Price Equation Research Articles

The Price equation and the causal analysis of evolutionary change.

Though the Price equation in itself is simply a statistical identity, biologists have often adopted a 'causal interpretation' of the equation, in the sense that its component terms have been supposed to correspond to distinct causal processes in evolution, such as natural selection and transmission bias. In this paper, we bring the issue of causal interpretation to the fore, by studying the conditions under which it is legitimate to read causal meaning into the Price equation. We argue that only if substantive assumptions about causal structure are made, which can be represented in the form of a causal model, can the component terms of the Price equation be interpreted as causally meaningful. We conclude with a reflection on the epistemic uses of the Price equation, emphasizing the difference between the description, explanation and prediction of evolutionary change. This article is part of the theme issue 'Fifty years of the Price equation'.

The Price equation and evolutionary epidemiology.

The Price equation has found widespread application in many areas of evolutionary biology, including the evolutionary epidemiology of infectious diseases. In this paper, we illustrate the utility of this approach to modelling disease evolution by first deriving a version of Price's equation that can be applied in continuous time and to populations with overlapping generations. We then show how this version of Price's equation provides an alternative perspective on pathogen evolution by considering the epidemiological meaning of each of its terms. Finally, we extend these results to the case where population size is small and generates demographic stochasticity. We show that the particular partitioning of evolutionary change given by Price's equation is also a natural way to partition the evolutionary consequences of demographic stochasticity, and demonstrate how such stochasticity tends to weaken selection on birth rate (e.g. the transmission rate of an infectious disease) and enhance selection on mortality rate (e.g. factors, like virulence, that cause the end of an infection). In the long term, if there is a trade-off between virulence and transmission across parasite strains, the weaker selection on transmission and stronger selection on virulence that arises from demographic stochasticity will tend to drive the evolution of lower levels of virulence. This article is part of the theme issue 'Fifty years of the Price equation'.

The gene's eye view, the Gouldian knot, Fisherian swords and the causes of selection.

The biological units-of-selection debate has centred on questions of which units experience selection and adaptation. Here, I use a causal framework and the Price equation to develop the gene's eye perspective. Genes are causally special in being both replicators and interactors. Gene effects are tied together in a complex Gouldian knot of interactions, but Fisher deployed three swords to try to cut the knot. The first, Fisher's average excess, is non-causal, so not fully satisfactory in that respect. The Price equation highlights Fisher's other two swords, choosing to model only selection, and only the part that is transmissible across generations. The models developed here show that many causes of organismal fitness do not cause Gouldian complications. Only two kinds of elements must be added to the focal gene for a causal explanation of its selective change: co-replicators that are associated with the focal gene and co-interactors that interact non-additively with the focal gene. Identical equations for co-replication and co-interaction describe interactions between gene copies at a single locus or at separate loci, and also for genes situated within the same individual or in different individuals. These results resolve some of the objections to the gene's eye view. This article is part of the theme issue 'Fifty years of the Price equation'.

Selection, adaptation, inheritance and design in human culture: the view from the Price equation.

For decades, parts of the literature on human culture have been gripped by an analogy: culture changes in a way that is substantially isomorphic to genetic evolution. This leads to a number of sub-claims: that design-like properties in cultural traditions should be explained in a parallel way to the design-like features of organisms, namely with reference to selection; that culture is a system of inheritance; and that cultural evolutionary processes can produce adaptation in the genetic sense. The Price equation provides a minimal description of any evolutionary system, and a method for identifying the action of selection. As such, it helps clarify some of these claims about culture conceptually. Looking closely through the lens of the Price equation, the differences between genes and culture come into sharp relief. Culture is only a system of inheritance metaphorically, or as an idealization, and the idealization may lead us to overlook causally important features of how cultural influence works. Design-like properties in cultural systems may owe more to transmission biases than to cultural selection. Where culture enhances genetic fitness, it is ambiguous whether what is doing the work is cultural transmission, or just the genetically evolved properties of the mind. I conclude that there are costs to trying to press culture into a template based on Darwinian evolution, even if one broadens the definition of ‘Darwinian’.This article is part of the theme issue ‘Fifty years of the Price equation’.

Different perspectives on non-genetic inheritance illustrate the versatile utility of the Price equation in evolutionary biology.

The diversity of genetic and non-genetic processes that make offspring resemble their parents are increasingly well understood. In addition to genetic inheritance, parent-offspring similarity is affected by epigenetic, behavioural and cultural mechanisms that collectively can be referred to as non-genetic inheritance. Given the generality of the Price equation as a description of evolutionary change, is it not surprising that the Price equation has been adopted to model the evolutionary implications of non-genetic inheritance. In this paper, we briefly introduce the heredity perspectives on which those models rely, discuss the extent to which these perspectives make different assumptions and place different emphases on the roles of heredity and development in evolution, and the types of empirical research programmes they motivate. The existence of multiple perspectives and explanatory aims highlight, on the one hand, the versatility of the Price equation and, on the other hand, the importance of understanding how heredity and development can be conceptualized in evolutionary studies. This article is part of the theme issue 'Fifty years of the Price equation'.

Group and individual selection during evolutionary transitions in individuality: meanings and partitions.

The Price equation embodies the 'conditions approach' to evolution in which the Darwinian conditions of heritable variation in fitness are represented in equation form. The equation can be applied recursively, leading to a partition of selection at the group and individual levels. After reviewing the well-known issues with the Price partition, as well as issues with a partition based on contextual analysis, we summarize a partition of group and individual selection based on counterfactual fitness, the fitness that grouped cells would have were they solitary. To understand 'group selection' in multi-level selection models, we assume that only group selection can make cells suboptimal when they are removed from the group. Our analyses suggest that there are at least three kinds of selection that can be occurring at the same time: group-specific selection along with two kinds of individual selection, within-group selection and global individual selection. Analyses based on counterfactual fitness allow us to specify how close a group is to being a pseudo-group, and this can be a basis for quantifying progression through an evolutionary transition in individuality (ETI). During an ETI, fitnesses at the two levels, group and individual, become decoupled, in the sense that fitness in a group may be quite high, even as counterfactual fitness goes to zero. This article is part of the theme issue 'Fifty years of the Price equation'.

The 'algebra of evolution': the Robertson-Price identity and viability selection for body mass in a wild bird population.

By the Robertson–Price identity, the change in a quantitative trait owing to selection, is equal to the trait's covariance with relative fitness. In this study, we applied the identity to long-term data on superb fairy-wrens Malurus cyaneus, to estimate phenotypic and genetic change owing to juvenile viability selection. Mortality in the four-week period between fledging and independence was 40%, and heavier nestlings were more likely to survive, but why? There was additive genetic variance for both nestling mass and survival, and a positive phenotypic covariance between the traits, but no evidence of additive genetic covariance. Comparing standardized gradients, the phenotypic selection gradient was positive, βP = 0.108 (0.036, 0.187 95% CI), whereas the genetic gradient was not different from zero, βA = −0.025 (−0.19, 0.107 95% CI). This suggests that factors other than nestling mass were the cause of variation in survival. In particular, there were temporal correlations between mass and survival both within and between years. We suggest that use of the Price equation to describe cross-generational change in the wild may be challenging, but a more modest aim of estimating its first term, the Robertson–Price identity, to assess within-generation change can provide valuable insights into the processes shaping phenotypic diversity in natural populations.This article is part of the theme issue ‘Fifty years of the Price equation’.

Fifty years of the Price equation.

Fifty years of the Price equation.

Risk-Based Premiums for Pension Insurance

The present paper develops a model to compute risk-based premiums for the USA pension insurance administered by the public Pension Benefit Guaranty Corporation (PBGC). Pension insurance is shown to he analogous to a financial put option, and pricing equations and their analytical solutions are obtained. The model includes costly audits that follow a Poisson process, whose average frequency is determined by the policymaker in order to attain Pareto-optimality. The model is estimated for a sample of us firms for the period ¡982-1986. The main policy implication is that risk-based premium rales increase at an increasing rate with the level of underfunding, in contrast with the current law of flat premium rates after certain level of underfunding.

DEBT LAFFER CURVE ANALYSIS: A CASE STUDY OF HEAVILY INDEBTED POOR COUNTRIES

This study explores that whether the debt financing or debt forgiving would be suitable for the Highly Indebted Poor Countries (HIPCs). Debt Laffer curve theory has been tested in 21 HIPCs by applying price equation of debt, maximized value of debt and price elasticity approach over the period of 1980 to 2014. The maximized value of debt criterion implies that Chad is not eligible for the debt write-off strategy in comparison with the rest of the countries. By applying price elasticity approach, it is observed that only Burkina Faso, Cameroon, Chad, and Republic of Congo are eligible for debt financing while the remaining countries should adopt debt write-off facility. The crux of the study is that overall debt forgiveness is suitable for the HIPCs. Moreover, it is also in the favor of both the creditor countries and various international financial institutions such as World Bank and IMF and HIPCs itself. The study suggests that the creditors should continue to be financing along with improving structural policies and institutions of the HIPCs

Predator–prey model for stock market fluctuations

We present a dynamical model for the price evolution of financial assets. The model is based on a two-level approach: In the first stage, one finds an agent-based model that describes the current state of investors’ beliefs, perspectives or strategies. The dynamics is inspired by a model for describing predator–prey population evolution: Agents change their mind through self- or mutual interaction, and the decision is adopted on a random basis, with no direct influence of the price itself. One of the most appealing properties of such a system is the presence of large oscillations in the number of agents sharing the same perspective, what may be linked with the existence of bullish and bearish periods in financial markets. In the second stage, one has the pricing mechanism, which will be driven by the relative population in the different groups of investors. The price equation will depend on the specific nature of the species, and thus, it may change from one market to the other: We will present a simple model of excess demand in the first place and then consider a more elaborate liquidity model. The outcomes of both models are analyzed and compared.

Elemental composition and moisture prediction in manure by portable X-ray fluorescence spectroscopy using random forest regression.

Manure elemental composition determination is essential to develop farm nutrient budgets and assess environmental risk. Portable X-ray fluorescence (PXRF) spectrometers could facilitate hazardous waste-free, rapid, and cost-effective elemental concentration determinations. However, sample moisture is a problem for elemental concentration determination by X-ray methods. The objective of this study was to quantify the effect of sample moisture content, predict moisture content, and correct for moisture effect on elemental concentration determinations in livestock manure. Oven-dried manure samples (n=40) were ground and adjusted to five moisture ranges of (w/w moisture)<10%, 10-20%, 20-30%, 40-50%, and 60-70%. Samples were scanned by PXRF for 180 s using a vacuum (<1,333 Pa) and without a filter. The presence of moisture negatively affected elemental determination in manure samples. Calibrations (n=200) were prepared using random forest regression with detector channel counts as independent variables. A three-step validation was performed using all the data, random cross-validation and external validation. The back end of the spectrum (14-15keV) had strong predictive power (r2 =.98) for moisture content. The random forest approach increased r2 between PXRF and wet chemical methods from <.66 to >.90 for P, K, and Mg and from .78 to .98 for Fe, compared with linear, nonlinear, and Lucas-Tooth and Price equations. These results indicated that elemental concentration can accurately be measured in dried and moist manure samples using PXRF and expands the potential applications of PXRF to in situ elemental determinations for agricultural and environmental samples.

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path‐dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black‐Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.

Environmental innovations, income distribution, international competitiveness and environmental policies: a Kaleckian growth model with a balance of payments constraint

The article integrates a baseline Kaleckian model with a balance of payments constrained growth for addressing the interaction among: income distribution, international competitiveness and environmental innovation taking into account the proactive role of environmental policies The crucial link is given by intermediate inputs (raw materials and energy) in the Kaleckian price equation, which impacts on real wages, the wage share, the rate of capital utilization (and hence growth) and international competitiveness (through changes in domestic prices for a given international inflation rate). It is shown that an economy which has higher energy efficiency will attain at the same time a higher wage share and higher growth. Being green not only helps attain a more sustainable growth path, but also a more inclusive one both in the short run and the medium-run. In the long-run, within a balance of payments constraint, the combination of green taxes with an industrial policy aimed at increasing the energy efficiency of production and the diversification of the economy is crucial to raise the wage share and sustain growth.

Diffusion Normalized Least Mean M-estimate Algorithms: Design and Performance Analysis

This work proposes diffusion normalized least mean M-estimate algorithm based on the modified Huber function, which can equip distributed networks with robust learning capability in the presence of impulsive interference. In order to exploit the system's underlying sparsity to further improve the learning performance, a sparse-aware variant is also developed by incorporating the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm of the estimates into the update process. We then analyze the transient, steady-state and stability behaviors of the algorithms in a unified framework. In particular, we present an analytical method that is simpler than conventional approaches to deal with the score function since it removes the requirements of integrals and Price's theorem. Simulations in various impulsive noise scenarios show that the proposed algorithms are superior to some existing diffusion algorithms and the theoretical results are verifiable.

An accurate solution for the generalized Black-Scholes equations governing option pricing

Today industries related to finance are essentially implementing advanced mathematical tools. In 1973, Fisher Black and Myron Scholes developed an eminent stochastic model which later coined as Black-Scholes differential equations for option pricing. This paper illustrates a convenient time integration scheme based on the generalized trapezoidal formulas (GTF $[\alpha = \frac{1}{3}]$) introduced by Chawla et al. in 1996. GTF is applied for the temporal discretization along with the classical finite difference schemes in space direction. The proposed scheme yields the (uniform) stability employing the uniform bound of the inverse operator, as well as second-order spatial accuracy and third-order temporal accuracy under reasonable conditions. Finally, the numerical illustrations and comparison with existing schemes demonstrate the stability and accuracy of the method.

Classical Option Pricing and Some Steps Further

This paper considers the asset price p as relations C=pV between the value C and the volume V of the executed transactions and studies the consequences of this definition for the option pricing equations. We show that the classical BSM model implicitly assumes that value C and volume V of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. We show that agents expectations those approve execution of transactions can further increase the dimension of the BSM model. We study the case when agents expectations may depend on the option price data and show that such assumption can lead to the nonlinear BSM-like equations. We reconsider the Heston stochastic volatility model for the price determined by the value and the volume and derive 3-dimensional BSM-like model with stochastic value volatility and constant volume volatility. Variety of the BSM-like equations states the problem of reasonable balance between the accuracy and the complexity of the option pricing equations.

Semiparametric Estimation of Latent Variable Asset Pricing Models

This paper studies semiparametric identification and estimation of the stochastic discount factor in consumption-based asset pricing models with latent state variables. We model consumption, dividends, and a multiplicative discount factor component via unknown functions of Markovian states describing aggregate output growth. For the case of affine state dynamics and polynomial approximation of the measurement and pricing equations, we provide rank conditions for identification and tractable algorithms for filtering, smoothing, and likelihood estimation. Empirically, we find sizable nonlinearities and interactions in the impacts of expected growth and volatility on the price-dividend ratio and the discount factor.

Fat Tails Arise Endogenously in Asset Prices From Supply/Demand, With or Without Jump Processes

We show that the quotient of Levy processes of jump-diffusion type has a fat-tailed distribution. An application is to price theory in economics, with the result that fat tails arise endogenously from modeling of price change based on an excess demand analysis resulting in a quotient of arbitrarily correlated demand and supply whether or not jump discontinuities are present. The assumption is that supply and demand are described by drift terms, Brownian (i.e., Gaussian) and compound Poisson jump processes. If P^(−1)dP/dt (the relative price change in an interval dt) is given by a suitable function of relative excess demand, (D − S) /S (where D and S are demand and supply), then the distribution has tail behavior F (x) ∼ x(−ζ) for a power ζ that depends on the function G in P^(−1)dP/dt = G (D/S). For G (x) ∼ |x|^(1/q) one has ζ = q. The empirical data for assets typically yields a value, ζ=˜ 3, or ζ ∈ [3, 5] for many financial markets. Many theoretical explanations have been offered for the disparity between the tail behavior of the standard asset price equation and empirical data. This issue never arises if one models price dynamics using basic economics methodology, i.e., generalized Walrasian adjustment, rather than the usual starting point for classical finance which assumes a normal distribution of price changes. The function G is deterministic, and can be calibrated with a smaller data set. The results establish a simple link between the decay exponent of the density function and the price adjustment function, a feature that can improve methodology for risk assessment. The mathematical results can be applied to other problems involving the relative difference or quotient of Levy processes of jump-diffusion type.

Moonshots, Investment Booms, and Selection Bias in the Transmission of Cultural Traits

Biased information about the payoffs received by others can drive innovation, risk-taking, and investment booms. We study this cultural phenomenon using a model based on two premises. The first is a tendency for large successes, and the actions that lead to them, to be more salient to onlookers than small successes or failures. The second premise is selection neglect—the failure of observers to adjust for biased observation. In our model, each firm in sequence chooses to adopt or to reject a project that has two possible payoffs, one positive and one negative. The probability of success is higher in the high state of the world than in the low state. Each firm observes the payoffs received by past adopters before making its decision, but there is a chance that an adopter's outcome will be censored, especially if the payoff was negative. Failure to account for biased censorship causes firms to become overly optimistic, leading to irrational booms in adoption. Booms may eventually collapse, or may last forever. We describe these effects as a form of cultural evolution with adoption or rejection viewed as traits transmitted between firms. Evolution here is driven not only by differential copying of successful traits, but also by cognitive reasoning about which traits are more likely to succeed—quantified using the Price Equation to decompose the effects of mutation pressure and evolutionary selection. This account provides a new explanation for investment booms, merger and IPO waves, and waves of technological innovation. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.