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.


  1. Thanks Steve!

    1. If I were forced to take P(T) literally, I would say there is a currency reform at time T, where all the old money is redeemed for real goods at a price P(T), and then a new money is issued.

    Why doesn't the government/central bank do something like this every period, to pin down the price level? Because it's a costly hassle; and it doesn't need to. Simply the threat that it might do it, eventually, if needed, should be enough (provided monetary policy is sensible). Fiscal policy has other jobs to do, and can't do those jobs if it is also being used to target inflation (or whatever).

    But I would prefer not to take it that literally. What we are trying to see is how robust are the claimed results of any model with respect to people's long-term expectations. If the results are very fragile, even in the limit when "long-term" means infinitely long term, I wouldn't trust that model.

    2. That last bit, about optimal monetary policy to kill off indeterminacy, is interesting. I'm trying to get the intuition. Does it amount to announcing a very price-elastic money supply rule, at some (implicit) price level target? So the central bank increases M if the price of money rises above that implicit target, and reduces M if the price of money falls below that target?

    3. Empirically, I think the mechanism that drives the results in Azariades' model is too small to matter. When the young expect higher inflation, the higher inflation tax causes them to supply less labour, because their wages will be worth less when they spend them. This only applies to currency, which pays 0% interest. And since wages can be spent quickly, and labour supply elasticity is low, it would take Zimbabwean levels of expected inflation for this effect to matter much. (I *think* we probably saw it in Zimbabwe.)

    4. The original argument was about the Neo-Fisherian claim. The US and Canadian economies do have money in them (they are not "cashless" in any sense). And yet the Fed and BoC *claim* to use a nominal interest rate instrument. We can re-phrase the Neo-Fisherian question this way: if the Fed or BoC wanted to increase the growth rate of M, should it raise or lower its nominal interest rate instrument today?

    But this was a worthwhile post, even if I disagree.

    1. "What we are trying to see is how robust are the claimed results of any model with respect to people's long-term expectations."

      Here, expectations are endogenous. In an rational expectations equilibrium you don't get to change expectations exogenously. If we're allowed to do that, it's clear we can get all kinds of behavior. That's not fragility, it's a different way of looking at the world. Change assumptions, you change results. We either think it's useful or not.

      2. Basically, the money growth rule offsets incipient changes in money demand.

      3. In Costas's model, you get sunspot equilibria when the income effect is large enough.

      4. It's a claim about inflation. Want inflation to go up? What should the central bank do to its interest rate target?

    2. 3. I think it's more an intertemporal consumption/leisure substitution effect, for the young workers (old consumers don't have any choices). But in either case, the income (wealth) effect from currency, which is around 5% of NGDP in countries like Canada, won't be very big.

    3. This was where this literature went. Models like this needed a big income effect to get sunpsot equilibria, some people argued this wasn't plausible, and some other people explored other types of models. In some of Farmer's work, for example, you need sufficient increasing returns.

  2. I believe Kocherlakota’s critique simply regards transversality conditions. What he seems to imply is that if the usual transversality makes sense when you select a certain solution (equilibrium) it is not obvious that you can select another solution simply forgetting about checking the validity of the transversality condition.

  3. Hi Stephen, Equilibria respond to context... the environment in which they are found. The context we have now is very high corporate profit rates with very low nominal rates.
    According to a model that I am developing with a mathematician. Inflation is in a rut because of this unusual situation of high profit rates and low nominal rates. We are exploring equations for the forces acting within this context.
    Here is a post that talks about a 3% inflation target.

    The model also implies that a rise the Fed rate would move the data points to the left from the current extreme on the lower right. There would be enough profit rate to absorb the rise in the Fed rate. The question is... Do the dynamics of the forces then create a situation where corporations will start to raise prices without causing a freefall toward a 0% real profit rate. The real profit rate is currently around 8% which is historically still very high.

    I hope that this model will place some proper context on how the Fisher effect would work now, but why it wouldn't have worked in previous business cycles.
    My concern with your theories is that you are trying to apply them to all business cycles. Yet I think the Fisher effect will only work when there is a very wide positive difference between profit rates and nominal rates.
    This situation exists in Japan and other places where inflation has fallen into a low rut. Taking nominal rates even lower increases the spread between profit rates and nominal rates and keeps inflation in its low rut. Corporations need to be "disciplined" with a higher rate to bring down their profit rates to a point where they begin to feel a need for firming up prices. Real profit rates are high enough to absorb higher nominal rates, but in past biz cycles they were not.
    Nick Rowe has always looked to the past to explain why the Fisher effect would not work. But the context has changed, and it looks like it would work now solely because of the very high spread between profit rates and nominal rates.

  4. Another great post! Once you stopped blogging about Paul Krugman you became the best macro-blogger in the business...hehe :-)

    1. Well, Paul would never really engage, which was no fun. Sorry this a bit technical for most people, but if no one else can figure it out, at least I'm learning something - I think.

  5. I am not an economist. But clearly, demand for treasury bonds, rising price and falling yields, have nothing to do with the real economy, since the yields have been declining since Greenspan took office, in good and bad times. So, bonds have a demand of their own, regardless of other factors. They have demand as collateral, starting with the Salomon Brothers, and now the shorter duration bonds have demand as essential bank capital required by Basel. So, it would seem, nothing is going to get in the way of bond demand except for the occasional tantrum, the process of shaking weak hands out so they will sell their bonds.

    More economists should focus on the supply and demand of bonds themselves. The demand is increasing and it appears that the supply is decreasing. JMO from a non economist.

  6. Prof, I have a question for you. Why did Bernanke save big business but let the RE industry decline and crash in 2008? Was it because the bonds backing RE were inferior to the bonds backing big business? I noticed big business bonds are used as collateral more and more, and I don't know if MBSs are used again as collateral.

    If that was the case, I am a bit disappointed in the Fed because the Fed adopted the flawed David X Li Copula that led to so many MBSs going bad. I would have thought the Fed had a responsibility to fix that mistake by saving the Commercial Paper market that was the foundation of subprime lending during that time.

    I guess that leads to my second question: why didn't the Fed save RE because of Basel 2's involvement in backing mispriced risk in the MBSs in the first place?