Determinancy of Rates and Quantities

Monetarists such as Nick Rowe and Scott Sumner are sceptical about the use of interest rates as a monetary policy tool.  As I understand it, the argument goes something like this.  If monetary policy involves targeting some variable with a $ value, such as the money supply or the exchange rate, then the value of this variable in, say, ten years time will provide a solid peg for the price level at that time.  We might not be able to predict the exact ratio of one to the other, but that ratio must be somehow determinate.  On the other hand, knowing the interest rate in ten years time tells us nothing.  Any price level could be consistent with a given interest rate.

I don't like this argument because it seems to ignore the necessary accounting implications of the path that takes us from today to the future time.  I want to illustrate this with a simple model of a closed service economy. 

In this economy, there is a single consumer service produced in quantity c at price p.  There is a government sector which imposes a fixed per head tax and spends on government services incurring a fixed deficit D.  The only assets are claims on the government being money, M, and an amount b of single period bonds.  The bonds are issued at price q and redeemed at 1. 

The government budget constraint is

b.q + M = M_-1 + b_-1 + D

The marginal benefit of holding money for liquidity purposes is a function L of the ratio of the money supply to nominal consumption.  In equilibrium, this is equal to the nominal yield on bonds:

L( M / p.c ) = 1 / q -1

Finally, households allocate their resources, being net income (p.c + D) and financial assets (b_-1 + M_-1), between consumption (p.c) and the new level of asset holdings (b.q + M).  We assume that they do this according to some preference rule, which satisfies the following function V for the new level of wealth*:  

b.q + M = V( p.c , D, b_-1+M_-1 )

For each period, these three equations determine p.c, b, and then either M or q.  We can either set M and get q, or set q and get M.

This means that if we know all the values of q from now until time t and we know the starting values of M and b, then we know all the values of M from now till time t.  Alternatively, we could note that knowing the value of M in ten years time is insufficient to determine p.c.  To determine p.c, we need to also know the value of M_-1 and b_-1, and once we know the value of those, then knowing q is as good as knowing M.

So it is absolutely correct that we need a nominal value to provide our peg.  But knowledge of one current nominal value is insufficient.  We also need to know preceding values.  And once we know preceding values, we have our peg and we no longer need to know the current value.  An interest rate will suffice.

Knowledge of the starting nominal values and the subsequent history of rates gives us as much information as the starting values and the subsequent history of the money supply.

* I have not included the real interest rate in this function, to avoid getting into issues which would greatly complicate the analysis.  I do not believe it is relevant to what I am saying here.

[Edit.  This post was prompted by an exchange with Nick Rowe here.)

Income Distribution and the Savings Ratio

Unfortunately, I missed the exchange in the comments section of this Interfluidity post.  It raises some issues on the measurement and interpretation of savings ratios and inequality that I found rather interesting.  I thought it would be useful to go through some of these points

The issue is to do with whether the rich save more than the poor and it should be seen in the context of the question of whether the distribution of income affects overall consumption.  Much of the difficulty arises over whether capital gains should be included in household income when looking at this.

For the purposes of this discussion, I am going to treat gains as being the increase in the value of net assets, regardless of whether or not they are realised.  I am going to use the term disposable income to refer to what is normally treated as disposable income in the NIPA accounts, i.e. wages, benefits, interest, dividends, etc..  I will use the term HS income to refer to Haig Simons income, which is disposable income plus gains.

So first of all I want to assume a simple consumption function based on disposable income and gains.  I am going to assume that households have the same consumption function whatever their income, but that richer households have a greater proportion of their HS income in the form of gains.  So our consumption function is:

C_i = α_y . YD_i + α_a . A_i

where C is consumption, YD is disposable income and A is gains, for each household i.  The difference is that the rich have a greater ratio of A to YD.  What happens if we assume that households do not spend any of their gains (i.e. α_a = 0)?  Then, we will find the following:

• The aggregate savings rate of the economy (being 1 - C / YD ) will be equal be equal to 1 - α_y
• If we measure the savings ratios of rich and poor as for the aggregate (i.e. using disposable income) their savings ratios will both be the same as the aggregate measure, 1 - α_y
• If we measure savings ratios by reference to HS income (i.e. using 1 - C / [ Y + A ] ), then the savings ratio of the rich will be higher than the aggregate savings ratio and higher than that of the poor.
• Neither a transfer of earned income, nor of gains, from rich to poor will change the aggregate savings ratio of the economy (conventionally measured), nor the level of consumption.

Because the savings ratio is conventionally measured by reference to disposable income, we must do the same (ignoring gains) for individual households if want to get comparable measures.  And in this special case, that is consistent with the result that redistribution of income and gains has no effect on consumption.  However, I think this misses some important points.

So far we have just looked at the household consumption function based on income and gains that accrue directly to households.  But we need to be aware that this is not equivalent to GDP.  In the simplest case (ignoring flows with public and foreign sectors), disposable income will differ from GDP to the extent of earnings retained by firms.  This part of GDP is accruing to the benefit of shareholders, but is not part of their disposable income as it is not paid to them.  It is likely that retained earnings will increase the gains of households owning those shares, but this is certainly not a one-for-one relationship.

However, it is important to take this into account, because it is relevant to the issue of how income distribution is affecting consumption expenditure.  If we find greater profit retention by corporations, we are likely to find less consumption, even though households may still be benefiting in the form of gains.  But it is important to recognise that the effect here shows up as a fall in household disposable income rather than a rise in the savings ratio.  Furthermore, on our original assumed consumption function, it still makes no difference if we transfer income or gains from rich to poor.  The only thing that is making a difference is the earnings retention level.

Given our assumed consumption function, it is true to say that the rich save more, at least if we consider HS income.  However, this is really just a consequence of our assumptions that everybody saves more of their gains than their disposable income and that the rich have more gains.  That is why, on this assumption, re-distribution has no effect.

We might alternatively consider a consumption function where the propensity to spend out of gains is non-zero but that the rich have a lower propensity to spend out of disposable income.  So the rich may be consuming the same amount as with our previous assumption, but based on different inputs.  We could assume different functions for rich and poor, so the function for the rich is:

C_r = α_ry . YD_r + α_ra . A

and for the poor:

C_p = α_py . YD_p + α_pa . A

So even with α_ry < α_py, we might still find that the savings ratio of the rich (measured as 1 - C / YD ) is the same as that of the poor, because they are also spending partly out of gains.  And as before, we have to measure individual savings ratios that way if we want the ratios to be comparable to the aggregate savings ratios, simply because that is the way the aggregate savings ratio is measured.   

However, we now find that a redistribution of disposable income between rich and poor will change that aggregate savings ratio.  This is due to the fact that there is a different propensity to spend out that type of income.  The difference with our previous result is due to the distinction between average and marginal rates.

All I have intended to do here is highlight some of issues involved in measurement and interpretation.  I am not suggesting that actual household behaviour is actually like any of the scenarios I have used here.  On the whole, I believe there is useful truth in the proposition that the rich save more, but I think under-consumption theories are more problematic.  I think growing inequality, particularly wealth inequality is a big problem, but not necessarily for reasons of under-consumption.

It's still something I'm grappling with though, and hope to return to later.

(Ramanan also has a good post on this, put up after I wrote this but before I got round to posting it.)

Capital Arbitrage and BIS Risk Weightings

A recent article on VOX looked at some of the issues with risk weightings for bank regulatory capital under the revised BIS rules.

These rules require that assets with different perceived credit risk are valued differently for the purposes of determining a bank's minimum capital requirement.  Originally, the rules were very basic with assets falling into a small number of categories, with a different weighting assigned to each.  The system has now become much more complex, allowing detailed criteria against which each asset can be assessed.

One issue which has attracted much comment is the fact that banks are allowed to design their own criteria for determining risk weightings, known as the internal ratings based (IRB) approach.  Although the criteria have to be approved by the regulators, it is argued that they are not well placed to second guess the banks.  This is therefore likely to result in weaker criteria than would be designed by an outside party.

Another issue, more relevant to the VOX article, is capital arbitrage.  The problem here is that the actions of banks are not independent of the risk weighting criteria.  To some extent that's fine.  If banks choose to invest in safer assets on the basis that it has a lower capital requirement, then things are operating as intended.  The difficulty is that however detailed the criteria, they are merely a rough approximation of true risk.  There will always be an imperfect match with elements that don't fit well and it's those elements towards which business tends to gyrate.

A good way to illustrate this process is by looking at the world of asset-backed securities and ratings criteria.  When the ratings agents rate corporate borrowers, they assess the different companies as they are and decide the ratings.  Within any particular rating, there will therefore naturally be better and worse borrowers - some who are nearly at the next rating up and some who are barely above the rating below.

In rating asset-backed transactions, the rating agents use a set of criteria to determine what rating to give.  But now the game is different, because the structurer has the ability to design his transaction.  Improving credit risk costs money, so of the structurer's objective is to get away with as little credit enhancement as possible, whilst still achieving the desired rating.  In other words, he is always aiming for the very bottom of the range of credit risk for that rating.  And because the criteria are only really rules of thumb, that can mean a risk that is below what would normally be expected for that rating.

So, a similar issue can arise in banks that are subject to capital requirements based on a set of rules.  One aspect of this that was highly relevant to the financial crisis was the banking book / trading book distinction.  But the problem arises, albeit it in smaller way, even within the normal loan portfolio.  And making the rules more and more granular is unlikely to solve the problem in a practical way.  This is one of the main reasons why alternative capital requirements, such as the Leverage Ratio, are so important.

A Simplified Look at Samuelson's Overlapping Generations Model

I've talked quite a bit recently about overlapping generation models, in particular Samuelson.  I thought it would be useful to put Samuelson's model into some simple illustrations, because I believe the process by which it creates a functional relationship between income and wealth is a very important one.

In this model, agents live for three periods.  They work and earn for the first two periods, but do not work for the third.  They spread their spending however, so that in the first two periods they are spending less than they earn and are therefore saving, which they do by acquiring assets.  In the last period, they realise their savings and use the proceeds to fund their spending.  The life cycle of each agent is shown below.

Some accounting identities are being used here.  Saving is the difference between income and spending.  Saving or dis-saving then changes the balance of assets held.  The value of assets may also be affected by changes in the price; the older generation may find it can sell the assets for more than it paid for them.  More on this later.

After the third period, each agent "dies" but a new generation is born to repeat the process.  There is therefore a constant chain of newer generations saving and older generations dis-saving.  It is useful in this model to think of savings as being held in the form of some kind of financial instrument, in fixed supply.  We therefore have a permanent market where the younger generations (1&2) are buying assets to facilitate their saving, and the older generation (3) is selling them.  This is shown below:

We have said nothing about how agents spread their spending.  The chart suggests they are spending equally in each period, but this need not be so.  They may wish to spend more earlier in life or even to defer spending.  This will change the numbers slightly, but the overall pattern will stay the same.  We will still get a position whereby each generation accumulates wealth and then runs it down.  Furthermore, if the time preferences do not change and aggregate income stays the same then the aggregate balance of assets will stay the same.  Each individual's balance rises then falls, but the saving cancels out the dis-saving.

The market for assets consists then of younger generations as buyers and older generations as sellers.  This is where Samuelson's results come from.   In his simple case, with no growth, the number of buyers and sellers is the same from one period to the next, so the price of assets stays constant.  However, if we have a situation where each new generation is larger than the last, the number of buyers is growing and the value of assets must rise.  With the quantity of assets in fixed supply, the price must grow in line with the population in order to be able to fulfill the growing demand for savings.  This growth in price creates a yield on the asset - Samuelson's rate of interest.

Samuelson's result should be viewed with caution.  For the interest  rate to be equal to the growth rate requires that there is no outside sectors (government or overseas) and that there is no internal transfer element to the return on assets.  However, I do believe that the idea of viewing the demand for assets as a function of a natural life-cycle and a trade between different generations gives us important insights into things we see in the real world.