We build a simple model of leveraged asset purchases with margin calls. Investment funds use what is perhaps the most basic financial strategy, called “value investing,” i.e., systematically attempting to buy underpriced assets. When funds do not borrow, the price fluctuations of the asset are approximately normally distributed and uncorrelated across time. This changes when the funds are allowed to leverage, i.e., borrow from a bank, which allows them to purchase more assets than their wealth would otherwise permit. During good times funds that use more leverage have higher profits, increasing their wealth and making them dominant in the market. However, if a downward price fluctuation occurs while one or more funds are fully leveraged, the resulting margin call causes them to sell into an already falling market, amplifying the downward price movement. If the funds hold large positions in the asset this can cause substantial losses. This in turns leads to clustered volatility: Before a crash, when the value funds are dominant, they damp volatility, and after the crash, when they suffer severe losses, volatility is high. This leads to power law tails which are both due to the leverage-induced crashes and due to the clustered volatility induced by the wealth dynamics. This is in contrast to previous explanations of fat tails and clustered volatility, which depended on “irrational behavior,” such as trend following. A standard (supposedly more sophisticated) risk control policy in which individual banks base leverage limits on volatility causes leverage to rise during periods of low volatility, and to contract more quickly when volatility gets high, making these extreme fluctuations even worse.
Conventional economics supposes that agents value the present vs. the future using an exponential discounting function. In contrast, experiments with animals and humans suggest that agents are better described as hyperbolic discounters, whose discount function decays much more slowly at large times, as a power law. This is generally regarded as being time inconsistent or irrational. We show that when agents cannot be sure of their own future one-period discount rates, then hyperbolic discounting can become rational and exponential discounting irrational. This has important implications for environmental economics, as it implies a much larger weight for the far future.
The use of equilibrium models in economics springs from the desire for parsimonious models of economic phenomena that take human reasoning into account. This approach has been the cornerstone of modern economic theory. We explain why this is so, extolling the virtues of equilibrium theory; then we present a critique and describe why this approach is inherently limited, and why economics needs to move in new directions if it is to continue to make progress. We stress that this shouldn’t be a question of dogma, but should be resolved empirically. There are situations where equilibrium models provide useful predictions and there are situations where they can never provide useful predictions. There are also many situations where the jury is still out, i.e., where so far they fail to provide a good description of the world, but where proper extensions might change this. Our goal is to convince the skeptics that equilibrium models can be useful, but also to make traditional economists more aware of the limitations of equilibrium models. We sketch some alternative approaches and discuss why they should play an important role in future research in economics.
We provide a theory of pricing for emerging asset classes, like emerging markets, that are not yet mature enough to be attractive to the general public. Our model provides an explanation for the volatile access of emerging economies to international financial markets and for several stylized facts we identify in the data during the 1990’s. We present a general equilibrium model with incomplete markets and endogenous collateral and an extension encompassing adverse selection. We show that contagion, flight to liquidity and issuance rationing can occur in equilibrium during what we call global anxious times.
We prove the existence of monetary equilibrium in a finite horizon economy with production. We also show that if agents expect the monetary authority to significantly decrease the supply of bank money available for short term loans in the future, then the economy will fall into a liquidity trap today.
Keywords: Central bank, Inside money, Outside money, Incomplete assets, Production, Monetary equilibrium, Liquidity, Liquidity trap
We introduce grading into games of status. Each player chooses effort, producing a stochastic output or score. Utilities depend on the ranking of all the scores. By clustering scores into grades, the ranking is coarsened, and the incentives to work are changed.
We first apply games of status to grading exams. Our main conclusion is that if students care primarily about their status (relative rank) in class, they are often best motivated to work not by revealing their exact numerical exam scores (100,99,…,1), but instead by clumping them into coarse categories (A,B,C).
When student abilities are disparate, the optimal grading scheme is always coarse. Furthermore, it awards fewer A’s than there are alpha-quality students, creating small elites. When students are homogeneous, we characterize optimal grading schemes in terms of the stochastic dominance between student performances (when they shirk or work) on subintervals of scores, showing again why coarse grading may be advantageous.
In both the disparate case and the homogeneous case, we prove that absolute grading is better than grading on a curve, provided student scores are independent.
We next bring games of money and status to bear on the optimal wage schedule: workers can be motivated not merely by the purchasing power of wages, but also by the status higher wages confer. How should the employer combine both incentive devices to generate an optimal pay schedule?
When workers’ abilities are disparate, the optimal wage schedule creates different grades than we found with status incentives alone. The very top type should be motivated solely by money, with enormous salaries going to a tiny elite. Furthermore, if the population of workers diminishes as we go up the ability ladder and their disutility for work does not fall as fast, then the optimal wage schedule exhibits increasing wage differentials, despite the linearity in production.
When workers are homogeneous, the same status grades are optimal as we found with status incentives alone. A bonus is paid only to scores in the top status grade.
We show that if agents are risk neutral, prizes outperform wages if and only if there is sufficient pride and envy relative to the noisiness of performance. If agents are risk averse, prizes are a necessary supplement to wages (as bonuses).
We show that if students care primarily about their status (relative rank) in class, they are best motivated to work not by revealing their exact numerical exam scores (100,99,…,1), but instead by clumping them in broad categories (A,B,C). If their abilities are disparate, the optimal grading scheme awards fewer A’s than there are alpha-quality students, creating small elites. If their abilities are common knowledge, then it is better to grade them on an absolute scale (100 to 90 is an A, etc.) rather than on a curve (top 15% is an A, etc.). We develop criteria for optimal grading schemes in terms of the stochastic dominance between student performances.
We define liquidity as the flexibility to move goods (money) from one project (investment) to another. We show that credit constraints on demand by themselves can cause an under-supply of liquidity, without the uncertainty, intermediation, asymmetric information or complicated international financial framework used in other models in the literature. In this respect liquidity is like a commodity: according to our offsetting distortions principle, a distortion in the demand for any good can often be understood as an inefficiency of supply.
We show that the liquidity under-supply is a non-monotone function of the credit constraint. This result is also a particular case of a more general principle applying to any commodity with supply alternatives: second best supply inefficiency is non-monotone in the demand distortion. Defining liquidity as flexibility ensures that there will be alternatives, and thus non monotonicity. If we interpret the credit constraints as the degree of financial development in the economy, our second proposition suggests that when financial markets are very undeveloped, as in some emerging markets, financial innovation may paradoxically make government intervention (taxation) more necessary.
Finally, we think about the magnitude of the under-supply in the context of a specific demand distortion. We model the credit constraint by assuming that borrowers will default unless their promises are covered by collateral. Further, we assume that only an exogenous proportion beta of a durable good can serve as collateral. This parameter will represent the degree of financial development of the economy. We show that when the price of the collateral is endogenous, the magnitude of the under supply can be much larger. Any policy intervention that affects the interest rate in equilibrium will have two effects on the borrowing constraint: a direct effect, also present in the case when the credit constraint is exogenous, and an indirect effect through the price of the collateral. We explore our findings by solving and simulating a particular case in which utilities for the consumption good and collateral are quadratic.
We show that very little is needed to create liquidity under-supply in equilibrium. Credit constraints on demand by themselves can cause an under-supply of liquidity, without the uncertainty, intermediation, asymmetric information or complicated international financial framework used in other models in the literature. We show that the under-supply is a non-monotone function of the demand distortion that causes it, a result that may have interesting implications for emerging markets economies. Finally, when we make the credit constraint endogenous, the inefficiency can be large due to the presence of a multiplier.
The classical Fisher equation asserts that in a nonstochastic economy, the inflation rate must equal the difference between the nominal and real interest rates. We extend this equation to a representative agent economy with real uncertainty in which the central bank sets the nominal rate of interest. The Fisher equation still holds, but with the rate of inflation replaced by the harmonic mean of the growth rate of money. Except for logarithmic utility, we show that on almost every path the long-run rate of inflation is strictly higher than it would be in the nonstochastic world obtained by replacing output with expected output in every period. If the central bank sets the nominal interest rate equal to the discount rate of the representative agent, then the long-run rate of inflation is positive (and the same) on almost every path. By contrast, the classical Fisher equation asserts that inflation should then be zero. In fact, no constant interest rate will stabilize prices, even if the economy is stationary with bounded i.d.d. shocks. The central bank must actively manage interest rates if it wants to keep prices bounded forever. However, not even an active central bank can keep prices exactly constant.
Keywords: Inflation, Equilibrium, Control, Interest rate, Central bank, Harmonic Fisher equation
We build a finite horizon model with inside and outside money, in which interest rates, price levels and commodity allocations are determinate, even though asset markets are incomplete and asset deliveries are purely nominal.
Keywords: Central bank, Inside money, Outside money, Incomplete assets, Monetary equilibrium, Determinacy
Arrow’s original proof of his impossibility theorem proceeded in two steps: showing the existence of a decisive voter, and then showing that a decisive voter is a dictator. Barbera replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step. I give three brief proofs, all of which turn on replacing the decisive/pivotal voter with an extremely pivotal voter (a voter who by unilaterally changing his vote can move some alternative from the bottom of the social ranking to the top), thereby simplifying both steps in Arrow’s proof.
My first proof is the most straightforward, and the second uses Condorcet preferences (which are transformed into each other by moving the bottom alternative to the top). The third (and shortest) proof proceeds by reinterpreting Step 1 of the first proof as saying that all social decisions are made the same way (neutrality).
We build a finite horizon model with inside and outside money, in which interest rates, price levels and commodity allocations are determinate, even though asset markets are incomplete and asset deliveries are purely nominal.
Keywords: Central bank, Inside money, Outside money, Incomplete assets, Monetary equilibrium, Real determinacy