Integrated Assessment Models (IAMs) require parameterization of both economic and climatic processes. The latter includes Equilibrium Climate Sensitivity (ECS), or the temperature response to doubling CO2 levels, and Ocean Heat Uptake (OHU) efficiency. ECS distributions in IAMs have been drawn from climate model runs that lack an empirical basis, and in Monte Carlo experiments may not be constrained to consistent OHU values. Empirical ECS estimates are now available, but have not yet been applied in IAMs. We incorporate a new estimate of the ECS distribution conditioned on observed OHU efficiency into two widely used IAMs. The resulting Social Cost of Carbon (SCC) estimates are much lower than those from models based on simulated ECS parameters. In the DICE model, the average SCC falls by approximately 40–50% depending on the discount rate, while in the FUND model the average SCC falls by over 80%. The span of estimates across discount rates also shrinks substantially.

We explore the implications of recent empirical findings about CO 2 fertilization and climate sensitivity on the social cost of carbon (SCC) in the FUND model. New compilations of satellite and experimental evidence suggest larger agricultural productivity gains due to CO 2 growth are being experienced than are reflected in FUND parameterization. We also discuss recent studies applying empirical constraints to the probability distribution of equilibrium climate sensitivity and we argue that previous Monte Carlo analyses in IAMs have not adequately reflected the findings of this literature. Updating the distributions of these parameters under varying discount rates is influential on SCC estimates. The lower bound of the social cost of carbon is likely negative and the upper bound is much lower than previously claimed, at least through the mid-twenty-first century. Also the choice of discount rate becomes much less important under the updated parameter distributions.

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a population distribution of heterogeneity doesn’t yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-form expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-form expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the multivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), population distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithms) are hence directly available. Two simulation studies demonstrate the efficacy of our approach. Copyright Springer Science + Business Media, LLC 2006Closed-Form Bayesian Inferences, Logit model, Generalized Multivariate Gamma Distribution,

Reference prices have long been studied in applied economics and business research.One of the classic formulations of the reference price is in terms of an iterative function of past prices. There are a number of limitations of such a formulation, however. Such limitations include burdensome computational time to estimate parameters, an inability to truly account for customer heterogeneity, and an estimation procedure that implies a misspecified model. Managerial recommendations based on inferences from such a model can be quite misleading. We mathematically reformulate the reference price by developing a closed-form expansion that addresses the aforementioned issues, enabling one to elicit truly meaningful managerial advice from the model. We estimate our model on a real world data set to illustrate the efficacy of our approach. Our work is not only useful from a modeling perspective, but also has important behavioral and managerial implications, which modelers and non-modelers alike would find useful.

This supplemental material section contains a robustness analysis where we examined the posterior estimate's sensitivity to choice of priors. In addition, we include a number of tables and figures to supplement our report. RobustnessWe examined the robustness of our Bayesian model by checking the extent to which our marginal posterior estimates depend on the specification of prior distribution parameters. Specifically, we re-estimated the model using significantly more diffuse normally distributed prior distributions by increasing the variance parameter by a factor of 100 in equations 10 to 14. We again ran the sampler in WinBUGS using the same MCMC specifications and computed marginal posterior estimates. Marginal posterior estimates are depicted in Figures