Samuelson's Consumption Loan Model with Inside Money

One of the big questions of any theory of money is why a people accept useless bits of paper in exchange for useful goods.  The MMT guys would point to the role of taxation.  The state's requirement for money in return for extinguishing tax liabilities creates sufficient demand to create a market for money.  I would be inclined to agree that use of a currency by the state is sufficient to make it valuable.

In Samuelson's consumption loan model, people want to hold money because it the only available store of value.  In this model, there is a continual flow of new people wanting to save, so people will accept payment in money, hoping that they will always be able to pass it on to someone who wants it as a store of value. 

However, I find it a slightly odd feature of this model that the money is a financial asset, but without any obvious party on the liability side.  There is no government in Samuelson to whom the money can be remitted.   Samuelson's money can never be extinguished or destroyed.  It must be passed round forever.

This means that the value of money in this model is purely based on the hope that it will have value in the future.  This has caused some people to think of it as a kind of bubble asset.

So I thought it would be interesting to look at how Samuelson might work with inside money, where the private sector is borrowing as well as saving.  To do this, we need a reason for people to borrow, so I'm also going to add in an extra asset - land.  So the economy would look something like the following:

Households live for two periods then die (I'm simplifying it to two periods from Samuelson's three).  They work in the first and save some of their earnings to spend in the second period, when they do not work at all.  A new cohort of households is born at the start of each period, replacing the households that die.

There is a bank.  The bank makes loans by crediting household deposit accounts.  The deposit accounts serve as money.  Loans are used to buy land.  Land is held for personal enjoyment rather than for producing goods, so both workers and retireds hold land.  Retireds also hold deposits, whereas workers have to fund their land holdings with debt.  The national balance sheet at the start of each period looks something like this:

	Worker	Retired	Bank
Loans	- L		L
Deposits		D	- D
Land	pa . Aw	pa . Ar

This is expressed in monetary units, where p^a is the price of land.

All transactions take place at the end of each period.  At this point, retireds are still alive and new households have just been born, so all three generations can transact.  Retireds then die, workers retire and newborns start work.  For simplicity, we are assuming the rate of interest on loans and deposits is zero and that there are no credit constraints.  The flow of funds between entities is shown in the schematic below (inserting a land market and a separate production, for ease of illustration):

At this point, newborns are borrowing to buy the land they will use as workers.  Retireds are selling their land and spending their deposits to pay for consumption.  Workers are doing lots of things.  They are repaying the loans they took out as newborns.  They are possibly adjusting their land holdings.  They are consuming and they are retaining the balance of their earnings as deposits.  A more formal flow of funds is set out below:

 

	Newborn	Worker	Retired	Bank
Loans	L	- L t-1		L t-1 - L
Earnings		pc . C
Land	- pa . Aw	pa . ( Aw - Art-1 )	pa . Art-1
Deposits		- D	D t-1	D - D t-1
Consumption		- pc .Cw	- pc . Cr
Total	0	0	0	0

There are a few interesting things that this model brings out.

1. The demand for money now partly arises from the need of workers to repay their loans.  In Samuelson's version, people held money as savings in the hope and belief that future generations would also want to hold money as savings.  Now they can hold money in the knowledge that someone else will at some point need it to repay a debt.  Ultimately, the need to repay debts exactly matches the funds available to do so.  The supply of money creates its own demand.

2. The general price level, depends on the volume of lending.  We are assuming that the bank decides how much to lend and therefore the nominal value of deposits in the economy.  Yet the optimal real value of deposits is determined by household preferences - how much to save for retirement, how to split asset holdings between land and deposits.  If prices are perfectly flexible, an increase in lending will lead to a rise in prices[1].  But a change in preferences can also effect the price level.

3. This result does not depend on the fact that deposits are used as the medium of exchange.  The price level is driven by the demand and supply of deposits as a store of value.  We could imagine the consumption good being used as the medium of exchange (and as the unit of account) and we would still get the same result.

I prefer this model to Samuelson's as the money seems to be much closer to what we actually have in the real world (although I have not included state money).  For me, the most useful thing it illustrates is the way the value of the currency is determined by the long and short positions that people have on it.  Where most borrowing and a significant part of investment take place in instruments that are highly correlated with the currency, then saving and borrowing habits are key to determining its value.

---

[1] We can do an experiment with this model.  We assume that each generation aims to consume the same amount in retirement as when working, and to keep a constant ratio between consumption and land holding.  With zero growth, this means C^w = C^r and A^w = A^r = A^r_t-1 in all periods.  If the bank then decides to increase lending by 20%, then constant production requires a rise of 20% in the price of land and of goods.  However, whereas the price of land adjusts immediately, the price of goods rises by 10% in the first period and the rest in the second.

Money and the Structure of Financial Balances

There's no shortage of theories of money, but I'd thought I'd add mine.  Actually, this isn't really a theory of money, it's just something I've been thinking about, to do with what money is and how it fits in with other things.  I think it makes sense.

There's nothing really ground-breaking here, but perhaps the logical order is unconventional.  Rather than seeing money as a fundamental base on which the broader structure of financial assets and liabilities is built, I go the other way.  So I start with the observation that an economy needs financial assets generally.  I then look at how these are most effectively used as an exchange medium.

The Role of Financial Assets

So the first step is to recognise the benefits of financial assets.  As a starting point, we can take it that people sometimes want to save, consuming less than they earn, which implies that they will be holding some form of asset for a period of time.  They could hold real assets, such as land or machinery.  However, people who wish to save for the purposes of deferring consumption are not necessarily interested in actually using the physical attributes of the real asset.  There is therefore a benefit in having a different person hold the real asset whilst giving the saver some kind of claim on future resources.

At this point, we don't need to worry about the fact that we haven't provided for any unit of account, because it doesn't matter if these contracts are for the delivery of specified real goods.  Even if they were, they would still constitute a form of financial asset.

The existence of such contracts as a means of saving also allows some agents to have a negative net worth.  For example, someone might own no assets, but have contracted to deliver something in the future.  This is not possible where the only assets are real assets.  The state is the main example of an entity that is likely to have a significant negative net worth.

Creating a Homogenous Exchange Medium

If these financial claims are transferable, they may themselves provide a useful medium of exchange, as they are portable and non-perishable.  However, the efficiency can be improved by some kind of pooling.

This is where a large number of claims are held by a central depository (the "bank").  Savers ("depositors") then hold deposits entitling them to a share of the pool.  Rather than transferring underlying claims between each other, depositors would transfer their deposits, with the bank maintaining a record of each depositor's balance.

The main advantage of this is that provides for an homogenous medium of exchange.  Where the underlying claims ("loans") are being used as the medium, there is the problem that they will all be different.  In particular, the "borrowers" under the loans will be different, so each time a party received a "payment", it would need to make a separate credit assessment.  With deposits, what they receive is identical each time.

De-pegging

Pooling also means that the deposits can have an indefinite life, even when the loans have a finite term.  This allows some further features.  In particular:

1. The deposits can be non-convertible.  This means that the only way they can be used is by transferring them to someone else (or by settling a loan - see point 2).[1] 

2. The loans no longer need to be for delivery of goods or real assets.  Instead, they can be specified to be settled only by cancellation against deposits.  This also allows loans held directly, i.e. not by the bank, to be settled by transfer of deposits.

These two steps then mean that loans and deposits do not need to be pegged in some way to the value of any particular good or real asset.  They can form their own unit of account.  Their exchange value against real goods is ultimately determined by demand and supply.   Note that this needs to match demand and supply for all assets denominated this way, not just deposits.

Removing convertibility also means that the (nominal) rate of interest can be set by the bank.  It is no longer dictated by the demand and supply of funds.

What we have now is a model of modern banking, provided that we interpret the bank as representing the entire banking sector, including the central bank. 

Liquidity

So far, I have said nothing about liquidity.  This is because I have only looked at a single bank, and when we have a banking monopoly with no convertibility, liquidity is not an issue.  With a monopoly, the bank is indifferent to payments between depositors, because the deposits cannot go anywhere else.  The bank can afford to allow any depositor to pay away the entire amount of its deposit.  The bank therefore does not need to apply any restrictions on payments and the entire deposit base can constitute the medium of exchange.

With competition, payments between depositors may involve the liability side of the deposit moving between banks.  Banks can then no longer afford to allow depositors to pay away their entire balance.  Some deposit amounts therefore need to be locked in for minimum periods.  This gives us sight deposits and time deposits.  The split depends on the requirements of banks and their depositors and represents a kind of market in liquidity, which extends into the willingness of banks to provide short term credit.

The above is not intended to be historical, in that I'm not suggesting this is how money actually evolved.  It's just meant to set out what I see as the logical order.  Money rests on the back of the wider network of financial claims, rather than the other way round.  Financial assets serve an important role in an economy.  Once we accept that role, it is then natural to use them (or at least a subset of them) as an exchange medium.  We need to understand the function of financial assets before we can understand how money fits in.

---

[1] We can allow for depositors to exchange some or all of their registered deposits for bearer notes, or back again, but the total balance of registered and bearer would remain the same.  These notes would then function as currency.

Preference Functions and the Balanced Budget Multiplier

One of the things that fascinates me in economics is how different economists manage to come up with completely different conclusions about the results of particular policy measures.  Where this happens, I think it provides an excellent opportunity to examine how results flow from assumptions and to appreciate how fragile certain conclusions might be.

One case of this that struck me recently is the New Keynesian position that a balanced budget increase in public expenditure, which is expected to be permanent, has no impact on GDP.  This result, which contrasts with the old Keynesian result, is said to be down to the assumption of rational inter-temporally maximising agents.  However, what is perhaps not so well appreciated is that it is also contingent on some rather ad hoc assumptions about household preferences.

To look at this, I'm going to assume a closed economy with no investment, so GDP (Y) comprises simply public spending (G) and household consumption (C) (all in real terms, assuming a single good):

(1)  Y = G + C

I'm going to go with the standard assumption that public spending is funded with lump sum taxes (T).  I don't like this assumption very much, partly because I think it sometimes skews the results, but it makes it easier to be clear about what constitutes a balance budget change in spending.  So household disposable income (YD) is given:

(2)  YD = Y - T

Now, in the standard New Keynesian analysis, consumption is based on lifetime income over an infinite life.  So, if it is assumed that price flexibility will ensure full employment GDP in the long run, then the permanent increase in taxes will reduce permanent disposable income.  Expected future consumption will therefore fall and so households will also cut current consumption.

However, this result arises only because of our assumption about household preferences.  We could take a different approach. 

For example, we could assume that household maximise over a finite lifetime.  In its simplest form, assume each household lives for two periods and then dies.  It works and earns only in the first period but splits its income so as to consume the same amount in each period (so we are assuming, for simplicity, zero elasticity of substitution between consumption in the two periods.)  We will also assume an equal number of new households is "born" in every period.  Each period, we therefore have half our households spending their savings from the previous period and half spending some fraction of their current earnings.  In equilibrium, these households will be saving half their income, so the stock of savings at any time will be equal to 50% of disposable income. [Edit - to clarify, equilibrium here means a steady-state non-growth equilibrium when prices have been able to fully adjust to remove any output gap.]

It will help at this stage to put some numbers on this.  I am going to assume that full employment GDP is 100 and the current real stock of savings is 40.  The savings take the form of public debt, but I am going to assume that public spending and taxes are both currently zero.  Lastly, prices are assumed to be sticky in the current period and the next, but to be fully flexible thereafter.

So in this instance, we find that expenditure falls short of what is needed for full employment.  The table below shows the general price level for goods (p), and savings, consumption, public spending, GDP, taxes and disposable income, all in real terms.  In addition to identities (1) and (2), we have the real value of savings (A) is given as:

(3)  A_t = (YD_t-1 - Cw_t-1) / ( p_t / p_t-1 )

where Cw is workers' consumption.  Consumption by retireds is equal to their savings.  Workers in one period are retireds in the next, so inter-temporal optimising under rational expectations requires that workers' consumption in one period is equal to retireds' consumption in the next.

Period	1	2	3
Price level	1.00	1.00	0.80
Value of savings	40	40	50
Workers' consumption	40	50	50
Retireds' consumption	40	40	50
Public spending	0	0	0
GDP	80	90	100
Taxes	0	0	0
Household disposable income	80	90	100

Each subsequent period will be the same as period 3.  So, in this instance, the sticky prices are creating a temporary shortfall in GDP.  What is the effect of a permanent balanced budget increase in public spending?  This is shown in the table below.

Period	1	2	3
Price level	1.00	1.00	1.00
Value of savings	40	40	40
Workers' consumption	40	40	40
Retireds' consumption	40	40	40
Public spending	20	20	20
GDP	100	100	100
Taxes	20	20	20
Household disposable income	80	80	80

Again, each subsequent period will be the same as period 3.  The effect has been an elimination of the temporary output gap.  The longer run effect is that the general deflation that occurred in the first example does not occur here.

It helps to understand what's going on here if you realise that the assumptions about household preferences imply a steady state ratio between consumption and the real value of savings.  In our example, the output gap arises because the real value of savings is too low.  GDP is prevented from rising, because if disposable income were to rise, households would try to save more and with the balanced budget assumption they cannot achieve this.  For the same reason, consumption stays the same when there is a balanced budget increase in public spending.  However, the fact that there is no price deflation in later periods keeps consumption at its existing level.

I am not arguing here that one particular assumption about household preferences is better than the other.  I think the latter approach is more consistent with heterodox views, and I find it more plausible, but I suspect what actually happens is much more complicated.  All I wanted to do here was illustrate what I thought was an interesting case of where conflicting conclusions arise from different assumptions, even if both involve rational inter-temporal maximising agents. 

(I should point out that I am aware that the result in my overlapping generation case also depends on how the tax is levied.  I have assumed the tax to be paid out of workers' income, but if the retired bear all or some of the burden there is a different result.  This is another reason why I am wary of unrealistic assumptions about the structure of taxation.)