The Price of Everything, the Value of the Economy: A Clark Medal for Emi Nakamura!

Fantastic and well-deserved news this morning with the Clark Medal being awarded to Emi Nakamura, who has recently moved from Columbia to Berkeley. Incredibly, Nakamura’s award is the first Clark to go to a macroeconomist in the 21st century. The Great Recession, the massive changes in global trade patterns, the rise of monetary areas like the Eurozone, the “savings glut” and its effect on interest rates, the change in openness to hot financial flows: it has been a wild twenty years for the macroeconomy in the two decades since Andrei Schleifer won the Clark. It’s hard to imagine what could be more important for an economist to understand than these patterns.

Something unusual has happened in macroeconomics over the past twenty years: it has become more like Industrial Organization! A brief history may be useful. The term macroeconomics is due to Ragnar Frisch, in his 1933 article on the propagation of economic shocks. He writes,

“The macro-dynamic analysis…tries to give an account of the whole economic system taken in its entirety. Obviously in this case it is impossible to carry through the analysis in great detail. Of course, it is always possible to give even a macro-dynamic analysis in detail if we confine ourselves to a purely formal theory. Indeed, it is always possible by a suitable system of subscripts and superscripts, etc., to introduce practically all factors which we may imagine…Such a theory, however, would have only a rather limited interest. It would hardly be possible to study such fundamental problems as the exact time shape of the solution, [etc.]. These latter problems are just the essential problems in business cycle analysis. In order to attack these problems on a macro-dynamic basis…we must deliberately disregard a considerable amount of the details of the picture.

And so we did. The Keynesians collapsed the microfoundations of the macroeconomy into a handful of relevant market-wide parameters. The Lucas Critique argued that we can collapse some things – many agents into a representative agent, for instance – but we ought always begin our analysis with the fundamental parameters of tastes, constraints, and technologies. The neoclassical synthesis combined these raw parameters with nominal ridigities – sticky prices, limited information, and so on. But Frisch’s main point nonetheless held strong: to what use are these deeper theoretical parameters if we cannot estimate their value and their effect on the macroeconomy? As Einstein taught us, the goal of the scientist should be to make things as simple as possible, but no simpler.

What has changed recently in macroeconomics is twofold. First, computational power now makes it possible to estimate or calibrate very complex dynamic and stochastic models, with forward looking agents, with price paths in and out of equilibrium, with multiple frictions – it is in this way that macro begins to look like industrial organization, with microeconomic parameters at the base. But second, and again analogous to IO, the amount of data available to the researcher has grown enormously. We now have price scanner data that tells us exactly when and how prices change, how those changes propagate across supply chains and countries, how they interact with taxes, and so on. Frisch’s problem has in some sense been solved: we no longer have the same trade-off between usefulness and depth when studying the macroeconomy.

Nakamura is best known for using this deep combination of data and theory to understand how exactly firms set prices. Price rigidities play a particularly important role in theories of the macroeconomy that potentially involve inefficiency. Consider a (somewhat bowdlerized) version of real business cycle theory. Here, shocks hit the economy: for instance, an oil cartel withholds supply for political reasons. Firms must react to this “real” supply-side shock by reorganizing economic activity. The real shock then propagates across industries. The role of monetary policy in such a world is limited: a recession simply reflects industries reacting to real change in the economic environment.

When prices are “sticky”, however, that is no longer true. The speed by which real shocks propagate, and the distortion sticky prices introduce, can be affected by monetary policy, since firms will react to changes in expected inflation by changing the frequency in which they update prices. Famously, Golosov and Lucas in the JPE argued, theoretically and empirically, that the welfare effects of “sticky prices” or “menu costs” are not terribly large. Extracting these welfare effects is quite sensitive to a number of features in the data and in the theory. To what extent is there short-term price dispersion rather than an exogenous chance for all firms in an industry to change their prices? Note that price dispersion is difficult to maintain unless we have consumer search costs – otherwise, everyone buys from the cheapest vendor – so price dispersion adds a non-trivial technical challenge. How much do prices actually change – do we want to sweep out short-term sales, for example? When inflation is higher, do firms adjust prices equally often but with bigger price jumps (consider the famous doubling of the price of Coca-Cola), or do they adjust prices more often keeping the percentage change similar to low-inflation environments? How much heterogeneity is there is the price-setting practice across industries, and to what extent do these differences affect the welfare consequences of prices given the links across industries?

Namakura has pushed us very far into answering these questions. She has built insane price datasets, come up with clever identification strategies to separate pricing models, and used these tools to vastly increase our understanding of the interaction between price rigidities and the business cycle. Her “Five Facts” paper uses BLS microdata to show that sales were roughly half of the “price changes” earlier researchers has found, that prices change more rapidly when inflation is higher, and that there is huge heterogeneity across industries in price change behavior. Taking that data back to the 1970s, Nakamura and coauthors also show that high inflation environments do not cause more price dispersion: rather, firms update their prices more often. Bob Lucas in his Macroeconomic Priorities made a compelling argument that business cycle welfare costs are much smaller than the costs of inflation and inflation costs are themselves much smaller than the costs of tax distortions. As Nakamura points out, if you believe this, no wonder you prioritize price stability and tax policy! (Many have quibbled with Lucas’ basic argument, but even adding heterogeneous agents, it is tough to get business cycles to have large economic consequences; see, e.g., Krusell et al RED 2009.) Understanding better the true costs of inflation, via the feedback of monetary expansion on pricesetting, goes a great deal toward helping policymakers calibrate the costs and benefits of price stability vis-a-vis other macroeconomic goals.

Though generally known as an empirical macroeconomist, Nakamura also has a number of papers, many with her husband Jon Steinsson, on the theory of price setting. For example, why are prices both sticky and also involve sales? In a clever paper in the JME, Nakamura and Steinsson model a firm pricing to habit-forming consumers. If the firm does not constrain itself, it has the incentive to raise prices once consumers form their habit for a given product (as a Cheez-It fan, I understand the model well – my willingness to pay for a box shipped up from the US to the Cheez-It-free land of Canada is absurdly high). To avoid this time inconsistency problems, firms would like to commit to a price path with some flexibility to respond to changes in demand. An equilibrium in this relational contract-type model involves a price cap with sales when demand falls: rigid prices plus sales, as we see in the data! In a second theoretical paper with Steinsson and Alisdair McKay, Nakamura looks into how much communication about future nominal interest rates can affect behavior. In principle, a ton: if you tell me the Fed will keep the real interest rate low for many years (low rates in the future raise consumption in the future which raises inflation in the future which lowers real rates today), I will borrow away. Adding borrowing constraints and income risk, however, means that I will never borrow too much money: I might get a bad shock tomorrow and wind up on the street. Giving five years of forward guidance about interest rates rather than a year, therefore, doesn’t really affect my behavior that much: the desire to have precautionary savings is what limits my borrowing, not the interest rate.

Nakamura’s prize is a well-deserved award, going to a leader in the shift in macro toward a more empirical, more deeply “microeconomic” in its theory, style of macro. Her focus is keenly targeted toward some of the key puzzles relevant to macroeconomic policymakers. There is no way to cover such a broad field in one post – this is not one of those awards given for a single paper – but luckily Nakamura has two great easily-readable summaries of her core work. First, in the Annual Review of Economics, she lays out the new empirical facts on price changes, the attempts to identify the link between monetary policy and price changes, and the implications for business cycle theory. Second, in the Journal of Economic Perspectives, she discusses how macroeconomists have attempted to more credibly identify theoretical parameters. In particular, external validity is so concerning in macro – remember the Lucas Critique! – that the essence of the problem involves combining empirical variation for identification with theory mapping that variation into broader policy guidance. I hesitate to stop here since Nakamura has so many influential papers, but let us take just more quick tasters that are well worth your more deep exploration. On the government spending side, she uses local spending shocks and a serious model to figure out the national fiscal multiplier from government spending. Second, she recently has linked the end of large-scale increases in female moves from home production to the labor force has caused recessions to last longer.

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Kenneth Arrow Part II: The Theory of General Equilibrium

The first post in this series discussed Ken Arrow’s work in the broad sense, with particular focus on social choice. In this post, we will dive into his most famous accomplishment, the theory of general equilibrium (1954, Econometrica). I beg the reader to offer some sympathy for the approximations and simplifications that will appear below: the history of general equilibrium is, by this point, well-trodden ground for historians of thought, and the interpretation of history and theory in this area is quite contentious.

My read of the literature on GE following Arrow is as follows. First, the theory of general equilibrium is an incredible proof that markets can, in theory and in certain cases, work as efficiently as an all-powerful planner. That said, the three other hopes of general equilibrium theory since the days of Walras are, in fact, disproven by the work of Arrow and its followers. Market forces will not necessarily lead us toward these socially optimal equilibrium prices. Walrasian demand does not have empirical content derived from basic ordinal utility maximization. We cannot rigorously perform comparative statics on general equilibrium economic statistics without assumptions that go beyond simple utility maximization. From my read of Walras and the early general equilibrium theorists, all three of those results would be a real shock.

Let’s start at the beginning. There is an idea going back to Adam Smith and the invisible hand, an idea that individual action will, via the price system, lead to an increase or even maximization of economic welfare (an an aside, Smith’s own use of “invisible hand” trope is overstated, as William Grampp among others has convincingly argued). The kind of people who denigrate modern economics – the neo-Marxists, the back-of-the-room scribblers, the wannabe-contrarian-dilletantes – see Arrow’s work, and the idea of using general equilibrium theory to “prove that markets work”, as a barbarism. We know, and have known well before Arrow, that externalities exist. We know, and have known well before Arrow, that the distribution of income depends on the distribution of endowments. What Arrow was interested in was examining not only whether the invisible hand argument “is true, but whether it could be true”. That is, if we are to claim markets are uniquely powerful at organizing economic activity, we ought formally show that the market could work in such a manner, and understand the precise conditions under which it won’t generate these claimed benefits. How ought we do this? Prove the precise conditions under which there exists a price vector where markets clear, show the outcome satisfies some welfare criterion that is desirable, and note exactly why each of the conditions are necessary for such an outcome.

The question is, how difficult is it to prove these prices exist? The term “general equilibrium” has had many meanings in economics. Today, it is often used to mean “as opposed to partial equilibrium”, meaning that we consider economic effects allowing all agents to adjust to a change in the environment. For instance, a small random trial of guaranteed incomes has, as its primary effect, an impact on the incomes of the recipients; the general equilibrium effects of making such a policy widespread on the labor market will be difficult to discern. In the 19th and early 20th century, however, the term was much more concerned with the idea of the economy as a self-regulating system. Arrow put it very nicely in an encyclopedia chapter he wrote in 1966: general equilibrium is both “the simple notion of determinateness, that the relations which describe the economic system must form a system sufficiently complete to determine the values of its variables and…the more specific notion that each relation represents a balance of forces.”

If you were a classical, a Smith or a Marx or a Ricardo, the problem of what price will obtain in a market is simple to solve: ignore demand. Prices are implied by costs and a zero profit condition, essentially free entry. And we more or less think like this now in some markets. With free entry and every firm producing at the identical minimum efficient scale, price is entirely determined by the supply side, and only quantity is determined by demand. With one factor, labor where the Malthusian condition plays the role of free entry, or labor and land in the Ricardian system, this classical model of value is well-defined. How to handle capital and differentiated labor is a problem to be assumed away, or handled informally; Samuelson has many papers where he is incensed by Marx’s handling of capital as embodied labor.

The French mathematical economist Leon Walras finally cracked the nut by introducing demand and price-taking. There are household who produce and consume. Equilibrium involves supply and demand equating in each market, hence price is where margins along the supply and demand curves equate. Walras famously (and informally) proposed a method by which prices might actually reach equilibrium: the tatonnement. An auctioneer calls out a price vector: in some markets there is excess demand and in some excess supply. Prices are then adjusted one at a time. Of course each price change will affect excess demand and supply in other markets, but you might imagine things can “converge” if you adjust prices just right. Not bad for the 1870s – there is a reason Schumpeter calls this the “Magna Carta” of economic theory in his History of Economic Analysis. But Walras was mistaken on two counts: first, knowing whether there even exists an equilibrium that clears every market simultaneously is, it turns out, equivalent to a problem in Poincare’s analysis situs beyond the reach of mathematics in the 19th century, and second, the conditions under which tatonnement actually converges are a devilish problem.

The equilibrium existence problem is easy to understand. Take the simplest case, with all j goods made up of the linear combination of k factors. Demand equals supply just says that Aq=e, where q is the quantity of each good produced, e is the endowment of each factor, and A is the input-output matrix whereby product j is made up of some combination of factors k. Also, zero profit in every market will imply Ap(k)=p(j), where p(k) are the factor prices and p(j) the good prices. It was pointed out that even in this simple system where everything is linear, it is not at all trivial to ensure that prices and quantities are not negative. It would not be until Abraham Wald in the mid-1930s – later Arrow’s professor at Columbia and a fellow Romanian, links that are surely not a coincidence! – that formal conditions were shown giving existence of general equilibrium in a simple system like this one, though Wald’s proof greatly simplified by the general problem by imposing implausible restrictions on aggregate demand.

Mathematicians like Wald, trained in the Vienna tradition, were aghast at the state of mathematical reasoning in economics at the time. Oskar Morgenstern absolutely hammered the great economist John Hicks in a 1941 review of Hicks’ Value and Capital, particularly over the crazy assertion (similar to Walras!) that the number of unknowns and equations being identical in a general equilibrium system sufficed for a solution to exist (if this isn’t clear to you in a nonlinear system, a trivial example with two equations and two unknowns is here). Von Neumann apparently said (p. 85) to Oskar, in reference to Hicks and those of his school, “if those books are unearthed a hundred years hence, people will not believe they were written in our time. Rather they will think they are about contemporary with Newton, so primitive is the mathematics.” And Hicks was quite technically advanced compared to his contemporary economists, bringing the Keynesian macroeconomics and the microeconomics of indifference curves and demand analysis together masterfully. Arrow and Hahn even credit their initial interest in the problems of general equilibrium to the serendipity of coming across Hicks’ book.

Mathematics had advanced since Walras, however, and those trained at the mathematical frontier finally had the tools to tackle Walras’ problem seriously. Let D(p) be a vector of demand for all goods given price p, and e be initial endowments of each good. Then we simply need D(p)=e or D(p)-e=0 in each market. To make things a bit harder, we can introduce intermediate and factor goods with some form of production function, but the basic problem is the same: find whether there exists a vector p such that a nonlinear equation is equal to zero. This is the mathematics of fixed points, and Brouwer had, in 1912, given a nice theorem: every continuous function from a compact convex subset to itself has a fixed point. Von Neumann used this in the 1930s to prove a similar result to Wald. A mathematician named Shizuo Kakutani, inspired by von Neumann, extended the Brouwer result to set-valued mappings called correspondences, and John Nash in 1950 used that result to show, in a trivial proof, the existence of mixed equilibria in noncooperative games. The math had arrived: we had the tools to formally state when non-trivial non-linear demand and supply systems had a fixed point, and hence a price that cleared all markets. We further had techniques for handling “corner solutions” where demand for a given good was zero at some price, surely a common outcome in the world: the idea of the linear program and complementary slackness, and its origin in convex set theory as applied to the dual, provided just the mathematics Arrow and his contemporaries would need.

So here we stood in the early 1950s. The mathematical conditions necessary to prove that a set-valued function has an equilibrium have been worked out. Hicks, in Value and Capital, has given Arrow the idea that relating the future to today is simple: just put a date on every commodity and enlarge the commodity space. Indeed, adding state-contingency is easy: put an index for state in addition to date on every commodity. So we need not only zero excess demand in apples, or in apples delivered in May 1955, but in apples delivered in May 1955 if Eisenhower loses his reelection bid. Complex, it seems, but no matter: the conditions for the existence of a fixed point will be the same in this enlarged commodity space.

With these tools in mind, Arrow and Debreu can begin their proof. They first define a generalization of an n-person game where the feasible set of actions for each player depends on the actions of every other player; think of the feasible set as “what can I afford given the prices that will result for the commodities I am endowed with?” The set of actions is an n-tuple where n is the number of date and state indexed commodities a player could buy. Debreu showed in 1952 PNAS that these generalized games have an equilibrium as long as each payoff function varies continuously with other player’s actions, the feasible set of choices convex and varies continuously in other player’s actions, and the set of actions which improve a player’s payoff are convex for every action profile. Arrow and Debreu then show that the usual implications on individual demand are sufficient to aggregate up to the conditions Debreu’s earlier paper requires. This method is much, much different from what is done by McKenzie or other early general equilibrium theorists: excess demand is never taken as a primitive. This allows the Arrow-Debreu proof to provide substantial economic intuition as Duffie and Sonnenschein point out in a 1989 JEL. For instance, showing that the Arrow-Debreu equilibrium exists even with taxation is trivial using their method but much less so in methods that begin with excess demand functions.

This is already quite an accomplishment: Arrow and Debreu have shown that there exists a price vector that clears all markets simultaneously. The nature of their proof, as later theorists will point out, relies less on convexity on preferences and production sets as on the fact that every agent is “small” relative to the market (convexity is used to get continuity in the Debreu game, and you can get this equally well by making all consumers infinitesimal and then randomizing allocations to smooth things out; see Duffie and Sonnenschein above for an example). At this point, it’s the mid-1950s, heyday of the Neoclassical synthesis: surely we want to be able to answer questions like, when there is a negative demand shock, how will the economy best reach a Pareto-optimal equilibrium again? How do different speeds of adjustment due to sticky prices or other frictions affect the rate at which optimal is regained? Those types of question implicitly assume that the equilibrium is unique (at least locally) so that we actually can “return” to where we were before the shock. And of course we know some of the assumptions needed for the Arrow-Debreu proof are unrealistic – e.g., no fixed costs in production – but we would at least like to work out how to manipulate the economy in the “simple” case before figuring out how to deal with those issues.

Here is where things didn’t work out as hoped. Uzawa (RESTUD, 1960) proved that not only could Brouwer’s theorem be used to prove the existence of general equilibrum, but that the opposite was true as well: the existence of general equilibrium was logically equivalent to Brouwer. A result like this certainly makes one worry about how much one could say about prices in general equilibrium. The 1970s brought us the Sonnenschein-Mantel-Debreu “Anything Goes” theorem: aggregate excess demand functions do not inherit all the properties of individual excess demand functions because of wealth effects (when relative prices change, the value of one’s endowment changes as well). For any aggregate excess demand function satisfying a couple minor restrictions, there exists an economy with individual preferences generating that function; in particular, fewer restrictions than are placed on individual excess demand as derived from individual preference maximization. This tells us, importantly, that there is no generic reason for equilibria to be unique in an economy.

Multiplicity of equilibria is a problem: if the goal of GE was to be able to take underlying primitives like tastes and technology, calculate “the” prices that clear the market, then examine how those prices change (“comparative statics”), we essentially lose the ability to do all but local comparative statics since large changes in the environment may cause the economy to jump to a different equilibrium (luckily, Debreu (1970, Econometrica) at least generically gives us a finite number of equilibria, so we may at least be able to say something about local comparative statics for very small shocks). Indeed, these analyses are tough without an equilibrium selection mechanism, which we don’t really have even now. Some would say this is no big deal: of course the same technology and tastes can generate many equilibria, just as cars may wind up all driving on either the left or the right in equilibrium. And true, all of the Arrow-Debreu equilibria are Pareto optimal. But it is still far afield from what might have been hoped for in the 1930s when this quest for a modern GE theory began.

Worse yet is stability, as Arrow and his collaborators (1958, Ecta; 1959, Ecta) would help discover. Even if we have a unique equilibrium, Herbert Scarf (IER, 1960) showed, via many simple examples, how Walrasian tatonnement can lead to cycles which never converge. Despite a great deal of the intellectual effort in the 1960s and 1970s, we do not have a good model of price adjustment even now. I should think we are unlikely to ever have such a theory: as many theorists have pointed out, if we are in a period of price adjustment and not in an equilibrium, then the zero profit condition ought not apply, ergo why should there be “one” price rather than ten or a hundred or a thousand?

The problem of multiplicity and instability for comparative static analysis ought be clear, but it should also be noted how problematic they are for welfare analysis. Consider the Second Welfare Theorem: under the Arrow-Debreu system, for every Pareto optimal allocation, there exists an initial endowment of resources such that that allocation is an equilibrium. This is literally the main justification for the benefits of the market: if we reallocate endowments, free exchange can get us to any Pareto optimal point, ergo can get us to any reasonable socially optimal point no matter what social welfare function you happen to hold. How valid is this justification? Call x* the allocation that maximizes some social welfare function. Let e* be an initial endowment for which x* is an equilibrium outcome – such an endowment must exist via Arrow-Debreu’s proof. Does endowing agents with e* guarantee we reach that social welfare maximum? No: x* may not be unique. Even if it unique, will we reach it? No: if it is not a stable equilibrium, it is only by dint of luck that our price adjustment process will ever reach it.

So let’s sum up. In the 1870s, Walras showed us that demand and supply, with agents as price takers, can generate supremely useful insights into the economy. Since demand matters, changes in demand in one market will affect other markets as well. If the price of apples rises, demand for pears will rise, as will their price, whose secondary effect should be accounted for in the market for apples. By the 1930s we have the beginnings of a nice model of individual choice based on constrained preference maximization. Taking prices as given, individual demands have well-defined forms, and excess demand in the economy can be computed by a simple summing up. So we now want to know: is there in fact a price that clears the market? Yes, Arrow and Debreu show, there is, and we needn’t assume anything strange about individual demand to generate this. These equilibrium prices always give Pareto optimal allocations, as had long been known, but there also always exist endowments such that every Pareto optimal allocation is an equilibria. It is a beautiful and important result, and a triumph for the intuition of the invisible hand it its most formal sense.

Alas, it is there we reach a dead end. Individual preferences alone do not suffice to tell us what equilibria we are at, nor that any equilibria will be stable, nor that any equilibria will be reached by an economically sensible adjustment process. To say anything meaningful about aggregate economic outcomes, or about comparative statics after modest shocks, or about how technological changes change price, we need to make assumptions that go beyond individual rationality and profit maximization. This is, it seems to me, a shock for the economists of the middle of the century, and still a shock for many today. I do not think this means “general equilibrium is dead” or that the mathematical exploration in the field was a waste. We learned a great deal about precisely when markets could even in principle achieve the first best, and that education was critical for the work Arrow would later do on health care, innovation, and the environment, which I will discuss in the next two posts. And we needn’t throw out general equilibrium analysis because of uniqueness or stability problems, any more than we would throw out game theoretic analysis because of the same problems. But it does mean that individual rationality as the sole paradigm of economic analysis is dead: it is mathematically proven that postulates of individual rationality will not allow us to say anything of consequence about economic aggregates or game theoretic outcomes in the frequent scenarios where we do not have a unique equilibria with a well-defined way to get there (via learning in games, or a tatonnament process in GE, or something of a similar nature). Arrow himself (1986, J. Business) accepts this: “In the aggregate, the hypothesis of rational behavior has in general no implications.” This is an opportunity for economists, not a burden, and we still await the next Arrow who can guide us on how to proceed.

Some notes on the literature: For those interested in the theoretical development of general equilibrium, I recommend General Equilibrium Analysis by Roy Weintraub, a reformed theorist who now works in the history of thought. Wade Hands has a nice review of the neoclassical synthesis and the ways in which Keynesianism and GE analysis were interrelated. On the battle for McKenzie to be credited alongside Arrow and Debreu, and the potentially scandalous way Debreu may have secretly been responsible for the Arrow and Debreu paper being published first, see the fine book Finding Equilibrium by Weintraub and Duppe; both Debreu and McKenzie have particularly wild histories. Till Duppe, a scholar of Debreu, also has a nice paper in the JHET on precisely how Arrow and Debreu came to work together, and what the contribution of each to their famous ’54 paper was.

Yuliy Sannikov and the Continuous Time Approach to Dynamic Contracting

The John Bates Clark Award, given to the best economist in the United States under 40, was given to Princeton’s Yuliy Sannikov today. The JBC has, in recent years, been tilted quite heavily toward applied empirical microeconomics, but the prize for Sannikov breaks that streak in striking fashion. Sannikov, it can be fairly said, is a mathematical genius and a high theorist of the first order. He is one of a very small number of people to win three gold medals at the International Math Olympiad – perhaps only Gabriel Carroll, another excellent young theorist, has an equally impressive mathematical background in his youth. Sannikov’s most famous work is in the pure theory of dynamic contracting, which I will spend most of this post discussing, but the methods he has developed turn out to have interesting uses in corporate finance and in macroeconomic models that wish to incorporate a financial sector without using linearization techniques that rob such models of much of their richness. A quick warning: Sannikov’s work is not for the faint of heart, and certainly not for those scared of an equation or two. Economists – and I count myself among this group – are generally scared of differential equations, as they don’t appear in most branches of economic theory (with exceptions, of course: Romer’s 1986 work on endogenous growth, the turnpike theorems, the theory of evolutionary games, etc.). As his work is incredibly technical, I will do my best to provide an overview of his basic technique and its uses without writing down a bunch of equations, but there really is no substitute for going to the mathematics itself if you find these ideas interesting.

The idea of dynamic contracting is an old one. Assume that a risk-neutral principal can commit to a contract that pays an agent on the basis of observed output, with that output being generated this year, next year, and so on. A risk-averse agent takes an unobservable action in every period, which affects output subject to some uncertainty. Payoffs in the future are discounted. Take the simplest possible case: there are two periods, an agent can either work hard or not, output is either 1 or 0, and the probability it is 1 is higher if the agent works hard than otherwise. The first big idea in the dynamic moral hazard of the late 1970s and early 1980s (in particular, Rogerson 1985 Econometrica, Lambert 1983 Bell J. Econ, Lazear and Moore 1984 QJE) is that the optimal contract will condition period 2 payoffs on whether there was a good or bad outcome in period 1; that is, payoffs are history-dependent. The idea is that you can use payoffs in period 2 to induce effort in period 1 (because continuation value increases) and in period 2 (because there is a gap between the payment following good or bad outcomes in that period), getting more bang for your buck. Get your employee to work hard today by dangling a chance at a big promotion opportunity tomorrow, then actually give them the promotion if they work hard tomorrow.

The second big result is that dynamic moral hazard (caveat: at least in cases where saving isn’t possible) isn’t such a problem. In a one-shot moral hazard problem, there is a tradeoff between risk aversion and high powered incentives. I either give you a big bonus when things go well and none if things go poorly (in which case you are induced to work hard, but may be unhappy because much of the bonus is based on things you can’t control), or I give you a fixed salary and hence you have no incentive to work hard. The reason this tradeoff disappears in a dynamic context is that when the agent takes actions over and over and over again, the principle can, using a Law of Large Numbers type argument, figure out exactly the frequency at which the agent has been slacking off. Further, when the agent isn’t slacking off, the uncertainty in output each period is just i.i.d., hence the principal can smooth out the agent’s bad luck, and hence as the discount rate goes to zero there is no tradeoff between providing incentives and the agent’s dislike of risk. Both of these results will hold even in infinite period models, where we just need to realize that all the agent cares about is her expected continuation value following every action, and hence we can analyze infinitely long problems in a very similar way to two period problems (Spear and Srivistava 1987).

Sannikov revisited this literature by solving for optimal or near-to-optimal contracts when agents take actions in continuous rather than discrete time. Note that the older literature generally used dynamic programming arguments and took the discount rate to a limit of zero in order to get interested results. These dynamic programs generally were solved using approximations that formed linear programs, and hence precise intuition of why the model was generating particular results in particular circumstances wasn’t obvious. Comparative statics in particular were tough – I can tell you whether an efficient contract exists, but it is tough to know how that efficient contract changes as the environment changes. Further, situations where discounting is positive are surely of independent interest – workers generally get performance reviews every year, contractors generally do not renegotiate continuously, etc. Sannikov wrote a model where an agent takes actions that control the mean of output continuously over time with Brownian motion drift (a nice analogue of the agent taking an action that each period generates some output that depends on the action and some random term). The agent has the usual decreasing marginal utility of income, so as the agent gets richer over time, it becomes tougher to incentivize the agent with a few extra bucks of payment.

Solving for the optimal contract essentially involves solving two embedded dynamic optimization problems. The agent optimizes effort over time given the contract the principal committed to, and hence the agent chooses an optimal dynamic history-dependent contract given what the agent will do in response. The space of possible history-dependent contracts is enormous. Sannikov shows that you can massively simplify, and solve analytically, for the optimal contract using a four step argument.

First, as in the discrete time approach, we can simplify things by noting that the agent only cares about their continuous-time continuation value following every action they make. The continuation value turns out to be a martingale (conditioning on history, my expectation of the continuation value tomorrow is just my continuation value today), and is basically just a ledger of my promises that I have made to the agent in the future on the basis of what happened in the past. Therefore, to solve for the optimal contract, I should just solve for the optimal stochastic process that determines the continuation value over time. The Martingale Representation Theorem tells me exactly and uniquely what that stochastic process must look like, under the constraint that the continuation value accurately “tracks” past promises. This stochastic process turns out to have a particular analytic form with natural properties (e.g., if you pay flow utility today, you can pay less tomorrow) that depend on the actions the agents take. Second, plug the agent’s incentive compatibility constraint into our equation for the stochastic process that determines the continuation value over time. Third, we just maximize profits for the principal given the stochastic process determining continuation payoffs that must be given to the agent. The principal’s problem determines an HJB equation which can be solved using Ito’s rule plus some effort checking boundary conditions – I’m afraid these details are far too complex for a blog post. But the basic idea is that we wind up with an analytic expression for the optimal way to control the agent’s continuation value over time, and we can throw all sorts of comparative statics right at that equation.

What does this method give us? Because the continuation value and the flow payoffs can be constructed analytically even for positive discount rates, we can actually answer questions like: should you use long-term incentives (continuation value) or short-term incentives (flow payoffs) more when, e.g., your workers have a good outside option? What happens as the discount rate increases? What happens if the uncertainty in the mapping between the agent’s actions and output increases? Answering questions of these types is very challenging, if not impossible, in a discrete time setting.

Though I’ve presented the basic Sannikov method in terms of incentives for workers, dynamic moral hazard – that certain unobservable actions control prices, or output, or other economic parameters, and hence how various institutions or contracts affect those unobservable actions – is a widespread problem. Brunnermeier and Sannikov have a nice recent AER which builds on the intuition of Kiyotaki-Moore models of the macroeconomy with financial acceleration. The essential idea is that small shocks in the financial sector may cause bigger real economy shocks due to deleveraging. Brunnermeier and Sannikov use the continuous-time approach to show important nonlinearities: minor financial shocks don’t do very much since investors and firms rely on their existing wealth, but major shocks off the steady state require capital sales which further depress asset prices and lead to further fire sales. A particularly interesting result is that exogenous risk is low – the economy isn’t very volatile – then there isn’t much precautionary savings, and so a shock that hits the economy will cause major harmful deleveraging and hence endogenous risk. That is, the very calmness of the world economy since 1983 may have made the eventual recession in 2008 worse due to endogenous choices of cash versus asset holdings. Further, capital requirements may actually be harmful if they aren’t reduced following shocks, since those very capital requirements will force banks to deleverage, accelerating the downturn started by the shock.

Sannikov’s entire oeuvre is essentially a graduate course in a new technique, so if you find the results described above interesting, it is worth digging deep into his CV. He is a great choice for the Clark medal, particularly given the deep and rigorous application he has applied his theory to in recent years. There really is no simple version of his results, but his 2012 survey, his recent working paper on moral hazard in labor contracts, and his dissertation work published in Econometrica in 2007 are most relevant. In related work, we’ve previously discussed on this site David Rahman’s model of collusion with continuous-time information flow, a problem very much related to work by Sannikov and his coauthor Andrzej Skrzypacz, as well as Aislinn Bohren’s model of reputation which is related to the single longest theory paper I’ve ever seen, Sannikov and Feingold’s Econometrica on the possibility of “fooling people” by pretending to be a type that you are not. I also like that this year’s JBC makes me look like a good prognosticator: Sannikov is one of a handful of names I’d listed as particularly deserving just two years ago when Gentzkow won!

“Ranking Firms Using Revealed Preference,” I. Sorkin (2015)

Roughly 20 percent of earnings inequality is not driven by your personal characteristics or the type of job you work at, but by the precise firm you work for. This is odd. In a traditional neoclassical labor market, every firm should offer to same wage to workers with the same marginal productivity. If a firm doesn’t do so, surely their workers will quit and go to firms that pay better. One explanation is that since search frictions make it hard to immediately replace workers, firms with market power will wind up sharing rents with their employees. It is costly to search for jobs, but as your career advances, you try to move “up the job ladder” from positions that pay just your marginal product to positions that pay a premium: eventually you wind up as the city bus driver with the six figure contract and once there you don’t leave. But is this all that is going on?

Isaac Sorkin, a job market candidate from Michigan, correctly notes that workers care about the utility their job offers, not the wage. Some jobs stink even though they pay well: 80 hour weeks, high pressure bosses, frequent business travel to the middle of nowhere, low levels of autonomy, etc. We can’t observe the utility a job offers, of course, but this is a problem that always comes up in demand analysis. If a Chipotle burrito and a kale salad cost the same, but you buy the burrito, then you have revealed that you get more utility from the former; this is the old theory of revealed preference. Even though we rarely observe a single person choosing from a set of job offers, we do observe worker flows between firms. If we can isolate workers who leave their existing job for individual reasons, as distinct from those who leave because their entire firm suffers a negative shock, then their new job is “revealed” better. Intuitively, we see a lot of lawyers quit to run a bed and breakfast in Vermont, but basically zero lawyers quitting to take a mining job that pays the same as running a B&B, hence the B&B must be a “better job” than mining, and further if we don’t see any B&B owners quitting to become lawyers, the B&B must be a “better job” than corporate law even if the pay is lower.

A sensible idea, then: the same worker may be paid different amounts in relation to marginal productivity either because they have moved up the job ladder and luckily landed at a firm with market power and hence pay above marginal product (a “good job”), or because different jobs offer different compensating differentials (in which case high paying jobs may actually be “bad jobs” with long hours and terrible work environments). To separate the two rationales, we need to identify the relative attractiveness of jobs, for which revealed preference should work. The problem in practice is both figuring out which workers are leaving for individual reasons, and getting around the problem that it is unusual to observe in the data a nonzero number of people going from firm A to firm B and vice versa.

Sorkin solves these difficulties in a very clever way. Would you believe the secret is to draw on the good old Perron-Frebonius theorem, a trusted tool of microeconomists interested in network structure? How could that be? Workers meet firms in a search process, with firms posting offers in terms of a utility bundle of wages plus amenities. Each worker also has idiosyncratic tastes about things like where to live, how they like the boss, and so on. The number of folks that move voluntarily from job A to job B depends on how big firm A is (bigger firms have more workers that might leave), how frequently A has no negative productivity shocks (in which case moves are voluntary), and the probability a worker from A is offered a job at B when matched and accepts it, which depends on the relative utilities of the two jobs including the individual idiosyncratic portion. An assumption about the distribution of idiosyncratic utility across jobs allows Sorkin to translate probabilities of accepting a job into relative utilities.

What is particularly nice is that the model gives a linear restriction on any two job pairs: the relative probability of moving from A to B instead of B to A depends on the relative utility (abstracting from idiosyncratic portions) adjusted for firm size and offer probability. That is, if M(A,B) is the number of moves from A to B, and V(A) is a (defined in the paper) function of the non-idiosyncratic utility of job A, then

M(A,B)/M(B,A) = V(B)/V(A)

and hence

M(A,B)V(A) = M(B,A)V(B)

Taking this to data is still problematic because we need to restrict to job changes that are not just “my factory went out of business”, and because M(A,B) or M(B,A) are zero for many firm pairs. The first problem is solved by estimating the probability a given job switch is voluntary using the fact that layoff probability is related to the size and growth rate of a firm. The second problem can be solved by noting that if we sum the previous equation over all firms B not equal to A, we have

sum(B!=A)M(A,B)*V(A) = sum(B!=A)M(B,A)*V(B)

or

V(A) = sum(B!=A)M(B,A)*V(B)/sum(B!=A)M(A,B)

The numerator is the number of hires A makes weighted for the non-idiosyncratic utility of firms the hires come from, and the denominator is the number of people that leave firm A. There is one such linear restriction per firm, but the utility of firm A depends on the utility of all firms. How to avoid this circularity? Write the linear restrictions in matrix form, and use the Perron-Frebonius theorem to see that the relative values of V are determined by a particular eigenvector as long as the matrix of moves is strongly connected! Strongly connected just means that there is at least one chain of moves between employers that can get me from firm A to B and vice versa, for all firm pairs!. All that’s left to do now is to take this to the data (not a trivial computation task, since there are so many firms in the US data that calculating eigenvectors will require some numerical techniques).

So what do we learn? Industries like education offer high utility compared to pay, and industries like mining offer the opposite, as you’d expect. Many low paying jobs offer relatively high nonpay utility, and many female-dominated sectors do as well, implying the measured earnings inequality and gender gaps may be overstating the true extent of utility inequality. That is, a teacher making half what a miner makes is partly reflective of the fact that mining is a job that requires compensating differentials to make up for long hours in the dark and dangerous mine shaft. Further, roughly two thirds of the earnings inequality related to firms seems to be reflecting compensating differentials, and since just over 20% of earnings inequality in the US is firm related, this means that about 15% of earnings inequality is just reflecting the differential perceived quality of jobs. This is a surprising result, and it appears to be driven by differences in job amenities that are not easy to measure. Goldman Sachs is a “good job” despite relatively low pay compared to other finance firms because they offer good training and connections. This type of amenity is hard to observe, but Sorkin’s theoretical approach based on revealed preference allows the econometrician to “see” these types of differences across jobs, and hence to more properly understand which jobs are desirable. This is another great example of a question – how does the quality of jobs differ and what does that say about the nature of earnings inequality – that is fundamentally unanswerable by methodological techniques that are unwilling to inject some theoretical assumptions into the analysis.

November 2015 Working Paper. Sorkin has done some intriguing work using historical data on the minimum wage as well. Essentially, minimum wage changes that are not indexed to inflation are only temporary in real terms, so if it costly to switch from labor to machines, you might not do so in response to a “temporary” minimum wage shock. But a permanent increase does appear to cause long run shifts away from labor, something Sorkin sees in industries from apparel in the early 20th century to fast food restaurants. Simon Jäger, a job candidate from Harvard, also has an interesting purely empirical paper about friction in the labor market, taking advantage of early deaths of German workers. When these deaths happen, working in similar roles at the firm see higher wages and lower separation probability for many years, whereas other coworkers see lower wages, with particularly large effects when the dead worker has unusual skills. All quite intuitive from a search model theory of labor, where workers are partial substitutes for folks with the same skills, but complements for folks with firm-specific capital but dissimilar skills. Add these papers to the evidence that efficiency in the search-and-matching process of labor to firms is a first order policy problem.

“Designing Efficient College and Tax Policies,” S. Findeisen & D. Sachs (2014)

It’s job market season, which is a great time of year for economists because we get to read lots of interesting papers. The one, by Dominik Sachs from Cologne and his coauthor Sebastian Findeisen, is particularly relevant given the recent Obama policy announcement about further subsidizing community college. The basic facts of marginal college students are fairly well-known: there is a pretty substantial wage bump for college grads (including ones who are not currently attending but who would attend if college was a little cheaper), many do not go to college even given this wage bump, there are probably externalities both in the economic and social realm from having a more education population though these are quite hard to measure, borrowing constraints bind for some potential college students but don’t appear to be that important, and it is very hard to design policies which benefit only marginal college candidates without also subsidizing those who would go whether or not the subsidy existed.

The naive thought might be “why should we subsidize college in the absence of borrowing constraints? By revealed preference, people choose not to go to college even given the wage bump, which likely implies that for many people studying and spending time going to class gives negative utility. Given the wage bump, these people are apparently willing to pay a lot of money to avoid spending time in college. The social externalities of college probably exist, but in general equilibrium more college-educated workers might drive down the return to college for people who are currently going. Therefore, we ought not distort the market.”

However, Sachs and Findeisen point out that there is also a fiscal externality: higher wages equals higher tax revenue in the future, and only the government cares about that revenue. Even more, the government is risk-neutral, or at least less risk-averse than individuals, about that revenue; people might avoid going to college if, along with bumping up their expected future wages, college also introduces uncertainty into their future wage path. If a subsidy could be targeted largely to students on the margin rather than those currently attending college, and if those marginal students see a big wage bump, and if government revenue less transfers back to the taxpayer is high, then it may be worth it for the government to subsidize college even if there are no other social benefits!

The authors write a nice little structural model. People choose to go to college or not depending on their innate ability, their parent’s wealth, the cost of college, the wage bump they expect (and the variance thereof), and their personal taste or distaste for studying as opposed to working (“psychic costs”). All of those variables aside from personal taste and innate ability can be pulled out of U.S. longitudinal data, performance on the army qualifying test can proxy for innate ability, and given distributional assumptions, we can identify the last free parameter, personal taste, by assuming that people go to college only if their lifetime discounted utility from attendance, less psychic costs, exceeds the lifetime utility from working instead. A choice model of this type seems to match data from previous studies with quasirandom variation concerning the returns to college education.

The direct benefit to the government from higher tax revenue from a subsidy policy, then, is the cost of the subsidy times the number subsidized, minus the proportion of subsidized students who would not have gone to college but for the subsidy times the discounted lifetime wage bump for those students times government tax revenue as a percent of that wage bump. The authors find that a general college subsidy program nearly pays for itself: if you subsidize everyone there aren’t many marginal students, but even for those students the wage bump is substantial. Targeting low income students is even better. Though the low income students affected on the margin tend to be less academically gifted, and hence to earn a lower (absolute) increase in wages from going to college, subsidies targeted at low income students do not waste as much money subsidizing students who would go to college anyway (i.e., a large percentage of high income kids). Note that the subsidies are small enough in absolute terms that the distortion on parental labor supply, from working less in order to qualify for subsidies, is of no quantitative importance, a fact the authors show rigorously. Merit-based subsidies will attract better students who have more to gain from going to college, but they also largely affect people who would go to college anyway, hence offer less bang for the buck to government compared to need-based grants.

The authors have a nice calibrated model in hand, so there are many more questions they ask beyond the direct partial equilibrium benefits of college attendance. For example, in general equilibrium, if we induce people to go to college, the college wage premium will fall. But note that wages for non-college-grads will rise in relative terms, so the net effect of the grants discussed in the previous paragraph on government revenue is essentially unchanged. Further, as Nate Hilger found using quasirandom variation in income due to layoffs, liquidity constraints do not appear to be terribly important for the college making decision: it is increasing grants, not changing loan eligibility, that will do anything of any importance to college attendance.

November 2014 working paper (No IDEAS version). The authors have a handful of other very interesting papers in the New Dynamic Public Finance framework, which is blazing hot right now. As far as I understand the project of NDPF, essentially we can simplify the (technically all-but-impossible-to-solve) dynamic mechanism problem of designing optimal taxes and subsidies under risk aversion and savings behavior to an equivalent reduced form that essentially only depends on simple first order conditions and a handful of elasticities. Famously, it is not obvious that capital taxation should be zero.

Labor Unions and the Rust Belt

I’ve got two nice papers for you today, both exploring a really vexing question: why is it that union-heavy regions of the US have fared so disastrously over the past few decades? In principle, it shouldn’t matter: absent any frictions, a rational union and a profit-maximizing employer ought both desire to take whatever actions generate the most total surplus for the firm, with union power simply affecting how those rents are shared between management, labor and owners. Nonetheless, we notice empirically a couple of particularly odd facts. First, especially in the US, union-dominated firms tend to limit adoption of new, productivity-enhancing technology; the late adoption of the radial tire among U.S. firms is a nice example. Second, unions often negotiate not only about wages but about “work rules”, insisting upon conditions like inflexible employee roles. A great example here is a California longshoremen contract which insisted upon a crew whose sole job was to stand and watch while another crew did the job. Note that preference for leisure can’t explain this, since surely taking that leisure at home rather than standing around the worksite would be preferable for the employees!

What, then, might drive unions to push so hard for seemingly “irrational” contract terms, and how might union bargaining power under various informational frictions or limited commitment affect the dynamic productivity of firms? “Competition, Work Rules and Productivity” by the BEA’s Benjamin Bridgman discusses the first issue, and a new NBER working paper, “Competitive Pressure and the Decline of the Rust Belt: A Macroeconomic Analysis” by Alder, Lagakos and Ohanian covers the second; let’s examine these in turn.

First, work rules. Let a union care first about keeping all members employed, and about keeping wage as high as possible given full employment. Assume that the union cannot negotiate the price at which products are sold. Abstractly, work rules are most like a fixed cost that is a complete waste: no matter how much we produce, we have to incur some bureaucratic cost of guys standing around and the like. Firms will set marginal revenue equal to marginal cost when deciding how much to produce, and at what price that production should be sold. Why would the union like these wasteful costs?

Let firm output given n workers just be n-F, where n is the number of employees, and F is how many of them are essentially doing nothing because of work rules. The firm chooses price p and the number of employees n given demand D(p) and wage w to maximize p*D(p)-w*n, subject to total production being feasible D(p)=n-F. Note that, as long as total firm profits under optimal pricing exceed F, the firm stays in business and its pricing decision, letting marginal revenue equal marginal cost, is unaffected by F. That is, the optimal production quantity does not depend on F. However, the total amount of employment does depend on F, since to produce quantity D(p) you need to employ n-F workers. Hence there is a tradeoff if the union only negotiates wages: to employ more people, you need a lower wage, but using wasteful work rules, employment can be kept high even when wages are raised. Note also that F is limited by the total rents earned by the firm, since if work rules are particularly onerous, firms that are barely breaking even without work rules will simply shut down. Hence in more competitive industries (formally, when demand is less elastic), work rules are less likely to imposed by unions. Bridgman also notes that if firms can choose technology (output is An-F, where A is the level of technology), then unions will resist new technology unless they can impose more onerous work rules, since more productive technology lowers the number of employees needed to produce a given amount of output.

This is a nice result. Note that the work rule requirements have nothing to do with employees not wanting to work hard, since work rules in the above model are a pure waste and generate no additional leisure time for workers. Of course, this result really hinges on limiting what unions can bargain over: if they can select the level of output, or can impose the level of employment directly, or can permit lump-sum transfers from management to labor, then unionized firms will produce at the same productivity at non-unionized firms. Information frictions, among other worries, might be a reason why we don’t see these types of contracts at some unionized firms. With this caveat in mind, let’s turn to the experience of the Rust Belt.

The U.S. Rust Belt, roughly made up of states surrounding the Great Lakes, saw a precipitous decline from the 1950s to today. Alder et al present the following stylized facts: the share of manufacturing employment in the U.S. located in the Rust Belt fell from the 1950s to the mid-1980s, there was a large wage gap between Rust Belt and other U.S. manufacturing workers during this period, Rust Belt firms were less likely to adopt new innovations, and labor productivity growth in Rust Belt states was lower than the U.S. average. After the mid-1980s, Rust Belt manufacturing firms begin to look a lot more like manufacturing firms in the rest of the U.S.: the wage gap is essentially gone, the employment share stabilizes, strikes become much less common, and productivity growth is similar. What happened?

In a nice little model, the authors point out that output competition (do I have lots of market power?) and labor market bargaining power (are my workers powerful enough to extract a lot of my rents?) interact in an interesting way when firms invest in productivity-increasing technology and when unions cannot commit to avoid a hold-up problem by striking for a better deal after the technology investment cost is sunk. Without commitment, stronger unions will optimally bargain away some of the additional rents created by adopting an innovation, hence unions function as a type of tax on innovation. With sustained market power, firms have an ambiguous incentive to adopt new technology – on the one hand, they already have a lot of market power and hence better technology will not accrue too many more sales, but on the other hand, having market power in the future makes investments today more valuable. Calibrating the model with reasonable parameters for market power, union strength, and various elasticities, the authors find that roughly 2/3 of the decline in the Rust Belt’s manufacturing share can be explained by strong unions and little output market competition decreasing the incentive to invest in upgrading technology. After the 1980s, declining union power and more foreign competition limited both disincentives and the Rust Belt saw little further decline.

Note again that unions and firms rationally took actions that lowered the total surplus generated in their industry, and that if the union could have committed not to hold up the firm after an innovation was adopted, optimal technology adoption would have been restored. Alder et al cite some interesting quotes from union heads suggesting that the confrontational nature of U.S. management-union relations led to a belief that management figures out profits, and unions figure out to secure part of that profit for their members. Both papers discussed here show that this type of division, by limiting the nature of bargains which can be struck, can have calamitous effects for both workers and firms.

Bridgman’s latest working paper version is here (RePEc IDEAS page); the latest version of Adler, Lagakos and Ohanian is here (RePEc IDEAS). David Lagakos in particular has a very nice set of recent papers about why services and agriculture tend to have such low productivity, particularly in the developing world; despite his macro background, I think he might be a closet microeconomist!

“Wall Street and Silicon Valley: A Delicate Interaction,” G.-M. Angeletos, G. Lorenzoni & A. Pavan (2012)

The Keynesian Beauty Contest – is there any better example of an “old” concept in economics that, when read in its original form, is just screaming out for a modern analysis? You’ve got coordination problems, higher-order beliefs, signal extraction about underlying fundamentals, optimal policy response by a planner herself informationally constrained: all of these, of course, problems that have consumed micro theorists over the past few decades. The general problem of irrational exuberance when we start to model things formally, though, is that it turns out to be very difficult to generate “irrational” actions by rational, forward-looking agents. Angeletos et al have a very nice model that can generate irrational-looking asset price movements even when all agents are perfectly rational, based on the idea of information frictions between the real and financial sector.

Here is the basic plot. Entrepreneurs get an individual signal and a correlated signal about the “real” state of the economy (the correlation in error about fundamentals may be a reduced-form measure of previous herding, for instance). The entrepreneurs then make a costly investment. In the next period, some percentage of the entrepreneurs have to sell their asset on a competitive market. This may represent, say, idiosyncratic liquidity shocks, but really it is just in the model to abstract away from the finance sector learning about entrepreneur signals based on the extensive margin choice of whether to sell or not. The price paid for the asset depends on the financial sector’s beliefs about the real state of the economy, which come from a public noisy signal and the trader’s observations about how much investment was made by entrepreneurs. Note that the price traders pay is partially a function of trader beliefs about the state of the economy derived from the total investment made by entrepreneurs, and the total investment made is partially a function of the price at which entrepreneurs expect to be able to sell capital should a liquidity crisis hit a given firm. That is, higher order beliefs of both the traders and entrepreneurs about what the other aggregate class will do determine equilibrium investment and prices.

What does this imply? Capital investment is higher in the first stage if either the state of the world is believed to be good by entrepreneurs, or if the price paid in the following period for assets is expected to be high. Traders will pay a high price for an asset if the state of the world is believed to be good. These traders look at capital investment and essentially see another noisy signal about the state of the world. When an entrepreneur sees a correlated signal that is higher than his private signal, he increases investment due to a rational belief that the state of the world is better, but then increases it even more because of an endogenous strategic complementarity among the entrepreneurs, all of whom prefer higher investment by the class as a whole since that leads to more positive beliefs by traders and hence higher asset prices tomorrow. Of course, traders understand this effect, but a fixed point argument shows that even accounting for the aggregate strategic increase in investment when the correlated signal is high, aggregate capital can be read by traders precisely as a noisy signal of the actual state of the world. This means that when when entrepreneurs invest partially on the basis of a signal correlated among their class (i.e., there are information spillovers), investment is based too heavily on noise. An overweighting of public signals in a type of coordination game is right along the lines of the lesson in Morris and Shin (2002). Note that the individual signals for entrepreneurs are necessary to keep the traders from being able to completely invert the information contained in capital production.

What can a planner who doesn’t observe these signals do? Consider taxing investment as a function of asset prices, where high taxes appear when the market gets particularly frothy. This is good on the one hand: entrepreneurs build too much capital following a high correlated signal because other entrepreneurs will be doing the same and therefore traders will infer the state of the world is high and pay high prices for the asset. Taxing high asset prices lowers the incentive for entrepreneurs to shade capital production up when the correlated signal is good. But this tax will also lower the incentive to produce more capital when the actual state of the world, and not just the correlated signal, is good. The authors discuss how taxing capital and the financial sector separately can help alleviate that concern.

Proving all of this formally, it should be noted, is quite a challenge. And the formality is really a blessing, because we can see what is necessary and what is not if a beauty contest story is to explain excess aggregate volatility. First, we require some correlation in signals in the real sector to get the Morris-Shin effect operating. Second, we do not require the correlation to be on a signal about the real world; it could instead be correlation about a higher order belief held by the financial sector! The correlation merely allows entrepreneurs to figure something out about how much capital they as a class will produce, and hence about what traders in the next period will infer about the state of the world from that aggregate capital production. Instead of a signal that correlates entrepreneur beliefs about the state of the world, then, we could have a correlated signal about higher-order beliefs, say, how traders will interpret how entrepreneurs interpret how traders interpret capital production. The basic mechanism will remain: traders essentially read from aggregate actions of entrepreneurs a noisy signal about the true state of the world. And all this beauty contest logic holds in an otherwise perfectly standard Neokeynesian rational expectations model!

2012 working paper (IDEAS version). This paper used to go by the title “Beauty Contests and Irrational Exuberance”; I prefer the old name!

Dale Mortensen as Micro Theorist

Northwestern’s sole Nobel Laureate in economics, Dale Mortensen, passed overnight; he remained active as a teacher and researcher over the past few years, though I’d be hearing word through the grapevine about his declining health over the past few months. Surely everyone knows Mortensen the macroeconomist for his work on search models in the labor market. There is something odd here, though: Northwestern has really never been known as a hotbed of labor research. To the extent that researchers rely on their coworkers to generate and work through ideas, how exactly did Mortensen became such a productive and influential researcher?

Here’s an interpretation: Mortensen’s critical contribution to economics is as the vector by which important ideas in micro theory entered real world macro; his first well-known paper is literally published in a 1970 book called “Microeconomic Foundations of Employment and Inflation Theory.” Mortensen had the good fortune to be a labor economist working in the 1970s and 1980s at a school with a frankly incredible collection of microeconomic theorists; during those two decades, Myerson, Milgrom, Loury, Schwartz, Kamien, Judd, Matt Jackson, Kalai, Wolinsky, Satterthwaite, Reinganum and many others were associated with Northwestern. And this was a rare condition! Game theory is everywhere today, and pioneers in that field (von Neumann, Nash, Blackwell, etc.) were active in the middle of the century. Nonetheless, by the late 1970s, game theory in the social sciences was close to dead. Paul Samuelson, the great theorist, wrote essentially nothing using game theory between the early 1950s and the 1990s. Quickly scanning the American Economic Review from 1970-1974, I find, at best, one article per year that can be called game-theoretic.

What is the link between Mortensen’s work and developments in microeconomic theory? The essential labor market insight of search models (an insight which predates Mortensen) is that the number of hires and layoffs is substantial even in the depth of a recession. That is, the rise in the unemployment rate cannot simply be because the marginal revenue of the potential workers is always less than the cost, since huge numbers of the unemployed are hired during recessions (as others are fired). Therefore, a model which explains changes in churn rather than changes in the aggregate rate seems qualitatively important if we are to develop policies to address unemployment. This suggests that there might be some use in a model where workers and firms search for each other, perhaps with costs or other frictions. Early models along this line by Mortensen and others were generally one-sided and hence non-strategic: they had the flavor of optimal stopping problems.

Unfortunately, Diamond in a 1971 JET pointed out that Nash equilibrium in two-sided search leads to a conclusion that all workers are paid their reservation wage: all employers pay the reservation wage, workers believe this to be true hence do not engage in costly search to switch jobs, hence the belief is accurate and nobody can profitably deviate. Getting around the “Diamond Paradox” involved enriching the model of who searches when and the extent to which old offers can be recovered; Mortensen’s work with Burdett is a nice example. One also might ask whether laissez faire search is efficient or not: given the contemporaneous work of micro theorists like Glenn Loury on mathematically similar problems like the patent race, you might imagine that efficient search is unlikely.

Beyond the efficiency of matches themselves is the question of how to split surplus. Consider a labor market. In the absence of search frictions, Shapley (first with Gale, later with Shubik) had shown in the 1960s and early 1970s the existence of stable two-sided matches even when “wages” are included. It turns out these stable matches are tightly linked to the cooperative idea of a core. But what if this matching is dynamic? Firms and workers meet with some probability over time. A match generates surplus. Who gets this surplus? Surely you might imagine that the firm should have to pay a higher wage (more of the surplus) to workers who expect to get good future offers if they do not accept the job today. Now we have something that sounds familiar from non-cooperative game theory: wage is based on the endogenous outside options of the two parties. It turns out that noncooperative game theory had very little to say about bargaining until Rubinstein’s famous bargaining game in 1982 and the powerful extensions by Wolinsky and his coauthors. Mortensen’s dynamic search models were a natural fit for those theoretic developments.

I imagine that when people hear “microfoundations”, they have in mind esoteric calibrated rational expectations models. But microfoundations in the style of Mortensen’s work is much more straightforward: we simply cannot understand even the qualitative nature of counterfactual policy in the absence of models that account for strategic behavior. And thus the role for even high micro theory, which investigates the nature of uniqueness of strategic outcomes (game theory) and the potential for a planner to improve welfare through alternative rules (mechanism design). Powerful tools indeed, and well used by Mortensen.

“The Great Diversification and Its Unraveling,” V. Carvalho and X. Gabaix (2013)

I rarely post about macro papers here, but I came across this interesting result by Carvalho and Gabaix in the new AER. Particularly in the mid-2000s, it was fashionable to talk about a “Great Moderation” – many measures of economic volatility fell sharply right around 1983 and stayed low. Many authors studied the potential causes. Was it a result of better monetary policy (as seemed to be the general belief when I working at the Fed) or merely good luck? Ben Bernanke summarized the rough outline of this debate in a 2004 speech.

The last few years have been disheartening for promoters of good policy, since many measures of economic volatility have again soared since the start of the financial crisis. So now we have two facts to explain: why did volatility decline, and why did it rise again? Dupor, among others, has pointed out the difficulty of generating aggregate fluctuations from sectoral shocks: as the number of sectors increase, then independence of shocks will generate little aggregate volatility as the number of sectors grows large. Recent work by the authors of the present paper get around that concern by noting the granularity of important sectors (Gabaix), or the network structure of economic linkages (Carvalho).

In this paper, the authors show theoretically that a measure of “fundamental volatility” in total factor productivity should be linked to volatility in GDP, and that that measure of volatility is essentially composed of three factors: the ratio of sectoral gross output to value added, a diversification effect where volatility declines if the economy where value-added shares of the economy are spread across more sectors, and a compositional effect, where the economy contains fewer sectors with high gross output to GDP. They then compare their measure of fundamental volatility, constructed using data on 88 sectors, to overall GDP volatility, and find that it fits the data well, both in the US and overseas.

What, then, caused the shifts in fundamental volatility? The decline in the early 1980s appears to be heavily driven by a declining share of the economy in machinery, primary metals and the like. These sectors are heavily integrated into other areas of the economy as users and producers of intermediate inputs, so it is no surprise that a decline in their share of the economy will reduce overall volatility. The rise in the 2000s appears to result almost wholly from the increasing importance of the financial sector, an individually volatile sector. Given that the importance of this sector had been rising since the late 1990s, a measure of fundamental volatility (or, better, a firm-level measure, which is difficult to do given currently existing data) could have provided an “early warning” that the Great Moderation would soon come to an end.

August 2012 working paper (IDEAS version). Final paper in AER 103(5), 2013.

Financial Crisis Reading Lists

The Journal of Economic Literature is really a great service to economists. It is a journal that publishes up-to-date literature reviews and ideas for future research in many small subfields that would otherwise be impenetrable. A recent issue published two articles trying to help us non-macro and finance guys catch up on the “facts” of the 2007 financial crisis. The first is Gorton and Metrick, “Getting up to Speed on the Global Financial Crisis: A One Weekend Reader’s Guide,” and the second is Andrew Lo’s “Reading About the Financial Crisis: A Twenty-One Book Review.”

There is a lot of popular confusion about what caused the financial crisis, what amplified it, and what the responses have been. Roughly, we can all agree that first there was a massive rise in house prices, not only in the US; second, there was new, enormous pools of institutional investments looking for safe returns, with many of these pools being operated by risk-averse Asian governments; third, house prices peaked in the US and elsewhere in 2006; fourth, in August 2007, problems with mortgage-related bonds led to interbank repo funding problems, requiring massive liquidity help from central banks; fifth, in September 2008, Lehman Brothers filed for bankruptcy, leading a money market fund to “break the buck” and causing massive flight away from assets related to investment banks or assets not explicitly backed by strong governments; sixth, a sovereign debt problem has arisen in a number of periphery countries since, particularly in Europe.

Looking through the summaries provided in these two reading lists, I only see four really firm additional facts. First, as Andrew Lo has pointed out many times, leverage at investment banks was not terribly high by global standards. Second, arguments that the crisis was caused by investment banks packaging worthless securities and then fooling buyers, while containing a grain of truth, does not explain the crisis: indeed, the bigger problem was how many of these worthless securities were still on bank’s own balance sheets in 2008, explicitly or implicitly through CDOs and other instruments. Third, rising total leverage across an economy as a whole is strongly related to banking crises, a point made best in Reinhart and Rogoff’s work, but also in a new AER by Schularick and Taylor. Fourth, the crisis in the financial sector transmitted to the real economy principally via restrictions on credit to real economy firms. Campello, Graham and Harvey, in a 2010 JFE, used a large-scale 2008 survey of global CFOs to show how firms who were credit constrained before the financial crisis were much more likely to have to cut back on hiring and investment spending, regardless of their profitability or the usefulness of their investment opportunities. That is, savings fled to safety because of the uncertain health of banking intermediaries, which led banks to cut back commercial lending, which led to a recession in the real economy.

What’s interesting is how little the mortgage market, per se, had to do with the crisis. I was at the Fed before the crisis, and remember coauthoring in early 2007 an internal memo about the economic effects of a downturn in the housing sector. The bubble in housing prices was obvious to (almost) everyone at the Fed. But the size of the mortgage market (in terms of wealth) and the construction and home-improvement sector (in terms of employment) was simply not that big; certainly, the massive stock losses after the dot-com bubble had more real effects because of declines in total wealth. I can only imagine that everyone on Wall Street also knew this. What was unexpected was the way in which these particular losses in wealth harmed the financial health of banks; in particular, the location of losses because of the huge number of novel derivatives was really opaque. And we know, both then because of the very-popular-at-the-Fed theoretical work of Diamond and Dybvig, and now because of the empirical work of Reinhart and Rogoff, that bank runs, whether in the proper or in the “shadow” banking system, have real effects that are very difficult to contain.

If you’ve got some free time this weekend, particularly if you’re not a macroeconomist, it’s worth looking through the references in the Lo and Gorton/Metrick papers.

Lo’s “Twenty-One Book Review” (2012) (IDEAS version). Gorton and Metrick’s “Getting Up to Speed” (IDEAS version).