### Archive

Archive for the ‘Money’ Category

## Is zero inflation optimal?

Discussion on the ‘costs of inflation’ is a staple of A level Economics courses. You have probably heard these arguments before:

Shoe leather costs – as inflation rises and money becomes worth less, you have to take more trips to the bank so are wearing out your ‘shoe leather’ (really the cost here is the cost of your time rather than your shoes unless you are wearing really flimsy shoes).
Menu costs – as inflation rises and firms have to change their prices more often, they incur costs reprinting menus, catalogues etc.
Tax distortions – the classic case is ‘fiscal drag’ – governments usually set and fix tax bands at the start of the financial year, so if prices are rising sharply that has a distortionary effect, eg if higher rate income tax kicks in at £40000 a year, and you are earning £35000 a year in an environment where there is 20% inflation, your employer might give you a pay rise to keep pace with inflation putting you on £42000 – in real terms that is just the same as earning £35000 a year ago, but now you have suddenly become a ‘high rate’ tax payer. This also happens for capital gains tax, when people sell assets like houses that have appreciated in value due to high inflation, the tax paid becomes disproportionately high.

These answers will probably get you marks in an A level exam but in the modern economy these aren’t major issues unless you get into hyperinflation territory. There is so much electronic banking these days that shoe leather costs are minimised, and many businesses, especially in retail, change prices all the time anyway regardless of inflation, cycling offers and discounts to attract customers. Many do business on the internet as well where it’s easy to change your menus. So the menu costs are not massive. As for tax distortions they are a real nuisance but they could easily be dealt with if the government just changes their tax rules to be indexed to inflation. The fact they don’t is probably because the government is usually the one who benefits from fiscal drag anyway so they will just take the extra tax revenue (in an environment of high inflation they want to take all the benefits they can get).

Here are the big problems that come from inflation being high:

Money illusion – price signals break down when people lose their ability to judge and compare prices over time. This sounds like such a basic fact but it underpins the whole market economy. Lets say for example you decide to go shopping for some Levis jeans. Say last time you went looking for jeans was eight months ago, you saw Levis were around £80 so you will have this kind of idea in your mind as to the price of Levis. If we are in a low inflation environment (say 2%) and you see some Levis in a shop at £90 then you will have an instinctive feel for the fact that is probably pricey and you can get it elsewhere. If the market price was £80 eight months ago and inflation is 2% per year then that means that every month prices will be rising $1.02^\frac{1}{12}=1.00165$ in other words they are rising 0.165% per month, so after eight months you would expect the market price to be $80(1.00165^8 )=81.06$. This is basically a stable price, so your ‘feel’ for the market price as being £80 will be more or less right.

Now think of that scenario but in an environment where inflation is running at 60%. Say you see some jeans on offer for £105, is that a good deal or a bad deal. That is quite hard to do in your head – you have to make an ‘educated guess’ and you could be right or wrong.

If you actually had a calculator, and knew what you were doing in terms of compound interest, you could say that 60% inflation means $1.6^\frac{1}{12}=1.0399$ ie prices are rising 3.99% per month, so after eight months you would expect the market price to be $80(1.0399^8 )=109.44$. So in this case an offer of £105 is good, they are below the market price in real terms. But you aren’t going to do that in your head so you have lost the instinctive feel for prices.

This is what happens when inflation gets high, and the higher inflation is the more people lose their ability to compare prices, so it undermines the whole ability of firms to compete with each other on prices, and prices lose their ability to signal scarcity.

Inflation variability – generally low inflation means stable inflation and high inflation means variable inflation from one year to the next. Inflation in the UK (the RPI) from 1997 to 2005 was 3.3%, 2.4%, 2%, 2.7%, 1.3%, 2.9%, 2.6%, 3.2%, 2.4%. Compare this to the period between 1973 and 1981: 12%, 19.9%, 23.4%, 16.6%, 9.9%, 9.3%, 18.4%, 13%. Instead of varying by 1-2% per year, it is jumping around in variations of 5-10% per year. And this is the UK, which has traditionally been a low or medium inflation economy, the figures for some Latin American countries would dwarf that for variability. Variability is a problem because businesses need to plan for inflation when they set their wage levels, set their prices, sign contracts with suppliers etc. If you get it wrong by a percent or so its not too bad but if you think inflation will be 10% higher than it ends up being, and you have set your prices accordingly, you could be in for a big loss. The result is when inflation is high, firms do less investment and think more ‘short term’ rather than planning for the long term.

On the other hand there are some ways in which having some inflation is preferable to none.

Wage flexibility – generally wages are ‘sticky downwards’, which means that you very rarely get pay cuts in nominal terms, apart from in the more flexible parts of the private sector. Trying to cut workers’ wages in nominal terms stirs up a hornets nest of legal challenges and trade union action, so a pay freeze is about as strict as you usually get. Sometimes in a benevolent economic environment, especially when there is an unsustainable boom, wages can jump ahead of productivity, which means there will be problems down the line, not least in the context of international trade where a country with firms paying wages above its level of productivity will lose competitiveness to international competitors. So you need wages to adjust downwards, which is a slow and painful process – and the lower inflation is, the harder this is. You don’t actually need high inflation to carry out an adjustment, just a medium level. In the UK for instance, the government announced a three year public sector pay freeze in 2010. If inflation during that period runs at 4% then that means in real terms wages will fall by close to 12%, so that is an effective way of bringing wages back down in line with productivity. But if inflation was creeping around 1-2% it would take much longer to get the same adjustment.

Option of real negative interest rates – this is a big one when the economy is in trouble, particularly in the face of a liquidity trap. The issue here is that nominal interest rates can’t fall below 0%, but what influences investment is not nominal but real interest rates. As $r \approx i - \pi$. you can stimulate investment by pushing down real interest rates through higher inflation, when nominal interest rates are 0%, real interest rates can be $- \pi$. Again you don’t want to trigger hyperinflation in order to break out of a liquidity trap, but you might find inflation of 5-6% useful, rather than 1-2%.

Seignorage revenue -fraught with danger this one, because it can easily be misused and trigger an inflationary spiral. However for a developing country where it is difficult to collect taxes, seignorage revenue can be an important part of government revenue.

So what is the optimal level of inflation? There isn’t a set guide and it depends on opinion. By and large most Western democracies aim for low levels around about 1.5-2%. There are some that argue you should try to push inflation down to as close to 0% as possible, and there are others that argue some of the excessively tight monetary policy used to keep inflation low is counterproductive, and that there is nothing really wrong with inflation at 4-6%. Certainly the real negatives of inflation don’t kick in till you get higher rates (eg double figures).

The big danger with inflation though is its tendency to accelerate out of control, so once you get to 10%, it can quickly start to creep up towards 15-20% and beyond, where negatives do start to mount up.

Categories: Macro, Money

## Seignorage – a tax on real money balances

Governments in most developed countries typically finance a budget deficit (the gap between government spending and tax receipts) through borrowing from the domestic or foreign private sector. Occasionally, if they can’t raise enough revenue through selling bonds to the private sector, they can get the central bank to print money to buy the bonds. This is debt monetization. This is quite rare in developed countries but it is more likely in developing countries where there can be crises (eg wars) that lead to a collapse in the ability to collect taxes, so deficits can rise beyond the government’s ability to raise revenue through borrowing. It can also happen when lenders fear sovereign default, and begin to demand ates of interest on government borrowing that the government cannot afford.

The revenue gained by government by printing money is called seignorage. It is effectively a tax on real money balances.

The seignorage revenue received by the government is $Seignorage = \frac{\Delta M}{P}$. We can multiply the expression by $\frac{M}{M}$ just to give us an alternative way of writing it.

$Seignorage = \frac{\Delta M}{P}=\frac{\Delta M}{M}\frac{M}{P}$, in other words it is the rate of money growth multiplied by real money balances.

Inflation approximately equals nominal money growth minus output growth $\pi = g_M - g_Y$, so in the short run where there is no output growth, growth $\pi = g_M=\frac{\Delta M}{M}$.

So we can write the expression for seignorage as being $Seignorage = \pi \frac{M}{P}$, ie it is inflation multiplied by the amount of real money balances in the economy (hence it being a tax on real money balances).

This is pretty effective tax because you can’t avoid it. Anybody who holds money effectively pays the tax because their money becomes worth a little bit less but the government is getting free money to spend on its spending programmes.

There is a complicating factor here, because the demand for money declines as inflation rises. You can express money demand in the form $\frac{M}{P}=YL(i)$, ie the demand for real money balances is a function of income (or output), and peoples liquidity preference schedule (how downward sloping the money demand curve is). Note that it is a function that is increasing in terms of income (the richer people are the more money they want to hold) and declining in terms of nominal interest rate (when interest rates are higher, people want to hold less money and more bonds or other forms of illiquid assets).

You can write this in terms of real interest rates as $\frac{M}{P}=YL(r+ \pi^e)$.

Over time, income, the real interest rate, and expected inflation can all change. But it is useful to think of what would happen in the ‘super short run’ (like month by month timescales) when modelling seignorage, because the big risk with seignorage (as will be explained soon) is that it triggers very high inflation, where inflation changes very quickly over a time scale where the real interest rate and income are more or less static.

So to model this we will assume that income and real interest rate stay constant and the variable factor is expected inflation: $\frac{M}{P}=\bar{Y}L(\bar{r}+ \pi^e)$, where L (demand for money) and hence seignorage, is declining in $\pi^e$.

This implies during times of high inflation, money demand depends mainly on expected inflation. As expected inflation rises, money demand falls. This is basically because as money is losing its value quickly, you don’t want to hold it for very long – higher expected inflation increases the opportunity cost of holding money.

However, in practice the real interest rate may become very negative because the nominal interest rate does not keep up with inflation, so you are not always better off holding bonds either! And it is hard to put your money in bonds fast enough, it may not be practical. So instead people start bartering goods, they start demanding wages more often (eg twice a week), or they start using a hard currency (eg dollarization). As inflation rises rapidly, people do whatever they can to avoid holding cash, and demand for money collapses.

Think about what is going on here. On the one hand, increasing money growth, is increasing the rate of the inflation tax (so seignorage is rising), on the other hand, increasing money growth is increasing inflation and decreasing money demand and so the amount of real money balances being held in the economy (so seignorage is falling). There are two effects working against each other here.

We have two equations: $Seignorage = \frac{\Delta M}{M} \frac{M}{P}$ and $\frac{M}{P} = \bar{Y}L(\bar{r}+\pi^e)$.

We can combine them to get: $Seignorage = \frac{\Delta M}{M}\bar{Y}L(\bar{r}+\pi^e)$, where L (demand for money) and hence seignorage, is declining in $\pi_e$.

Now think about what would happen if we had constant money growth. Over time, inflationary expectations would adjust to the constant level of money growth, they would catch up with it, $\frac{\Delta M}{M}=\pi^e$ so $Seignorage = \frac{\Delta M}{M} \bar{r} +\frac{\Delta M}{M}$. Here $\frac{\Delta M}{M}$ is entering the equation twice. Inflation is increasing in it directly and declining in it indirectly via inflationary expectations and falling money demand. At the start the indirect effect is small but it becomes large quite quickly, eventually outweighing the direct effect. There is therefore a hump shape like a Laffer curve, in terms of the amount of ‘tax’ (seignorage) that can be collected in this way. There will be a rate of constant money growth which optimises the amount of seignorage revenue.

Example: suppose the economy has GDP of 100. The real money stock is given by the money demand equation $\frac{M}{P} =\bar{Y}[1-(\bar{r}+\pi^e)]$ where $\bar{Y}=100$ and $\bar{r}=0.03$ so $\frac{M}{P} =100[1-({0.03}+\pi^e)]$.

To find the rate of constant nominal money growth that would maximise seignorage we start with the equation $Seignorage = \frac{\Delta M}{M}100[1-({0.03}+\pi^e)]$ remembering that in the case of constant nominal money growth, $\frac{\Delta M}{M}=\pi^e$.

So $Seignorage = \frac{\Delta M}{M}100[1-(0.03+\frac{\Delta M}{M}]=\frac{\Delta M}{M}100[0.97-\frac{\Delta M}{M}]=97\frac{\Delta M}{M}-100(\frac{\Delta M}{M})^2$.

So to optimise this we differentiate $\frac{d(Seignorage)}{d\frac{\Delta M}{M}}=97-200\frac{\Delta M}{M}$ and set equal to zero so $0=97-200(\frac{\Delta M}{M}) \Rightarrow \frac{97}{200}=\frac{\Delta M}{M} \Rightarrow 0.485=\frac{\Delta M}{M}$.

This tells us that the optimising rate of nominal money growth is 48.5%. With this level of money growth we could raise seignorage revenue of $Seignorage = 97(0.485)-100(0.485)^2 = 23.523$ ie we can raise maximum income of 23.523 through seignorage. Given that GDP is 100, this means the maximum budget deficit we could cover through seignorage would be 23.523% through seignorage.

Now what would happen if the budget deficit was actually 28%, ie we needed to raise income of 28 through seignorage. We can’t do this keeping constant money growth, the only way we could do this is to hike up nominal money growth above the level of expected inflation. We can get away with this in the short run, but inflationary expectations will catch back up with our new level of expected inflation.

Remember that in our short run with no growth, $\pi^e = \frac{\Delta M}{M}$, so when we have money growth of 48.5%, we already have inflation of 48.5% (not a good situation to be in). So inflationary expectations will be 48.5%.

What rate of nominal money growth could get us seignorage revenue of 28 with inflationary expectations of 48.5%?

$28 = \frac{\Delta M}{M}100[1-(0.03+0.485)] \Rightarrow 28=\frac{\Delta M}{M}(48.5) \Rightarrow \frac{\Delta M}{M} = 0.5773$. So we need money growth of 57.73% which means inflation of 57.73%. In the short run we can get the required level of seignorage with this higher level of inflation, but what happens when inflationary expectations catch back up to the new level of money growth? We will have to do the equation again, with 57.73% as the new value for expected inflation. This will imply that we need an even higher rate of money growth, and hence inflation, to hit our seignorage target.

The moral of the story here is that there is an optimal rate of seignorage revenue that you can generate (depending on the parameters of the money demand equation) through constant money growth.

If you want/need to raise seignorage revenue higher than that, you have to do it through increasing money growth. This means you are going to trigger an inflationary spiral, and this is how you end up with hyperinflation.

Categories: Macro, Money

## The Quantity Theory of Money

The quantity theory of money basically explains how the quantity of money in the economy affects the price level. The quantity theory is usually explained simply in the identity MV = PT. This means quantity of money (M) multiplied by velocity (V) equals price level (P) multiplied by the number of transactions (T). The velocity of money describes the rate at which money circulates round the economy – for instance if you were to follow the life of one pound coin, velocity tells us how many times that pound changes hands in a given time period.

Because this is an identity, the left hand side always equals the right. Suppose you had an economy existing solely of cars, and there were 100 cars bought and sold in a year at a price of £3000 each, and the overall quantity of money in the economy was £50000

The number of transactions (T) is 100, the average price level of transaction is (P) is £3000 per car, so PT = £300000. M is £50000, so MV = PT means 50000V = 300000, so V = 6. This means the velocity of money is 6, each pound must change hands 6 times a year.

Usually when using the quantity theory of money model we make a couple of assumptions:

1 – The number of transactions is proportional to the total output in the economy. This is fairly logical – if the economy grows then more stuff is produced so more stuff changes hands. So we can roughly substitute output (Y) for transactions (T) and express the quantity theory identity in a more useful form: MV = PY. You can think of velocity now as being the income velocity (number of times a pound enters someone’s income in a given period of time).

2 – Velocity of money is constant. This is a simplification of reality because the money demand function will change depending on the interest rate. The velocity of money is related to the money demand functon – by assuming the velocity is constant we are assuming a money demand function of $\frac{M}{P}=kY$ where k is a constant. This can be rewritten as $\frac{M}{k}=PY$ which is the same as MV = PY when V = (1/k). If k is large then people have a high demand for money and want to hold a lot of money for each pound of income, so money does not change hands very much, V is small. If k is small then people have a low demand for money and want to hold little money for each pound of income, in this case money changes hands quickly and V is large. Assuming k is constant means V is constant which makes the identity a lot more informative as if one part is fixed it allows to see how certain elements of the identity will adjust in response to changes to the others.

So if we call the quantity theory of money $M\bar{V}=PY$ where V is now fixed, then it leads us to important conclusions:

– If the quantity of money in the economy (M) increases faster than output (Y) increases, then the price level (P) has to rise.
– If the quantity of money (M) increases slower than output (Y) increases, then the price level (P) has to fall.

In the real world prices tend to adjust upwards more easily than they adjust downwards (known as being sticky downwards) so if the price level cannot adjust downwards quickly enough, then output itself may decrease in response to an excessively slow increase in M. In this case you have a recession rooted in monetary factors – people are not spending due to a lack of available money to facilitate transactions, so demand drops. This type of recession can be addressed by increasing the money supply.

Categories: Macro, Money

## The LM relation

The downward sloping money demand curve can be written like this: $\frac {M}{P}=YL(i)$. What this means is real money balances = the level of income (Y) in the economy multiplied by the liquidity preference (L). The liquidity preference is basically the steepness of the curve, it shows how much people prefer to hold bonds (or other illiquid assets) rather than money, at a higher interest rate, if the liquidity preference schedule is steep then it means demand for money is less elastic with respect to interest rate, if it is fairly flat then it means demand for money is elastic with respect to interest rate (ie a small fall in interest rate means a large increase in the amount of money people want to hold rather than holding bonds). The reason there is an (i) in brackets after the L means that liquidity preference is a function of i, the nominal interest rate. It will of course depend negatively on i, because the higher the interest rate, the lower is M.

L(i) determines the steepness of the curve, and Y determines where the curve is, if you increase Y, then you push the whole curve up. This is the principle behind the LM relation. We can basically think of a relationship between i, the nominal interest rate, and Y, the level of income in the economy. As Y increases, then you shift up the money demand curve, so if you keep the money supply constant (the vertical line) then the point of intersection on the diagram is higher. This means that the new interest rate which brings the money market into equilibrium (money supply = money demand) is higher. So if money supply is constant, increasing Y means you get an increase in i; decreasing Y means you get a decrease in i. This is the LM relation – and it is upward sloping:

Think of movements along the LM curve as being what happens when you shift money demand curve up and down in the money supply/money demand diagram. Higher Y means money demand shifts up so you get higher i in equilibrium, that’s a movement up the LM curve.

What shifts the LM curve up or down is movements in the money supply curve. If you shift the money supply curve out (to the right) on the money supply/money demand diagram then you will get a lower i in equilibrium, at all levels of income, that’s the equivalent to shifting the LM curve down. If you shift the money supply curve in (to the left) on the money supply/money demand diagram then you will get a higher i in equilibrium at all levels of income, that’s the equivalent to shifting the LM curve up.

Remember that prices also affect the position of the money supply curve, because we are thinking about real money balances (M/P). As P is on the denominator, an increase in P makes (M/P) smaller, so an increase in P (rising prices) has the same effect as a decrease in M (reducing nominal money supply), so it moves the money supply curve to the left.

So to sum up:
The LM curve is upward sloping, as income in the economy increases, the interest rate increases
The LM curve shifts down when there is an increase in nominal money supply (a monetary expansion) or a fall in prices (rare!)
The LM curve shifts up when there is a reduction in the nominal money supply (a monetary contraction) or a rise in prices (inflation)

The Central Bank can stop inflation from pushing the LM curve up and increasing interest rates, by increasing nominal money supply in a proportionate amount that keeps (M/P) constant when P is rising.

Categories: ISLM, Macro, Money

## Instruments of monetary policy

The Central Bank has a few tools it can use to control the money supply.

The most well-known method in modern economies is open market operations. The Central Bank will hold a stock of illiquid assets like bonds. When it wants to decrease the supply of money in the economy, it will sell some of its bonds, when retail banks buy them, it simply deducts the value in cash from their reserve accounts at the Central Bank. That means those banks hold bonds where they used to have cash reserves, and the amount of high-powered money has fallen. Alternatively, if it wants to increase the amount of high-powered money, it can go to the markets and buy some bonds from retail banks. It pays for it by simply increasing the value of those banks’ reserve accounts held at the Central Bank, so bank reserves rise, and bank reserves are a component of high-powered money.

It can change the required reserve ratio, $\theta$, by forcing retail banks to hold a higher proportion of their liabilities to depositors in the form of reserves. This raises the money multiplier, $\frac{1}{c+\theta (1-c)}$. So it means any increase in high-powered money that the Central Bank introduces, will have a more magnified effect on the total amount of money in the economy. Not all countries have a legally required reserve ratio for banks.

As the Central Bank is the ‘lender of last resort‘ when banks are short on liquidity and need a short-term loan it can control the rate at which it charges interest to other banks. This will obviously have a knock on to banks’ market rates. If the Central Bank pushes up the rates at which it lends to other banks, they will charge their borrowers higher rates themselves. But if the Central Bank cuts its rates, other banks will take advantage of the ability to borrow cheaply from the Central Bank, by cutting their rates on lending to try and compete with each other for potential borrowers, who are all chasing the most competitive rates.

Categories: Macro, Monetary Policy, Money

## Fractional reserve banking

Fractional reserve banking is at the heart of the way money is created. It means that banks do not keep cash reserves equal to the balances of their depositors, instead they just hold a fraction of their depositors’ balances in reserve. This allows them to use the rest of the depositors’ money to make loans to others. It also relies on the gamble that depositors are not going to want to withdraw all their money at once but will only demand access to a small amount of their deposits at any one time.

For instance if you had a bank that had £1000 of deposits from customers, that £1000 is a liability to the bank because it owes £1000 to its customers should they wish to withdraw it. If the bank keeps £1000 in reserves then it is fully covered, but has no money free to make loans to others – this would be 100% reserve banking. If instead the bank said that it would operate a reserve ratio of 10% (or 0.1) then it would just keep £100 in reserves and then make £900 of loans to other customers. This £900 would then be an asset to the bank as it is owed by the customers to the bank – of course the bank would charge interest on this loan as well.

Now what would happen if that bank then received a new deposit of £100? It now has £1100 of deposits from customers and £200 in reserves. But if it is just keeping a reserve ratio of 0.1, then for £1100 of deposits it only needs to keep £110 in reserves, so it can reduce its reserves by lending out another £90. So from £100 deposit the bank creates £90 of new loans. The reserve ratio of 0.1 means that the bank lends out 0.9 of its new deposits.

If that £90 gets lent to a customer that goes and deposits it in their bank (or spends it and it ends up deposited in another customer’s bank). Their bank will keep 0.1 x £90 = £9 extra in reserves and lend out 0.9 x £90 = £81 to another customer.

And so the pattern repeats itself. That customer deposits the £81 which goes into another bank. When their bank gets it they keep back 0.1 x £81 = £8.10 in reserves and lend out 0.9 x £81 = £72.90. The original deposit of £100 is creating new loans through every round of lending and depositing but each round gets smaller.

Where will it end? We can think of it in terms of a geometric series. If we call the initial deposit $d$ and the reserve ratio $\theta$ then the rounds of spending look like this:

$d + d(1-\theta) + d(1-\theta)^2 + d(1-\theta)^3 +....d(1-\theta)^n = d[1 + (1-\theta) + (1-\theta)^2 + (1-\theta)^3 +....(1-\theta)^n ]$. This is a geometric series with geometric ratio $1-\theta$.

As it’s a geometric series it will sum to $d[\frac{1-(1-\theta)^{n+1}}{1-(1-\theta)}] = d[\frac{1-(1-\theta)^{n+1}}{\theta}]$ (see here for why). As $0<(1-\theta)<1$ then it means when $n \rightarrow \infty$, $(1-\theta)^{n+1} \rightarrow 0$ so $d[\frac{1-(1-\theta)^{n+1}}{\theta}] \rightarrow d[\frac{1}{\theta}]$.

This is the bank multiplier, $\frac{1}{\theta}$. The total amount of money created from the initial deposit is found by multiplying the initial deposit by $\frac{1}{\theta}$.

We can extend this model by taking account of the fact that customers will hold some of their wealth in cash, they won’t just deposit everything straight back into the bank.

Before making the model lets lay out all the definitions we need.
$M$ is the total amount of money people have.
$D$ is the total amount customers deposit with banks.
$CU$ is the total amount of currency people hold in cash.
$c$ is the currency ratio, the proportion of their total money that they hold in cash, so $cM = CU$ and $(1-c)M = D$
$R$ are the cash reserves banks keep in order to keep enough liquidity to cover the expected withdrawals of consumers.
$\theta$ is the reserve ratio, ie $R = \theta D$

Now if we extend the original model by saying there is a reserve ratio of 0.1 and a currency ratio of 0.4, this time the initial £100 meant again that the bank could lend out 0.9 x £100 = £90 to a customer.

That now has £90, but will hold 0.4 x £90 = £36 in cash and deposit 0.6 x £90 = £54 into a bank. The bank will then lend out 0.9 x £54 = £48.60 to another customer who will deposit 0.6 x £48.60 = £29.16 into theirs.

The pattern continues as before but this time the multiplier is smaller. The geometric series is:
$d[1 + [(1-\theta)(1-c)] + [(1-\theta)(1-c)]^2 + [(1-\theta)(1-c)]^3 +....[(1-\theta)(1-c)]^n ]$, the geometric ratio this time is $(1-\theta)(1-c)$.

So this geometric series will sum to $d[\frac{1-[(1-\theta)(1-c)]^{n+1}}{1-[(1-\theta)(1-c)]}] = d[\frac{1-[(1-\theta)(1-c)]^{n+1}}{1-[1-\theta-c+\theta c]}]=d[\frac{1-[(1-\theta)(1-c)]^{n+1}}{\theta+c-\theta c}]$.

Again, as $0<\theta<1$ and $0 then when $n \rightarrow \infty$, $[(1-\theta)(1-c)]^{n+1} \rightarrow 0$. So $d[\frac{1-[(1-\theta)(1-c)]^{n+1}}{\theta+c-\theta c}] \rightarrow d[\frac{1}{\theta+c-\theta c}]=d[\frac{1}{c+\theta(1-c)}]$

This is the money multiplier, $\frac{1}{c+\theta(1-c)}$. It is smaller than the bank multiplier.

Categories: Macro, Money

## The money multiplier

If the interest rate is determined by the interaction between money supply and money demand, how can we model this in a simple way?

We can look at it through defining the concept of Central Bank money, or high-powered money (also known as the monetary base).

Firstly think in terms of assets and liabilities. Retail banks (the type individual consumers bank with) will hold as assets the loans they make (eg mortgages, personal loans), that need to be repaid back to them, they will hold bonds and shares etc, and they will keep a certain amount of liquid assets (like cash) as reserves in their account with the Central Bank. They need these reserves to have liquidity available to meet the daily needs of customers’ withdrawals, when customers want to withdraw money they want cash not bonds, so the bank needs to have a ready supply of it.

Retail banks effectively ‘bank’ with the Central Bank and they have their own accounts there, this is effectively what happens if you buy something with a debit card, say you bank with Barclays and the shop banks with NatWest, when you pay £30 for something on a debit card, then the electronic transaction deducts £30 from Barclays’ account at the Central Bank and credits NatWest’s account at the Central Bank with £30. At the same time, Barclays will deduct £30 from your current account balance, while NatWest will credit the shop’s balance by £30.

The liabilities the retail banks have are the customers’ deposits. When you use your debit card or withdraw cash from the ATM, the bank has to give you money (up to the sum of your current account balance) so that is a liability to it – something it owes.

Now the Central Bank will also have assets and liabilities. It holds foreign exchange reserves and bonds, shares, gold etc as its assets, and its liabilities are the retail banks balances (their reserves) in their accounts with the central bank, plus the money the Central Bank has ‘issued’ (ie coins and notes in circulation).

So high-powered money, or the monetary base, is equal to retail banks’ reserves plus currency in circulation.

With this in mind we can make a few definitions to make a simplifed model:

$M^d$, the total demand for money in the economy, is made up partly of currency $CU$(coins and notes floating around in circulation) and partly of current account deposits $D$. If we denote the proportion of total money made up of currency as $c$ then the demand for currency is $CU^d = cM^d$ and the demand for current account deposits is $D^d = (1-c)M^d$.

$H^d$, the demand for high-powered money (or the monetary base) is the equal to the demand for currency, $CU^d = cM^d$ plus demand for retail bank reserves, $R^d$. Consumers are depositing $D$ among of deposits in current accounts with the retail banks, but the banks won’t hold all of this as reserves, they operate a system of fractional reserve banking. This means that they know that (or are banking on the fact that) customers won’t suddenly all demand to withdraw their deposits at once, so they only need to keep a proportion of their total deposit liabilities to customers in reserve, to meet day to day withdrawals…and they will use the rest to lend out to other people in order to make interest on it. So if we denote the proportion as the ‘reserve ratio’, $\theta$, then the demand for retail bank reserves, will be $R^d=\theta D^d = \theta (1-c)M^d$

So putting that together, we have two expressions:
Total demand for money in the economy: $M^d = CU^d + D^d = cM^d + (1-c)M^d$
Demand for Central Bank money: $H^d = CU^d + R^d = cM^d + \theta (1-c)M^d$

From the second equation we can say that $H^d = M^d (c + \theta (1-c))$ So $M^d = \frac{1}{(c + \theta (1-c))}H^d$

This expression basically expresses how total demand for money relates to demand for high-powered money, ie demand for total money will always be higher than demand for high-powered money.

But remember that the total demand for money is determined by two things, the interest rate and overall incomes in the economy. If we assume that incomes are fixed in the short run then it will be the interest rate that determines the demand for money, when interest rates are high there will be a lower demand for both parts of high-powered money, currency and retail bank reserves (because customers have lower demand for current account deposits when interest rates are high, they are instead putting their money in illiquid forms of saving like bonds).

The money market will head into equilibrium, ie the demand and supply for money will come into equilibrium because the interest rate will adjust to get there, just like any other market comes into equilibrium due to adjustments in the price. When we get to the equilibrium we can say that $M^s = M^d =M$ and $H^s = H^d =H$ so $M = \frac{1}{(c + \theta (1-c))}H$

Here we have a money multiplier of $\frac{1}{(c + \theta (1-c))}$

The multiplier shows us how changes in high-powered money translate into changes in the overall amount of money in the economy. This is where the name high-powered comes from, the monetary base is a type of money that has magnified effects on the overall amount of money: if you increase the monetary base by £1, you get an increase in the overall amount of money in the economy of £$\frac{1}{(c + \theta (1-c))}$

As an example, suppose the total amount of high powered money is £1000000, the reserve ratio is 0.1 and the proportion of money which people hold as currency as being 0.2, then the total amount of money is $M = \frac{1}{(0.2 + 0.1(1-0.2))}1000000 = 3571428.57$. Our multiplier here is $\frac{1}{(0.2 + 0.1(1-0.2))} = 3.571429$

The Central Bank controls the amount of high-powered money in the economy. If it decides to increase the amount of high-powered money by £10000, then the multiplier implies that it will increase the total amount of money by $3.571(10000)=35714.29$. The new total amount of money in the economy is $M = \frac{1}{(0.2 + 0.1(1-0.2))}1010000 = 3607142.86$

Categories: Macro, Money