We consider the problem of valuing inputs in a production environment in which input supply is uncertain. Inputs can be workers in a firm, risk factors for a disease, securities in a financial market, or nodes in a networked economy. Each input takes its values from a finite set, and uncertainty is modeled as a probability distribution over this set. First, we provide an axiomatic solution to our valuation problem, defining three intuitive axioms which we use to uniquely characterize a valuation scheme that we call the a priori Shapley value.Second, we solve the problem of valuing inputs a posteriori -that is, after observing output. This leads to the Bayesian Shapley value.Third, we consider the problem of rationalizing uncertainty when the inputs are rational workers supplying labor in a non-cooperative production game in which payoffs are given by the Shapley wage function. We find that probability distributions over labor supply that can be supported as mixed strategy Nash equilibria always exist. We also provide an intuitive condition under which we prove the existence of a pure strategy Nash equilibrium. We present several applications of our theory to real-life situations. JEL classification codes: C70, D20, D80, J30.

We consider the problem of valuing inputs in a production environment in which input supply is uncertain. Inputs can be workers in a firm, risk factors for a disease, securities in a financial market, or nodes in a networked economy. Each input takes its values from a finite set, and uncertainty is modeled as a probability distribution over this set. First, we provide an axiomatic solution to our valuation problem, defining three intuitive axioms which we use to uniquely characterize a valuation scheme that we call the a priori Shapley value.Second, we solve the problem of valuing inputs a posteriori -that is, after observing output. This leads to the Bayesian Shapley value.Third, we consider the problem of rationalizing uncertainty when the inputs are rational workers supplying labor in a non-cooperative production game in which payoffs are given by the Shapley wage function. We find that probability distributions over labor supply that can be supported as mixed strategy Nash equilibria always exist. We also provide an intuitive condition under which we prove the existence of a pure strategy Nash equilibrium. We present several applications of our theory to real-life situations. JEL classification codes: C70, D20, D80, J30.

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We study the effects of race, network centrality, and policies that tolerate some level of virus spread (laissez-faire) on COVID-19 deaths in nursing homes in the United States. Our analysis uses unique data on nursing home networks and calibration-based estimates of states' preferences for health relative to short-term economic gains. Our findings suggest that laissez-faire policies increase deaths. Nursing homes with a larger share of Black residents experience more deaths, but they are less vulnerable to laissez-faire policies, especially when not central in social networks. Our findings highlight significant interactions between COVID-19 policies, race, and network structure among US seniors.

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