Understanding the economy
I was never really that interested in economics until after 2008 when suddenly technical terms like structural deficit started appearing in the news. Oddly, with anything relating to macro economics, journalists have to pretend they were born understanding all the concepts perfectly and they're not going to patronize you by explaining them. This is in stark contrast to anything related science where they have to pretend to understand even less than they do! Somehow this attitude has leaked out to the wider world, so that friends and colleagues down the pub - or politicians on Question time - will b******t eternally about the effect of interest rate rises, but happily or even boastfully admit to knowing nothing about how the rest of the universe works. I, on the other hand, knew I didn't know anything about economics, but thought it was less important than all the other stuff I didn't know. But when economics stories started to become the main content of the news rather than a footnote, I decided I needed to try to understand things a bit better. So I did some maths and sent myself my an email so I wouldn't forget what I'd come up with.
From: me
To: me
Date: 01/May/2010
Subject: understanding the economy
I've been trying to work out how the economy works so that I can understand the news, but I didn't want to read any turgid books so I decided to have a go at doing it from first principles using some simple calculus. I feel I've learnt a lot from the exercise. I've probably made some howling mistakes....
g = \frac{\delta G}{G} \tag{1}
$$
I = rD \tag{2}
$$
S + rD - T = \delta D \tag{3}
$$
\delta\left(\frac{D}{G}\right) = \left(\frac{S}{G} - \frac{T}{G}\right) - \frac{D}{G}\left(g - r\right) \tag{4}
$$
What we are ultimately interested in is $\frac{D}{G}$ so let's try to solve (4) for $\frac{D}{G}$. Let's assume that $\frac{T}{G}$ and $\frac{S}{G}$ are fixed and that $\frac{S}{G} - \frac{T}{G}$ is positive as historically this has almost always been the case. Then
Now under normal conditions you would expect case C to be true. Why? Because gilts are "safe" and so the return they offer, $r$, should be lower than the return offered by the wider economy, $g$. At this point it helps to make some more definitions:
\text{"government overspend"}/(g - r) \tag{5}
$$
If we want to reduce the value of $\frac{D}{G}$ that we are tending towards then we need either to reduce "government overspend" or increase $g$, or decrease $r$.
If $g$ and $r$ are close or if $r$ is greater than $g$ (such as when in or just coming out of a recession) then the value (5) is much more sensitive to changes in $g$ and $r$ than in "government overspend". Therefore if reducing "government overspend", either by decreasing spending $\frac{S}{G}$ or increasing the tax rate $\frac{T}{G}$, damages growth $g$ it will do more harm than good.
This seems to back the government's policy and that of the Lib Dems and be counter to the policy of the Conservatives, which is to reduce the government overspend immediately. [NB: I wrote this just before the May general election in 2010]
From (5) it is clear that when $(g - r)$ starts to stabilize that is when reducing the "government overspend" becomes a sensible approach for reducing the target $\frac{D}{G}$. But whilst $g$ is rising fast, such as when the economy comes out of recesssion that is a bad idea.
Another observation is that you can keep the "government overspend" the same whilst reducing or increasing the "size of state". Neither increasing nor decreasing the "size of state" affect the target $\frac{D}{G}$ except in as much as they affect $g$, or $r$. However, it seems obvious that in the short term at least:
Broadly speaking Labour supports a larger state and argues that this positively affects growth $g$. The Conservatives support a smaller state and argue that this positively affects growth $g$.
To me it seems that during and immediately after a recession the rich are not ready to invest in the economy. By increasing the size of the state the government can start projects that increase growth (and protect the poor). By decreasing the size of the state the government give a small amount back to the rich, which given their lack of confidence in the economy they are more likely to sit on than invest in growth. So increasing the size of the state during a recession and contracting it in the good times seems a sensible policy.
Observation: Explaining the boom and bust cycle.
The argument above concentrated on case C, $g > r$. This is most likely to be the case when the economy is healthy. However, the longer the economy remains healthy the more investors will be willing to risk money they previously invested in safe - but low interest - gilts in the wider economy. Eventually the government will have to raise the return on gilts $r$ to attract buyers. At this point $\frac{D}{G}$ will start rising unless "government overspend" is capped. If $\frac{D}{G}$ rises too high fear of default will frighten away buyers of gilts so eventually "government overspend" is reduced and projects are cancelled and growth will fall. Eventually the same investors who originally switched to the wider economy return to gilts and the government has money to restart growth with.
This cycle continues. It looks like a 7 - 10 year cycle historically and I wonder if the fact that gilts are usually 15 year bonds defines this.
Observation: Explaining quantitive easing
During a recession case A holds, i.e. $r > g$. To prevent $\frac{D}{G}$ growing exponentially the Bank of England was forced to try to return to case C, $g > r$. It did this by printing money to buy gilts. This increased the "price" of gilts, i.e. it enabled the government to auction them off at a lower rate of interest. This reduces $r$ and returns the economy to the healthier condition where $g > r$.
And to discover how WWII and Bretton Woods led to this state of affairs, read Third World War by Elido Fazi.
From: me
To: me
Date: 01/May/2010
Subject: understanding the economy
I've been trying to work out how the economy works so that I can understand the news, but I didn't want to read any turgid books so I decided to have a go at doing it from first principles using some simple calculus. I feel I've learnt a lot from the exercise. I've probably made some howling mistakes....
Idea:
It seems to me that the deficit, the national debt, and growth are not in themselves important. The really important thing is the national debt as a proportion of GDP. Why? Because GDP reflects the ability of the government to generate taxes and therefore its ability to avoid defaulting on interest payments on the debt. The moment this ratio goes too high there's a risk that we will not be able to sell gilts to fund the budget at which point severe and immediate cuts or tax rises or both must be made. Therefore it's interesting to work out what factors affect this ratio of debt to GDP. That way we can work out how to avoid it getting too high.Definitions:
Let $g$ be growth
Let $G$ be GDP
Let $\delta G$ be the change in GDP in one year
Let $G$ be GDP
Let $\delta G$ be the change in GDP in one year
Theorem:
$$g = \frac{\delta G}{G} \tag{1}
$$
Proof:
by definition
Definitions:
Let $D$ be government debt
Let $I$ be the interest on government debt paid in one year
Let $r$ be the average rate of interest on existing gilts
Let $I$ be the interest on government debt paid in one year
Let $r$ be the average rate of interest on existing gilts
Theorem:
$$I = rD \tag{2}
$$
Proof:
by definition
Definitions:
Let $S$ be the government spending (not including debt interest $I$)
Let $T$ be the sum raised by taxation
Let $\delta D$ be the deficit
Let $T$ be the sum raised by taxation
Let $\delta D$ be the deficit
Theorem:
$$S + rD - T = \delta D \tag{3}
$$
Proof:
what we spend our money on:
total spending = $S + I$, therefore
total spending = $S + rD$
where we get the money from:
total spending = $T$ + change in balance, i.e.
total spending = $T + \delta D$
so $S + rD = T + \delta D$ and subtracting $T$ from both sides gives the result
total spending = $S + I$, therefore
total spending = $S + rD$
where we get the money from:
total spending = $T$ + change in balance, i.e.
total spending = $T + \delta D$
so $S + rD = T + \delta D$ and subtracting $T$ from both sides gives the result
Definition:
$\delta\left(\frac{D}{G}\right)$ is the yearly change in the debt to GDP ratio
Theorem:
$$\delta\left(\frac{D}{G}\right) = \left(\frac{S}{G} - \frac{T}{G}\right) - \frac{D}{G}\left(g - r\right) \tag{4}
$$
Proof:
Differentiating $\frac{D}{G}$ we get
$$
\frac{d}{dt}\frac{D}{G} = \frac{dD}{dt}\frac{1}{G} - \frac{D}{G^2}\frac{dG}{dt}
$$
multiplying both sides by $G$ and letting $dt = 1$ (year) we get
$$
G\delta\left(\frac{D}{G}\right) = \delta D - \frac{D}{G}\delta G
$$
substituting in (1) we get
$$
G\delta\left(\frac{D}{G}\right) = \delta D - Dg
$$
substituting in (3) we get
$$
\begin{align}
G\delta\left(\frac{D}{G}\right) &= S + rD - T - Dg\\
&= (S - T) - D(g - r)
\end{align}
$$
and dividing both sides by $G$ gives the result
$$
\frac{d}{dt}\frac{D}{G} = \frac{dD}{dt}\frac{1}{G} - \frac{D}{G^2}\frac{dG}{dt}
$$
multiplying both sides by $G$ and letting $dt = 1$ (year) we get
$$
G\delta\left(\frac{D}{G}\right) = \delta D - \frac{D}{G}\delta G
$$
substituting in (1) we get
$$
G\delta\left(\frac{D}{G}\right) = \delta D - Dg
$$
substituting in (3) we get
$$
\begin{align}
G\delta\left(\frac{D}{G}\right) &= S + rD - T - Dg\\
&= (S - T) - D(g - r)
\end{align}
$$
and dividing both sides by $G$ gives the result
Observations:
In equation (4) the term $\frac{T}{G}$ is the taxation as a proportion of GDP. This figure is largely a policy decision, it roughly equates to "average taxation on transactions". So if we're considering the effect of a particular economic policy it is reasonable to take this to be a constant. Also the term $\frac{S}{G}$, or spending as a proportion of GDP is roughly speaking a policy decision and can be taken as a constant for a particular economic policy.What we are ultimately interested in is $\frac{D}{G}$ so let's try to solve (4) for $\frac{D}{G}$. Let's assume that $\frac{T}{G}$ and $\frac{S}{G}$ are fixed and that $\frac{S}{G} - \frac{T}{G}$ is positive as historically this has almost always been the case. Then
[case A]
if $g < r$ :
$\frac{D}{G}$ rises at an ever increasing rate
[case B]
if $g = r$:
$\frac{D}{G}$ rises at a constant rate
[case C]
if $g > r$:
$\frac{D}{G}$ tends towards $\left(\frac{S}{G} - \frac{T}{G}\right)/(g - r)$
if $g < r$ :
$\frac{D}{G}$ rises at an ever increasing rate
[case B]
if $g = r$:
$\frac{D}{G}$ rises at a constant rate
[case C]
if $g > r$:
$\frac{D}{G}$ tends towards $\left(\frac{S}{G} - \frac{T}{G}\right)/(g - r)$
Now under normal conditions you would expect case C to be true. Why? Because gilts are "safe" and so the return they offer, $r$, should be lower than the return offered by the wider economy, $g$. At this point it helps to make some more definitions:
Definitions:
Let "size of state" be the midpoint between $\frac{S}{G}$ and $\frac{T}{G}$
Let "government overspend" be the difference of $\frac{S}{G}$ less $\frac{T}{G}$
Let "economic policy" be a choice of values for $\frac{T}{G}$ and $\frac{S}{G}$
Let "government overspend" be the difference of $\frac{S}{G}$ less $\frac{T}{G}$
Let "economic policy" be a choice of values for $\frac{T}{G}$ and $\frac{S}{G}$
Theorem:
For a given "economic policy", if the debt as a proportion of GDP $\frac{D}{G}$ stabilizes then it tends towards
$$\text{"government overspend"}/(g - r) \tag{5}
$$
Proof:
By definition
Observations:
If we want to reduce the value of $\frac{D}{G}$ that we are tending towards then we need either to reduce "government overspend" or increase $g$, or decrease $r$.
If $g$ and $r$ are close or if $r$ is greater than $g$ (such as when in or just coming out of a recession) then the value (5) is much more sensitive to changes in $g$ and $r$ than in "government overspend". Therefore if reducing "government overspend", either by decreasing spending $\frac{S}{G}$ or increasing the tax rate $\frac{T}{G}$, damages growth $g$ it will do more harm than good.
This seems to back the government's policy and that of the Lib Dems and be counter to the policy of the Conservatives, which is to reduce the government overspend immediately. [NB: I wrote this just before the May general election in 2010]
From (5) it is clear that when $(g - r)$ starts to stabilize that is when reducing the "government overspend" becomes a sensible approach for reducing the target $\frac{D}{G}$. But whilst $g$ is rising fast, such as when the economy comes out of recesssion that is a bad idea.
Another observation is that you can keep the "government overspend" the same whilst reducing or increasing the "size of state". Neither increasing nor decreasing the "size of state" affect the target $\frac{D}{G}$ except in as much as they affect $g$, or $r$. However, it seems obvious that in the short term at least:
- increasing the "size of state" benefits the poor and makes the rich worse off
- decreasing the "size of state" benefits the rich and makes the poor worse off
- keeping the "size of state" the same but reducing "government overspend" makes everyone worse off but reduces $\frac{D}{G}$
Broadly speaking Labour supports a larger state and argues that this positively affects growth $g$. The Conservatives support a smaller state and argue that this positively affects growth $g$.
To me it seems that during and immediately after a recession the rich are not ready to invest in the economy. By increasing the size of the state the government can start projects that increase growth (and protect the poor). By decreasing the size of the state the government give a small amount back to the rich, which given their lack of confidence in the economy they are more likely to sit on than invest in growth. So increasing the size of the state during a recession and contracting it in the good times seems a sensible policy.
Observation: Explaining the boom and bust cycle.
The argument above concentrated on case C, $g > r$. This is most likely to be the case when the economy is healthy. However, the longer the economy remains healthy the more investors will be willing to risk money they previously invested in safe - but low interest - gilts in the wider economy. Eventually the government will have to raise the return on gilts $r$ to attract buyers. At this point $\frac{D}{G}$ will start rising unless "government overspend" is capped. If $\frac{D}{G}$ rises too high fear of default will frighten away buyers of gilts so eventually "government overspend" is reduced and projects are cancelled and growth will fall. Eventually the same investors who originally switched to the wider economy return to gilts and the government has money to restart growth with.
This cycle continues. It looks like a 7 - 10 year cycle historically and I wonder if the fact that gilts are usually 15 year bonds defines this.
Observation: Explaining quantitive easing
During a recession case A holds, i.e. $r > g$. To prevent $\frac{D}{G}$ growing exponentially the Bank of England was forced to try to return to case C, $g > r$. It did this by printing money to buy gilts. This increased the "price" of gilts, i.e. it enabled the government to auction them off at a lower rate of interest. This reduces $r$ and returns the economy to the healthier condition where $g > r$.
POSTSCRIPT
Most comedy is just entertaining, but this routine - The History of Oil by Robert Newman - is enlightening! In it he explains what it means to say that the dollar is the world's reserve currency, how that means America can spend more than it earns, and how trade in oil keeps maintains this status quo.
And to discover how WWII and Bretton Woods led to this state of affairs, read Third World War by Elido Fazi.
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