Simulation Model for Samuelson and Inside Money

I know some people like the simulation models that I sometimes put on my blog, so I thought I'd do one using the Samuelson / inside money model that I described last time.

The balance sheet and flow of funds for this are those set out last time.  The equation listing and parameter values are given at the end of the post.

Most of the model is actually governed by accounting identities.  The only assumptions are as follows:

1. Workers attempt to balance their consumption so that they spend the same amount when working as in retirement.  

2. The price of goods is sticky.  Price adjustment responds to the rate of loan growth (as an indicator of long term inflation) and the output gap.

3. The utility of holding land is such that at some point the marginal utility is zero.  This presumes some level of disutility from holding land, perhaps from having to maintain it.  (This simply allows me to have a zero interest rate on loans and deposits).

4. The price of land adjusts to equate the expected return on land to that on deposits for retiring workers.  The expected return on land for newborns, exceeds their cost of loans, but they are assumed to be subject to credit constraints. (The expected return on land for newborns is not modelled here - it's not necessary because of the credit constraint assumption -but for other simulations, it might be necessary to include it.)

5. The simulations are run using adaptive expectations.

The graphs below show a simulation of a change in the nominal growth rate in loan volume from zero to 2%.

Price stickiness allows a temporary increase in production.  Rising prices start to increase the expected returns on land relative to deposits, which makes it more attractive to hold land in retirement.  The balance of housing ownership therefore moves towards retireds.  This requires that the real value of loans must fall, even though it was initiated by a rise in the nominal value.

Model Listing

Retireds consume the full value of their assets:

Cr = ( Ar_t-1 . pa + D_t-1 ) / pc

Workers consume an amount equal to what they expect to consume in retirement

Cw = E[Cw_t+1]

subject to an inter-temporal budget constraint.

Cw . pc + E[Cw_t+1] . E[pc_t+1] = C . pc + Aw_t-1 . pa - L_t-1 + Ar . ( E[pa_t+1] - pa )

These last two equations are rearranged to eliminate E[Cw_t+1], and give an expression for Cw.

Output is total consumption

C = Cw + Cr

Deposits equals loans

D = L

Workers' housing is total housing less retireds' housing

Aw = A* - Ar

Retireds' housing is given by their budget constraint.

Ar = ( C . pc + Aw_t-1 . pa - L_t-1 - Cw . pc - D ) / pa

The following equations then determine price setting.  The price of goods adjusts at the nominal growth rate of loans, adjusted for a measure of the output gap.

pc = pc_t-1 . g . ( C / Cn )^σ

The price of land is determined by the following expression, which equates the marginal expected return on land with the expected return on deposits (zero).  The first part of the LHS is the assumed marginal utility of holding land, the second is the expected capital gain.

( α . Ar^β - λ ) + ( E[pa_t+1] / pa - 1 ) = 0

This is rearranged as an expression for pa.

Finally, expectations are adaptive, factoring in the growth rate for nominal loans.

E[pc_t+1] = E[pc_t]_t-1 . g . ( pc / E[pc_t]_t-1 )^ε

E[pa_t+1] = E[pa_t]_t-1 . g . ( pa / E[pa_t]_t-1 )^ε

Variables and Parameters

Variable	Description	Opening Value
Ar	Land held by retireds	100
Aw	Land held by workers	100
A*	Total land	200
C	Total consumption	400
Cn	Stable price level of consumption	400
Cr	Retireds' consumption	200
Cw	Workers' consumption	200
D	Deposits	100
L	Loans	100
g	Growth parameter for nominal loans	1.00
Pa	Price of land	1.00
Pc	Price of consumer goods	1.00

The expression E[X_t+1] means the value that agents in period t expect X to be in period t+1.

In the simulation the value of g is increased to 1.02 from period 2 onwards. 

Parameter	Description	Value
α	Land utility parameter	10.0
β	Land utility parameter	-0.50
λ	Land utility parameter	1.00
ε	Adjustment rate of expectations	0.75
σ	Adjustment rate of consumer prices	0.50

[Edit - amended for errors noticed by Terry H and Anton]

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.