# Category Archives: Macroeconomic Theory

## Understanding John Taylor

There has been a great deal of debate regarding Taylor rules recently. The U.S. House of Representatives recently proposed a bill that would require the Federal Reserve to articulate their policy in the form of a rule, such as the Taylor Rule. This bill created some debate about whether or not the Federal Reserve should adopt the Taylor Rule or not. In reality, the bill did not require the Federal Reserve to adopt the Taylor Rule, but rather used the Taylor Rule as an example.

In addition, John Taylor has been advocating the Taylor Rule as a guide to policy recently as well as attributing the recent financial crisis/recession to the deviations from the Taylor Rule. While it should not surprise anyone that Taylor has been advocating a rule of his own design and which bears his name, he has faced criticism regarding his recent advocacy of the rule and his views on the financial crisis.

Those who know me know that I am no advocate of Taylor Rules or the Taylor Rule interpretation of monetary policy (see here, here, and here). Nonetheless, a number of people have simply dismissed Taylor’s arguments because they think that he is either (a) deliberately misleading the public for ideological reasons, or (b) mistaken about the literature on monetary policy. Neither of these views is charitable to Taylor since they imply that he is either being deliberately obtuse or does not understand the very literature that he is citing. I myself am similarly puzzled by some of Taylor’s comments. Nonetheless, it seems to me that an attempt to better understand Taylor’s position can not only help us to understand Taylor himself, but it might also clarify some of the underlying issues regarding monetary policy. In other words, rather than simply accept the easy (uncharitable) view of Taylor, let’s see if there is something to learn from Taylor’s position. (I am not going to link the dismissive views of Taylor. However, I will address some of the substantive criticism raised by Tony Yates later in the post.)

Let’s begin with Taylor’s position. This is a lengthy quote from Taylor’s blog, but I think that this a very explicit outline of Taylor’s ideas regarding monetary policy history:

Let me begin with a mini history of monetary policy in the United States during the past 50 years. When I first started doing monetary economics in the late 1960s and 1970s, monetary policy was highly discretionary and interventionist. It went from boom to bust and back again, repeatedly falling behind the curve, and then over-reacting. The Fed had lofty goals but no consistent strategy. If you measure macroeconomic performance as I do by both price stability and output stability, the results were terrible. Unemployment and inflation both rose.

Then in the early 1980s policy changed. It became more focused, more systematic, more rules-based, and it stayed that way through the 1990s and into the start of this century. Using the same performance measures, the results were excellent. Inflation and unemployment both came down. We got the Great Moderation, or the NICE period (non-inflationary consistently expansionary) as Mervyn King put it. Researchers like John Judd and Glenn Rudebush at the San Francisco Fed and Richard Clarida, Mark Gertler and Jordi Gali showed that this improved performance was closely associated with more rules-based policy, which they defined as systematic changes in the instrument of policy — the federal funds rate — in response to developments in the economy.

[…]

But then there was a setback. The Fed decided to hold the interest rate very low during 2003-2005, thereby deviating from the rules-based policy that worked well during the Great Moderation. You do not need policy rules to see the change: With the inflation rate around 2%, the federal funds rate was only 1% in 2003, compared with 5.5% in 1997 when the inflation rate was also about 2%. The results were not good. In my view this policy change brought on a search for yield, excesses in the housing market, and, along with a regulatory process which broke rules for safety and soundness, was a key factor in the financial crisis and the Great Recession.

[…]

This deviation from rules-based monetary policy went beyond the United States, as first pointed out by researchers at the OECD, and is now obvious to any observer. Central banks followed each other down through extra low interest rates in 2003-2005 and more recently through quantitative easing. QE in the US was followed by QE in Japan and by QE in the Eurozone with exchange rates moving as expected in each case. Researchers at the BIS showed the deviation went beyond OECD and called it the Global Great Deviation. Rich Clarida commented that “QE begets QE!” Complaints about spillover and pleas for coordination grew. NICE ended in both senses of the word. World monetary policy now seems to have moved into a strategy-free zone.

This short history demonstrates that shifts toward and away from steady predictable monetary policy have made a great deal of difference for the performance of the economy, just as basic macroeconomic theory tells us. This history has now been corroborated by David Papell and his colleagues using modern statistical methods. Allan Meltzer found nearly the same thing in his more detailed monetary history of the Fed.

My reading of this suggests that there are two important points that we can learn about Taylor’s view. First, Taylor’s view of the Great Moderation is actually quite different than the New Keynesian consensus — even though he seems to think that they are quite similar. The typical New Keynesian story about the Great Moderation is that prior to 1979, the Federal Reserve failed to follow the Taylor principle (i.e. raise the nominal interest rate more than one-for-one with an increase in inflation, or in other words, raise the real interest rate when inflation rises). In contrast, Taylor’s view seems to be that the Federal Reserve became more rule-based. However, a Taylor rule with different parameters than Taylor’s original rule can still be consistent with rule-based policy. So what Taylor seems to mean is that if we look at the federal funds rate before and after 1979, it seems to be consistent with his proposed Taylor Rule in the latter period, but there are significant deviations from that rule in the former period.

This brings me to the second point. Taylor’s view about the importance of the Taylor Rule is one based on empirical observation. What this means is that his view is quite different from those working in the New Keynesian wing of the optimal monetary policy literature. To see how Taylor’s view is different from the New Keynesian literature, we need to consider two things that Taylor published in 1993.

The first source that we need to consult is Taylor’s book, Macroeconomic Policy in a World Economy. In that book Taylor presents a rational expectations model and in the latter chapters uses the model to compare monetary policy rules that look at inflation, real output, and nominal income. He finds that the preferred monetary policy rule in the countries that he considers is akin to what we would now call a Taylor Rule. In other words, the policy that reduces the variance of output and inflation is a rule that responds to both inflation and the output gap.

However, the canonical Taylor Rule and the one that John Taylor now advocates does not actually appear in the book (the results presented in the book suggest different coefficients on inflation and output). The canonical Taylor Rule in which the coefficient on inflation is equal to 1.5 and the coefficient on the output gap is equal to 0.5 appears in Taylor’s paper “Discretion versus policy rules in practice”:

Thus, as we can see in the excerpt from Taylor’s paper, the reason that he finds this particular policy rule desirable is that it seems to describe monetary policy during a time in which policymakers seemed to be doing well.

However, Taylor is also quick to point out that the Federal Reserve needn’t adopt this rule, but rather that the rule should be one of the indicators that the Federal Reserve looks at when conducting policy:

Indeed, Taylor’s views on monetary policy do not seem to have changed much from his 1993 paper. He still advocates using the Taylor Rule as a guide to monetary policy rather than as a formula required for monetary policy.

However, what is most important is the following distinction between Taylor’s 1993 book and Taylor’s 1993 paper. In his book, Taylor shows using evidence from simulations that a feedback rule for monetary policy in which the central bank responds to inflation and the output gap (rather than inflation itself or nominal income) is the preferable policy among the three alternatives he considers. In contrast, in his 1993 paper, we begin to see that Taylor views the version of the rule in which the coefficient on inflation is 1.5 and the coefficient on the output gap is 0.5 as a useful benchmark for policy because it seems to describe policy well between the period 1987 – 1992 — a period that Taylor would classify as good policy. In other words, Taylor’s advocacy of the conventional 1.5/0.5 Taylor Rule seems to be informed by the empirical observation that when policy is good, it also tends to coincide with this rule.

This is also evident in Taylor’s 1999 paper entitled, “A Historical Analysis of Monetary Policy Rules.” In this paper, Taylor does two things. First, he estimates reaction functions for the Federal Reserve to determine the effect of inflation and the output gap on the federal funds rate. In doing so, he shows that the Greenspan era seems to have produced a policy consistent with the conventional 1.5/0.5 version of the Taylor Rule whereas for the pre-1979 period, this was not the case. Again, this provides Taylor with some evidence that when Federal Reserve policy is approximately consistent with the conventional Taylor Rule, the corresponding macroeconomic outcomes seem to be better.

This is best illustrated by the second thing that Taylor does in the paper. In the last section of the paper, Taylor plots the path of the federal funds rate if monetary policy had followed a Taylor rule and the actual federal funds rate for the same two eras described above. What the plots of the data show is that during the 1970s, when inflation was high and when nobody would really consider macroeconomic outcomes desirable, the Federal Reserve systematically set the federal funds rate below where they would have set it had they been following the Taylor Rule. In contrast, when Taylor plots the federal funds rate implied by the conventional Taylor Rule and the actual federal funds rate for the Greenspan era (in which inflation was low and the variance of the output gap was low), he finds that policy is very consistent with the Taylor Rule.

He argues on the basis of this empirical observation that the deviations from the Taylor Rule in the earlier period represent “policy mistakes”:

…if one defines policy mistakes as deviations from such a good policy rule, then such mistakes have been associated with either high and prolonged inflation or drawn-out periods of low capacity utilization, much as simple monetary theory would predict. (Taylor, 1999: 340).

Thus, when we think about John Taylor’s position, we should recognize that Taylor’s position on monetary policy and the Taylor Rule is driven much more by empirical evidence than it is by model simulations. He sees periods of good policy as largely consistent with the conventional Taylor Rule and periods of bad policy as inconsistent with the conventional Taylor Rule. This reinforces his view that the Taylor Rule is a good indicator about the stance of monetary policy.

Taylor’s advocacy of the Taylor Rule as a guide for monetary policy is very different from the related New Keynesian literature on optimal monetary policy. That literature, beginning with Rotemberg and Woodford (1999) — incidentally writing in the same volume as Taylor’s 1999 paper, which was edited by Taylor — derives welfare criteria using the utility function of the representative agent in the New Keynesian model. In the context of these models, it is straightforward to show that the optimal monetary policy is one that minimizes the weighted sum of the variance of inflation and the variance of the output gap.

I bring this up because this literature reached different conclusions regarding the coefficients in the Taylor Rule. For example, as Tony Yates explains:

…if you take a modern macro model and work out what is the optimal Taylor Rule – tune the coefficients so that they maximise social welfare, properly defined in model terms, you will get very large coefficients on the term in inflation. Perhaps an order of magnitude greater than JT’s. This same result is manifest in ‘pure’ optimal policies, where we don’t try to calculate the best Taylor Rule, but we calculate the best interest rate scheme in general. In such a model, interest rates are ludicrously volatile. This lead to the common practice of including terms in interest rate volatility in the criterion function that we used to judge policy. Doing that dials down interest rate volatility. Or, in the exercise where we try to find the best Taylor Rule, it dials down the inflation coefficient to something reasonable. This pointed to a huge disconnect between what the models were suggesting should happen, and what central banks were actually doing to tame inflation [and what John Taylor was saying they should do]. JT points out that most agree that the response to inflation should be greater than one for one. But should it be less than 20? Without an entirely arbitary term penalising interest rate volatility, it’s possible to get that answer.

I suspect that if one brought up this point to Taylor, he would suggest that these fine-tuned coefficients are unreasonable. As evidence in favor of his position, he would cite the empirical observations discussed above. Thus, there is a disconnect between what the Taylor Rule literature has to say about Taylor Rules and what John Taylor has to say about Taylor Rules. I suspect the difference is that the literature is primarily based on considering optimal monetary policy in terms of a theoretical model whereas John Taylor’s advocacy of the Taylor Rule is based on his own empirical observations.

Nonetheless, as Tony pointed out to me in conversation, if that is indeed the position that Taylor would take, then quotes like this from Taylor’s recent WSJ op-ed are misleading, “The summary is accurate except for the suggestion that I put the rule forth simply as a description of past policy when in fact the rule emerged from years of research on optimal monetary policy.” I think that what Taylor is really saying is that Taylor Rules, defined generally as rules in which the central bank adjusts the interest rate to changes in inflation and the output gap, are consistent with optimal policy rather than arguing that his exact Taylor Rule is the optimal policy in these models. Nonetheless, I agree with Tony that this statement is misleading regardless of what Taylor meant when he wrote it.

But suppose that we give Taylor the benefit of doubt and suggest that this statement was unintentionally misleading. There is still this bit about the financial crisis to discuss and it is on this subject that there are questions that need to be asked of Taylor.

In Taylor’s book Getting Off Track, he argues that deviations from the Taylor Rule caused the financial crisis. To demonstrate this, he first shows that from 2003 – 2006, the federal funds rates was approximately 2 percentage points below the rate implied by the conventional Taylor Rule. He then provides empirical evidence regarding the effects of the deviations from the Taylor Rule on housing starts. He constructs a counterfactual to suggest that if the Federal Reserve had followed the Taylor Rule, then then housing starts would have been between 200,000 – 400,000 units lower each year between 2003 and 2006 than what we actually observed. He also shows that the deviations from the Taylor Rule in Europe can explain changes in housing investment in for a sample that includes Germany, Austria, Italy, the Netherlands, Belgium, Finland, France, Spain, Greece, and Ireland.

Taylor therefore argues that by keeping interest rates too low for too long, the Federal Reserve (and the ECB by following suit with low interest rates) created the housing boom that ultimately went bust and led to a financial crisis.

In a separate post, Tony Yates responds to this hypothesis by making the following points:

2. John’s rule was shown to deliver pretty good results in variations on a narrow class of DSGE models. The crisis has cast much doubt on whether this class is wide enough to embrace the truth. In particular, it typically left out the financial sector. Modifications of the rule such that central bank rates respond to spreads can be shown to deliver good results in prototype financial-inclusive DSGE models. But these models are just a beginning, and certainly not the last word, on how to describe the financial sector. In models in which the Taylor Rule was shown to be good, smallish deviations from it don’t cause financial crises, therefore, because almost none of these models articulate anything that causes a financial crisis. How can you put a financial crisis in real life down to departures from a rule whose benefits were derived in a model that had no finance? There is a story to be told. But it requires much alteration of the original model. Perhaps nominal illusion; misapprehension of risk, learning, and runs. And who knows what the best monetary policy would be in that model.

3. In the models in which the TR is shown to be good, the effects of monetary policy are small and relatively short-lived. To most in the macro profession, the financial crisis looks like a real phenomenon, building up over 2-2.5 decades, accompanying relative nominal stability. Such phenomena don’t have monetary causes, at least not seen through the spectacles of models in which the TR does well. Conversely, if monetary policy is deduced to have two decade long impulses, then we must revise our view about the efficacy of the Taylor Rule.

Thus, we are back to the literature on optimal monetary policy. Again, I suspect that if one raised these points to John Taylor, he might argue that (i) his empirical evidence on the financial crisis trumps the optimal policy literature (which admittedly has issues — like the lack of a financial sector in my of these models), (ii) his empirical analysis suggests that a Taylor Rule might be optimal in a properly modified model, or (iii) regardless of whether the conventional Taylor Rule is optimal, the deviation from this type of policy is harmful as evident by the empirical evidence.

Nonetheless, this brings me to my own questions about/criticisms of Taylor’s approach:

1. Suppose that Taylor believes that point (i) is true. If this is the case, then citing the optimal monetary policy literature as supportive of the Taylor Rule in the WSJ is not simply innocently misleading the readers, it is deliberately misleading the readers by choosing to only cite this literature when it fits with his view. One should not selectively cite literature when it is favorable to one’s view and then not cite the same literature when it is no longer favorable.

2. As Tony Yates points out, point (ii) is impossible to answer.

3. Regarding point (iii), the question is whether or not empirical evidence is sufficient to establish the Taylor Rule as a desirable policy. For example, as the work of Athanasios Orphanides demonstrates, the conclusions about whether the Federal Reserve following the Taylor principle (i.e. having a coefficient on inflation greater than 1) in the pre- and post-Volcker era are dependent upon the data that one uses in their analysis. When one uses the data that the Federal Reserve had in real-time, the problems associated with policy have more to do with the responsiveness of the Fed to the output gap than they do with the rate of inflation. In other words, the Federal Reserve does not typically do a good job forecasting the output gap in real-time. This is a critical flaw in the Taylor Rule because it implies that even if the Taylor Rule is optimal, the central bank might not be able to set policy consistent with the rule.

In other words, if the deviations from the Taylor Rule have such a large effect on economic outcomes and it is very difficult for the central bank to maintain a policy consistent with the Taylor Rule, then perhaps this isn’t a desirable policy after all.

4. One has to stake out a position regarding where they stand on models and the data. Taylor’s initial advocacy of this type of rule seems to be driven by the model simulations that he has done. However, his more recent advocacy of this type of rule seems to be driven by the empirical evidence in his 1993 and 1999 papers and his book, Getting Off Track. But the empirical evidence should be consistent with the model simulations and it is not clear that this is true. In other words, one should not make statements about the empirical importance of a rule when the outcome from deviating from that rule is not even a feature of the model that was used to do the simulations.

5. In addition, the Taylor Rule lacks the intuition of, say, a money growth rule. With a money growth rule, the analysis is simply based on quantity theoretic arguments. If one targets a particular rate of growth in the monetary aggregate (assuming that velocity is stable), we have a good idea about what nominal income growth (or inflation) will be. In addition, the quantity theory is well known (if not always well understood) and can be shown to be consistent with a large number of models (even models with flexible prices). This sort of rule for policy is intuitive. If you know that in the long run money growth causes inflation then the way to prevent inflation is to limit money growth.

It is not so clear what the intuition is behind the Taylor Rule. It says that we need to tighten policy when inflation rises and/or when real GDP is above potential. That part is fairly intuitive. But what are the “correct” parameters? And why is Taylor’s preferred parameterization such a good rule? Is it solely based on his empirical work because the optimal monetary policy literature suggests alternatives?

6. Why did things change between the 1970s and the early 2000s. In his 1999 paper, Taylor argues that the Federal Reserve kept interest rates too low for too long and we ended up with stagflation. In his book Getting Off Track, he implies that when the Federal Reserve kept interest rates too low for too long we ended up with a housing boom and bust. But why wasn’t there inflation/stagflation? Why was there such a different response to having interest rates too low in the early 2000s as opposed to the 1970s? These are questions that empirics alone cannot answer.

In any event, I hope that this post brings some clarity to the debate.

## Interest Rates and Investment

The conventional way of discussing monetary policy is by referencing the interest rate target of the central bank. This is also the way that monetary policy is communicated in the basic New Keynesian model. The idea is that the transmission of monetary policy is primarily through the interest rate. I would like to argue in this post that this is a problematic way of thinking about monetary policy and that the transmission mechanism of policy is unclear.

In the New Keynesian model, the real interest rate affects the time path of consumption through the consumption Euler equation. In particular, when the real interest rate falls, the household would want to save less and therefore would want to consume more. This increases real economic activity in the current period. If we add capital to the model, a lower interest rate encourages a greater investment in capital. Thus, if monetary policy can affect the real interest rate in the short run, then the interest rate target of the central bank can be used as a stabilization tool.

This investment mechanism, however, is questionable. It ignores how investment is actually done in the real world. We can illustrate this lesson with a simple example.

Suppose that there is a firm. The firm produces a product and is deciding whether to build a new factory to increase its production. Let V(t) denote the value of the factory at time t. The initial value of the project is $V(0) = V_0$. Now suppose that the value to the firm of building the factory is growing over time:

${{\dot{V}}\over{V}} = a$

It follows that the value of the factory at some arbitrary date in the future, say time T, is

$e^{aT} V_0$

Now suppose that the cost to build the factory is some fixed cost, $F$. The firm’s objective is to choose the optimal point in time to build the factory so as to maximize the expected discounted net value of the project:

$\max\limits_{T} e^{-rT} [e^{aT}V_0 - F]$

where $r > a$ is the real interest rate. The maximization problem implies that

$T^* = max\bigg[{{1}\over{a}} ln\bigg({{rI}\over{(r-a)V_0}}\bigg),0\bigg]$

Assuming that $T^* > 0$ (i.e. the optimal time to invest is not immediately), it is straightforward to see that when the real interest rate declines, it is beneficial to put off the investment further into the future.

We can understand the intuition behind this result as follows. In a standard model with capital, the marginal product of capital (net of some adjustment cost) is equal to the real interest rate. Thus, when the real interest rate falls, the firm wants to increase its investment in capital, but because it is costly to adjust that capital, it takes time for the capital stock to reach the firm’s desired level. In contrast, the framework presented above suggests that investment is an option and the firm has to decide when to exercise that option. In that case, a lower the real interest rate means that the future is more important (all else equal). But if the future is more important, then that increases the opportunity cost of exercising the option today. So the firm would want to wait to exercise the option.

So which way is best to think about interest rates and investment? The empirical evidence on the issue (albeit somewhat dated) seems to suggest that price variables, like the real interest rate, are not particularly useful in explaining investment (at least compared to other variables). So is this really the mechanism that should be emphasized in the conduct of monetary policy?

[I should note that this insight is (at least I thought) well known. This example is precisely the example provided by Dixit and Pindyck (1994). Countless other examples can be found in Stokey (2008).]

## Interest on Reserves and the Federal Funds Rate

The payment of interest on reserves is supposed to put a floor beneath the federal funds rate. Since banks can lend to one another overnight at the federal funds rate, they have a choice. The bank can either lend excess reserves to another bank at the federal funds rate or they can hold the reserves at the Federal Reserve and collect the interest the Fed pays on reserves. In theory, this means that the federal funds rate should never go below the interest rate on reserves. The reason is simple. No bank should have the incentive to lend at a lower rate than they would receive by not lending.

However, the effective federal funds rate has been consistently below the interest rate on reserves. How can this be so? Marvin Goodfriend explains:

The interest on reserves floor for the federal funds rate failed, and continues to fail to this day, because non-depository institutions (such as government-sponsored enterprises (GSEs) Fannie Mae and Freddie Mac, and Federal Home Loan Banks (FHLBs)) are authorized to hold overnight balances at the Fed, but are not eligible to receive interest on those balances. Hence, the GSEs and FHLSs [sp] have an incentive to try to earn interest on their overnight balances at the Fed by lending them to depositories eligible to receive interest on their reserve balances. The federal funds rate is thereby driven below interest on reserves to the point that depositories are willing to borrow from the GSEs and the FHLBs, deposit the proceeds at the Fed, and earn the spread between interest on reserves and the federal funds rate.

More here.

## What Does It Mean for the Natural Rate of Interest to Be Negative?

Talk of the zero lower bound has permeated the debate about monetary policy in recent years. In particular, there is one consistent story across a variety of different thinkers involving the difference between the natural rate of interest and the market rate of interest. Specifically, the argument holds that if the market rate of interest is higher than the natural rate of interest then monetary policy is too tight. With regards to the current state of the world, this is potentially problematic is the market rate of interest is zero, but needs to be lower.

I find this way of thinking about monetary policy to be quite odd for several reasons. First, conceivably when one talks about the natural rate of interest, the reference is to a real interest rate. New Keynesians, for example, clearly see the natural rate of interest as a real rate of interest (at least in their models). Second, the market rate of interest is a nominal rate. Thus, it is odd to say that the market rate of interest is above the natural rate of interest when one is nominal and one is real. I suppose that what they mean is that given the nominal interest rate and given the expectations of inflation, the implied real market rate is too high. But this seems to be an odd way to describe what is going on.

Regardless of this confusion, what advocates of this approach appear to be saying is this: when the market rate of interest is at the zero lower bound and the natural rate of interest is negative, unless inflation expectations rise, there is no way to equate the real market rate of interest with the natural rate.

But this brings me to the most important question that I have about this entire argument: Why is the natural rate of interest negative?

It is easy to imagine a real market interest rate being negative. If inflation expectations are positive and policymakers drive a nominal interest rate low enough, then the implied real interest rate is negative. It is NOT, however, easy to imagine the natural rate of interest being negative.

To simplify matters, let’s consider a world with zero inflation. The central bank uses an interest rate rule to set monetary policy. The nominal market rate is therefore equal to the real market interest rate. Thus, assuming that the central bank is pursuing a policy to maintain zero inflation, they are effectively setting the real rate of interest. Thus, the optimal policy is to set the interest rate equal to the natural interest rate. Also, since everyone knows the central bank will never create inflation, this makes the zero lower bound impenetrable (i.e. you cannot even use inflation expectations to lower the real rate when the nominal rate hits zero). I have therefore created a world in which a central bank is incapable of setting the market rate of interest equal to the natural rate of interest if the natural rate is negative. My question is, why in the world would we ever reach this point?

So let’s consider the determination of the natural rate of interest. I will define the natural rate of interest as the real rate of interest that would result with perfect markets, perfect information, and perfectly flexible prices (the New Keynesian would be proud, I think). To determine the equilibrium real interest rate, we need to understand saving behavior and we need to understand investment behavior. The equilibrium interest rate is then determined by the market in our perfect benchmark world. So let’s set up a really simple model of saving and investment.

Time is continuous and infinite. A representative household receives an endowment of income, y, and can either consume the income or save it. If they save it, they earn a real interest rate, r. The household generates utility via consumption. The household utility function is given as

$\int_0^{\infty} e^{-\rho t} u[c(t)] dt$

where $\rho$ is the rate of time preference and $c$ is consumption. The household’s asset holdings evolve according to:

$\dot{a} = y - c + ra$

where $a$ are the asset holdings of the individual. In a steady state equilibrium, it is straightforward to show that

$r = \rho$

The real interest rate is equal to the rate of time preference.

Now let’s consider the firm. Firms face an investment decision. Let’s suppose for simplicity that the firm produces bacon. We can then think of the firm as facing a duration problem. They purchase a pig at birth and they raise the pig. The firm then has to decide how long to wait until they slaughter the pig to make the bacon. Suppose that the duration of investment is given as $\theta$. The production of bacon is given by the production function:

$b = f(\theta)$

where f’,-f”>0 and b is the quantity of bacon produced. The purchase of the pig requires some initial outlay of investment, $i$, which is assumed to be exogenously fixed in real terms and then it just grows until it is slaughtered. The value of the pig over the duration of the investment is given as

$p = \int_{-\theta}^0 e^{-rt} i dt$

Integration of this expression yields

${{p}\over{i}} = {{1}\over{r}}(e^{r\theta} - 1)$

Let’s normalize the amount of investment done to 1. Thus, we can write the firm’s profit equation as

$\textrm{Profit} = f(\theta) - e^{r\theta}$

The firm’s profit-maximizing decision is therefore given as

$f'(\theta) = re^{r\theta}$

Given that the firm makes zero economic profits, it is straightforward to show that

$r = {{f'(\theta)}\over{f(\theta)}}$

So let’s summarize what we have. We have an inverse supply of saving curve that is given as

$r = \rho$

Thus, the saving curve is a horizontal line at the rate of time preference.

The inverse investment demand curve is given as

$r = {{f'(\theta)}\over{f(\theta)}}$

The intersection of these two curves determine the equilibrium real interest rate and the equilibrium duration of investment. Since the supply curve is horizontal, the real interest rate is always equal to the rate of time preference. So this brings me back to my question: How can we explain why the natural rate of interest would be negative?

You might look at the equilibrium conditions and think “sure the natural rate of interest can be negative, we just have to assume that the rate of time preference is negative.” While, this might mathematically be true, it would seem to imply that people value the future more than the present. Does anybody believe that to be true? Are we really to believe that the the zero lower bound is a problem because the general public’s preferences change such that they suddenly value the future more than the present?

But suppose you are willing to believe this. Suppose you think it is perfectly reasonable to assume that people woke up sometime during the recession and their rate of time preference was negative. There are two sides to the market. So what would happen to the duration of investment if the real interest rate was negative? From our inverse investment demand curve, we see that the real interest rate is equal to the ratio of the marginal product of duration over total production. We have made the standard assumption that the marginal product is positive, so this would seem to rule out any equilibrium in which the real interest rate was negative. But suppose at a sufficiently long duration, the marginal product is negative. We could always write down a production function with this characteristic, but how generalizable would this production function be? And why would a firm choose this actually duration when they could have chosen a shorter duration and had the same level of production?

Thus, the only way that one can believe that the natural rate of interest is negative is if they believe that individuals suddenly value the future more than the present and that in a perfect, frictionless world firms would prefer to undertake dynamically inefficient investment projects. And not only that, advocates of this viewpoint also think that the problem with policy is that we cannot use our policy tools to get us to a point consistent with these conditions!

Finally, you might argue that I have simply cherry-picked a model that fits my conclusion. But the model I have presented here is just Hirshleifer’s attempt to model the theories of Bohm-Bawerk and Wicksell, the economists that came up with the idea of the natural rate of interest. So this would seem to be a good starting point for analysis.

P.S. If you are interested in evaluating monetary policy within a framework like this, you should check out one of my working papers, written with Alex Salter.

## More on Germany

Tony Yates has a new post speculating on Germany’s attitude toward fiscal policy. Tony’s assumption is that the German’s opposition to a deal with Greece is rooted in their beliefs about the macroeconomy. In particular, the New Keynesian consensus never really took hold in Germany and therefore they don’t tend to look favorably on stabilization policies. Some might view Tony’s post as a condemnation of Germany, but I don’t view it as such. If you’re a New Keynesian, you probably think this is a bad thing. If you’re not a New Keynesian, or if you’re otherwise opposed to stabilization policy, you probably think this is a good thing. But regardless of whether it is good or bad, this is a possible explanation. Nonetheless, I don’t find this explanation all that convincing.

As I wrote in my previous post on the topic, I think that macroeconomists tend to look at the negotiations between Germany (and the EU more broadly) and Greece all wrong. Naturally, as macroeconomists, we tend to think about these negotiations in terms of the “big picture”. In other words, when macroeconomists look at the negotiations, they often think about the macroeconomic effects of a financial market collapse in Greece and the possible damage that results from various interlinkages. Those who see these costs as being very large then tend to advocate the importance of coming to an agreement. In addition, those who advocate stabilization policies think that such an agreement should also allow Greece to defer the costs until they have a chance to improve their economy through (you guessed it) stabilization policies.

I think that this is the wrong way to think about the negotiations. The negotiations have to be viewed in the context of game theory. The Germans and the Greeks are playing a dynamic game. Thus, Germany has an incentive to make sure that any deal that is reached between the EU and Greece is one that prevents these types of negotiations from happening in the future. In other words, Germany is trying to minimize the costs associated from moral hazard in the future. This isn’t just about Greece, this is about setting a precedent for all future negotiations. When you think about the negotiations in this context, I think that you come up with a better understanding of Germany’s behavior.

But I also think that once you think about the negotiations in terms of game theory, the supposed German opposition to fiscal stabilization doesn’t hold up very well as an explanation of the German response. For example, suppose that the Germans do believe in fiscal stabilization policies. Wouldn’t it then make sense to use austerity as a punishment mechanism for the bailout? If Germany’s true desire is to prevent such negotiations from taking place in the future, then they have an incentive to enact some sort of punishment on Greece in any deal that they reach. Imposing austerity would be one possible punishment. Even if the Germans don’t believe in fiscal stabilization, they know that the Greeks do. As a result, this threat is still a credible way of imposing costs in the game theoretic context because the expected costs to Greece are conditional on Greece’s expectations.

In short, whether or not Germany believes in economic stabilization policy, they are likely to pursue the same strategy within a game theoretic context. Thus, viewing this strategy on the part of the Germans doesn’t necessarily tell us anything about German beliefs.

## Resolving the Glasner-Sumner Dispute

David Glasner and Scott Sumner are arguing about whether saving = investment is an identity or an equilibrium condition. So I thought I would step in and resolve this dispute. Instead of using textbook accounting identities, let’s consider a framework everyone is familiar with — a two-period consumption model.

1. Consider a Robinson Crusoe economy. There is one guy on an island with production opportunities, but no market opportunities. For simplicity, think of a two-period model. In the first period, the individual receives an endowment, Y. The individual can invest that endowment to generate future production or consume the endowment. The individual transforms Y into P1, production now, and P2, production later. It follows that investment is defined as I = Y – P1. Savings is defined as S = Y – C1, where C1 is consumption in the first period. Since there is only one guy on the island, it must be true that P1 = C1. These decisions are both determined by the individual’s rate of time preference. Thus, S = I is an identity.

2. Consider the same guy on an island, but who now has market opportunities. Now we have the same definitions for saving and investment. Saving is

S = Y – C1
I = Y – P1

Note that with exchange opportunities, it is very unlikely that C1 = P1. Thus, at the individual level, savings probably doesn’t equal investment. Combining these conditions, we get

S = I + P1 – C1

for the individual. Now sum across all terms and we get

$\sum S = \sum I + \sum P1 - \sum C1$

Now in equilibrium, market-clearing requires that total production equals total consumption. Thus, market clearing implies that total savings is equal to total investment:

$\sum S = \sum I$

Saving = Investment is therefore an equilibrium condition.

3. Finally, David’s issue is that he doesn’t think that gross domestic income and gross domestic expenditure are the same thing. Empirically, he’s correct. This is why we have GDP Plus.

## Germany, Greece, and Rent-Seeking

Greece is currently seeking a bailout from the European Union. However, negotiations (at least as I write this) are at a standstill. Greece wants a bailout, but the new Greek government has indicated that it is unwilling to enact so-called austerity reforms. During the negotiations, the European Central Bank has given emergency funding to the Greek financial system. However, this funding is conditional on the negotiations between Greece and the EU. The finance ministers of various EU countries want Greece to commit to reducing their debt in line with their 2012 commitments. Since Greece appears unwilling to meet those requirements, the support from the ECB is likely to stop by the end of the month. Thus, Greece could face a significant financial crisis if no deal is reached.

In the midst of these negotiations, some have argued that the EU, and Germany specifically, do not appear to understand the magnitude of the situation. Paul Krugman, for instance, writes

As long as it stays on the euro, then, Greece needs the good will of the central bank, which may, in turn, depend on the attitude of Germany and other creditor nations.

But think about how that plays into debt negotiations. Is Germany really prepared, in effect, to say to a fellow European democracy “Pay up or we’ll destroy your banking system?”

[…]

Doing the right thing would, however, require that other Europeans, Germans in particular, abandon self-serving myths and stop substituting moralizing for analysis.

This last statement is particular telling. Many commentators agree with Krugman and view the Germans and other members of the EU as moralizing. In other words, they are not using economic analysis, but rather relying on their own views about right and wrong and how a government should operate. However, I would submit that rather than assuming that the Germans and other members of the EU are vindictive moralizers, an understanding of economics can actually teach us why EU members have taken their current position. But to understand why, we need to know something about rent-seeking.

Suppose that there are two countries, Germany and Greece. In addition, suppose that each of these countries have an endowment of resources, $R_i$, where $i=1$ will refer to Germany and $i=2$ will refer to Greece. Now let’s assume that each country can devote some amount of resources to production, $P_i$ and some amount of time to fighting with each other, $F_i$. It follows that each country has a resource constraint:

$R_i = P_i + F_i$

Now, let’s assume that the total production between the two countries is given as

$Y = (P_1^{1/s} + P_2^{1/s})^s$

where $s \geq 1$ is a measure of complementarity in production. One way to think about $s$ is that the higher its value, the most closely linked the two countries are in terms of international trade, production, etc.

Thus, we see that the countries can commit their resources to production or to fighting. The more each country contributes to production, the higher the total level of production. However, fighting with one another can also provide benefits (this obviously doesn’t have to refer to actual fighting, it could refer to negotiations like those that are ongoing). However, whereas increased production will cause an increase in the amount of production/income that is generated, fighting will only have an effect on the distribution of income.

Thus, each country faces a trade-off. The more resources they commit to production, the higher the level of income that is generated. The more resources they commit to fighting, the greater the distribution of the existing income they receive (but there is less income as a result). Thus each country has to choose the share of resources that they want to commit to production and fighting.

We will assume that there is a contest success function (as in Tullock, 1980) that is a function of the amount of resources that each country commits to fighting. For Germany, we assume that the contest success function is given as

$\mu_1 = {{F_1^m}\over{F_1^m + F_2^m}}$

where $m$ is an index of the decisiveness of the conflict. Correspondingly, for Greece $\mu_2 = 1 - \mu_1$.

Given these definitions, we can then define the distribution of income:

$Y_1 = \mu_1 Y$
$Y_2 = \mu_2 Y$

Now let’s assume that each country wants to maximize their own income $Y_i$, taking what the other country is doing as given. Thus, each country wants to maximize the following

$\max\limits_{P_i, F_i} {{F_i^m}\over{F_i^m + F_j^m}} [(P_1^{1/s} + P_2^{1/s})^s]$
$\textrm{s.t.} \hspace{2mm} R_i = P_i + F_i$

In equilibrium, it follows that

${{F_2 P_2^{(1-s)/s}}\over{F_1^m}} = {{F_1 P_1^{(1-s)/s}}\over{F_2^m}}$

Now let’s use this equilibrium condition to understand the interaction between Germany and Greece. Suppose that we simply choose resource endowments of $R_1 = 100$ and $R_2 = 50$ thereby assuming that Germany has twice as many resources as Greece. Now let’s consider the implications under two scenarios. First, we will consider the scenario in which $s = 1$. In this case, total production is just the sum of German and Greek production. It follows from equilibrium that

$F_1 = F_2$

Thus, in this case, Germany and Greece devote the same amount of resources to the fighting. However, since Germany has twice as many resources as Greece, it follows that Greece is devoting a larger percentage of their resources to fighting. In devoting resources to fighting, the two countries produce less total production. To see this, consider that if each country devoted all of their resources to production, total production would be 100 + 50 = 150. If we assume that $m = 1$, then $F_1 = F_2 = 37.5$. Thus, total production is 62.5 + 12.5 = 75. And yet this is their optimal choice given what they expect the other country to do!

As a result of the percentage of resources devoted to fighting, Greece gets 1/2 of the resulting production, despite only having 1/3 of the total resources. It should therefore be straightforward to understand why Greece is devoting so many resources to getting a larger bailout without bearing the costs of so-called austerity measures. In addition, it is important to note that it is in Germany’s best interest to devote resources to fighting, given what they expect Greece to do.

It is straightforward to show two other results with this framework. First, as $s$ increases, the weaker side has less of an incentive to fight whereas the stronger side has a greater incentive to fight. In other words, when production becomes more complementary, then the poorer side has less of an incentive to fight because of the linkages in production between their production effort and the other country. While the richer country has an incentive to fight more, the total resources devoted to fighting will fall.

Again, this informs the discussion about Germany and Greece in comparison to other countries. If you want to understand why there seems to be a greater conflict between Germany and Greece than between Germany and other EU countries, consider that Greece is Germany’s 40th largest trading partner, just after Malaysia and just ahead of Slovenia. All else equal the model described above suggests that we should expect more resources devoted to conflict.

The second characteristic has to do with the decisiveness of conflict. As $m$ increases, the conflict between the two countries can be considered more decisive. As the conflict becomes more decisive, the model predicts that the sides will have more of an incentive to devote to fighting. Thus, if Germany believes that this is (or should be) the last round of negotiations with Greece, then we would expect them to devote more resources to fighting.

So what is the point of this exercise?

The entire point of this exercise is to understand that Germany is, in fact, acting in their own economic interest given what they expect Greece to do. If we want to understand the positions of Germany and Greece, we need to understand strategic behavior. Germany’s refusal to simply give in to Greece’s demands and “do what’s best for Europe” ignores a lot of the aspects of what is going on here. Those who think that Germany is being too harsh ignore some of the key aspects of the framework discussed above.

First, one should note that the distribution of income in the framework above is always more equal to the distribution of resources. Thus, it pays for Greece to fight. However, Germany knows that it pays for Greece to fight and therefore Germany’s best strategic decision is to devote resources to fighting as well.

Second, by claiming that Germany is failing to do what is in the “best interests of Europe”, the critics are presuming that they know the social welfare function that needs to be maximized. Perhaps they are correct. Perhaps they are not. But even if these critics are correct, this doesn’t mean that Germany’s behavior is incomprehensible or that it is based on something other than economics. Rather the confusion is on the part of the critics who fail to understand that a Nash equilibrium may or may not be the socially desirable equilibrium. Germany’s rhetoric might be moralizing, but we can understand their behavior through an understanding of economics. And this is true whether you like Germany’s behavior or not.

[This post is an application of Hirshleifer’s Paradox of Power model. See here.]