A New Keynesian toy model

I’ve been keeping a collection of “toy models” on my computer. I do this for two reasons. First, building them is a lot of fun and useful as a kind of intellectual work-out to develop the “model-building” regions of my brain. Second, I think they help clarify my own thinking about economic issues.

I’d like to share one of my favorite toy models with you. I learnt it from Cedric Tille when I was at the IfW Kiel. The purpose of this model is to show the basic intuition behind a strand of literature called “New Keynesian” macroeconomics. The NK approach can be thought of as a combination of the techniques of the “Real Business Cycle”  literature (rational expectations, continuous market clearing, dynamically optimizing agents) with “old” Keynesian economics (monetary policy has real effects, government spending has a multiplier effect, etc.). The model is simple enough to be taught to first-year econ students and at the same time rich enough to provide a basis for discussion of the effects of monetary policy, technology shocks, fiscal policy, the distinction between expected and unexpected shocks and more. It is also much closer to current macroeconomic research than the usual AS-AD model contained in most textbooks. The model has a natural extension to an open economy setting, which is contained in this paper by Corset & Pesenti.

Here goes.

Technology. An economy’s output (Y) is produced by labor (L) alone. The aggregate production function is

Y = A*L,                           (1)

where A is the technology parameter (labor productivity).

Households. Households consume output and supply labor. They trade off the marginal utility from consumption against the marginal disutility of working. Under usual assumptions about the shape of utility functions, consumption will be an increasing function of the real wage. Denoting the nominal wage by W and the price level by P, let household consumption (C) be given by

C = k*(W/P),                 (2)

where k is a positive parameter. The basic intuition behind this consumption function is that a higher real wage induces people to substitute consumption for leisure (substitution effect) and raises their real income (income effect). Both effects act to increase consumption, while the effect on labor supply is ambiguous.*

In order to purchase goods, households must hold money. Money demand (M) is a function of nominal consumption spending:

M = (1/v)*P*C,             (3)

where v is the (exogenous) velocity of money. Note that this is just a versions of the quantity theory of money. The money supply is set by the central bank and exogenous to the model. We will think of M as describing the stance of monetary policy.

Firms. Firms compete in a monopolistic way, i.e. each firm has a monopoly over the specific kind of consumption good it produces, but there is a large number of close substitutes. It can be shown that under this kind of competition, the aggregate price level will be set as a mark-up over marginal costs of production. Nominal marginal costs are equal to W/A — it takes 1/A hours to produce one unit of output and each hour costs W euros.

Crucially, firms must set prices before learning the labor productivity and the monetary policy stance. Hence, they must form expectations about nominal marginal costs. Let z be the mark-up, which indicates the market power of firms (which in turn depends on how “tough’’ competition is in the goods market). Then the price level is given by

P = z*E(W/A),               (4)

where E() denotes the expected value conditional on information available to firms when they set prices.

Closing the model. The model is closed by the goods market clearing condition:

Y = C.                              (5)

This is a model with five endogenous variables (Y, C, L, W, and P) and two exogenous variables (M and A). Let’s find the general equilibrium of this economy. First, combine (2) and (3) to get
W = (v/k)*M.                (6)
Taking expectations and inserting into (4) yields
P = (z*v/k)*E(M/A).               (7)
Next, combine (3), (5) and (6) to get
Y = (k/z)*M/E(M/A).              (8)
Re-inserting this into (1) yields
L = (k/z)*[M/A]/[E(M/A)].   (9)

Equilibrium. Suppose that, in the long run, expected values equal actual values, i.e. E(M/A)=M/A. This is just the rational expectations assumption which in this context means that firms don’t make persistent, systematic mistakes in forming expectations about productivity and monetary policy. With this assumption, (8) reduces to

Yn = (k/z)*A,

which we can call the natural rate of output or full-employment output. It increases in productivity and decreases in the degree of monopolistic distortions. The long-run (“natural”) level of employment is given via (9) by

Ln = k/z.

Using these results in (8) yields

Y/Yn = [M/A]/[E(M/A)].

This equation relates the ratio of actual to natural output (the output gap) to the monetary stance and the state of technology. What exactly does this mean?

  • An unexpected increase in money supply raises output over its natural level. The reason is that an increase in M while P is fixed makes households spend more which raises output and employment.
  • An unexpected increase in labor productivity reduces the output below its natural level. The reason is that a higher A increases potential output, but does nothing to stimulate household spending. Hence output stays the same while labor demand (and therefore employment) falls. So a positive technology shock produces underemployment in the short run.
  • Expected changes in monetary policy or technology have no effect on the output gap. In the long run, money is completely neutral with respect to Y and L.
  • If the central bank has a way of knowing A in advance (for instance, because they employ competent economists who can forecast A perfectly), they could set M in such a way as to completely stabilize the economy at the natural output level. They “simply” have to set M=b*A.

Fiscal policy. How do we get fiscal policy into the model? Easy. Just add government spending into the goods market clearing condition:

Y = C + G                                     (5*)

and assume for simplicity that the government makes spending proportional to total output G=g*Y. (You also must assume that the government finances its expenditure by lump-sum taxes on households only so that firms’ pricing decisions and households’ labor supply are not distorted.) In this case natural output becomes

Yn =(k/z)*A/(1-g),

which increases in g. Government spending doesn’t affect the output gap, though, because it moves actual and potential output by the same amount.

 

*) A utility function which gives rise to such a consumption function is U(C,L) = log(C) — (1/k)*L.

Draghis Nullzinspolitik, Friedmans Regel und die deutsche Presse

Die deutsche Presse ist völlig aus dem Häuschen. Nein, nicht wegen der andauernden Flüchtlingskrise, auch nicht wegen Griechenland, nicht einmal Fußball ist der Grund des Aufruhrs. Der Grund ist die jüngste Entscheidung der Europäischen Zentralbank den Hauptrefinanzierungssatz (vulgo Leitzins) von 0,05% auf 0,00% zu senken.

Na mehr brauchst’ nicht.

Die Süddeutsche Zeitung titelt „Draghi kennt keinen halt mehr“, die Welt legt noch eins drauf: „Mario Draghi raubt der Welt des Geldes das Fundament“, „Ist das Mario Draghis letzte Schlacht?“ fragt Spiegel-Online und die FAZ raunt: „Wie geht es weiter mit dem Euro?“

Nun, wie ich auf diesem Blog schon früher einmal festgestellt habe, macht Geld eben verrückt – sogar die biedere deutsche Wirtschaftspresse. Aber hier scheint mir das Maß der monetären Manie neue Höchststände zu erreichen. Nehmen wir den „Welt“-Artikel her. Dieser wirft Draghi vor Inflationserwartungen zu schüren – in deutschen Augen die schlimmste Sünde für einen Geldpolitiker – und erklärt ohne jeden Anschein von Ironie nur wenige Zeilen davor, dass die Niedrigzinspolitik weitgehend wirkungslos gewesen sei. Er beklagt, dass die EZB den Banken jetzt kostenlos Geld leiht, und wirft ihr gleichzeitig vor, dass sie die „Profitabilität der Geldhäuser massiv unter Druck“ bringe. Die Nullzinspolitik, so die „Welt“, setze alle Regeln des Marktes außer Kraft.

Wie reagiert der gute Ökonom auf solchen Unsinn? Ein guter Anfang ist wie immer bei Milton Friedman zu finden.

Jeder weiß, dass Friedman den Monetarismus begründet hat. Das ist jene Doktrin, der zufolge die Zentralbank für ein möglichst konstantes Wachstum der Geldmenge zu sorgen hat. Wenige wissen, dass Friedmans Monetarismus eine einfache Regel für den optimalen Nominalzins impliziert. Das optimale Zinsniveau beträgt – die Spannung steigt – null.

Das Argument, warum der Nullzins optimal ist, sollte nicht schwer zu verstehen sein. Die Regel für die optimale Bereitstellung von Geld ist dieselbe wie die für die optimale Bereitstellung von Wiener Schnitzeln. Die privaten Grenzkosten des Schnitzelkonsums (also die Menge an anderen Gütern, auf die die einzelne Konsumentin verzichten muss, wenn sie ein zusätzliches Schnitzel isst) muss gleich sein den sozialen Grenzkosten der Schnitzelproduktion (die Menge an anderen Gütern, auf die die Gesellschaft verzichten muss, wenn sie ein zusätzliches Schnitzel produziert). Die optimale Geldmenge ist erreicht, wenn die privaten Kosten der Geldhaltung gleich den sozialen Kosten der Geldproduktion sind. Die privaten Kosten der Geldhaltung sind die nominalen Zinserträge, auf die ich verzichte, wenn ich mein Vermögen in Form von Geld halte anstatt in Anleihen und andere Wertpapiere zu investieren. Die sozialen Kosten der Geldproduktion sind praktisch null. Euroscheine zu drucken kostet fast nichts, digitales Buchgeld zu schaffen genau nichts. Ergo sollte die Zentralbank genau so viel Geld bereitstellen, dass der Nominalzins auf null sinkt.

Aber was ist mit Inflation? Heizt eine Nullzinspolitik nicht die Preissteigerung an? Nein. Die Inflation wird, zumindest langfristig, vom Wachstum der Geldmenge bestimmt und nicht von ihrem Niveau. Eine Nullzinspolitik ist vereinbar mit einer wachsenden, fallenden oder gleichbleibenden Geldmenge und daher mit Inflation, Deflation oder perfekter Preisstabilität. (Friedmans ursprüngliche Analyse verlangt im Optimum eine leichte Deflation.)

Und die Banken? Werden die durch die Nullzinspolitik nicht zu immer riskanteren Investitionen gedrängt? Wieder daneben. Ich bin eine Bank. Investition A garantiert eine Rendite von 4% jährlich. Investition B bringt 10% oder 0% mit gleichen Wahrscheinlichkeiten. Wenn ich, solange der Leitzins bei 1% lag, Investition A gegenüber Investition B bevorzugt habe, warum sollte ich meine Präferenz ändern wenn der Leitzinssatz auf 0% sinkt?

Damit hier kein falscher Eindruck entsteht sollte ich vielleicht darauf hinweisen, dass Friedmans Regel eher wenig mit der jüngsten Zinsentscheidung der EZB zu tun hat. Mario Draghi weiß bestimmt, dass diese Regel, obwohl hilfreich als eine erste Annäherung an gute Geldpolitik, in unserer komplexen Realität nicht ganz optimal ist.

Wenn er an eine Regel denkt, dann wohl eher an die Taylor-Regel, die grob besagt, dass der Nominalzinssatz sich an der Inflation und der „Outputlücke“ (Differenz zwischen tatsächlichem BIP und seinem „natürlichen“, d.h. idealen Niveau) orientieren sollte. Gemäß der Taylor-Regel sollte der Leitzinssatz schon seit geraumer Zeit nicht null, sondern negativ sein. Weil aber der Nominalzins nicht negativ sein kann (der Beweis dieser Aussage ist dem geneigten Leser überlassen!), ist null die nächstbeste Alternative.

Wie dem auch sei, die EZB-Politik der Nullzinsen steht durchaus im Einklang mit der ökonomischen Lehrmeinung. Das heißt selbstverständlich nicht automatisch, dass sie auch richtig ist. Aber die Hysterie, mit der sie in deutschen Medien diskutiert wird, basiert weitgehend auf ökonomischem Analphabetismus.

(Die andere geldpolitische Entscheidung der EZB, den Aufkauf von Staatsanleihen auszuweiten, steht auf wesentlich dünnerem Eis – aber davon ein andermal.)

How is NGDP targeting different from inflation targeting?

This is basically a note to myself, but may be of interest to some of you. I have been thinking lately about nominal GDP targeting. The idea in itself is old, but it has received a lot of attention in the blogosphere post 2009. I want to know under which conditions NGDP targeting produces different results than an inflation targeting regime.

So, to get a first intuition about this, I took the standard ASAD model from the shelf. I transformed everything into growth rates or, if you want to look at it differently, into deviations from the steady state equilibrium. Then I plugged in simple characterizations of the two different policy regimes. And the result is this: A minimalist ASAD analysis of NGDP targeting.

The basic insight is that NGDP targeting requires that any shock to aggregate demand (positive or negative) be completely offset by monetary policy while supply side shocks be neglected. In other words, NGDP targeting fixes the position of the AD curve and doesn’t worry about the position of the AS curve. Inflation targeting, on the other hand, demands that monetary policy takes into account both AD and AS shocks. As my little modelling exercise shows, there is a tight correspondence between the two regimes in the sense that for every given NGDP target there is an inflation target that produces the exact same path of real output and inflation as the NGDP target, and vice versa. I also show that if there were no supply side shocks, only demand shocks, the two regimes are equivalent. The difference between the regimes lies only in how they react to AS shocks: an inflation target tends to amplify the effects of AS shocks on output, the NGDP target doesn’t. Hence NGDP targeting tends to produce smaller fluctuations in real output growth than an inflation target.

So, if you believe macroeconomic shocks are mainly due to aggregate demand, you should be indifferent between the two regimes.

The Bank of England on money creation

Money creation makes people go nuts. Everyone can verify that claim by searching youtube for “money creation” or “money multiplier”, which will turn out a long list of clips telling you the SHOCKING TRUTH about our FRACTIONAL RESERVE BANKING system. Interestingly the money creation craze can be found on both ends of the political spectrum. To socialists, the fact that banks create money out of thin air proves that they are the ultimate force of evil in the world, designed to enslave the working class in a never-ending spiral of debt and compound interest. Libertarians, on the other hand, go “Money out of thin air! Inflation! Theft!”.

Introductory economics textbooks are partly to blame for this confusion. Most of them present an oversimplified model of money creation that goes something like this: there is an initial deposit of $100 with bank A. Bank A keeps 10% (say) as reserves and lends out the remaining $90. This new loan ends up as a deposit with bank B. Bank B keeps 10% of it as reserves and lends out $81, which will end up in bank C, and so on. By the magic of infinite series, the initial deposit of $100 creates new money in the amount of $1000; it gets multiplied by a factor equal to the inverse of the reserve ratio.

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The ECB’s Sterilization Policy And Its Fiscal Effects

First of all, it’s obvious that I have horribly failed at something I always try to do when writing a new post: coming up with a title that hopefully makes people actually want to read it. Yet I still feel this is important, and I’m thankful to Max for insisting on continuing the discussion. In the comments section of my last post, originally meant more as a general monetary policy post, a vivid discussion has emerged on what the ECBs Outright Monetary Transactions Policy (OMT) entails and particularly in what way it would potentially lead to fiscal transfers between Eurozone members, potentially making it illegal under EU treaties. While writing my latest comment, I noticed it was getting way too long, so let me offer a response as a new post.

The way I see it, the main disagreement between Max and me involves the direction any possible fiscal transfers would go if the ECB would, some day, actually buy bonds under the OMT program. We don’t seem to disagree on the fact that any purchases of government bonds by the ECB would potentially prove legally problematic, but rather on what these purchases would entail economically with regards to possible fiscal transfers within the Union. Max argues that, through sterilization, i.e. the ECBs attempts to remove an equal amount of money from the market as it is injecting by buying government bonds of troubles periphery countries, it is substituting low-risk assets on its balance sheet for high-risk assets, making its entire balance sheet more risky and thus representing a real cost to the core, which gets their share of any interest payments accrued from these assets (and thus potentially stands to loose these due to their increased riskiness). However, it would seem that this is based on an inaccurate description of how the ECB conducts (and would conduct) said sterilization. That no OMT purchases have ever actually taken place does not really make the issue harder – for all intents and purposes, OMT is just a replacement for the Securities Markets Program (SMP) instituted by the ECB in 2010 and under which it has already bought around €200 billion worth of bonds of periphery countries (mostly Italy), most of which it still holds on its books. Although there might be some technical differences, conceptually it would seem to me that the main feature of the “change” is that OMT made this program open-ended (thus also reducing the actual need to buy the bonds in the first place). So we know pretty well how OMT as well as sterilization measures would work – so how would they?

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The ECB Isn’t Allowed To Directly Finance EU Governments – Who Said It Needs To?

A large part of the debate on whether or not the ECB can do quantitative easing revolves around the issue that the ECB statutes prohibit the central bank from “financing” any of the member governments directly (or, depending on what German courts say, indirectly as well). To a certain extent this policy makes sense – it avoids a lot of explicit moral hazard and essentially prevents the EU from ever getting stuck in a hyperinflationary situation where governments issue bonds to raise their spending and the central banks just acquiesces and goes on buying these bonds. Also, the Bundesbank has a price stability fetish because of something that happened over 80 years ago but for some reason they can’t seem to learn the correct lessons from. Somewhere else on this blog I have also argued that introducing Eurobonds would provide an instrument for the ECB to actually engage in straight-forward QE, even though just buying a reasonably weighted basket of national bonds would do the same trick (however, with potentially different fiscal implications). But why should buying government bonds be one of the go-to policy to try and gain traction in a liquidity trap in the first place?

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Could Leaning Against Asset Price Bubbles Exacerbate the Business Cycle?

In my last post I argued that using monetary policy to lean against asset price bubbles would only tend to hurt the real economy in the long-run if for some reason the central banks target is constructed in an asymmetric way, i.e. it reacts to asset price deviations to the upside but ignores them to the downside. If its asset price target, however defined, is correctly chosen and designed in a similar way as its inflation and/or output targets in e.g. a classic Taylor rule, then on average there will be as much leaning against asset prices as there will be propping up asset price. But what effects would such a policy have on the business cycle in general? More precisely, would a central bank that targets asset prices tend to have a stabilizing or destabilizing effect on the overall system?

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