## Tuesday, March 29, 2016

### A very simple neo-Fisherian model

A sharp colleague recently pushed me to write down a really simple model that can clarify the intuition of how raising interest rates might raise, rather than lower, inflation. Here is an answer.

(This follows the last post on the question, which links to a paper. Warning: this post uses mathjax and has graphs. If you don't see them, come back to the original. I have to hit shift-reload twice to see math in Safari. )

I'll use the standard intertemporal-substitution relation, that higher real interest rates induce you to postpone consumption, $c_t = E_t c_{t+1} - \sigma(i_t - E_t \pi_{t+1})$ I'll pair it here with the simplest possible Phillips curve, that inflation is higher when output is higher. $\pi_t = \kappa c_t$ I'll also assume that people know about the interest rate rise ahead of time, so $$\pi_{t+1}=E_t\pi_{t+1}$$.

Now substitute $$\pi_t$$ for $$c_t$$, $\pi_t = \pi_{t+1} - \sigma \kappa(i_t - \pi_{t+1})$ So the solution is $E_t \pi_{t+1} = \frac{1}{1+\sigma\kappa} \pi_t + \frac{\sigma \kappa}{1+\sigma\kappa}i_t$

Inflation is stable. You can solve this backwards to $\pi_{t} = \frac{\sigma \kappa}{1+\sigma\kappa} \sum_{j=0}^\infty \left( \frac{1}{1+\sigma\kappa}\right)^j i_{t-j}$

Here is a plot of what happens when the Fed raises nominal interest rates, using $$\sigma=1, \kappa=1$$:

When interest rates rise, inflation rises steadily.

Now, intuition. (In economics intuition describes equations. If you have intuition but can't quite come up with the equations, you have a hunch not a result.) During the time of high real interest rates -- when the nominal rate has risen, but inflation has not yet caught up -- consumption must grow faster.

People consume less ahead of the time of high real interest rates, so they have more savings, and earn more interest on those savings. Afterwards, they can consume more. Since more consumption pushes up prices, giving more inflation, inflation must also rise during the period of high consumption growth.

One way to look at this is that consumption and inflation was depressed before the rise, because people knew the rise was going to happen. In that sense, higher interest rates do lower consumption, but rational expectations reverses the arrow of time: higher future interest rates lower consumption and inflation today.

(The case of a surprise rise in interest rates is a bit more subtle. It's possible in that case that $$\pi_t$$ and $$c_t$$ jump down unexpectedly at time $$t$$ when $$i_t$$ jumps up. Analyzing that case, like all the other complications, takes a paper not a blog post. The point here was to show a simple model that illustrates the possibility of a neo-Fisherian result, not to argue that the result is general. My skeptical colleauge wanted to see how it's even possible.)

I really like that the Phillips curve here is so completely old fashioned. This is Phillips' Phillips curve, with a permanent inflation-output tradeoff. That fact shows squarely where the neo-Fisherian result comes from. The forward-looking intertemporal-substitution IS equation is the central ingredient.

Model 2:

You might object that with this static Phillips curve, there is a permanent inflation-output tradeoff. Maybe we're getting the permanent rise in inflation from the permanent rise in output? No, but let's see it. Here's the same model with an accelerationist Phillips curve, with slowly adaptive expectations. Change the Phillips curve to $c_{t} = \kappa(\pi_{t}-\pi_{t-1}^{e})$ $\pi_{t}^{e} = \lambda\pi_{t-1}^{e}+(1-\lambda)\pi_{t}$ or, equivalently, $\pi_{t}^{e}=(1-\lambda)\sum_{j=0}^{\infty}\lambda^{j}\pi_{t-j}.$

Substituting out consumption again, $(\pi_{t}-\pi_{t-1}^{e})=(\pi_{t+1}-\pi_{t}^{e})-\sigma\kappa(i_{t}-\pi_{t+1})$ $(1+\sigma\kappa)\pi_{t+1}=\pi_{t}+\pi_{t}^{e}-\pi_{t-1}^{e}+\sigma\kappa i_{t}$ $\pi_{t+1}=\frac{1}{1+\sigma\kappa}\left( \pi_{t}+\pi_{t}^{e}-\pi_{t-1} ^{e}\right) +\frac{\sigma\kappa}{1+\sigma\kappa}i_{t}.$ Explicitly, $(1+\sigma\kappa)\pi_{t+1}=\pi_{t}+\gamma(1-\lambda)\left[ \sum_{j=0}^{\infty }\lambda^{j}\Delta\pi_{t-j}\right] +\sigma\kappa i_{t}$

Simulating this model, with $$\lambda=0.9$$.

As you can see, we still have a completely positive response. Inflation ends up moving one for one with the rate change. Consumption booms and then slowly reverts to zero. The words are really about the same.

The positive consumption response does not survive with more realistic or better grounded Phillips curves. With the standard forward looking new Keynesian Phillips curve inflation looks about the same, but output goes down throughout the episode: you get stagflation.

The absolutely simplest model is, of course, just $i_t = r + E_t \pi_{t+1}$. Then if the Fed raises
the nominal interest rate, inflation must follow. But my challenge was to spell out the market forces
that push inflation up. I'm less able to tell the corresponding story in very simple terms.

#### 31 comments:

1. From a demographic POV if you raise rates to where retirees can get a safe and significant income you might encourage retirement. That would open up jobs for younger people. That's the demographic that forms households, buy homes, and spends money to furnish the homes. They also spend money on clothes, school supplies, toys, and minivans.

Low rates signal to potential retirees that they have to save more than planned. Between the increased savings by older people and the barriers to creating new households by younger people you have an economy where nobody is spending.

1. If only the world worked in that magical way.

Higher real rates for retirees have to be paid by someone, you can't get a net gain just by raising interest rates and getting people to retire (tried this idea in the 30s with SS- didn't work). You have to subtract out the losses from those paying the higher interest rates, which you ignored.

2. I don't know what planet you live on but on Earth even before the Depression the retired elderly in America were experiencing severe hardship. In the 1930s the older workers in America were the first to be fired, which threw the problem into high relief and could no longer be ignored.

There had been pension movements for decades before the New Deal. The Depression simply intensified the urgency. By the time Social Security was enacted over half the States had already implemented pension plans for the elderly.

I guess Father Coughlin, Upton Sinclair, and Huey Long were merely campaigning to get old people to retire and make way for the young workers. Is that your fairy tale?

Yes, someone has to pay for it with higher rates and I do believe that will cause deflation in prices. However, you can do that while you still have a semblance of control (however ephemeral) over the system or you can wait until confidence has totally collapsed and trillions in debt and CDS are blowing up. Either way, it's where we're heading. Do you want to shoot a cannon at the snow pack and have an avalanche on your terms or wait for nature to run its course and kill some people?

There simply isn't any other way out and the longer this ZIRP/NIRP insanity goes on the worse the consequences. The first thing you do when you realize you've dug yourself into a hole is stop digging.

To think there's an elegant and painless way out of 8 years of profound asset price distortion caused by artificially suppressing the cost of money is true magical thinking and it's the stuff that central bank can-kickers' wet dreams are made of.

Pilots call the situation we're in the "coffin corner".

3. The world I live in is one where economics teaches that growth isn't a zero sum game, and that we don't come up with ad hoc justifications for preferred policies (with no mention of potential costs!).

"There had been pension movements for decades before the New Deal....."

None of what you wrote is relevant. There is no correlation between retirement benefits and improved employment outcomes for younger demographics. Treating jobs as a stock to be filled has no support.

"However, you can do that while you still have a semblance of control (however ephemeral) over the system or you can wait until confidence has totally collapsed and trillions in debt and CDS are blowing up."

So a new justification. And what expert is judging this? What centrally planned organization has a track record of measuring the "snowpack" and determining that now is the best time to fire the cannon?

4. The discussion of the history of SS is relevant to the refutation of your nonsense "tried this idea in the 30s with SS- didn't work. You can't defend against historical fact so you brush it aside.

I share your skepticism of central planning, which is why I think it should stop. Maybe when you've dug yourself a hole your solution is to "dig smarter". The solution here is not smarter credit expansion or better central planning. It's to stop doing all that nonsense.

I agree with Fisher that monetary stimulus is a band-aid. I don't really see much of value in terms of Fisher's solution for deflation, which seem to be summed up on page 125 of "Booms and Depressions":

How much reflation is right? In other words, how far back by way of correction should we put the price level before starting to safeguard it?

The answer is: far enough back to repair, as nearly as
possible, the injustice to the creditor or the debtor, as the
case may be."

Ah, so Krugman is channeling Fisher. You stimulate until it's enough.

I'm more in the Mises camp: “There is no means of avoiding the final collapse of a boom brought about by credit expansion. The alternative is only whether the crisis should come sooner as the result of a voluntary abandonment of further credit expansion or later as a final and total catastrophe of the currency involved.”

I'd prefer "a voluntary abandonment of further credit expansion" to "a final and total catastrophe of the currency involved."

The crisis is here, it's been here and it's getting worse. Fisher seemed to like medical metaphors so let's just say putting band-aids on the brain tumor isn't working any more. I'd say we should fire the cannon in oh, about 2012, 2013, 2014, 2015, and now.

5. Baconbacon,

"Higher real rates for retirees have to be paid by someone, you can't get a net gain just by raising interest rates and getting people to retire (tried this idea in the 30s with SS- didn't work). You have to subtract out the losses from those paying the higher interest rates, which you ignored."

The federal government makes a lot of interest costs tax deductible for precisely that reason - to get people to borrow at positive real interest rates. It creates a win-win scenario for both borrower and lender (retiree). Lender (retiree) gets a positive real income and yet borrower pays less in after tax interest.

6. Frank, I think the government should refrain from that kind of behavior. It's not win-win. It creates market distortions. Anything you subsidize you get more of. Subsidize mortgages and you get more of them. That sounds good until you realize that when more people can buy a house the price of houses goes up and relatively speaking you're back where you started.

Tax deductions for certain kinds of debt is just another variation of central planning.

I think it's time for the "Stop F#cking with Everything" Party. People can optimize their positions if you just leave them alone and stop tinkering all the time. It's what all life forms do.

7. Michael,

"Frank, I think the government should refrain from that kind of behavior. It's not win-win. It creates market distortions."

And yet:

"From a demographic POV if you raise rates to where retirees can get a safe and significant income you might encourage retirement. That would open up jobs for younger people."

How were you planning to get borrowers at those higher rates so your retirees could retire comfortably? On the one hand, you want a market distortion to give retirees a higher income, and on the other hand you disdain market distortions - which is it?

8. Excellent point Frank.

In terms of what I said about rates, I probably should have phrased it instead as "stop artificially depressing rates".

The only concessions I'll make to central intervention in rates are what William McChesney Martin said:

1. The central bank should "lean into the wind" and,
2. "Take away the punch bowl just as the party gets going"

But that doesn't really release me from the contradictions you point out if we speak in absolute terms. As Walt Whitman said,

"Do I contradict myself?
Very well then I contradict myself;
(I am large, I contain multitudes.)"

I would like to claim 100% purity of philosophy but in thousands of years of civilization no one has ever come up with one. Every philosophical system ultimately runs into contradictions or unanswerable questions. Otherwise we'd all believe the same thing - The One True Belief System.

If you believe in free will and an omnipotent God you run into theological problems that end with silly statements like St Augustine: “we shall exercise our wills in the future because He has foreknowledge that we shall do so.” If you're a strict Determinist you really can't convict anyone of a crime.

Ayn Rand accepted Social Security payments.

Thus we have to live with approximations and adjustments because reality is messy.

I once wrote to Walter Williams about his definition of "rights". Williams said that for something to be a " right" it couldn't infringe on the rights of others. I asked him about the 5th and 14th Amendments. Those "rights" to due process mean the government can appropriate my time for jury duty.

Williams replied, "It's an imperfect world."

I agree and therefore strive to hew to my core beliefs as much as possible, but it's impossible to be 100% faithful, nor is it practical. So I will oppose central planning to the best of my ability and live with the concessions I have to make.

My apologies to Dr. Cochrane for straying far afield, but I think we'd all be a lot better off in our endeavours if we didn't demand "one size fits all" solutions.

9. P.S. if I had my way there wouldn't be central banks. If we have to have them I'll take Mr. Martin's version.

10. Michael,

Now we are getting somewhere.

"P.S. if I had my way there wouldn't be central banks. If we have to have them I'll take Mr. Martin's version."

William McChesney Martin was Fed President at a time when the federal government was constrained on the size of it's deficits (and debt) by the U. S. Treasury's gold holdings (all of the gold in Fort Nox). Those days are long gone. These days, leaning into the wind has some pretty serious fiscal ramifications.

Today we have interest payments on federal debt made from available tax revenue. The potential for conflict between Martin's ideal Fed policy and fiscal policy is pronounced. At the extreme, interest payments on federal debt can exceed available tax revenue.

That puts the central bank at a cross roads - either maintain Martin's fortitude or acquiesce to fiscal difficulties.

If we must "take Mr. Martin's version" to heart, then the federal government must never borrow (sell bonds) nor print / coin money to finance budget deficits. That leaves the federal government to either run perpetual balanced budgets / surpluses or seek alternative financing means.

11. Oh the horror of balanced budgets! We wouldn't have enough money to go to war or run a welfare state. Isn't that what drove Nixon to close the gold window? Between the Great Society and the Viet Nam War we were running out of gold, plus the Fed had increased money supply. First France and then other countries began redeeming dollars for gold. I remember my father refusing to order French wine in restaurants because he hated Charles de Gaulle.

You say interest payments on federal debt come from tax revenue. And tax revenue comes from where? The private sector.

And your point about how Martin's approach leads to interest payments exceeding tax revenue escapes me. Japanese bonds pay how much interest, and how much government spending is allocated to paying that piddly interest rate? The low rate merely enables rampant borrowing.

I don't see how Martin's advocacy of mild counter-cyclical monetary policy or taking away the punch bowl implies that the government can't borrow.

When Martin began running the Fed rates had been at 1% since early WW2 to reduce the federal debt burden. Despite the "constraints" of the gold standard debt/GDP was the highest in our history. Truman tried to pressure him to keep it that way but with inflation at 8% Martin did the right thing, eventually taking rates to 4‰.

Volcker had brass too. He acknowledged that raising rates would be painful but like Martin he took the long view. Then, as now, people were in favor of accommodating to the disease instead of treating it.

I only laud the manipulations of Martin and Volcker because they reversed the problems caused by previous manipulations, not as an endorsement of meddling. I wish the current Fed had half the courage to stop the insanity, for is it not written in "Joseph and the Amazing Technicolor Dreamcoat" that

"I was wandering along by the banks of the river
When seven fat cows came up out of the Nile, uh-huh
And right behind these fine healthy animals came
Seven other cows, skinny and vile, uh-huh
Well the thin cows ate the fat cows which I
Thought would do them good, uh-huh
But it didn't make them fatter like such
A monster supper should"

I think we are on our 7th fat cow.

12. @ Frank Restly

"The federal government makes a lot of interest costs tax deductible for precisely that reason - to get people to borrow at positive real interest rates. It creates a win-win scenario for both borrower and lender (retiree)."

Someone is still paying the higher rate. Tax advantages are generally transfers of tax burdens (some possible exceptions exist).

13. "The discussion of the history of SS is relevant to the refutation of your nonsense "tried this idea in the 30s with SS- didn't work. You can't defend against historical fact so you brush it aside."

No, its irrelevant. SS greatly expanded retirement benefits, it doesn't matter if it expanded from 0 or from some higher plane, it still provided an incentive to retire.

14. Michael,

"Oh the horror of balanced budgets!"

Should the federal government raise taxes and / or cut spending during a recession - aka pro-cyclical policy? If you believe so, try running for elected office under that banner - "Balanced Budgets 24/7".

"I don't see how Martin's advocacy of mild counter-cyclical monetary policy or taking away the punch bowl implies that the government can't borrow."

I didn't say the government can't borrow. To get to "Martin-esque" monetary policy, the government shouldn't borrow.

"Then, as now, people were in favor of accommodating to the disease instead of treating it."

I don't want to accommodate or treat the disease, I want to kill it permanently, meaning government doesn't borrow - ever.

2. I think there is little doubt that when interest rates are raised from very low levels they cause higher inflation. The three graphs on http://www.philipji.com/item/2016-03-10/does-raising-the-fed-funds-rate-raise-inflation seem to show that clearly.

However, I do not think the causation is through its effect on consumption. I think this can be seen by plotting the Effective Federal Funds Rate vs PCE as can be seen at https://research.stlouisfed.org/fred2/graph/?g=3Zy3

In 2004, for instance, the Fed starts to raise the Funds Rate but PCE has been rising for a long time before.

However, a much clearer cause and effect can be seen between the Fed Funds Rate and Industrial and Commercial Loans. See graph at https://research.stlouisfed.org/fred2/graph/?g=3I8H

The only way to explain this is to invoke financial asset markets. I would lay out the argument as follows:
1. Money is a medium of exchange, so it can be used to buy both financial assets and real goods and services.
2. Very low interest rates favour financial asset markets because by making leverage cheap it permits very high returns on financial assets
3. Raising interest rates harms financial asset markets and thus diverts money into real goods and services through industrial and commercial loans.
4. Hence raising rates from very low levels initially raises inflation (because inflation measures prices of real goods and services).
5. Eventually higher rates causes the collapse of financial asset markets and then causes a fall in consumption expenditures through its effect on consumers' net worth.

3. "In that sense, higher interest rates do lower consumption, but rational expectations reverses the arrow of time: higher future interest rates lower consumption and inflation today. "

This sounds like an argument for forward guidance since it must cut both ways: lower future interest rates raise consumption and inflation today.

4. John,

"People consume less ahead of the time of high real interest rates, so they have more savings, and earn more interest on those savings. Afterwards, they can consume more. Since more consumption pushes up prices, giving more inflation, inflation must also rise during the period of high consumption growth."

You are making several assumptions here that need to be pointed out:

1. The rise in the nominal interest rate has no influence on the demand for credit. If instead the increase in the nominal rate of interest leads to fewer borrowers, then interest rates may rise but interest income from savings may fall.

2. The quantity of goods produced does not rise after the nominal interest rate is increased. If good production instead rises in lockstep with the increase in interest income, then the inflation rate will not move.

3. Debt is held by savers / consumers instead of the central bank. If instead the central bank holds all the debt, then savers / consumers receive no interest income to spend.

5. On a very general level, I find it a bit disturbing that monetary policy seems to have had the incorrect sign for many many decades, which is what this and related posts imply. After central banks having profoundly impacted global economic affairs for a very long time, should we conclude that raising rates in fact raises inflation and not lowers it? Realizing that we have been conducting policy with the incorrect sign can only happen in pseudo sciences like economics. Deeply disturbing as monetary policy impacts the well being of every human on the planet. Possibly, monetary policy is to important to be left to economists.

6. "...higher future interest rates lower consumption and inflation today"

Why do we not see this in the chart? (e.g. an initial dip followed by an increase)

And, wouldn't the reverse of that be "lower the interest rates, higher the consumption today"? I thought this is what the FED was trying to do.

I guess the disconnect comes from what's happening at t vs t+1, how far apart are t and t+1.

7. Insightful as always - one clarifying question:

"Higher real interest rates induce you to postpone consumption".

Isn't that true only for net borrowers? So I understand that might be true overall in today's economy, given the growth of credit overall, but I just wanted to clarify whether that's true for those who are net savers (i.e. expect spending power to grow from higher real rates and thus are more willing to spend now).

Cheers

8. Doesn't it matter how the Fed raises rates? If they raise rates by contracting the money supply, and inducing a liquidity effect, then this seems counterintuitive. If instead they raise rates by (credibly) raising their inflation target, then this seems obvious. How do models like this distinguish between these two ways of the Fed raising rates?

9. I don't have an alternative model for inflation, but could you elaborate on how changes in consumption cause changes in inflation?

10. I went through this derivation a while back, but I'll do it again here. Start with the equation for exchange:

M * V = P * Q = Real GDP (RGDP) * (1 + Inflation Rate (INF%) )

We want to bring credit (debt) into this equation and so first thing:

Money Supply (M) = Debt (D) * Bank Loan Retainage (RET%) * (1 - Interest Rate (INT%))

M = D * RET% * ( 1 - INT% )

This is a single bank, single interest rate model. The single (central) bank makes loans at an interest rate (INT%) and holds a percentage of those loans (RET%). They buy and sell those loans at par to hit their desired loan retainage.

And so the total money supply at any time is equal to the amount of debt created times the percentage of debt retained by the central bank minus the interest payments made back to the central bank.

Next we need an expression for income.

Income (INC) = Non-bank Interest Income (INT) + Income from sale of goods (P*Q or M*V)
INC = INT + M * V

The nonbank interest income can be expressed as follows:

INT = D * (1 - RET%) * INT%

And so:

INC = D * (1 - RET%) * INT% + M * V

Now we can rewrite the equation of equation:

M * V = INC * ( 1 - Liquidity Preference (LP) ) + Change in Debt (dD/dt)

With a liquidity preference of 1, no current income is spent on goods and all expenditures are financed with new debt. We assume that anyone that borrows money is going to immediately spend it on goods. This is a reasonable assumption - who borrows money to bury it in a hole?

M * V = INC * ( 1 - LP ) + dD/dt

Substituting our expression for income (INC)

M * V = ( D * (1 - RET%) * INT% + M * V ) * ( 1 - LP ) + dD/dt

Moving all the M * V terms to the left hand side of the equation:

M * V * LP = ( D * (1 - RET%) * INT% ) * ( 1 - LP ) + dD/dt

Substituting back into the equation our expression for the money supply:

D * RET% * ( 1 - INT% ) * V * LP = ( D * (1 - RET%) * INT% ) * ( 1 - LP ) + dD/dt

Now let the total debt be equation to an exponential function:

D = exp ( f(t) )
dD/dt = f'(t) * exp ( f(t) ) = f'(t) * D

This way we can eliminate D from all sides of the equation:

RET% * ( 1 - INT% ) * V * LP = ( (1 - RET%) * INT% ) * ( 1 - LP ) + f'(t)

Now we can solve for the velocity of money (V):

V = [ ( (1 - RET%) * INT% ) * ( 1 - LP ) + f'(t) ] / [ RET% * ( 1 - INT% ) * LP ]

Likewise:

M * V = D * [ ( (1 - RET%) * INT% ) * ( 1 - LP ) + f'(t) ] / LP = RGDP * ( 1 + INF% )

Rearranging terms:

RGDP / D = [ ( (1 - RET%) * INT% ) * ( 1 - LP ) + f'(t) ] / [ LP * ( 1 + INF% ) ]

Now we need to include productivity in our equation.

Productivity (PROD) = Real GDP (RGDP) / Debt (D)

PROD = [ ( (1 - RET%) * INT% ) * ( 1 - LP ) + f'(t) ] / [ LP * ( 1 + INF% ) ]

And now we can solve for the inflation rate (INF%):

INF% = [ ( (1 - RET%) * INT% ) * ( 1 - LP ) + f'(t) ] / [ LP * PROD ] - 1

It should be apparent that a rising nominal interest rate will have a positive effect on the inflation rate with the following assumptions:

1. Productivity ( PROD ) does not change
2. Liquidity preference ( LP ) does not change
3. Central bank loan retainage ( RET% ) does not change
4. Credit demand ( f'(t) ) does not change

1. Found a bit of a mistake. I need to subtract out debt purchases by the public from the income parameter. Basically whatever loans the central bank doesn't retain are purchased from income (INC).

INC = D * (1 - RET%) * INT% + M * V - dD/dt * (1 - RET%)

Here, new debt (dD/dt) purchased from the central bank by the public is subtracted from income to reflect income that can either be kept as a liquid asset or spent on goods. We make the assumption that the central bank has complete control over the amount of debt held by the public and always buys and sells debt at par.

Resolving for the velocity of money (V):

V = [ ( 1 - RET% ) * ( INT% - f'(t) ) * ( 1 - LP ) + f'(t) ] / [ RET% * ( 1 - INT% ) * LP ]

Again solving for the inflation rate (INF%):

INF% = [ ( 1 - RET% ) * ( INT% - f'(t) ) * ( 1 - LP ) + f'(t) ] / [ LP * PROD ] - 1

Let's do some central banking. The central bank has two levers at it's disposal, it can change the nominal interest rate (INT%) and it can change it's loan retainage (RET%). How does it hit an inflation target (for instance 2%)?

2% = [ ( 1 - RET% ) * ( INT% - f'(t) ) * ( 1 - LP ) + f'(t) ] / [ LP * PROD ] - 1

Let's say that liquidity preference ( LP ) is 5%, credit demand ( f'(t) ) is 5%, and productivity (PROD) is 25%.

[ 1.02 * 0.05 * 0.25 - 0.05 ] / 0.95 = ( 1 - RET% ) * ( INT% - 0.05 )

0.01342 = ( 1 - RET% ) * ( INT% - 0.05 )

If the interest rate is less than credit demand then people are spending more money buying each other's debt than they are receiving in interest payments and so to hit a positive inflation rate, the interest rate must be greater than credit demand. Here are a couple of solutions:

0.01342 = ( 1 - RET% ) * ( INT% - 0.05 )
INT% = 5.5%, RET% = -168%
INT% = 6.0%, RET% = -34%
INT% = 6.5%, RET% = 10.5%
INT% = 7.0%, RET% = 32.9%

Notice, that to hit a 2% inflation target, the central bank can select from any number of interest rates. Notice also that with two tools (interest rate and open market operations), the central bank can hit a real interest rate target with some clarifications.

1. The inflation data the central bank uses will be backward looking.
2. The central bank may need to simultaneously raise it's interest rate AND increase open market operations to hit a real interest rate target. This runs counter to experience.

11. The solution will always limit to the static solution. Looking at the first equation, it's very easy to see that the static solution is pi = i. That's why in the limit inflation reacts to the interest rate one-to-one. All the rest, the Phillips curve, the dynamics and all, is just confusing this simple fact. In other words, the conclusion is built into the model from the start. It's as realistic as the initial equation, which means not much.

12. That makes sense.

Certainly, empirically at least, a countries inflation rate is usually within a couple of percent of their interest rate. In addition, there is no empirical evidence that moving to NIRP creates inflation. Even in Japan, their sales tax rise had a much larger effect on inflation than cutting short rates.

Perhaps also inflation rates may in part follow interest rates because of discounting from future goods and services. For example, if I want to buy a durable good either now, or in a year's time, I have the choice of buying it today and foregoing interest at a rate of r or buying it in a year's time having inflated by i. My suggestion is that i and r need to be similar to prevent excessive bringing forward or pushing back of consumption.

13. "
People consume less ahead of the time of high real interest rates, so they have more savings, and earn more interest on those savings. Afterwards, they can consume more. Since more consumption pushes up prices, giving more inflation, inflation must also rise during the period of high consumption growth. "

This intuition doesn't match the graph. If the future inflation is caused by future increases in consumption due to higher savings then the graph has to have a decline in the inflation rate once that savings starts due to lower consumption now.

14. Is it possible to have a version of this model but with micro foundations? That is, with agents who have endowments, maximize utility etc, and with well specified institutional arrangements (so we know who trades with whom, how, etc)?

1. Yes. See my "money as stock" on my webpage for an example. It's really easy.

15. Julio Garin et al. flesh out the ideas in this post a little bit more in their short working paper: http://juliogarin.com/files/Raise_Rates_to_Raise_Inflation_NF.pdf. As you said, the key ingredient is the forward-looking nature of the IS curve. Interesting read and something you can certainly point your colleague to.

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