Uncertainty | Macro Paper Warehouse

Eliciting Multiple Prior Beliefs

Mon, 01 Jan 0001 00:00:00 +0000

Multiple prior decision models—in which beliefs are represented by a set of probability measures rather than a single measure, generating a probability interval for each event—have become increasingly important in economics, but choice-based incentive-compatible elicitation of probability intervals remains an open problem: existing scoring rules and matching-probability methods cannot recover probability intervals without assuming probabilistic sophistication that is precisely least warranted in settings where multiple priors are most relevant. This paper develops a preference-based identification of a subject’s probability interval for an event, and a method for eliciting it under weak decision-theoretic assumptions with no need for probabilistic sophistication. Three incentivized experiments on artificial and natural sources of uncertainty demonstrate that the elicited intervals are sensitive to the direction and amount of information, are typically consistent with objective probabilities where available, and exhibit a predominance of non-degenerate probability intervals that are wider when there is less information or predictability. On aggregate, the choice-based intervals are similar to stated probability intervals, providing behavioral foundations for the use of stated interval techniques in the field.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What is the key identification challenge for multiple prior elicitation?

The key challenge is that existing incentive-compatible elicitation methods—scoring rules and matching-probability approaches—confound a subject’s probability interval with their ambiguity attitude, so they cannot separately identify the probability interval without assuming probabilistic sophistication. Under the popular α-maxmin EU model, the matching probability of an event depends on both the subject’s probability interval and their ambiguity attitude parameter α; even eliciting both the event and its complement’s matching probabilities yields two equations in three unknowns. Probabilistic sophistication is least warranted precisely in settings with deep uncertainty where multiple priors are most relevant, making precision-laden methods unsuitable.

Q2. What is the paper’s elicitation solution?

The paper develops a preference-based method that identifies a subject’s probability interval under weak decision-theoretic assumptions—with no need for probabilistic sophistication—using a series of incentivized choices, and demonstrates its feasibility in three laboratory experiments. The approach comprises two components: (i) a preference-based identification theorem establishing the conditions under which the probability interval can be recovered from observable choices; and (ii) a concrete elicitation procedure that is incentive compatible and does not impose the precision-laden assumption of probabilistic sophistication.

Q3. What do the experiments show?

Three incentivized experiments on artificial and natural sources of uncertainty demonstrate that probability intervals elicited by the method are sensitive to the direction and amount of information, are typically consistent with objective probabilities where available, and predominantly non-degenerate—with intervals wider when there is less information or predictability. The sensitivity to information and consistency with objective probabilities provide external validation that the elicited intervals capture real beliefs rather than noise or confusion. The predominance of non-degenerate intervals (rather than point probabilities) indicates that subjects genuinely hold imprecise beliefs in the relevant settings.

Q4. What is the relationship between choice-based and stated probability intervals?

On aggregate, probability intervals elicited with the choice-based method are similar to those stated by subjects, suggesting that the new method can provide behavioral foundations for the use of stated probability-interval techniques that are widely used in field surveys but previously lacked incentive-compatible grounding. This convergence is informative because stated intervals are cognitively simpler and can be collected at large scale in surveys, while the choice-based intervals are theoretically grounded; the consistency between them justifies the use of simpler stated methods in field applications.

Key concepts

multiple priors : a model of beliefs in which a decision maker’s uncertainty is represented by a set of probability measures rather than a single measure; associated with the Gilboa-Schmeidler (1989) maxmin expected utility model and its generalizations; generates a probability interval for each event. probability interval : the interval [p(E), p̄(E)] of probability values a subject’s set of priors assigns to event E; non-degenerate (with width > 0) when the subject’s beliefs are genuinely imprecise. incentive-compatible elicitation : an elicitation procedure in which subjects’ optimal strategy is to report their true beliefs; for Bayesian single-prior beliefs, achieved by scoring rules and matching-probability methods, but these fail for multiple priors. probabilistic sophistication : the assumption that a multiple-prior agent’s set of priors is generated by precise probabilistic beliefs; existing methods require this assumption to disentangle the probability interval from ambiguity attitude, but the paper’s method does not.

Firm idiosyncratic risk and productivity investment: Macroeconomic implications

Mon, 01 Jan 0001 00:00:00 +0000

This paper quantifies how idiosyncratic firm-level risk affects aggregate output, TFP, and firm life-cycle growth in an environment where firm productivity evolves endogenously through risky investment. The paper embeds endogenous productivity investment into a Lucas span-of-control model with risk-averse firm owners and endogenous entry and exit, and studies the effects of mean-preserving increases in the variance of returns to productivity investment. A mean-preserving increase in the variance of firm productivity shocks that raises the firm exit rate by 10% (from 0.10 to 0.11) is estimated to cause a 0.73% decline in output, a 0.38% decline in measured TFP, and a 3.69% decline in firm productivity investment; these elasticities remain approximately constant in the empirically relevant range. The driving force is that risk-averse firm owners reduce their risky productivity investment as variance rises; if capital financing constraints are present—as is common in developing economies—these effects are amplified and the increase in uncertainty may also slow firm life-cycle growth. Previously circulated as “Uncertainty, Firm Lifecycle Growth, and Aggregate Productivity.”

Summary based on a working paper version, AI-assisted and human-reviewed. See the linked published article for the authoritative version.

In depth

Q1. What distinguishes this paper from standard models of firm misallocation?

Unlike the bulk of firm misallocation literature (Hsieh-Klenow 2009; Gopinath et al. 2017; Sraer-Thesmar 2023), which takes firm productivity as exogenous, this paper models productivity as an endogenous outcome of risky investment, so that idiosyncratic uncertainty affects allocative efficiency not only through selection effects but also through its discouragement of productivity investment by risk-averse owners. The paper incorporates endogenous productivity investment into a standard Lucas span-of-control model, allowing the model to capture how higher uncertainty reduces the incentive to invest in productivity, on top of any selection effects from the exit option.

Q2. What are the two opposing effects of higher idiosyncratic risk?

Higher idiosyncratic firm-level risk has two opposing effects on aggregate productivity: (i) a selection effect—a mean-preserving increase in variance leads to stronger selection and raises the productivity of survivors while reallocating exiters to alternative productive uses—that tends to raise average productivity; and (ii) a productivity investment effect—risk-averse owners reduce risky productivity investment in response to higher variance—that tends to reduce aggregate productivity and firm life-cycle growth. The paper shows quantitatively that the productivity investment effect dominates in the baseline calibration, so that higher idiosyncratic risk reduces output and TFP despite positive selection effects.

Q3. What are the main quantitative findings?

A mean-preserving increase in the variance of firm productivity shocks calibrated to raise the firm exit rate by 10% (from 0.10 to 0.11) results in a 0.73% decline in output, a 0.38% decline in measured TFP, and a 3.69% decline in firm productivity investment; these elasticities remain approximately constant in the empirically relevant range. The exit-rate increase from 0.10 to 0.11 is also associated with a 7.5% increase in the job destruction rate and a 14.6% increase in the standard deviation of firm growth rates—the latter is less than one-fifth of the increases in these risk measures observed when comparing India or Mexico to the U.S.

Q4. How do capital financing constraints interact with the results?

When firms face capital financing constraints—as is common in developing economies—the negative effects of higher idiosyncratic risk are amplified and the increase in uncertainty may also slow firm life-cycle growth. The mechanism is that constrained firms must rely more heavily on internal financing, making risk-averse owners even more sensitive to increases in variance. The paper implies that the macro-financial implications of idiosyncratic risk are more severe in developing economies where both idiosyncratic risk levels and financing constraints are greater—consistent with cross-country patterns of firm growth dynamics.

Key concepts

productivity investment : endogenous spending by firms on activities that shift their productivity process; in the model, this investment exposes firm owners to idiosyncratic risk via the innovation in the productivity process; the key margin through which higher uncertainty reduces aggregate productivity and output. mean-preserving increase in variance : a statistical experiment that increases the spread of the distribution of returns to productivity investment while leaving the mean unchanged; used here to isolate the pure risk effect on firm behavior and aggregate outcomes from any change in expected returns. span-of-control model : the Lucas (1978) model of firm size distribution with decreasing returns to scale in the entrepreneurial input; used as the production environment; extended here by adding endogenous productivity investment and endogenous entry and exit.