Coordination problems and discontinuities

Just a quickie. This is for Steve Randy Waldman, who says "Downward price stickiness is a coordination problem, plain and simple."

It is a coordination problem (I think), but it's a bit more than just a coordination problem. There must be something else too, like a discontinuity, or something, to get that coordination problem.

Let Loss(i) = a(P(i)-P)² + b(P(i)-P*)² where P is the mean P(i) across all firms. The parameter 'a' measures the extent to which each firm wants to follow whatever the other firms are doing.

Firm i minimises its loss by setting P(i) = (aP + bP*)/(a+b)

In a symmetric game, where all firms face the same loss function, the unique Nash Equilibrium is P(i)=P=P*. We only get multiple equilibria if b = 0, where it doesn't matter what they do, as long as they all do the same thing.

On the other hand, if there are only two possible values for P(i), for example P(t-1) and P*, then both P(t-1) and P* will be Nash Equilibria, if a > b. That's a coordination problem, because all firms can get stuck in the bad equilibrium P(t-1), unless someone convinces them to all jump to P* together.

For example, maybe it's better if we all drive on the left hand side of the road (because, in the northern hemisphere, it reduces the number of tornadoes) . But we can get stuck in the Nash equilibrium where we all drive on the right, because any individual who deviates will crash. That's a coordination problem. If we all start out driving on the right, we get stuck there. We need someone to tell us all to stop our cars and switch sides at precisely midnight, if we want to jump to the better equilibrium. There's a discontinuity in that example, because a compromise, like driving in the middle of the road, is never individually optimal.

Each firm's profit-maximising price is a function of all the other prices. And it's presumably an increasing function of other firms' prices. We call this "real rigidity" or "strategic complementarity". That helps us get price stickiness, but it isn't sufficient by itself. We need something more. We need some sort of discontinuity, like small menu costs, to get us stuck. Otherwise each individual firm will change its price just a little, when P* changes, even if it expected all other firms would hold their prices fixed. But if all firms change their prices just a little, it wouldn't make sense for each firm to expect all other firms to hold their prices fixed when P* changes. The only Nash equilibrium is for all firms to change their prices to match the new P*.

Real rigidities/strategic complementarities (large positive 'a') mean that the menu costs (or whatever) we need to assume to get sticky prices are smaller than they otherwise would be. But they aren't alone sufficient. We still need the menu costs, or something. Strategic complementarities (in pricing) are a (stickiness) multiplier, but they still need something (some stickiness) to multiply.

A load of us old guys were bashing our heads against this one back in the 1980s/90s. New Keynesian macro didn't start with Woodford. But try telling young people that today; they won't believe you.

Comments

[Benoit Essiambre]

What about the example I gave in Waldman's comments section:

Say a 5% wage cut in an industry would save people's jobs, workers would take it. They prefer losing 5% wages than losing their job. But often times, for this pay cut to save an employer when prices are dropping, all the workers in the business must take the wage cut. Not only that, but all the worker’s in the business’ suppliers must also cut at the same time, as well as the suppliers’ suppliers.

If only part of the supply chain takes a cut, the cost saving burden is too concentrated on them and they have to cut unrealistically deep to save their part of the industry. Unless employees can coordinate their pay cut not only within their own employer but also across other companies in the supply chain to convince them to do the same, it may not achieve much.

But who wants to be the first to cut their wages if it probably isn’t even going to save their job?

What employee organisation is going to reach out to others to synchronously bargain down their industry’s wages?

For individual workers then, the rational behavior, the Nash equilibrium, is to extract as much money as they can from their current employer before they are laid off even if this behavior kills the business and destroys their own job.

There's the same concern on the side of the employers. Going through the trouble of renegotiating a contract downward when they know it won't do much unless other players do the same at the same time, might make them feel the situation is hopeless and their only option is to close parts of the business until the market resets (suppliers close down and new ones open that offer lower pay and lower supply prices).

The only way I can think of to do sufficiently coordinated wage "cuts" is through monetary policy (Quotes because those cuts may not really cost anything to the workers in the long run on an after tax, after inflation basis, since individual tax burden will be lower if more people are paying taxes instead of receiving unemployment benefits).

[Majromax]

> Let Loss(i) = a(P(i)-P)2 + b(P(i)-P*)2 where P is the mean P(i) across all firms. The parameter 'a' measures the extent to which each firm wants to follow whatever the other firms are doing.

> In a symmetric game, where all firms face the same loss function, the unique Nash Equilibrium is P(i)=P=P*. We only get multiple equilibria if b = 0, where it doesn't matter what they do, as long as they all do the same thing.

I feel that you've averaged out over nominal debt, and I think that nominal debt plays a strong part in the discontinuity. If my hypothetical brewery has to buy barley and hops now (with cash) to make beer to sell next quarter, then even without debt markets I have a nominal debt to myself. I benefit if the price level increases, so that the beer I eventually sell more than makes up for the barley I purchased.

The farmer faces the opposite situation, where their barley sold buys next quarter's beer. They have a net savings position between harvest and beer-purchase, so they win if the price level decreases.

In reality, these agents could lend to each other to make their earnings indifferent to the price level, but do they really do this? Even if the loan is specified in advance, these agents are still subject to asymmetric returns from unexpected inflation.

(Real farmers are more closely in the brewer's position, since their income is lumped at harvest time and their expenses are year-round. However, in light of Frances's beer post I cannot resist.)

[Steve Roth]

@Benoit: "The only way I can think of to do sufficiently coordinated wage "cuts" is through monetary policy"

This doesn't address the possibility of hour cuts. It's how Germany in particular and Scandinavians to a lesser extent deal with the issue. Requires coordination, achieved in those places through a high-level system of work councils, etc. (Think of Elinor Ostrom's principles/best practices for effective institutional management of pooled resources.)

Think of it as replacing musical chairs with a flexible bench that can get wider/shorter, but also deeper/shallower.

Sociologists call this system "corporatist concertation." The redoubtable Lane Kenworthy has numbers to suggest that it's the single most powerful and reliable institutional correlate to national wealth:

http://www.asymptosis.com/labor-power-and-economic-growth.html

Sadly, for convoluted historical and political reasons, this system is illegal in the U.S.

[Nick Rowe]

Benoit and Majro: where's the *kink*? We need some sort of discontinuity in the objective function (or its derivative) to make it work. That's why I strained myself to do the math!! It wasn't just performance art!

Steve: red herring. Whether we are stuck in a bad equilibrium with too few hours per worker, or too few workers employed, doesn't change this question. We are still stuck in a bad equilibrium. Now, why should we be stuck there?

Dammit, I wish I could draw upward-sloping ovals in Paint. Put P(i) on the vertical axis, P on the horizontal, draw a 45 degree line, with point P* on the 45 degree line, then draw a whole slew of indifference curves, as ovals, centred on the P* point, then draw an upward-sloping reaction function through the points where the indifference curves are vertical.

[Jason Smith]

Enjoyable as always, Nick. However, I do have a way to achieve stickiness as a coordination problem that doesn't require a discontinuity. Wages are sticky in the same way glue is sticky -- i.e. it's not when there is only one molecule of glue, but a bunch of molecules are ...

http://informationtransfereconomics.blogspot.com/2014/10/wage-stickiness-is-entropic-force.html

[Majromax]

> Benoit and Majro: where's the *kink*? We need some sort of discontinuity in the objective function (or its derivative) to make it work. That's why I strained myself to do the math!! It wasn't just performance art!

The kink comes in the form of either debt (even self-owed) or cash-in-advance constraints.

Our "Calvo fairy" visits with the seasonal cycle. In fall, the farmers decide on the farm-gate price of barley; in spring the brewers decide on the price of beer. The barter optimum is fully stable, but the cash-in-advance monetary economy is not, as price increases redistribute more of that cash-in-advance to price-increasing party.

[Nick Rowe]

Majro: "The kink comes in the form of either debt (even self-owed) or cash-in-advance constraints."

Why does that create a kink? My profit = F(my price Pi, your price P, and something else P*)

Yes I know that F12 may be positive. But why is there a discontinuity in dF/dPi at Pi=P ??

[Steve Waldman]

Nick — I don't think you've fully understood the argument, which is undoubtedly a matter of my poor writing more than your poor reading.

Your loss function is inadequate to describe the payoffs faced by a firm with a leveraged capital structure and the possibility of insolvency. A limited liability firm will never set a price lower than what would bring in revenue sufficient to cover its obligations if it sells its full capacity. Setting a price lower than that ensures maximum loss to shareholders, while setting a price higher than the solvency threshold yields a nonzero chance that shareholders will retain control. Simple supply = marginal cost stories only hold under an assumption of infinite capital or unlimited liability. Shareholders effective supply curve is the upper envelope of a solvency constraint (a horizontal line) and marginal cost. With perfect capital markets, the solvency constraint may be unlikely to bind, as shortfalls in one periods revenues can be made up by continued borrowing while the capital structure is reorganized. With the imperfect capital markets real-world firms face, particularly when their products are under price pressure, solvency constraints are very likely to bind. Let's call the price at the solvency constraint P^. Perhaps the solvency constraint is the discontinuity you are looking for.

The simplest story to tell is that prices are rigid at P^ when MC < P^. And that's enough, as far as it goes, to explain price stickiness. If we imagine identical firms with the same P^, we follow the logic of your first term and let Bertrand competition take us down to P^, but P^ > P*. With identical firms, a deterministic solvency threshold, perfect competition, but a stochastic distribution of sales, all firms set P = P^ and the expected loss is a weighted average of the long-term equity value of the firm (assuming that's positive if it survives short-term stresses) and 0, with the weights being determined by the stochastic process that allocates market share. If quantity demanded at P^ is insufficient to accommodate all firms' capacity and we presume an equal distribution across identical firms, all firms fail. (But they would have failed with equal certainty at any price below P^.)

Fortunately, it's silly to assume such a perfect uniformity of sales or across firms. In real industries, success begets success, product differentiation strategies capture share or they don't, various forms of marketing, exploitation of relationships and vendor lock-in, render the distribution of sales unknown and contestable. Firms contest. With price locked by our assumption of identical capital structures at the same P^, the dimensions across which firms compete cannot be captured, as in your model, by price. In fact, the price dimension is excluded.

Of course, if I have "realistically" permitted nonuniformity of products and sales, I ought "realistically" allow P^ to differ across firms. (To be really realistic, I'd not let P^ be a scalar value, but would allow firms to face heterogenous values probably themselves understood as probability distributions.) But none of this essentially changes the story. All firms drop P to their effective P^, and try to contest the market. Firms with a lower P^ have an advantage, but since this is now monopolistic competition (we've permitted product differentiation, etc), there is no guarantee that the firm with the cleanest capital structure (lowest P^) will win. If there is one firm whose P^ is at or below the Marshallian P*, then at least one firm might set its price there, but it still won't take the whole market because monopolistic competition and capacity constraints. And if there are one or very few clean firms, they might collude to anchor their prices just below the solvency thresholds of their more leveraged peers. The capital structure of firms in the industry does most of the work of organizing a cartel. Only the few with the luxury of considering defection must actually collude.

It's worth pointing out, as an unsurprising empirical matter, capital structures (broadly defined, including obligations to labor etc) within industries are often clustered, fairly similar. Financial markets are competitive, Modigliani and Miller doesn't hold in the real world, and efficient production plans make similar demands on real capital and labor resources. Nominal demand contractions that are not simultaneously matched by repricings of the entire capital structure are often solvency threats for many or most firms in an industry. The usual outcome is not a repricing by all firms across the industry, but a lot of gambling for redemption, and eventually consolidation with the "strongest firms" (lowest solvency constraint, best access to external capital) enjoying an advantage, but a contestable advantage, in the contest to survive.

[Nick Rowe]

Steve: sorry, but I'm still not getting it.

Your first main paragraph tells me that firms don't maximise E(profits), they maximise Emax{0, profits} (or something similar), because if you go bust, you don't care if you go bust in a big way or a small way. OK. That's a kink in the objective function. But if profits = F(Pi, P, P*, random shock), we should still get a continuous reaction function for setting Pi = R(P, P*), unless the probability distribution of the random shock is somehow kinky in some very weird way. (And even then, without some sort of stickiness, they would make Pi contingent on the realisation of the shock.)

Second main paragraph: Assume elasticity of demand for an individual firm is constant everywhere. (Standard simplifying assumption in NK macro.) Then we get Pi = (1+k)MC, where k is a constant, related to elasticity of demand. Assume MPL = one everywhere, so MC=W. So all firms set P as a fixed markup over W, regardless of the level of output and employment. So W/P is pinned down at 1/(1+k). Now if the labour supply curve (for the economy as a whole) just happens to be perfectly elastic at W/P = 1/(1+k), then we do indeed get a multiplicity of equilibria. A leftward shift in the AD curve causes Y and L to fall, with no effect on P or W.

BUT. That result is a sheer fluke. Unless by sheer chance the labour supply curve is at exactly the right height, we get no equilibrium at all (except zero employment, or continuous full employment). And that is a very unreaslistic assumption about the labour supply curve. If it slopes up, we get a unique equilibrium level of L and Y.

ONE way to get what you are looking for is a kinked demand curve, where it's kinked at the price set by all other firms. And you can motivate that kink through search costs (if you raise price all your existing customers will hear about it, and some will go to your competitors, but if you cut price other firms' customers won't hear about it, so you gain few new customers). That gives you a kink in the profit function, and a discontinuity in the derivative of the profit function wrt Pi at Pi = P. That will give you a continuum of macroeconomic equilibria, over a certain range, even with a sensible labour supply function. (But even that result is rather fragile, so you can lose it if you make tiny changes in the assumptions.)

[Nick Rowe]

Here's another way to think about it. Assume symmetric firms, for simplicity. Put Pi on the vertical axis, P on the horizontal axis, and draw the reaction function Pi=R(P). Equilibrium, if it exists, is where the reaction function crosses the 45 degree line. "Coordination problem" means multiple eququilibria. The reaction function must cross the 4 degree line in more than one place. Either you need a curvy reaction function, so it crosses 3 times (middle equilibrium unstable, outer two stable), or you need a thick reaction function (continuum of equilibria).

You can get the curvy reaction function if you have very strong strategic complementarity around the middle unstable equilibria (so dPi/dP > 1 over some range). That means if other firms raise their prices by 1%, i want to raise my price by more than 1% in response.

[rsj]

There is a tension between the swashbuckling capitalism as practiced and the sacred-text capitalism as espoused.

In the swashbuckling capitalism, firms eat each other, they go out of business, they are displaced by more profitable firms. In order for that to happen, when a firm sees a decline in sales and asks for lower cost inputs, the landlord -- together with all other suppliers -- have to say "no, you need to go out of business and be replaced with a better firm that can buy my inputs." In the swashbuckling capitalism, goods are mispriced all the time, quantities are miscalculated all the time -- and competition, that is the process of driving workers and managers out of a job -- is the mechanism to adjust prices, quantities, and production plans.

In the sacred-text capitalism, no one is driven out of business but everyone is harmoniously producing in an eternal equilibrium of blessed universal optimality.

So the question of what small deviation from the blessed state is necessary in order to get price rigidity is a strange question -- by definition it has many answers, since optimality is different from actuality in many ways. Each of these answers will have something to contribute towards rationalizing observed pricing behavior. You can pick your favorite deviation and odds are good that there is a price rigidity explanation in there. But a better question might be to ignore the blessed economy and ask why price rigidity is necessary in the actual economy.

In capitalism-as-buccaneering, when a general downturn appears, and many (but not all) firms see a sharp drop in sales, they cannot go to their landlords and ask for lower rent payments. They need to be driven out of business because the system exists to drive them out of business. The decision making process of landlords is conditioned to say "no, you need to stop producing now and be replaced with a better firm". You can't condition all the actors in an economy to punish the unprofitable and then be surprised when they do that during a downturn. In the blessed-state capitalism, everyone simultaneously gives their customers a break so that the break goes away in real terms. In capitalism-as-buccannering, that would be a bad decision making process, because all other suppliers must be trusted to not defect and keep charging the original high prices. If some supplier does defect, then they get to capture a disproportionate share of the firm's revenue because they stood pat while other suppliers cut their price. In capitalism-as-buccaneering, these types of battles determine the share of revenue captured by each supplier. If your coordination mechanism is swashbuckling, then you cannot expect enlightened philosophy to suddenty take over during a downturn.

Certainly it may be possible, in theory, for there to be a set of prices and quantities in which all the prices are lower, but there is no way to get to that place without a prolonged period of large unemployment. The landlord has to see his tenant driven out and be unable to rent the space for a considerable time to any other best tenant before they agree to lower their rent. In a general downturn, that must be true of many tenants.

[Steve Waldman]

Nick — We're talking way past each other here. We're literally speaking different languages. I'm making a microeconomic argument about the source of price stickiness. You can, if you'd like, argue that there's a fallacy of composition in there somewhere, that my micro description of why agents would fail to let P=MC as demand contracts would be offset in the general equilibrium. There might be an argument there, although I've no seen a compelling case.

As soon as you talk about a labor supply curve, we are outside of my microeconomic framework. The very source of the coordination problem that I posit is that cost of the factors of production, including labor, real, and financial capital, are approximately fixed under the timescale of the demand shock. If the price of labor is continuously sensitive to demand, you have already solved the coordination problem by assumption. If you are going to posit a labor demand curve, we should also posit demand curves for credit and other factors of production, and if we let everything vary continuously, poof! the problem goes away. You win under the frameworks you describe as conventional (and I don't doubt your descriptions — I had the misfortune of working my way through NK models at one point, and left the exercise exhausted and unimpressed, but I'm sure you know what you are talking about). I am not trying to justify the use of Calvo or other pricing in a NK context. I could care less. I am trying to explain sticky prices as an empirical phenomenon from, um, microfoundations.

You are assuming your conclusion when you define the objective function as a continuous function of price. Under my analysis, price is not a variable, it is not what agents optimize. It is pinned by a solvency constraint. Firms optimize over other dimensions, dimensions perhaps excluded from conventional models. Equilibrium is not when P=Pi in our universe of symmetrical firms. Setting up a reaction function in terms of price is incoherent in my framework, however useful it may be in many other contexts. In my account, P = P^. Firms maximize the long-term value of their control of firms E[Value] = F( I | I1, I2... ) where I is a vector that describes product characteristics and marketing choices conditional on observed or expected choices made by other firms 1, 2... We can simplify this, if you like, by letting the vector I have zero length. Then firms have nothing whatsoever to do. P is still pinned at P^, and the distribution of outcomes is fully determined by the random process we define for market share. Even in this degenerate case, firms are not tempted to optimize over P, because P^ (weakly) dominates all other possible values for P.

(There's a potential exception for a potential subset of firms not subject to a solvency constraint, for whom the various standard models of oligopoly might apply, taking the pricing of the solvency constrained firms as either given or unknown. In the simplest case all firms are solvency constrained. In the price-flexible limit, no firms are solvency constrained. In between we get the complexities of oligopoly.)

I'm very glad to cop to the criticism that my framework is entirely unconventional. I think it's pretty accurate anyway. I'm open to an argument that it's "partial equilibrium" in a bad way, although as you have pointed out sometimes, the role of partial equilibrium thinking in economics is too complicated to be written off in a stereotyped diss, since economic agents reason in partial equilibrium ways and a general equilibrium has to be resilient to whatever conditions the behavior of the agents that compose it. Further, this is explicitly a model of short-term dysfunction, with no pretenses to wiggling around some stable equilibrium. My claim is that, because of coordination failures (the inability of the costs of factors of production to smoothly adjust with price, therefore imposing a solvency floor), we observe in real life deadweight costs — the cost of managing bankruptcies, loss of going concern-value of enterprises that might otherwise be long-term solvent, etc. There really is no pretension of general equilibrium in my account. It is simply a description of pathology.

If one wants to insist on an RBC/NK world in which there is a general equilibrium out there, all this amounts to is an explanation of the severity of the shocks. But as I said, I'm not all that interested in integrating my thinking with an RBC/NK perspective. If one must integrate, paleokeynesian or AS/AD might be a better choices. AS won't scale down nicely with nominal shifts in AD, so it's best to keep nominal AD growing and scale up.

In any case, it's off, I think, to critique the framework by insisting on agents who optimize over price when the assumptions of the framework pin price at a corner solution. If you want to argue that the assumptions of the framework are not useful simplifications of economic reality, and so the framework is not useful, that's fine, though I'll argue. I think solvency concerns very often do dictate pricing. But if you want to pick apart a theory, you have to take its assumptions as given. Under my assumptions, P is just stuck.

[Nick Rowe]

Steve: OK. Now I think we are maybe beginning to get somewhere.

I'm entirely OK with unconventional/weird assumptions about pricing/constraints/objectives. Or, at least, we can set that weirdness to one side, temporarily. But I really really do worry about the fallacy of composition thing.

It is very easy to rig up a model where an individual firm will not cut its price when demand falls, taking other firms' prices, and wages, as given. Like I said, just assume a production function Y=L, assume constant elasticity of demand, and Bertrand competition, and you've got it. (You can even get it to *raise* price when demand falls, by rigging the assumptions slightly differently.) But unless you have a very elastic labour supply curve, it won't aggregate up to the macro level. If AD falls, Y and L fall, so there should be excess supply of labour at the existing W/P, so W should fall, so P falls.

But even pinning price at a corner solution won't be sufficient for what you need. If that corner itself shifts when other prices and wages shift, you can have stickiness at the micro level, but none at the macro level. You need all prices and wages at a corner solution, and those corners must all line up, so that everyone is at a corner at once. With the kinked demand curve model we *can* get that, but not always. Self-promotion: I did it once. But I rigged it like mad.

But, it's one thing to say there exists a unique Nash equilibrium. It's quite another thing to say that agents will instantly learn where the new Nash equilibrium is, when there's a shock. That's one thing I worry about, and think is a potentially fruitful way of thinking about what looks like sticky prices and wages. But how the hell to model that search for Nash equilibrium.....That's hard.

[rsj]

"But how the hell to model that search for Nash equilibrium.....That's hard."

There are agent based modeling approaches to this. Most have some kind of genetic algorithm, e.g. you assume each agent has some strategy with parameters, and then apply a fitness criteria to see which agents (or really, strategies) survive.

See, for example:
http://www.mtakti.hu/file/download/korosi/2011/somogyi.pdf

It's interesting that things like cyclical output gaps, short run non-neutrality of money, phillips curves, and price rigidity naturally fall out of bottom up approaches that don't explicitly bake these conclusions in. See also http://www2.econ.iastate.edu/tesfatsi/ABMMacroBaselineModel.MLengnick2011.pdf

[Nick Rowe]

rsj: not a bad answer, IMO. Not the sort of answer I'd be any good at, but so what.

[Nick Rowe]

And to my mind an intellectually unsatisfying answer....but well. Maybe I'm biased.

[Steve Waldman]

Nick — There's a symmetry to my story, in that labor and firms are modeled analogously. To say that there must be an excess supply of labor and so W falls is to contradict the account by assuming it false. Workers also face solvency constraints, which is consistent with the fact that (famously) their wages do not in fact fall when their appears to be an excess supply.

I think the right way to simplify my story to examine whether it holds up in a GE context is to consider economic units, without any K/L distinction. From a unit's perspective, L is just another factor supplier for which it has nominally fixed contracts over the short-term, in the same way a supplier of K might be promised r in nominal terms. In both cases, a unit's rigidity is a function of the rigidity of its arrangements with suppliers of its factors of production, and there is no reason analytically to distinguish the two cases, at least for a first cut. (There are some differences between "insolvency" of a firm and "sharp adjustment" of a human that may experience if she cannot accommodate the capital structure of the life she has built. Those differences might or might not be relevant. But I think as a first approximation, we should emphasize the similarities, which permits a useful homogeneity.)

So we have units. Each unit has obligations to other units contracted in nominal terms. It is a Russian doll scenario, except Russian dolls define a neat hierarchy, while in our model there might be arbitrary networks. My factor provider's factor provider might have me as a factor provider.

Now the coordination problem is clear, I think. If I face nominal price pressures, my best, still bad option, is to not cut my price unless I can renegotiate with the units to which I am obliged. But those units are reluctant to cut price unless they can renegotiate with their factor providers. I am their factor provider, but I don't know that, the networks are too indirect. So I refuse to cut price until my factor providers do, my factor providers refuse to cut price, until I cut price, and we are deadlocked, no one cuts prices and we all endure a lottery in which some of us survive and the rest pay deadweight insolvency and adjustment costs. Deadlock and equilibrium are just different ways of describing the stasis that arises from lots of units making independently rational decisions. Deadlock is just a bad equilibrium. But it is the predicted one.

[rsj]

"And to my mind an intellectually unsatisfying answer...."

Well, our understanding of information and computation has changed. Calculation requires physical resources -- energy -- and there are limits on the amount of information that can be stored in a given volume of space, for example. The problem of efficiently allocating resources in general has also been shown to be an NP complete problem in many cases, so it consumes more resources than are being allocated. To me assuming you have infinite computational capacity -- e.g. infinite energy -- in order to allocate a finite set of resources is deeply intellectually unsatisfying. It is the ultimate answer that "doesn't add up", because you need to allocate scare resources to solving the allocation problem and if you over optimize, then you wont have any resources left to allocate and you still wouldn't have allocated your dwindling resources efficiently.

At the same time, another revolution that has been occurring: the use of adaptive monte carlo methods that simulate how organisms in nature solve complex problems over time, so it's a natural candidate to describe how people solve these problem as well. There are some very exciting things happening in this area that make classical assumptions of "let's assume we have infinite energy in order to distribute a finite amount of goods" seem so ridiculous as to not be taken seriously.

One corollary of this is that economics loses its universal status: since no one can spend enough resources to allocate efficiently, everyone must allocate inefficiently, and economics goes back to being a social science. Depending on the nature of the short cuts we take, the economy will evolve in different ways in response to identical stimuli and so there are no universal laws about what will happen given a specific policy. "Structural knowledge" doesn't exist, even in principle. It would violate the laws of physics for any sufficiently complex society to be able to follow a general equilibrium time path. Therefore the time path that is followed depends on the solution strategies employed, which may change over time, and that's just the way it has to be. So if you are looking for economic "truth" that is independent of any solution strategy, you wont find it as it cannot exist -- even in principle.

[Nick Rowe]

Steve: OK. Think of a model with symmetric agents, where each agent is a worker/producer, who buys the output of other agents to use as an input in his own output, along with his own labour. An agent's profit-maximising price depends very much on the prices set by other agents, because his Marginal Cost is 99% determined by the prices of those inputs bought from other agents, and only 1% by his own labour.

His reaction function for setting his price P(i) will be something like:

P(i) = 0.99P + 0.01P*

It still doesn't work. There is a unique (and stable) Nash equilibrium where, for all i, P(i)=P=P*. If P* falls by 10%, each agent cuts his own price by only 0.1%, given all other agents' prices. But then all agents cut their prices by 0.1%, which leads to a further fall, and so on.

If a fall in aggregate demand has *any effect at all* on the profit-maximising price set by each agent, given prices set by other agents, price stickiness unravels. And if a fall in AD meant he were underemployed, so the marginal subjective cost of his own labour did fall, then he would have an incentive to cut his own price, if only by one epsilon.

And if cutting his price by one epsilon in a recession did reduce his profits/utility, then by continuity he would want to raise his price if there were no recession. Which means you didn't start out in Nash Equilibrium in the first place.

***Yes it is true that no individual agent can escape the bad effects of a fall in AD by cutting his own price, if other agents do not cut their prices. And it may well be true (and will be true in your sort of model) that cutting his own price to match the fall in P* will be even worse than leaving his own price fixed. But that is not sufficient to give you what you want. You need to show that he will not want to cut price even by one epsilon.***

Can this hierarchy of production *exacerbate* any other form of price stickiness? Oh yes. But it is not alone sufficient.

The only way the hierarchy of production would be sufficient for price stickiness is if the reaction function is literally P(i)=P. Then you get a knife-edge multiplicity of equilibria for P. But that reaction function only makes sense if each agent simply resells the output of other agents, adding none of his own labour. Which doesn't make sense. (Or if you have pro-cyclical elasticity of demand, and rigged all the parameters exactly right).

Risk of power cut here.

Finished editing.

[Nick Rowe]

Had to post the above slightly prematurely, because it looks like T-storm here, and I didn't want to lose it if the hydro went out. So I'm going back to edit it a little.

[Tom Brown]

@rsj, that's very interesting. I recently had occasion to share a very long car ride with a coworker who's an expert on both genetic algorithms and simulated annealing. I asked him many questions on the subject, including what sort of problems are amenable to each... and he told me that after many years using both he's come to the conclusion that they are essentially equivalent. Given a problem, he thinks he can formulate it for a genetic algorithm and have it be just as efficient as one formulated for simulated annealing. I found that to be an interesting conclusion.

Regarding how organisms in nature solve complex problems, I was intrigued by this biophysicist from MIT (Jeremy England) who has been doing some interesting work on abiogenesis (the origin of life): he's come to the conclusion that pumping low entropy energy (e.g. sun light) into an environment like that of the Earth 3.5 billion years ago is actually likely to result in self replicating organisms. He has some support for that hypothesis mathematically and via simulation, and some preliminary experimental results to support it as well (as I understand it). This is exciting news as that's one of the unsolved problems in science. It'll be interesting to see what comes of his research.

[Nick Rowe]

OK, now let's think of a model with debt.

Start in symmetric equilibrium, where the worker/producers all have fixed nominal debts to each other, and must make annual payments to service that debt. And they don't like going bust.

It's exactly like a standard NK model, except we put a dummy variable "b", where b=1 or 0, in the utility function. (b=1 means the agent is bust.)

Let P* initially be at exactly that level that total debt service payments exactly equal total nominal income. Each agent both pays and receives PY in debt service payments. They are all just on the verge of going bust.

Then P* falls. Either P falls or Y falls. Either way, they all go bust. If an individual agent i holds his P(i) fixed, his Y(i) falls, regardless of what other agents do, and he goes bust. If an individual agent cuts his P(i) to hold his Y(i) fixed he goes bust, regardless of what other agents do.

Each individual agent is screwed, regardless of whether he cuts P(i) or cuts Y(i), regardless of what other agents do.

I can't see why it has any effect at all on his incentive to cut P(i) or cut (Yi). Unless there is some sort of non-seperability in his utility function, so that b affects the Marginal Rate of Substitution between consumption and leisure.

[Tom Brown]

... here's a bit more ([1], [2]) on Dr. England. Also you should check out Jason's link above.

[Steve Waldman]

Nick — Again, I think you are hiding mistakes in functional forms. In particular, the presumption of a constant nonzero coefficient on P* is mistaken if there are growing (or in the simple model, an all-or-none binary) insolvency costs that begin to obtain above P*.

Your followup comment is closer to the mark. In a nonstochastic symmetrical model with all or none insolvency, everybody does indeed go bust in a recession (and so nothing matters very much).

But if market share is stochastic, then it is possible that some survive if and only if P is set to at least P^ (the price at which units go bust event at full capacity). Thus the (weakly) dominant strategy for all agents is to put a lower bound on P at P^. If the stochastic process fails to distribute sufficient market share to keep you afloat, you've lost nothing. Certainly setting P = MC would not have helped, you'd have gone bust with certainty. But if the stochastic process is nonuniform (and real sales stochastic processes are closer to power law distributions than uniform distributions), then setting P at least P^ has given you a valuable lottery ticket.

If sales are distributed stochastically, it is not true that "each individual agent is screwed, regardless of whether he cuts P(i) or cuts Y(i), regardless of what other agents do." So each agent sets P high enough to have a chance ex ante. Ex post some win and some lose.

I'm thinking in terms of expected value maximizing economic units. Questions of consumption vs leisure are beyond the scope of the very simple model I'm proposing.

[Majromax]

@Nick Rowe:

I think you can still get sticky prices relatively easily if we require that firms not willingly take a nominal loss and that prices are announced in advance. We don't need debt proper, just liquidity constraints.

If I am a steel mill and the current price of ore is P1, then I will not purchase it for manufacture into steel if I cannot expect to sell the ore for P2 ≥ (P1 + operating costs). Unexpected inflation (is not a loss to me, it is a foregone gain. Deflation of any sort, however, is a loss.

Liquidity constraints at the firm level mean that losses and foregone profits are not equivalent. They would be if all firms had unlimited access to credit and a no-Ponzi scheme was magically enforced by the universe, but such unlimited access to credit means that the effective money supply is always at a market-clearing level. In such a universe we cannot have deflation caused by a shortage of money, which is what we really care about.

[rsj]

Tom, a nice summary of the complexity claims are here: http://theory.stanford.edu/~tim/papers/et.pdf

" there is no polynomial-time algorithm for computing Nash equilibria in bimatrix games, unless all problems in PPAD are solvable in polynomial
time. This raises the strong possibility [...] that there is no computationally
efficient, general-purpose algorithm for this problem. In turn, this casts doubt on
any interpretation of the Nash equilibrium concept that requires its computation — for example,
if PPAD [is not solvable in polynomial time) then there is no general and tractable procedure for learning Nash equilibria in a
reasonable amount of time, even with only two players and publicly known payoff matrices."

On the other hand:

"Does the fact the computing a Nash equilibrium is PPAD hard
indicate that we should discard the concept? Of course not. But it does suggest that more
tractable solutions concepts are likely to be more appropriate in some contexts[...] Gilboa
and Zemel [51] were the first to cite computational complexity as a possible reason to prefer the
correlated equilibrium [6] over the Nash equilibrium as a solution concept. They studied a number
of computational problems about equilibria in bimatrix games (Problem 2.8 and others) and
showed that all of them are NP-hard for Nash equilibria but polynomial-time solvable for correlated
equilibria."

Correlated equilibria (https://en.wikipedia.org/wiki/Correlated_equilibrium) means no coordination, or rather that everyone coordinates of a public signal sent by someone outside the game. Actors don't try to guess each other's moves and then think of countermoves based on what someone else might do. Instead, everyone coordinates based off of a public signal known to all.

Only these types of equilibria are computationally feasible.

[Nick Rowe]

Steve: OK.

If the utility-maximising markup of price over marginal cost is a decreasing function of aggregate Y, we can get an unstable equilibrium. (But that's not your case).

If the utility-maximising markup of price over marginal cost is a decreasing function of aggregate P, then weird things will happen. I *think* that's closer to your case.

Now *if* we can build a model where that happens, it will happen because of nominal debt, not because of the hierarchy of production. Because hierarchy of production is a real thing, and debt is a nominal thing.

Let N be NGDP (so N=PY)

Suppose we had a reaction function like P(i) = bP + (1-b)N - cP. So it's not homogenous in nominal variables. And b-c < 1, so it's stable, and the parameter c captures the effect of debt on markup. A fall in N would have real effects, but it would still cause P to fall.

What I'm thinking is, if we made the reaction function non-linear, so it's something like:

P(i) = bP + (1-b)N + c(D/P)^1/2, then *maybe*, as N falls, you eventually reach a point where P stops falling, because dP(i)/dP approaches one.

Dunno. But I'm beginning to think that *might* be the way to get something like your result.

[Nick Rowe]

I screwed up the math (of course). I think that should be:

P(i) = bP + (1-b)N + c(D/P)

So as N falls, P falls, but as N falls further, and P falls further, D/P rises, so the markup increases, so P eventually reaches a limit where it stops falling.

Something like that.

[Nick Rowe]

OK, there's a line up of firms for customers, and the firm's position in line is random. Those at the front get all the customers they want, and those at the back get none. And those at the front are just barely able to cover their debts. But are those at the back totally indifferent to how big they go bust? By cutting their price one epsilon, they can jump to the front of the line, and make smaller losses than staying at the back of the line.

Alternatively, they could set prices to maximise expected profits, and buy a lottery ticket on the side.

[Steve Waldman]

I think we're finding our way close to one another, though I think you are working much harder than me to get there. (I'm very grateful for the effort!) If you acknowledge that a long-term valuable (where the long term means after factor repricings) shareholder-controlled firm will never set an immediate-term price level that guarantees a firm's insolvency if there is a nonzero probability of solvency at a higher price, then there must be a floor for each firm's price reaction function. I think rather than posit a functional form ex ante with that behavior, the right thing to do would be to write down the short-term optimization problem of a simplified firm in this situation and derive the price reaction function. As we've discussed before, such an exercise would require modeling market share stochastically as a function of price, and no doubt our derived reaction function would depend upon our choice of distribution. (If our model is deterministic, as you've pointed out, our solvency-constrained identical firms all fail, deterministically.) If our price-parameterized market-share distribution varies steeply with relative price (i.e. if our firm's products are more commodity-like than differentiated), then for all firms P will be near the minimal price at which the firm survives given limited short-term capacity, and we can simplify (as I have above) by saying P is pinned at that minimal price P^ and there is no meaningful reaction function. If our market-share distribution is weakly parameterized by relative price, than we should derive a reaction function of the sort that you are working to define, one in which dPi/dP and dPi/dN approach zero as P approaches P^ from above.

(This would be an interesting exercise, we should try it, though in my experience it takes a lot of trial and error with no guarantee of success to get an analytical model and I don't have the slack right now to devote myself to the attempt.)

Whether the hierarchy of production is a "real thing" or a "nominal thing" depends entirely on how the contracts recruiting the factors of production are written. Leasing a facility or piece of equipment over a term at a nominally fixed rent is a form of financial leverage. Constantly rerenting the same facility or piece of equipment is quite different: in a demand shock, one can instantaneously choose to scale down, and one can benefit if there is a fall in the market rate. The social and legal institutions that recruit resources have real economic consequences, even as they bring the same real resources into the same technological production function. There is a continuum of arrangements between very flexible and very inflexble capital structures, not a binary. How long is the term of a lease? Obviously as the term goes to zero, the repricing risk embedded in the lease diminishes. Does the lease contain "elevator clauses" that recalibrate to a price index periodically? If so, it's a lot more flexible than a fixed price term lease, but a lot less flexible than constantly repurchasing at spot (as it remains costly to choose not to repurchase use of the leased item). Is labor best thought of as repurchased spot, or as leased for a term? That depends a great deal on institutional details. Is there a union contract or labor regulations that render firings would be costly? Are workers willing to renegotiate prices downward, and over what term? Was skilled labor recruited with a promise of generous severance benefits?

To abstract away from all of this heterogeneity, we can pull back to the Econ 101 notion of fixed versus variable costs. Some portion of the firm's cost structure is de facto fixed over a short production horizon. Some portion is variable. For a firm whose costs are all variable, the usual identity of MC and the supply curve holds. For firms whose solvency is not at issue, who have "money in the bank" or at least adequate credit lines, the usual identity of MC and the supply curve still holds, a few periods with high fixed costs means the firm must operate at a loss, but selling until P=MC will minimize the loss.

However, for a firm which will become insolvent over a short-time horizon if it fails to meet its fixed costs, the identity between supply curve and MC breaks down, because creditors liquidate or take control of the firm if the firm makes losses, while shareholders retain the long-term real perpetuity it will yield after factor price renegotiation if they can win a lottery in the current period.

[Nick Rowe]

Steve: this seems to be getting somewhere. But I need to take a break. My brain hurts. Back later.

[Nick Rowe]

Steve: "Whether the hierarchy of production is a "real thing" or a "nominal thing" depends entirely on how the contracts recruiting the factors of production are written."

If you rent one truck at a fixed nominal price and you can't break the lease, it's exactly like a fixed nominal debt. It's a fixed (and sunk) cost, and a fixed input. But if you can choose the number of trucks to rent, but the price is fixed, that's a sticky price, and it affects your marginal cost curve. But that begs the question: why doesn't the truck leasing firm cut its price when demand for its trucks falls so some of its trucks are unemployed?

[Nick Rowe]

Steve: "If our market-share distribution is weakly parameterized by relative price, than we should derive a reaction function of the sort that you are working to define, one in which dPi/dP and dPi/dN approach zero as P approaches P^ from above."

Willem Buiter (I think) coined the term "Panglossian expectations". If you are bust unless the very best thing happens, and you don't care how badly you go bust, you act as if you assumed the real world would be the best of all possible worlds. If there is an aggregate demand shock that firms observe (and can adjust their prices to after observing), plus a relative demand shock which you don't observe until after you have set price (but notice we are assuming sticky prices here, in one sense) then as P approaches P^ expectations would get more and more Panglossian, and individual firms would want to set higher and higher markups.

This is very similar to the case I talk about (in my next post) where demand gets less elastic in a recession, so desired markups over marginal cost increase, and you can get an unstable equilibrium.

With heterogenous firms, so that some are closer to P^ than others, expectations on average would be a weighted average of rational and Panglossian. The average markup would increase as P fell. The aggregate AS curve would be upward-sloping, as well as curved (becoming vertical at a high P). But would it ever be horizontal, at low enough P?

[Nick Rowe]

OK. I think I can build the model. Or sketch it anyway. May have a crack at it tonight. It's a choice between "A model for Steve" and "Panglossian expectations and the Phillips Curve" as a title. I think Dr Pangloss expects he will win.