Hang ’em with probability zero: Why does it not work?

A celebrated result in the economics of crime, which we call the Becker proposition (BP), states that it is optimal to impose the severest possible punishment (to maintain effective deterrence) at the lowest possible probability (to economize on enforcement costs). Several other applications, some unrelated to the economics of crime, arise when an economic agent faces punishments/ rewards with very low probabilities. For instance, insurance against low probability events, principal-agent contracts that impose punitive fines, seat belt usage and the usage of mobile phones among drivers etc. However, the BP, and the other applications mentioned above, are at variance with the evidence. The BP has largely been considered within an expected utility framework (EU). We re-examine the BP under rank dependent expected utility (RDU) and prospect theory (PT). We find that the BP always holds under RDU. However, under plausible scenarios within PT it does not hold, in line with the evidence.Behavioral economics; Illegal activity; Expected utility theory; Rank dependent expected utility; Prospect theory; Prelec and higher order Prelec probability weighting functions

An extension of the Becker proposition to non-expected utility theory

In a seminal paper, Becker (1968) showed that the most efficient way to deter crime is to impose the severest possible penalty (to maintain adequate deterrence) with the lowest possible probability (to economize on costs of enforcement). We shall call this the Becker proposition (BP). The BP is derived under the assumptions of expected utility theory (EU). However, EU is heavily rejected by the evidence.  A range of non-expected utility theories have been proposed to explain the evidence.  The two leading alternatives to EU are rank dependent utility (RDU) and cumulative prospect theory (CP). The main contributions of this paper are: (1) We formalize the BP in a more satisfactory manner. (2) We show that the BP holds under RDU and CP. (3) We give a formal behavioral approach to crime and punishment that could have applicability to a wide range of problems in the economics of crime.Crime and punishment; non-linear weighting of probabilities; cumulative prospect theory; rank dependent utility; probability weighting functions; punishment functions.

A value function that explains the magnitude and sign effects

Two of the anomalies of the exponentially discounted utility model are the 'magnitude effect' (larger magnitudes are discounted less) and the 'sign effect' (a loss is discounted less than a gain of the same magnitude). The literature has followed Loewenstein and Prelec (1992) in attributing the magnitude effect to the increasing elasticity of the value function and the sign effect to a higher elasticity for losses as compared to gains. We provide a simple, tractable, functional form that has these two properties, which we call the simple increasing elasticity value function (SIE). These functional forms underpin the main explanation of the magnitude and sign effects and may aid applications and further theoretical development.Anomalies of the exponentially discounted utility model; the magnitude effect; the sign effect; SIE value functions

Comparing the first-best and second-best provision of a club good: an example.

Excludable and congestible shared goods - club goods (e.g., internet access facilities) - are more prevalent than Samuelsonian public goods. Our example shows that, unlike the usual presumption with pure public goods, the optimal second-best supply of a club good might exceed its first-best level. We argue that this arises because user charges can be levied on club goods the government need not impose distortionary taxes to finance them. Thus, the first and second best in a club economy differ mainly because informational constraints prevent the government achieving the right income distribution in the latter.

Competitive Charitable Giving and Optimal Public Policy with Multiple Equilibria

Consider a large number of small individuals contributing to a charity or to a public good. We study the properties of a competitive equilibrium in giving and allow for multiple equilibria. Our proposed condition, aggregate strategic complementarity, is a necessary condition for multiple equilibria. Consider two equilibria with low (L) and high (H) levels of giving. Comparative statics at L could be perverse (subsidies reduce giving) while those at H could be normal (subsidies induce giving), which rules out the use of incentives at L. We demonstrate how public policy, in the form of temporary direct government grants to charity can engineer a move from L to H. We use a welfare analysis to determine the optimal mix of private and public contributions to charity. Our paper contributes to the broader and more fundamental question of using public policy to engineer moves between multiple equilibria. Multiple equilibria; privately supplied public goods; aggregate strategic substitutes and complements; competitive and non-cooperative equilibria; direct grants; charitable redistribution; voluntary contributions to public goods; optimal mix of public and private giving.

Insurance and Probability Weighting Functions

Evidence shows that (i) people overweight low probabilities and underweight high probabilities, but (ii) ignore events of extremely low probability and treat extremely high probability events as certain. Decision models, such as rank dependent utility (RDU) and cumulative prospect theory (CP), use probability weighting functions. Existing probability weighting functions incorporate (i) but not (ii). Our contribution is threefold. First, we show that this would lead people, even in the presence of fixed costs and actuarially unfair premiums, to insure fully against losses of sufficiently low probability. This is contrary to the evidence. Second, we introduce a new class of probability weighting functions, which we call higher order Prelec probability weighting functions, that incorporate (i) and (ii). Third, we show that if RDU or CP are combined with our new probability weighting function, then a decision maker will not buy insurance against a loss of sufficiently low probability; in agreement with the evidence. We also show that our weighting function solves the St. Petersburg paradox that reemerges under RDU and CP.Decision making under risk; Prelec’s probability weighting function; Higher order Prelec probability weighting functions; Behavioral economics; Rank dependent utility theory; Prospect theory; Insurance; St. Petersburg paradox

Optimal income taxation in the presence of tax evasion: Expected utility versus prospect theory

The predictions of expected utility theory (EUT) applied to tax evasion are flawed on two counts: (i) They are quantitatively in error by huge orders of magnitude. (ii) Higher taxation is predicted to lower evasion, which is at variance with the evidence. An emerging literature in behavioral economics, most notably based on prospect theory (PT), has shown that behavioral economics is much better at explaining tax evasion. We extend this literature to incorporate issues of optimal taxation. As a benchmark for a successful theory, we require that it should explain, jointly, the facts on the tax rate, tax gap and the level of government expenditure. We find that when taxpayers use EUT (respectively, PT) and the optimal tax is derived from a social welfare function that also uses EUT (respectively, PT), then, the calibration results are completely at odds with the facts. However, when taxpayers use PT but the social welfare function uses standard EUT, there is a very close match between the predictions and the facts. This has important implications for context dependent preferences but also for the newly emerging literature on liberalism versus paternalism in behavioral economics.Prospect theory; Expected utility theory; Tax evasion; Optimal taxation; Normative versus positive economics; Context dependent preferences; Liberalism; Paternalism

Composite Prospect Theory: A proposal to combine ‘prospect theory’ and ‘cumulative prospect theory’

Evidence shows that (i) people overweight low probabilities and underweight high probabilities, but (ii) ignore events of extremely low probability and treat extremely high probability events as certain. The main alternative decision theories, rank dependent utility (RDU) and cumulative prospect theory (CP) incorporate (i) but not (ii). By contrast, prospect theory (PT) addresses (i) and (ii) by proposing an editing phase that eliminates extremely low probability events, followed by a decision phase that only makes a choice from among the remaining alternatives. However, PT allows for the choice of stochastically dominated options, even when such dominance is obvious. We propose to combine PT and CP into composite cumulative prospect theory (CCP). CCP combines the editing and decision phases of PT into one phase and does not allow for the choice of stochastically dominated options. This, we believe, provides the best available alternative among decision theories of risk at the moment. As illustrative examples, we also show that CCP allows us to resolve three paradoxes: the insurance paradox, the Becker paradox and the St. Petersburg paradox.Decision making under risk; Composite Prelec probability weighting functions; Composite cumulative prospect theory; Composite rank dependent utility theory; Insurance; St. Petersburg paradox; Becker.s paradox

Non-Linearities, Large Forecasters And Evidential Reasoning Under Rational Expectations

Rational expectations is typically taken to mean that, conditional on the information set and the relevant economic theory, the expectation formed by an economic agent should be equal to its mathematical expectation. This is correct only when actual inflation is “linear” in the aggregate inflationary expectation or if it is non-linear then forecasters are “small” and use “causal reasoning”. We show that if actual in- flation is non-linear in expected inflation and (1) there are “large” forecasters, or, (2) small/ large forecasters who use “evidential reasoning”, then the optimal forecast does not equal the mathematical expectation of the variable being forecast. We also show that when actual inflation is non-linear in aggregate inflation there might be no solution if one identifies rational expectations with equating the expectations to the mathematical average, while there is a solution using the “correct” forecasting rule under rational expectations. Furthermore, results suggest that published forecasts of inflation may be systematically different from the statistical averages of actual inflation and output, on average, need not equal the natural rate. The paper has fundamental implications for macroeconomic forecasting and policy, testing the assumptions and implications of market efficiency and for rational expectations in general.Non-linearities; large forecasters; evidential reasoning; rational expectations; endogenous forecasts; classical and behavioral game theory