Sunday, July 31, 2016

Multiple Equilibria, Installment #2

The goal in this post is to provide some more illumination with respect to Narayana's note, and my previous post. As well, if I could eliminate Nick Rowe's confusion, that would be great.

The problem at hand is one of multiple equilibria. Sometimes multiple equilibrium models are used in an attempt to explain real-world phenomena. That's Roger Farmer's approach - maybe we're stuck in bad, suboptimal states because of self-filfilling low expectations. Sometimes policy rules can lead to multiple equilibria in models we study. That's considered problematic as, to analyze policy in a coherent fashion, we would like to have a unique mapping from policy rules to outcomes, so that the optimal policy problem we're solving is well-specified. That's the problem that comes up in New Keynesian models, but it's certainly not unique to that class of models, as we'll show in the example below.

For people who work in monetary economics, multiple equilibria are ubiquitous. In any model that builds up a role for valued fiat money from first principles, there is always an equilibrium in which money is not valued - if people believe that money will not have value at any date in the future, it will never have value. Fiat money has no intrinsic payoffs, so if people believe that others will not accept it in exchange, they will not accept it either - valued money is supported as an equilibrium because everyone has the self-confirming faith that it will always be valued. So in models of fiat money, there is an equilibrium in which money is not valued, and typically many equilibria in which it is.

One old workhorse of monetary economics is Samuelson's overlapping generations model. The specific example I'm going to use comes from Costas Azariadis's 1981 paper. Time is indexed by t = 1,2,3,..., and at t = 0 there are some old people endowed with M(0) units of money. In each period there are N two-period-lived people who work when they are young and consume when they are old. Each has preferences

(1) U[c(t+1),n(t)] = u[c(t+1)] - v[n(t)],

where c is consumption and n is labor supply. One unit of labor input produces one unit of consumption good. In equilibrium, the young work, purchase money from the old in exchange for goods, and then sell the money for goods when they're old. The government can inject money each period through lump sum transfers to the old. The money stock in period t is M(t). Assume preferences have standard properties: u is strictly concave and v is strictly convex, etc.

In equilibrium, everyone optimizes, and markets clear. There can be plenty of equilibria, including sunspot equilibria and cycles (see Azariadis's paper), but we'll focus on the deterministic ones. In general, we summarize equilibria as sequences {n(t)} that solve the difference equation

(2) [M(t)/M(t+1)]n(t+1)u'[n(t+1)] - n(t)v'[n(t)]=0,


(3) p(t) = n(t)/M(t),

where p(t) is the price of money - the inverse of the price level.

Here's an example. Let M(t)=1 for all time, and assume u has constant relative risk aversion a, with v just n to the power b. Here, a > 0 and b > 1. Then, if we write the difference equation (2) in logs (don't know how to deal with exponents in html), we get

(4) ln[n(t+1)] = [b/(1-a)]ln[n(t)]

So, if a < 1, then (4) looks like this:
And if a > 1, it looks like this:

In either case, there are two steady states: (i) n = 0, where money has no value forever, and nothing gets produced. You can't see that in the second picture, but it's an equilibrium nevertheless. (ii) n = 1. The second steady state is the quantity-theoretic equilibrium. The money growth rate is zero, the inflation rate is zero, the growth rate in output is zero, and the velocity of money is constant forever. But there are also other equilibria, depending on parameters.

First, suppose a < 1. In the first chart, there are many equilibria with 0 < n(0) < 1 which all converge in the limit to n = 0. These are hyperinflationary equilibria for which the inflation rate increases over time without bound. There are also many equilibria with n(0) > 1 for which n(t) grows over time without bound. These are hyperdeflationary equilibria, for which the inflation rate falls over time without bound.

So, those are all the equilibria for that case (I think there are no cyclical or sunspot equilibria either - see Costas's paper). What would Narayana's note say about this? He's interested in the limiting equilibria of finite horizon economies. If we looked for such equilibria here the search is not difficult. Suppose we fix the horizon at length T, where T is finite. Then p(t)=0 for all t. No one would want to hold money in any period, because it has no value in the final period. So, the only finite-horizon equlibrium is n = 0 for any T, so if I take the limit I get n = 0. So, Narayana's claim that a limiting equilibrium of the finite horizon economy is an equilibrium in the infinite horizon economy is correct, but we only found one equilibrium by this approach - the one where money has no value.

We could take a broader view, however. Take the infinite horizon economy, fix p(T), solve the difference equation (4) backward, then let T go to infinity. In this case, the difference equation is stable backward. So, this picks out two equilibria, n = 0 and n = 1. That's an equilibrium selection device which, if we took it seriously, would permit us to ignore all the non-steady-state equilibria that converge to n = 0 in the limit. But that approach shouldn't fill us with confidence. By conventional criteria, in this case n = 0 is "stable" and n = 1 is "unstable."

Next, consider the case 1 < a < 1 + b. In this case, the slope of the difference equation in the second figure is not too steep at n = 1. In addition to the two steady states, there are now many equilibria with n(0) > 0 that converge in the limit to n = 1. Again, literally following Narayana's advice gives one equilibrium, n = 0, but if we following our other limiting approach, the difference equation is unstable backward, and there are three limiting equilibria: (i) n = 0; (ii) n = 1; (iii) a two cycle {...,0,inf,0,inf,0,inf,...}. So that's an example for which Narayana's claim is not correct, as that's not an equilibrium of the infinite horizon economy, since n = 0 is a steady state.

One problem with the model I've specified is that it permits, under some conditions, hyperdeflations in which output grows without bound. A simple fix for that is to put an upper bound on labor supply, keeping preferences as we've specifed them. That will kill off all the hyperdeflationary equilibria, as well as the limiting two-cycle we get by the Narayana method. Then, the Narayana method, taken literally, gives us one equilibrium: n = 0. The Narayana method, taken liberally, gives us two equilibria: n = 0 and n = 1. Note that Narayana's NK model is misspecified in a similar way (see my previous blog post). Given his Phillips curve, he finds equilibria for which i = inf and i = -inf. But in the first such equilibrium, output is rising at an infinite rate, and in the second it is falling at an infinite rate. An upper bound on labor supply would put an upper bound on output, and kill the first equilibrium. As well, in Narayana's model, the Phillips curve is derived by assuming that a fraction of firms charge last period's average price. So, if i = -inf the sticky price firms sell no output, but the flexible price firms have to sell some output. This puts a lower bound on output, which kills off the i = -inf equilibrium. Thus, by the liberal Narayana method, there is only one equilibrium in his NK model - the Fisherian one.

What about the literal Narayana method in his NK model? Here we have a problem. In spite of the fact that this is a cashless model, nominal bonds are traded as claims to money. But in a finite horizon model, the value of money must be zero in the final period, and thus in all periods. So the price of nominal bonds is zero. Thus, we can't even start discussing the usual NK approach, which is assuming that the central bank can set the price of a nominal bond. The central bank is stuck with a price of zero.

Of course, we can wave our hands at this point, and claim that, in a finite horizon monetary model, the price of money is pegged in the last period through fiscal intervention. But that would be a different model, and we might ask why the fiscal authority doesn't do that intervention in every period - then we're done. The central bank should abandon its assigned job and hand it over to the fiscal authority.

Here's something interesting. In line with my previous blog post, there is an optimal monetary policy in this model that kills off indeterminacy. It looks like this:

M(t+1)/M(t) = {n(t+1)u'[n(t+1)]}/{n*v'[n(t)]}

where n* solves u'(n*)=v'(n*). In equilibrium, the money supply is constant, and the policy rule specifies out-of-equilibrium actions that eliminate the indeterminacy.

Question: Does Narayana have a point? Answer: Nah.

Monday, July 18, 2016

More Neo-Fisher

What follows is an attempt to make sense of Narayana's note on Neo-Fisherism. That discussion will lead into comments on a paper by George Evans and Bruce McGough.

Start with basics. What are Neo-Fisherite ideas anyway? Narayana says the absence of shocks, the equilibrium inflation rate should be constant if the nominal interest rate is pegged forever. The Fisher equation then implies that the inflation rate should move one for one with the nominal interest rate. This logic is sometimes referred to as “neo-Fisherian”.
I would actually call these New Keynesian (NK) claims. For example, in "Interest and Prices," Mike Woodford takes pains to address the concern, which came out of the previous macro literature, that nominal interest rate pegs are unstable. Woodford's claim is that a Taylor rule that conforms to the Taylor principle (a greater than one-for-one increase in the nominal interest rate in response to an increase in inflation) will imply determinacy. That is, if there are no shocks, then the nominal interest rate is pegged at a constant forever, and the inflation rate is a constant - the inflation target. Further, in the basic NK model, if Woodford's claim is correct then, in the absence of shocks, if the central bank wants to increase its inflation target, then the nominal interest rate should increase one-for-one with the increase in the inflation target, and actual inflation will respond accordingly. Under basic NK logic, this behavior is supported by promises to increase the nominal interest rate in response to higher inflation - and this inflation never materializes in equilibrium.

But, whatever we think Neo-Fisherite or New Keynesian ideas are, Narayana is making a particular argument in his note, and we want to get to the bottom of it. I don't think the analogy part is particularly helpful though. There are two problems considered in Narayana's note. One is an asset pricing problem, and the other has to do with the properties of a particular NK model. As far as I can tell, the extent of the commonality is that solving each problem can involve geometric series. Otherwise, understanding one problem won't help you much with the other.

The asset pricing problem looks like a trick question you might give to unwitting PhD students on a prelim exam. The equilibrium one-period real interest rate is negative and constant forever, and we're asked to price an asset that pays out a constant real amount each period forever. Question: Solve for the steady state price of the asset. Answer: Dummy, there is no steady state price for the asset. Since a rational economic agent in this world values future payoffs more than current payoffs, if we compute the present value of the payoffs, it will be infinite.

Well, so what? On to the second problem. Narayana uses a version of the standard NK model. We're in a world with certainty - no shocks. I'll change the notation so I don't have to use Greek letters. From standard asset pricing, and assuming constant relative risk aversion utility, we can take logs and get
Here, y is the output gap (the difference between actual output and efficient output), i is the inflation rate, R is the nominal interest rate, and r is the subjective discount rate (or the "natural real interest rate"). The second equation is a Phillips curve
This is the only difference from standard NK, as the Phillips curve doesn't have a term in anticipated inflation. This makes the solution easy, but I don't think it otherwise changes the basic mechanics.

In general, we can solve to get the difference equation
Then, an equilibrium involves finding a sequence of inflation rates that solves the difference equation (3) given some sequence of nominal interest rates, or some policy rule governing the central bank's choice of the nominal interest rate each period.

So, suppose that the nominal interest rate is a constant R forever, and suppose that, in period T the inflation rate is i(T). Then, we can solve the difference equation (3) forward to get
Similarly, we can solve (3) backward to get
So, for any real number i(T) equations (4) and (5) describe an equilibrium. Thus, there is a whole continuum of equilibria, indexed by i(T). In equation (4), the second term on the right-hand side converges to zero as n goes to infinity, for any i(T). Thus, all equilibria converge in the limit to an inflation rate of R-r. That's the long-run Fisher relation. In equation (5), the second term does not converge as n goes to infinity, i.e. as time runs backward to minus infinity. If i(T) < R - r, then inflation runs off to minus infinity as time runs backward, and if i(T) > R - r, then inflation runs off to infinity as time runs backward. This is typical of course - we have a difference equation that's stable if we solve it forward, and it's unstable if we solve it backward. Note that one equilibrium is i(t) = R - r in every period.

What Narayana does is to take equation (5), and let T go to infinity, so he's only looking at the backward solution. As should be clear, I hope, that's not describing all the equilibria. By any conventional notion of what we mean by convergence and stability, the nominal interest rate peg is stable, and all the equilibria converge in the limit to R - r. The Fisher relation holds in the long run. As a practical implication of this, I've heard many people argue that, if the central bank holds its nominal interest rate at zero, then surely inflation will eventually rise to the 2% inflation target. Well, they can't be thinking about this model then. In any equilibrium with R = 0 forever and with inflation initially lower than some inflation target i*, inflation either falls to -r in the limit, or rises to -r in the limit. If -r < i*, the central bank will never achieve its target by staying at zero.

But, with a nominal interest rate pegged at some value forever, we have an indeterminacy problem - there exists a plethora of equilibria. This makes it hard to make statements about what happens when the interest rate goes up or down. For example, it's certainly correct that, if we set T=0 in equation (4), and think of time running from zero to infinity, solving the difference equation (3) forward, then given i(0), the inflation rate will be higher along the whole equilibrium path, if R rises. But i(0) is not predetermined - it's not an initial condition, it's endogenous and the first step in only one equilibrium path. Who is to say that economic agents don't treat R as a signal and jump to another equilibrium path? We might also be tempted to set i(0) = R*-r, then solve for the equilibrium path given R = R**, and think of that as describing the effects of an increase in the nominal interest rate from R* to R**, since an inflation rate of R* - r is the long run inflation rate when R = R*. Though that's suggestive, it's not precise, due to the indeterminacy problem.

So what to do about that? If we follow the usual NK approach, we would specify a Taylor rule
In equation (6), the Taylor principle is d > 1, and Mike Woodford says that gives us determinacy. But what he means by that is local determinacy - that is, determinacy in a neighborhood of the inflation target i*. But this model is simple enough that it's easy to look at global determinacy - or indeterminacy, in this case. From equation (3) and (6), we get
And the picture looks like this:
D is the difference equation from (7). Note that the kink in the difference equation is where the nominal interest rate hits the zero lower bound (for low inflation rates). A is the desired steady state where the central bank hits its inflation target, and B is the undesired steady state in which the inflation rate is - r and the nominal interest rate is zero. A is an equilibrium, but it's unstable - there are many equilibria that converge in the limit to B. We won't discuss equilibria in which inflation increases without bound, as the model needs to be fixed a bit so that those make sense, but that's possible in a slightly modified model. These are well-known results - the Taylor principle has "perils," i.e. it yields indeterminacy, and there are many equilibria in which the central bank falls short of its inflation target forever - not great.

So, we might look for other policy rules that are better behaved. Here's one:
That rule implies a difference equation that looks like this:
The equilibrium is
The first part of the rule, (8), acts to offset effects of future inflation on current inflation, thus killing off equilibrium paths that will imply current inflation above target. (8) is only an off-equilibrium threat. The second part of the rule, (9), acts to bring inflation back to target next period. The equilibrium result is that inflation can be lower than the target in period 0, but the central bank hits its target in every future period. Further, note that the rule is neo-Fisherian, in more than one way. First, the central bank reacts to low inflation by increasing the nominal interest rate above its long-run level, temporarily. Second, the equilibrium satisfies the properties in the quote at the beginning of this post. After period 0, the nominal interest rate is constant forever, and inflation is constant. If the inflation target increases, then the nominal interest rate increases one-for-one in periods 1,2,3,... Narayana says those are Neo-Fisherian properties, and I stated above that I thought these were claims made of standard NK models under the Taylor principle. Seemingly, these are deemed by some people to be good properties of a monetary policy rule.

What Narayana seems to be getting at is that stickiness in expectations matters. In the example he gives in his note, fixed expectations in the infinite future can have very large effects today. You can see that in equation (5), for example, if we fix i(T) and solve backward. Indeed, it seems that conventional central banking wisdom comes from considering expectations as fixed, as is common practice in some undergraduate IS-LM/Phillips curve constructs. Take equation (1), fix all future variables, and an increase in the current nominal interest rate makes output and inflation go down. Indeed, sticky expectations is what George Evans and Bruce McGough have in mind. Here's their claim:
Following the Great Recession, many countries have experienced repeated periods with realized and expected inflation below target levels set by policymakers. Should policy respond to this by keeping interest rates near zero for a longer period or, in line with neo-Fisherian reasoning, by increasing the interest rate to the steady-state level corresponding to the target inflation rate? We have shown that neo-Fisherian policies, in which interest rates are set according to a peg, impart unavoidable instability. In contrast, a temporary peg at low interest rates, followed by later imposition of the Taylor rule around the target inflation rate, provides a natural return to normalcy, restoring inflation to its target and the economy to its steady state.
We can actually check this out in Narayana's model. Following Evans-McGough (E-M), we'll assume a form of adaptive expectations. Let e(t+1) denote the expected rate of inflation in period t+1 possessed by economic agents in period t. Assume that
So, h determines the degree of stickiness in inflation expectations - there is less expectational inertia as h increases. Using (1), (2), and (11) we can solve for current inflation and expected inflation for next period given the current nominal interest rate and expected inflation as of last period:
How this dynamic system behaves depends on parameters. To see some possibilities, consider extreme cases. If h=0, this is the fixed expectation case - expectations are so sticky that economic agents never learn. Letting e denote fixed inflation expecations,
That's the undergrad IS-LM/P-curve model. If you want inflation to go up, reduce the nominal interest rate. The other extreme is h = 1 which is essentially rear-mirror myopia - economic agents expect inflation next period to be what it was this period. This gives
That's extreme Neo-Fisherism. If you want inflation to go up by 1%, increase the nominal interest rate by 1%.

The question is, what happens for intermediate values of h? There are three cases: sticky expectations
medium-sticky expectations:
Not-so-sticky expecations:
The sticky expectations case gives the results that E-M are looking for. If the central banker follows a Taylor rule then, if inflation expectations are sufficiently low, the central banker goes to the zero lower bound, inflation increases, the Taylor rule eventually kicks in, and inflation converges in the limit to the inflation target i*. But, with medium-sticky or not-so-sticky expectations, from (12) increases in the nominal interest rate increase inflation. Further, if expectations are not-so-sticky there are Taylor rule perils. If d > 1, then there always exist equilibria converging to the zero lower bound with i = -r in the limit. In those equilibria the central bank undershoots its inflation target forever.

Under no circumstances is the standard Taylor rule with d > 1 well-behaved. At best, if inflation is initially below target, the inflation target is only achieved in the limit, and at worst the central banker gets stuck at the zero lower bound forever. But, there are other rules. Here's one:
Under this rule, the central banker hits the inflation target every period, provided initial inflation expectations are not too far below the inflation target. In the worst case, the central banker spends a finite number of periods at the zero lower bound when inflation expectations are too low. But, if inflation expectations are medium-sticky or not-so-sticky, the period at the zero lower bound exhibits inflation above the inflation target - i.e. a period at the zero lower bound can serve to bring inflation down.

The critical value for inflation expectations is
That is, under the rule (19), the central banker goes to the zero lower bound if inflation expections fall below e*. Note that e* is decreasing in h and goes to minus infinity as h goes to 1. As expectations become less sticky, the zero lower bound kicks in only for extreme anticipated deflations.

In their paper, E-M say
As we have shown, the adaptive learning viewpoint argues forcefully against the neo-Fisherian view and in support of the standard view.
As I hope I've made clear, that's overstated. I take the "standard view" to be (i) staying at the zero lower bound will eventually make inflation go up; (ii) a standard Taylor rule is the best the central bank can do. In Narayana's model, under adaptive learning, (i) is only correct under some parameter configurations - actual inflation and expectation inflation both have to be sufficiently sticky. Further, (ii) is never correct.