## A Monetary Policy Base Model

In this post I will (finally) lay out a mathematical model describing roughly how I see monetary policy functioning in view of the relationship between money and debt. This model will look a lot like other simple macro models but it will highlight a bit more, the role that debt plays in the process. I will then use it to tell some stories about depressions and liquidity traps. I am going for the simplest formulation possible which still makes my point here so I will use a lot of assumptions, many of which could be relaxed to get a model with more complexity.

The most important aspect of such a model is the treatment of credit markets and their role in money creation. I will assume that the following takes place simultaneously.

1. The central bank determines the quantity of reserves and the reserve requirement. I assume that all reserves are held by banks and that the reserve requirement is constant (changes in the requirement and the quantity of reserves are functionally equivalent).

2. People take out a certain amount of dollar-denominated loans from the banks and use it to buy stuff. This is where all of the “money” comes from.

3. People withdraw “money” from the economy (by selling goods or liquidating money “savings”) to pay off the debts which were accumulated in the previous period.

4. Any interest payments to the bank from the debt in the previous period is remitted back into the economy through dividend payments and/or operating expenses.

So the money supply in each period will be the quantity of money which remains in the economy after all of this and is carried over to the next period. I will assume that the economy starts with zero money so that in period 0, the quantity of money is equal to the amount of loans and I will assume that in this period, the banks are reserve constrained (the reserves ratio is at the minimum allowable level). This means that, given the initial supply of reserves (R) and reserve ratio (a), the initial “money supply” (M) and new loans (L) will be the following.

M_{0}=L_{0}=R_{0}/a

In the following period, this same quantity will need to be paid back plus interest, but the interest will flow back into the economy along with the quantity of money created by new loans so at any time t, the quantity of money will be given by the following.

M_{t}=L_{t}+L_{t-1}-L_{t-1}=L_{t}

In other words, the quantity of money in circulation is equal to the quantity of debt. Now we must model the market for debt.

Let the willingness to hold debt (demand for new loans) be a function of the real interest rate r and the value of real goods in the economy PY. (Y can be thought of as either real output or real wealth.) For the sake of exposition let this be the following.

L_{t}=P_{t}Y(1-r_{t})=Y(1-i_{t}+π^{e}) for i_{t}> π^{e}

L_{t}=P_{t}Y for i_{t}< π^{e}

For our purposes let us assume that Y is constant and determined by “real” factors. So *by assumption*, money is neutral. A more specific treatment of the business cycle would have to relax this assumption of course but I’m not going to do that here. Similarly let us assume that expected inflation is constant. For instance, assume that the central bank has an inflation target which everyone believes they will (and can) hit, at least on average. The exact specification of this function is chosen for its simplicity not its realism but the important points are that it is downward sloping in the real interest rate and that it cannot be greater than the total value of real goods. The maximum amount of loans possible does not have to be exactly equal to the total stock of real goods, it is likely to be somewhat less (I suspect an argument could be made for it somehow being more but this would be a bit more complicated to justify) but the important thing is that there is some maximum (the demand for loans does not go off to infinity as real rates (or alternatively, nominal rates) approach zero.

By assuming that inflation expectations are fixed, I am assuming that nominal rates move one-for-one with real rates. This is a simplifying assumption of course. Much of the complexities that arise in modeling monetary policy are related to the way in which these expectations are modeled but I don’t want to deal with that extensively here, though I will add some discussion of this issue at the end.

The supply of new loans as a function of the nominal rate will look like I describe in this post. Namely, it will be horizontal (perfectly elastic) at i=0 (or alternately at the rate of interest on reserves) up to the maximum quantity which is possible to create from the given quantity of reserves and the reserve requirement. At that point it will be vertical (perfectly inelastic). So in “normal times” (when nominal rates are above zero) the equilibrium quantity of loans in this market (and thus the money supply) will be equal to the maximum allowable quantity and the nominal interest rate will be given by the point at which this quantity intersects the demand curve.

M_{t}=R_{t}/a=L_{t}= P_{t}Y (1-i_{t}+π^{e})

i_{t}=1-R_{t}/(a P_{t}Y)+ π^{e}

Finally, let the price level in each period be determined by the equation of exchange.

MV=YP

And let velocity be fixed at v so that the price level in any period is given by the following.

P_{t}=vM_{t}/Y=vR_{t}/(aY)

The assumption of a fixed velocity is another drastic simplification. This, along with inflation expectations, represent the main points of this model which deserve more careful analysis, expecially because the theory I am working from is one which holds that money demand is essentially liquidity demand and that demand aught to be related to velocity in an important way. In other words, people find high velocity inconvenient and so they are willing to pay a price to acquire more money which lowers velocity (but likely increases prices). But digging into this would greatly complicate things and at this point would distract from my main point. So essentially, I am assuming that any increase in money goes entirely to prices. One could imagine however, that some of the increase is soaked up by a decrease in velocity.

This describes a complete model of the money supply and the economy. The path of the money supply, price level and interest rates, then depends entirely on the quantity of reserves provided by the central bank. So let us imagine that the central bank tries to hit its inflation target every period.

Assuming some initial condition for the quantity of reserves R_{0}, the initial quantity of money, price level, and interest rate will be the following.

M_{0}=R_{0}/a

P_{0}=vR_{0}/(aY)

i_{0}=1-R_{0}/(aP_{0}Y)+ π^{e}=1-1/v+ π^{e}

In the next period, the central bank will try to make P grow by a factor of (1+ π^{e}).

P_{1}=P_{0}(1+ π^{e})

This will require a proportionate increase in the money supply which requires a proportionate increase in the quantity of reserves.

M_{1}=(1+ π^{e}) R_{0}/a= R_{1}/a

R_{1}=(1+ π^{e}) R_{0}

Because the demand for loans is proportionate to the nominal value of output, this demand increases proportionately with the price level and so the nominal interest rate will be constant across time as well as the proportion of debt to nominal output.

i_{1}=1-R_{1}/(aP_{1}Y)+ π^{e}=1-(1+ π^{e}) R_{0}/(a(1+ π^{e})P_{0} Y)+ π^{e}=i_{0}=1-1/v+ π^{e}

The same thing will have to repeat in all following periods, so that at any time t, the state of the economy is described by the following dynamic equations.

R_{t}=(1+ π^{e})^{t} R_{0}

M_{t}=(1+ π^{e})^{t} R_{0}/a

P_{t}=P_{0}(1+ π^{e})^{t}

i_{t}=1-1/v+ π^{e}

This constitutes a “base scenario” in which, in order to hit its inflation target in every period, the central bank has to increase the supply of reserves exponentially which increases the supply of “money” by the same factor and also increases prices by the same factor. The nominal rate, real rate, and inflation rate are all constant.

Now some interesting points about this model.

**Recessions**

As I said, this does not explicitly model recessions but we can imagine how one would occur and what it would look like by imagining a sort of “off the equilibrium path” scenario. Consider what would happen if at some period, the money supply failed to hit the expected quantity. This may happen because the central bank fails to inject enough reserves or it could be because demand for new loans drops unexpectedly (which amounts to the same thing since the CB would have to fail to account for the drop).

If this happens, then prices will not rise as much as people had expected when they initiated their old loans. Since people borrow in anticipation of paying the loans back with income generated from the sale of goods and services in the future, their income will be less than they expected in nominal terms and so will their ability to repay those loans. But because they have real goods backing those loans, they will be competing more fiercely over the smaller-than-expected quantity of money left out in circulation. This is what causes prices to fall.

This condition of falling prices will be accompanied by some people (more than usual) being unable to repay their previous loans. This will cause houses to be foreclosed, businesses to close down etc. If people expect prices to continue to fall, they will be even more reluctant to undertake further debt (inflation expectations decrease which shifts loan demand to the left and lowers interest rates) and the process may be self-reinforcing to some degree.

This process of “deleveraging” will continue until enough people have defaulted, wiping away their debt without retiring the requisite quantity of money and therefore causing the ratio of debt to real goods to drop, (or until the government/central bank intervenes in some other way to increase the money supply) that the economy reaches a new equilibrium path.

This process of default can be avoided (at least in normal times) by central bank easing (increasing reserves) which allows interest rates to fall and more money to be created (monetary policy). Alternatively, if the CB refuses to do this for some reason, the government, because it has the ability to borrow with no collateral constraints, can add to the demand for new loans by borrowing and spending (fiscal policy). Either one of these should work equally well, at least if not in a liquidity trap (more on that below).

In this explanation, debt essentially plays the role that “sticky prices/wages” play in most other models (though they are not at all mutually exclusive). In my opinion, this is a *far* better explanation because debt is literally fixed nominally by the contract over very long time periods. The quantity of debt in the system is huge and it is directly linked to the quantity of money by the process of money creation. You don’t have to make any strained arguments about “labor’s” unwillingness to work for less even though it is in their interest, or businesses being unable to change their menu prices for years on end or anything like that.

**“Nominal rates are not a good indicator of the stance of monetary policy.” **

I put this in quotes as a nod to Sumner (and indirectly Friedman). There are two ways of interpreting this in the context of this model.

1. In a model with more realistic modeling of inflation expectations, the demand for loans would shift around with those expectations. So if the CB did something that increased the money base in the current period but also increased inflation expectations for the future, you would have loan demand shifting to the right along with the supply and you could see rates either rise or fall. So if we allow inflation expectations to change we can easily represent this phenomenon. (A similar thing could be done if Y were a measure of expected future wealth which depended on expectations).

2. Even without letting these things shift about, different specifications of this model could result in credit conditions (interest rates, and leverage ratio) that are not constant over time (more on this to come). If this were the case (and I suspect it is) then it may be the case that the interest rate associated with “neutral” monetary policy (producing the targeted inflation rate) is different in different periods. Specifically, I think a case can be made that this will tend to get lower, the longer this system goes on. If this is the case, one may mistakenly interpret low rates in one period as “looser” policy (more inflationary) than higher rates were in previous periods when in fact they may actually be “tighter” (less inflationary). This type of case will be left for a later post however.

**Liquidity traps**

In this model a “liquidity trap” occurs if the central bank finds that it cannot increase the quantity of loans (and hence the money supply) by increasing the quantity of reserves. In other words, it means banks have excess reserves and the supply/demand in the market for loans is such that demand crosses supply along the elastic section of the demand curve (at i=0 or the level of interest on reserves).

In the model presented here, this would occur if the real rate fell to zero or below. In this case all of the real property would essentially be mortgaged and so the CB could print more reserves and this would have no effect on the quantity of loans and therefore the quantity of money. Alternately, it could occur without all property being mortgaged if demand for loans hits the horizontal axis (or IOR) before this happens. If this is the case, the CB may be unable to create the quantity of money which is required to create the expected inflation and this could lead to a recession as described above which cannot be easily cured by “traditional” monetary policy.

**Market For Loans**

This may happen because, for some reason, demand for loans drops (inflation expectations drop, government deficits drop, a real shock to output occurs, appetite for debt declines, etc.) or it may be that the economy approaches such a state in a systematic, deterministic way. Whatever the reason, a wave of defaults and deleveraging will occur unless additional methods of increasing the supply of money are found.

One such method, as already described, is to have the government do additional borrowing and spending. Alternatively, the central bank can do a similar thing by printing money and buying other stuff (“quantitative easing”). Quantitative easing as we currently know it seems to be a method of increasing demand for loans by pushing down the longer end of the yield curve. Similarly, the CB can try to increase this demand through “forward guidance,” promising to be “looser” for longer. If this increases inflation expectations, it will increase demand for loans.

Of course, if just pushing down the long end of the yield curve is not sufficient to get out of the liquidity trap, the CB, in theory, could just print a bunch of money and use it to buy all kinds of stuff, pushing up prices and increasing the quantity of money in circulation without increasing debt. Of course, whether this would count as monetary or fiscal policy is debatable.

This should answer questions like “how does monetary policy work” or “how is inflation created” in the context of a credit-based theory of money. I have some ideas about why an economy may deviate from this scenario in a systematic way but I will try to present them as variations to this base model. Stay tuned for that.

I started thinking about how to design a computer program to carry out your model. Right away I ran into questions about the assets that people use as collateral for loans. New assets are created at some rate. There’s consumer goods that are immediately consumed, and then there are goods that are consumed slowly enough to consider depreciation. The latter can be assets that deserve loans.

Also sometimes assets gain in value — if somebody has a depreciating asset but you pay more for it than the previous owner thought it was worth, that sets its value higher. Unless there are customs set up that set the value some other way. Like, if you pay much more for a house than other people living nearby did recently, real estate assessors will say that your house is not worth as much as you paid for it, but they will say that neighboring houses are worth more than they were before that sale.

That sort of complication is beside the point for a monetary policy model, it would come in as external data that can affect the model’s behavior. For your monetary policy model, what sort of simple assumption would you want to make about the value of the assets that loans are built on? Would it be OK to just say they increase by, say, 3% per year?

OK, I got past that one. If you ever rewrite this, define price level earlier. You used P a good deal in formulas before you said what it meant.

Well in the model, I was basically assuming that they were growing at the rate of inflation. (The demand for loans which, the way I wrote it essentially takes the form of a percentage of the value of real assets, is proportional to the price level. A few times in there I talk about this as the percent of total assets or something like that. I think of this as the “leverage ratio” but I don’t think I bothered to define that in the post). Also I assumed that the “real” value of them was constant because that is the simplest thing to assume but you could imagine it growing at some rate (the net of production minus consumption/depreciation).

Let me see if I understand the basic idea.

The time interval is the central bank’s OODA loop, the time it takes the central bank to find out the current inflation etc. Then the central bank, knowing the amount of stuff sold Y (assumed constant for simplicity), the prices all that stuff is sold at P, and the money supply M, can caculate v (assumed constant for simplicity).

YP=vM.

to set P higher, P(t+1) = P(t)+inflation, calculate

Y(P+inflation)/v = M = L = R/a.

So set R(t+1) = a*Y*(P+inflation)/v where a, Y, P, and v are assumed known for the new time but in practice are estimated from their values in the last known time interval.

Once the central bank sets the new reserve R, everything else adapts except for the things that are held constant.

The banks set the interest rate at 1-1/v, and then the inflation is added on top of that, since banks know ahead of time how much inflation there will be.

The part I’m least clear on is L, “willingness to hold debt”. When the real interest rate is less than zero, people grab as much debt as they can on the assumption they can pay it off later for cheap, or get loans later to pay it cheaper.

When the real interest rate is more than zero, people borrow less. Rather than borrow enough to pay for everything that will be bought Y, they borrow enough less that the interest on their loans will buy the rest of everything that’s for sale.

I’m not sure about this last thing, it could be the interest from time t that gets spent by bank owners etc at time t+1. Or you could imagine that lots of short-term loans are being made during the t..t+1 interval and the interest is collected and spent in that time. That seems to me harder to deal with.

So what I suggest instead is M(t+1) = L(t+1) + N(t)

where N is total interest paid.

The time period is completely arbitrary, the most significant aspect of it in this case, (at least in my mind) is that it is the duration of all debt contracts. Obviously in reality, these contracts take on a wide variety of durations but can be refinanced. It is also the interval between monetary policy adjustments, though this only matters because of the structure I assumed for debt. One could imagine these contracts overlapping periods but this would get more complicated.

I’m not totally following the rest. Here are a few things.

The future price level is not estimated before the Fed sets policy (R) it is

causedby the policy in conjunction with all the other things (which presumably would be estimated in practice). So different R gives a different price level (and interest rate). I am simply assuming that the Fed is targeting the amount of inflation which is expected and then solving for the value of R that would produce that amount.Given my assumptions, the real interest rate cannot fall below zero because when it hits zero, all property is mortgaged. But this doesn’t happen in the base scenario unless you assume certain values of the parameters. But I’m not sure what you mean by this:

“When the real interest rate is more than zero, people borrow less. Rather than borrow enough to pay for everything that will be bought Y, they borrow enough less that the interest on their loans will buy the rest of everything that’s for sale.”

No matter how much they borrow, there is always enough to pay for everything that is bought. This is ensured by the equation of exchange (MV=PY). The price level is assumed to adjust to bring this into equilibrium.

The interest does not appear in the quantity of money equation because “the economy” is essentially paying it to itself. That money to pay that interest doesn’t exist, because the loans that created the money in the previous period only create money equal to the principle not the interest. Therefore, the money to pay the interest has to come from additional loans in the future. But by assuming that the money comes out of the new loans to pay the old interest and then goes

backinto the economy as dividends, it cancels itself out. Otherwise, the quantity of new loans would have to grow even faster and this would create a “built-in” liquidity trap.Incidentally, I suspect that liquidity traps may be built in and this may be the reason (or part of the reason) but there are a few possible explanations for it and they are all debatable.

Here’s how I understand it at this point:

1. At step 1, there has been an interval with Y stuff produced and sold at total price P. Interest N is owed on loans which have been otherwise paid off. The central bank wants total price during the next interval to be P+inflation.

The central bank estimates Y (kept constant in your case) and v (kept constant). Also CS (spending by bank and bank owners, the entire interest from last period in your case). In your model the estimates are all correct, since the result of incorrect estimates is not the point of the model.

vM=PY so M=PY/v we compute the money supply that will get the desired price.

M=L+CS There’s the money that got loaned, and then the interest from last interval that bank and owners spend.

M-CS = L = R/a

La=R The value the central bank sets to create the desired inflation.

2. The regular banks lend money. They set an interest rate of 1 – 1/v + inflation. I am baffled by this interest rate. Then banks attempt to lend R/a dollars.

Then debtors decide how much money they are willing to borrow. If i – inflation = inflation then they borrow less. I’m unclear about this part. Do they borrow less to account for the interest they already owe? So they won’t buy as much when the bank is buying part of Y? I just didn’t understand it.

3. At the end of the interval the loans are all repaid to the banks. The total money they can repay is L+CS but the total they owe is L(1+i) which may be different. I ignore this. At the beginning of the next interval the bank somehow has CS=Li = N, it has somehow acquired the rights to that spending, and the central bank is ready to decide how much lending should happen to meet its inflation goal for that interval.

I think fudging this gives plausible results for most variables. But if we demanded that debtors pay N-CS at the beginning of the interval, they take out a loan which they must repay without actually ever getting the money, it would affect L and i a little and the errors would build up over time.

1) You are still trying to add bank profits to the money supply despite my repeated attempts to explain that it cancels out. I don’t know how to explain it any better…

2) The interest rate is set by the intersection of supply and demand for loans. Supply is determined by the reserve ratio and the quantity of reserves, demand is the assumed function.

3)I’m not sure what you are doing here (or what you are “fudging”) but I think it is directly related to the problem in #1. It’s not as complicated as you are making it, in each period people owe (1+i)L from the previous period but only have L but I assume that the iL goes back to them so that it cancels out.

In reality, if the Fed is trying to decide what the inflation rate will be, then the Fed will not make a new decision until they get new information what the inflation rate is. If the Fed mainly uses CPI and PPI they both come out monthly. So the Fed can’t react quicker than once a month.

It isn’t realistic to have all debts last one month or one quarter, but that doesn’t change the qualitative results. The Fed won’t adjust the prime rate continuously though it does buy and sell bonds more or less continuously and can adjust that daily or even quicker, depending on the responses it gets.

As you say, it doesn’t affect the qualitative results so I don’t think this is really an important issue. But if you are trying to imagine the time periods involved, I would point out that it doesn’t have to be the minimum time possible that the Fed can make decisions, If debt contracts last a year and are all initiated at the same time, then it doesn’t matter whether the CB

canchange policy in the interim. Alternatively, if people can refinance their loans at any time, then the duration of those contracts relative to the “period” is not important for the reasons you are talking about (it may be important for other reasons since you would have loans with different interest rates overlapping and you would have some other issues) but I think all of this just distracts from the point I am trying to get across with the model which is that the money supply depends on the willingness to hold debt.I’m having a problem modeling this because I’m doing it in discrete time intervals. Some people like to do differential equations. That’s a bad idea, it tries to make things continuous that are in fact not continuous. It often results in difficult math. And the harder it is to deal with, the easier it is to make mistakes and not notice them.

A more correct approach is to have lots of single incidents at random times, and sum them. This is a big effort and the results tend to be tedious too. It’s worth doing when you want to get things right.

Doing things in discrete times is often a reasonable approximation and it’s easy.

So at one time the central bank must decide how much reserves to give to banks so they can lend the right amounts. (Alternatively it could change the reserve requirement.) it wants the total amount of money to be just right to get the right amount of inflation.

The banks lend that amount of money. Also they arrange that all the money they were owed last time is spent, either by the bank or by the bank’s owners.

So the money that is spent is both the loaned money, and also last interval’s bank net income.

At the beginning of the next interval all the loans are paid off, using both the money that was lent and also the money that the bank spent. But what if the interest owed is more than the money the bank spent last time? The bank could lend more money so it can be repaid, taking the money right away and letting the money plus interest be owed for later. Or it can be owed a dollar value in real stuff. This could amount to taking possession of some collateral.

But it inevitably happens each interval, if the bank charges interest. The amount of money that leaves the bank is the money loaned, plus the money spent by the bank plus bank owners. At the end of the interval, the money that enters the bank is the money loaned, plus this interval’s interest. Whenever the total amount of interested owed is bigger than the amount of interest owed last time (which is spent by bank and owners) then the debtors owe dollars they cannot get. The only way they can make up the difference is to get more dollars from the bank.

One debtor might pay off his debt. The debtors all together can never pay it off, unless the bank accepts barter.

“That’s a bad idea, it tries to make things continuous that are in fact not continuous”

Time, however, is continuous…

“It often results in difficult math. And the harder it is to deal with.”

That’s true but not always, sometimes continuous is simpler. However, in this case, that was my main motivation. You could probably make this continuous fairly easily though.

“That’s a bad idea, it tries to make things continuous that are in fact not continuous”

‘Time, however, is continuous…’

Time is continuous, but people usually don’t continuously take out new loans. One person pays off a loan, somebody else starts one. Somebody deposits money into his account and then starts buying things one at a time. It’s a whole bunch of dirac delta functions.

Sometimes it’s convenient to approximate that by something continuous. Sometimes it’s convenient to approximate it with what happens over intervals. If you care about the messy parts you can approximate by making events happen with a random number generator, and watch their behavior.

Horses for courses.

I think this is a major difference between economics and physical sciences. Models in economics aren’t usually designed for their quantitative predictive powers. Usually, if we can get the sign of a relationship correct we are happy. So economic models tend to be as simple as possible to get the point across and usually this means that either a continuous time “approximation” or a discreet time “approximation” is approximate enough.

First, banks and bank owners don’t have to spend all their income. You might want to look at what happens when they don’t.

Second, are you sure it cancels out? If the bank profits come from the interest from the previous interval, those might not cancel out except in equilibrium. And it isn’t equilibrium when inflation keeps rising.

Third, it might be interesting to count up bank profits. That might be a valuable measurement.

Fourth, if it really does cancel out then it won’t cause trouble to include it and watch it cancel out.

Yes! Supply is determined by the central bank. Demand — you want it to be enough to buy all the product Y at price P, given velocity v. The total money they need to do that is YP/v, ignoring the part of Y bought by banks and bank owners.

i1=1-R1/(aP1Y)+ πe=1-(1+ πe) R0/(a(1+ πe) Y)+ πe=i0=1-1/v+ πe <=== Typo? P0 needed?

i1=1-R1/(aP1Y)+ πe=1-(1+ πe) R0/(a(1+ πe)P0 Y)+ πe=i0=1-1/v+ πe

I'm missing something. When v=1, in the time allowed the debtors buy once from each other, and then they each owe all their money plus interest to the bank. As near as I can tell, the interest they owe is only inflation and the bank gets nothing more. That isn't much.

When v=2, they on average buy once and then use the money they receive to buy again. The interest rate is then 50% plus inflation. That seems like a lot.

As v goes up, interest rapidly approaches 1.

In your later discussion you talk about assets that serve as collateral for loans. I was treating Y as products that trade hands each period, and the model leaves out assets owned by debtors. It makes sense to me that higher interest rates should reduce demand for loans, and limited collateral should put a limit on loans just as limited reserves does. Since you have v constant, you get constant interest rates apart from inflation and exactly how the interest rate is set doesn't matter much except that v must be a small constant. I don't intend to criticize you. I just want to get the model to work, and if there are little problems to fix in the first draft that happens to everybody. Or if I've misunderstood I want to understand where I went wrong.

Rt=(1+ πe)t R0

Mt=(1+ πe)t R0/r <=== Typo? R0/a?

Pt=P0(1+ πe)t

It=1-1/v+ πe

It’s probably better to take care of the picky accounting. You can leave out whole concepts that don’t interest you at the moment, but it’s better that the model doesn’t leak money.

People owe (1+i)L from the previous interval but they have only L. They could give up assets that we haven’t considered. They could take out a larger loan to pay the interest from the previous interval. They could barter L/(1+πe) quantity of product from Y to the bank. Pick some way to do it and you don’t have to assume it cancels without making it cancel.

Also, since L(t+1) = (1+πe)L(t) ignoring the unpaid interest, iL(t+1) != iL(t).

Really and truly I think you’re on the way toward a rigorous model that will reveal interesting qualitative results. I’m not interested in finding fault with what you’ve done, as if you were claiming it was the final word on everything and any detail that wasn’t perfect meant you were wrong. I just want to make it computable.

“Bank owners don’t have to spend all of their income.”

The degree to which owners spend the dividend is built into the equation of exchange (MV=PY). If some of the profits stay on banks’ balance sheets then there is a different thing going on which is something I am aware of (I actually originally assumed that they kept all profits on the balance sheets and added the dividend assumption to remove the highly charged implication that comes out of a model like that. I think it is appropriate to start here as a base-case.)

“Are you sure it cancels out?”

Yes because I am basically assuming that whatever they pay in interest goes right back to them. (By “them” I mean the economy, not necessarily the same individuals.) The amount of these payments doesn’t matter.

“It might be interesting to count up bank profits.”

Indeed, I imagine somebody is counting those. In the model, in any given period it would i*L(t-1).

“Fourth, if it really does cancel out then it won’t cause trouble to include it and watch it cancel out.”

That is why I did it that way, though it seems it has caused some confusion…(haha)

Typos:

You got me on both counts. I originally had “r” for the reserve ration but changed it to “a” when I ended up putting the real interest rate in the demand function. I thought I got them all but missed on apparently. If there are any other suspect rs, it is safe to assume they should be as.

“I’m missing something. When v=1, in the time allowed the debtors buy once from each other, and then they each owe all their money plus interest to the bank. As near as I can tell, the interest they owe is only inflation and the bank gets nothing more. That isn’t much.”

Yes that is right but since we are assuming parameters, the parameters we assume matter. If you assume v=1, then you get a zero real interest rate. This just means that a higher v is probably more realistic in most cases. (Of course, in reality v is endogenous but that opens up a whole other can of worms.) Similar thing if you assume v=2 you just went too far in the other direction. These are both issues of “calibration.”

Regarding Y, basically what I am assuming is that the only “assets” people own are those which are produced in that period. Admittedly, I didn’t explain this very carefully. But assume that each period Y is produced and it is all exchanged using money and then it is either consumed or used up (depreciated) in the production of next period’s Y. This assumption could, of course be relaxed but would add another dimension of complication.

“It’s better that the model does not leak money.”

It doesn’t (or at least, I am still convinced it doesn’t). I think the confusion here comes from me assuming that everything in the period happens simultaneously. Instead try thinking about it in this order.

1. New loans are made [money supply increases by Lt]

2. Dividends are paid using “newly printed” money [money supply increases by i*Lt-1]

3. Old loans are repaid out of the stock of existing money AND new money [money supply decreases by (1+i)Lt-1]

$. Goods are exchanged using the new quantity of money [Lt-1+Lt+i*Lt-1-(1+i)Lt-1=Lt]

No offense taken, it’s good to have another eye to help work out the bugs.

“

since we are assuming parameters, the parameters we assume matter. If you assume v=1, then you get a zero real interest rate. This just means that a higher v is probably more realistic in most cases. (Of course, in reality v is endogenous but that opens up a whole other can of worms.) Similar thing if you assume v=2 you just went too far in the other direction. These are both issues of “calibration.””We have an arbitrary interval size. To be realistic maybe it should be around a month, but if you set the various parameters right it could be a year or a day or whatever.

I get the impression you have an ad hoc equation here. You set the interest rate to (v-1)/v because it gives you vaguely the right sort of slope or something. When it only works within a limited range, that determines the range we can reasonably set for a time interval. A time interval that’s too short will give us v=1 and a bad outcome. A time interval that’s too long will give us v=2 and a bad outcome.

If you specify how you want the interest rate to behave, then you can design a function for it which works the way you want.

My natural thought (which unfortunately does not work) would be something like this:

1. The bank has a minimum interest rate ilow it will accept on top of the inflation rate, and will not lend below that. In reality it might determine ilow from the frequency of defaults, but of course there are no overt defaults in the model. The bank would prefer to maximize interest rate.

2. Bank customers have a maximum interest rate ihigh that they will accept on top of the inflation rate, and will not borrow above that. In theory they should choose this based on their predicted ability to repay. They should never take on a loan that they think will drive them into bankruptcy and loss of their collateral. But of course in this model they can never repay their loans and must inevitably fall deeper into debt.

3. As a first approximation, when ilow>ihigh, split the difference.

Second try. Supply and demand. Bank has R/a. Customers need YP/v.

If R/a > YP/v, set i=ilow.

If R/a<YP/v, i=ihigh.

If R/a=YP/v, i=(ihigh+ilow)/2

Highly inelastic on both sides.

Third try.

if R/a1.1*YP/v i=ilow buyer’s market

if 0.9*YP/v <= R/a <= 1.1*Y^P/v

i=ilow + (ihigh-ilow)*(YPa/vR-.9)/.2 linear between limits

I've only desk-checked this so it could be wrong. But the idea is, if interest gets so high or low that the deal's off, set it at the limit. In between, do a weighted average.

Since your central bank sets R/a=YP/v, i will always equal (ilow+ihigh)/2 until you change the model.

You can do it however you want it. Linear between limits is at least simple, and it makes a kind of sense.

They are all ad-hoc equations.

You are missing the point by focusing on the length of the interval. The interval doesn’t determine v, v is assumed to be exogenous.

1. This makes me chuckle. One day all of this is going to result in you having an epiphany about how competitive markets work and then your world is going to be turned upside down and we are going to have nothing to argue about haha. Banks would like to maximize the interest rate but they can’t because there are lots of them competing with each other. There is a minimum they are willing to accept it is (by assumption) 0. You could make it higher if you wanted (see the post I linked to about banks lending excess reserves) but it wouldn’t change the structure of the model. Either way, the rate is determined by supply and demand, you’re just changing the shape of the supply curve.

2. Same thing but with the demand curve…

2.

Horses for courses. When you care about random fluctuations and how the model system handles them, as is important for market models, then you need to actually model the variations. You can take shortcuts and assume that things fit particular statistical distributions and then do the math on the distributions. That’s valuable when you lack computing time, which used to be a big issue 40 years ago.

Do whatever’s most useful to you at the moment. You can get insights from a quick model and then refine it later and maybe get more insights. The models are tools to help your insight, so use whatever is cost-effective for that.