## The Fisher Paradox

There is a bit of a paradox underlying much of monetary economics. If real rates are independent of monetary factors, then a reduction in the nominal rate should be accompanied by a reduction in the expected rate of inflation (or vice-versa). Yet we typically observe*, at least in the short run*, that if the central bank lowers its interest rate target, it causes a higher rate of inflation. Of course, both old monetarists and market monetarists reconcile this by saying “never reason from a price change” (always good advice) and instead, reason from a change in the money supply (and expected future money supply), assuming sticky prices in the short run and then separate the effects on interest rates into the well-known liquidity, income and Fisher effects which allows for the real rate to change in the short run and for the nominal rate to go either way.

That’s all perfectly reasonable but lately there has been a school of thought emerging known as “neo Fisherites” who are bringing this issue back into the discussion. Nick Rowe (for one) has recently been taking them to task(here, here and here).

Now let me say for starters that I suspect everything Nick says about these papers is correct, and I’m not trying to defend them. I agree that denying that lowering rates raises inflation is contrary to all observations, and I suspect (though I haven’t read them yet) that his analysis of the specific papers as lacking in economic intuition and relying on strange assumptions to “rig” the results in favor of their prior beliefs is most likely spot on. That is how I feel about most modern economic papers I read, sadly. However, I think beneath the snow job and the tiny pebble of wrongness, there is actually a kernel of insight (or at least the pebble started out as a kernel before it got all mangled and turned to the dark side) and it is closely related to the stuff I have been trying to say. So I will try to flesh it out a little bit in a way that does not contradict everything we know about how monetary policy actually works.

Note that this actually began as a discussion of monetary and “fiscal” policy, which I intend to get to but I will put that off for a future post since just dealing with this Fisher paradox will be enough to fill a lengthy post by itself, but keep in mind that adding that piece in will be important for making this model look like the real world. (And also keep in mind that I don’t mean what other people mean when I say “fiscal policy.” Frankly, it’s almost tongue-in-cheek. All macro is monetary.)

**The Base Model**

Start in a world where there is some form of “hard” money like gold. Let the quantity of this money be denoted G and assume that it is fixed and constant over time. Note, of course, that this would not be the case with hard money since it would have alternative uses and would be consumed at some rate and the demand for it in alternate uses would determine the rate at which it was consumed so that the change in quantity over time would be endogenous and so would the liquidity premium. However, this will complicate the model so let us assume that it never gets used up (or, if you prefer, that the rate at which it is used up just happens to be equal to the rate at which it is produced/extracted at all times in equilibrium) so that the quantity always stays the same.

Also assume that people get some satisfaction from holding this money over and above what they would get from holding other assets and call this satisfaction “liquidity.” Because of this there is a liquidity premium which one must pay for holding money relative to other assets and it is the difference between the rate of return on money and the rate of return on other assets. The real rate of return on other assets is r and the rate of return on money is the negative of the inflation rate—or the deflation rate–so let me call this d. Then we can write the Fisher equation as:

i=r-d

where i represents the liquidity premium which we commonly call the nominal interest rate.

Now assume that we are in a steady state where output is constant, population is constant, the real interest rate is constant and assume that these are all determined by “real” factors and are independent of monetary factors. Assume, which amounts to the same thing, that all markets are perfectly competitive and prices are perfectly flexible. By doing this, I am assuming away the “short run” which is the focus of nearly all monetary economics. I’m doing that on purpose.

Also, assume that what people care about is the “real” value of money that they hold (M/P), not the absolute quantity and that money demand is given by the following equation.

MD/P=200-400i

Where i is the nominal rate. This is also the liquidity premium. Now since the money supply is constant over time and we are in a steady state, the price level will have to be constant over time which means that the liquidity premium will have to equal the real rate. Let’s assume that the real rate is .10 and that the quantity of “hard” money G is 100. This will imply a price level of .625. This will be the price level at which people are indifferent between holding money and other assets on the margin given our money demand function.

**Central Bank**

Now let’s put a central bank in. The central bank issues credit in “dollars” a unit which they make up out of thin air. For now, let us assume that there is a fixed conversion rate between dollars and gold which is 1 unit of gold (perhaps oz.) per dollar and that the central bank will trade gold for dollars or dollars for gold. This is not necessary (see this post) but makes things simpler. Also assume that gold and dollars are perfect substitutes so we can just treat “the money supply” as the sum of dollars and gold where G is gold in circulation and D is dollars in circulation.

M=G+D

The bank works by offering to *lend* dollars at a particular nominal interest rate which it can set at whatever level it wants. So the bank sets some interest rate i and then the quantity of money people borrow is endogenously determined. And assume, for simplicity, that everyone believes the bank will keep the nominal rate at i forever.

Finally, let us assume that the bank has no operating costs and never consumes any real goods. However, let us also assume that any dollars paid back to the bank are destroyed and any gold paid back to the bank is either also destroyed or is kept in a vault from which it can only escape if the bank lends it in the future or uses it to redeem dollars at face value. Otherwise it will just sit there forever, since the bank promises never to spend any money on real goods.

How much will people borrow and what will happen to prices and inflation?

First of all, if the bank sets a nominal rate greater than or equal to that which prevailed without the bank (.10), then nobody will borrow from the bank and nothing will change, since the money market already clears without it. (If the bank is willing to borrow at rate i as well, then something different will happen but that’s not worth talking about.)

If, on the other hand, the bank sets a rate lower than that which would otherwise prevail, people will desire to hold more money. But they will also desire to invest more at current prices, since they can borrow at a lower rate and they will *also* desire to consume more today (relative to the future) for the same reason. So how does all of this work itself out?

Three things need to happen. First, people will be net borrowers from the central bank. This will expand the money supply. But they are not just borrowing in order to hold more cash, they are borrowing in order to purchase more consumption goods, more investment goods *and* more cash since lowering the interest rate makes all of these things cheaper in terms of future money (which is what they are trading for them by borrowing).

Now if you are a Keynesian or a monetarist (or basically anyone else besides an Austrian or a “neo Fisherite”), you say that output increases to meet the extra “aggregate demand.” However, that is a short-run story and I don’t want to talk about the short run, so I am assuming that markets are perfect and there is no short run. (Please note that assuming is different from asserting.) So* if* markets were perfect, what would happen?

Two things would happen. First, the increase in demand would cause prices to rise. How far would they rise? Well, they have to rise until people are willing to hold the new quantity of money. But if the real rate doesn’t change, and the nominal rate is fixed, then people will always want to borrow more money to invest which will drive the price level up to infinity. *Unless*, that is, the rate of inflation changes. If the real rate is .1 and the bank sets the nominal rate at .05, for instance, the rate of inflation will have to be -.05, or to put it another way, the rate of deflation (d), representing the return on holding money, will need to be .05.

This, in a nutshell is the core Fisherian (neo or otherwise) insight. But how can we reconcile that with a scenario in which the money supply is increasing and prices are rising? Actually, it’s pretty simple, you just have to separate the initial effect from the effect going forward.

When faced with the new, lower nominal rate, people will borrow new money from the central bank. This new money is created “out of thin air” but has a debt contract attached to it. If you borrow it, then at some point in the future you need to repay either dollars or gold (or forfeit some other real goods). This means that the expansion of the money supply today puts a burden on the future money supply. But that is actually exactly what is necessary to bring this system into equilibrium.

So let B be the quantity of dollars borrowed in any given period. Then the change in the money supply from one period to the next will be the change in borrowing minus the interest paid on the amount borrowed in the former period.

Mdot=Bdot-iB [1]

And since everyone expects the nominal rate to be constant forever, an equilibrium has to have M/P constant at 180 according to the money demand curve. And since the real rate is constant at .1, we know that the price level has to fall by .05 each period. And this means that the money supply must also fall by the same factor.

Pt=P0(1-.05)^t

Mt=M0(1-.05)^t

So an equilibrium will be a path for B and an initial price level for which everything above is true every period.

Now it turns out that there are a multitude of possible equilibria to this model as it is specified so far. Any initial price level (within some range) can be supported by some path for B and any initial value of B can be supported by some price level and future path of B. A unique solution would be possible if we had a more detailed utility function for our representative agents, as I tried to do here. But instead of doing that, let’s just make a simplifying assumption that the real value of borrowing B/P in each period is constant. (If you just put in an arbitrary number for B0 you can get a smooth path for prices but it will tend to require B to jump all over the place over time which isn’t very realistic. Since we are in a steady state, whatever makes sense in one period, should make sense in all periods in a real sense.)

This assumption implies that the nominal value of B will need to fall at the same rate as the price level which is the same rate as the overall money supply.

Bdot/B=-d=i-r

Mdot/M=-d=i-r

With this in hand, we can get an equilibrium “credit ratio” (ratio of borrowing to the total money supply) by solving the two equations above for Bdot and Mdot respectively and plugging them into equation [1] which will then give us the following.

B/M=d/(i+d)=(r-i)/r

So for the values postulated so far which, if you forgot, are the following:

G=100

r=.10

i=.05

the equilibrium credit ratio will be ½. This means (if we assume that D=B at all times, which is equivalent to assuming people pay back principle with dollars and interest with gold) that at each period, half of the money supply will be made up of credit money or “dollars” while the other half will be made up of “hard” money or “gold.” So in the first period, people will borrow 100 dollars and the money supply will be 200. This will push the price up to 10/9 (according to the money demand curve). Note that this is higher than 10/16, which was the price level with no central bank (or if the bank had set i=.10).

So did lowering the nominal rate increase or decrease inflation? The answer depends on what you mean by “inflation.” On the one hand, the expected rate of inflation in equilibrium *going forward* falls. This must be the case if the real rate remains unchanged, that is the Fisherian insight (if you will) and that part is pretty straightforward. But the way you arrive at that new equilibrium with a lower inflation rate is important and the way we do that here is by the money supply expanding and prices *rising *immediately. So the immediate effect is higher prices. And of course, if you have sticky prices or money illusion or something like that, this effect which is immediate in the model with perfect everything might play out over some period of time which we might call the “short run.”

So even if we believe in stable real rates in the long run, we shouldn’t be surprised when the people who actually conduct monetary policy tell us that turning the wheel left makes the car go to the left (to continue with Nick Rowe’s analogy). It does. But this raises a different sort of question. Can you get the short run effect and an increase in inflation expectations over a long-run timeframe?

I think most MMers will say “yes” but that is because they are imagining an exogenous path for the money supply so that the monetary authority can commit to increase it *at a faster rate* forever. That is not possible in this model. In this model, the size of the money supply and the inflation rate are endogenous* in the long run*. This is different from what most people who gas on about “endogenous money” on the internet have in mind. In the short run, there is no important difference between assuming that interest rates are exogenous or that the money supply is exogenous. Any value of either determines some value of the other. But if expanding the money supply requires people to borrow, then this rules out some time-paths for M and therefore P, which means that (rational) inflation expectations are tethered to what you might call a “Fisher path.”

So in this model, if they wanted to make the money supply expand forever, they would have to either raise rates and shrink the money supply today by borrowing money and then paying interest on it (and whether this would work or not in the long run would be questionable because then you really would have a situation where the new money was not backed by any real assets which you do *not* have with the above scenario, but I digress…) or they would need to continuously lower the nominal interest rate in order to make the real value of borrowing increase perpetually over time.

At this point please note that over the last thirty years nominal rates have been steadily decreasing and leverage has been steadily increasing (though I don’t have an ideal data series for the latter at this point so maybe you can argue with that part). However, if this is the case, then whether or not it is sustainable indefinitely depends on the shape of the “money demand” curve around the zero lower bound and the shape of it is not what everyone thinks because it is really a demand for debt curve (or at least they are bound up together) and everyone is ignoring that and just talking about “liquidity.” (But I digress again….)

At any rate, monetarists and Keynesians each have their own particular solution to this problem which both amount to the same thing. Keynesians have “fiscal policy” where the government does extra borrowing to keep the money supply going and prices rising and monetarists have the “buy the whole world” answer where the central bank just prints money and keeps buying stuff in order to keep the money supply and the price level rising. The only difference is that Keynesians pretend that the government will repay the debt some day and so they call it “fiscal” while the monetarists just assume a permanent increase in the money supply and call it “monetary.”

Of course, this makes monetarists less wrong than Keynesians in the grand scheme of things (at this point notice that for the past thirty years federal debt/GDP has been steadily rising) but in the “short run” they both have the same effect so long as the markets realize that the Keynesians are lying.

But then you are left with the question of how much of this “fiscal” or “monetary” policy will be required in the long run to sustain an equilibrium in which the price level, and the money supply are perpetually growing? And that question I will save till next time.

Here’s the backing theory explanation: The natural rate is 5% and the central bank starts lending at the low rate of 4%. This reduces the bank’s net worth and there is therefore less backing per dollar, so inflation results, at the same time that the low rate leads people to line up at the central bank asking for loans. If the central bank accommodates the loan demand, then money supply rises a lot and quantity theorists will mistakenly conclude that the high money growth caused the inflation. If the central bank reacts to the high loan demand by rationing loans to a low level, then the money supply does not grow as much. There will still be inflation, but not as much much as there was when the central bank accommodated the high loan demand. The bank’s rationing will also cause a tight money condition and unemployment will rise.

Occasionally, the money supply will grow by 10%, but since central bank assets grow by 10% at the same time, the money will hold its value. Quantity theorists will not understand that there are 10% more assets backing 10% more dollars, so they will think that prices are sticky, when the truth is that there was no reason for prices to change at all..

Mike,

I don’t buy that theory. I’m trying to do something different here.