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]