begin{equation} Y = C + S + T - U - Z + V \label{income-gdp} \end{equation}

and then equating the right hand sides of (\ref{income-exp}) and (\ref{income-gdp}) we have

\begin{equation} C + G + I + X - M = C + S + T - U - Z + V \label{balance-int} \end{equation}

and this can be simplified to give

\begin{equation} (G + U - T - V) + (I - S) + (Z + X - M) = 0 \label{balance-final} \end{equation}

As with all economics identities, (\ref{balance-final}) is a statement of the fact that if someone parts with money, someone else must receive it (even if either party doesn't realise it!).

However, this identity does allow us to understand how the public, private and foreign sectors are entwined and offers a rock-solid understanding of various deficits.

The first term in brackets is the deficit of the public sector, more usually referred to as the fiscal or budget deficit. It is the total of money leaving the public sector $$G+U$$, less revenue received $$T+V$$. This is the most famous deficit.

The second term could be called the deficit of the private sector, but since private sectors are rarely in deficit, $$S-I$$ is more usually referred to as net private sector savings. Again, this is somewhat at odds with the everyday notion of savings.

The third term is the current account balance — monies received from abroad less the amount we send abroad. The $$X-M$$ bit is the trade balance (or surplus) and its negative is referred to as the trade deficit.

## So what?

I set myself a time limit to batter this out into a blog post and my time is nearly up so there's no time for me to bring this to an elegant conclusion. Perhaps I'll write a follow-up blog post.

But if you'd like to get your thoughts flowing towards an interesting conclusion, then consider the situation of the UK just now: the current account balance $$Z+X-M$$ is well into the negative and the private sector is about breaking even, i.e. $$I-S$$ is small or close to zero. Given this, what must the fiscal deficit be according to (\ref{balance-final})? I'll leave this as an exercise for the reader, but you can see a visual answer in this post on Neil Wilson's 3spoken blog.

Right, I'm off for a wee dram and I'll toast you, dear reader, for getting this far... :)