FormalPara Definition

A stochastic dynamic optimization problem displays ‘certainty equivalence’ when the problem’s solution can be found by solving a similar non-stochastic problem in which the uncertain state variable at every point in time is assumed to be known and equal to its expected value.

Most decisions are made under uncertainty and have dynamic consequences. For example, the decision to buy and maintain a house rests, at least in part, on expectations as to the future course of housing prices, and is subject to wealth constraints that may depend on past decisions as well as exogenous conditions. The purchase decision may also rest on expectations of how home ownership will affect future incentives in alternative possible future situations. If housing prices fall significantly below the purchase price, there may be an incentive to renege on a mortgage commitment, which may entail a set of additional consequences (such as denial of credit or bankruptcy). Because these possibilities can interact with the incentives to buy, sell and maintain a property, uncertainty about the future can fundamentally change housing decisions and strategies.

Such decision problems can be analysed using dynamic programming (otherwise known as stochastic optimal control), a form of mathematical optimization theory. However, even in non-strategic environments under conditions of certainty, dynamic programming can be analytically demanding, and explicit closed-form solutions are known for only certain limited classes of problems (Bertsekas 1976; Hansen and Sargent 2007). While other problems may be analysed with numerical methods, these solutions can be difficult to generalize. If one additionally postulates that the decision maker is uncertain about the future environment that will determine the outcomes of their choices, optimal dynamic decisions may depend, crucially, on the nature of that uncertainty, and, consequently, characterizing such strategies can become quite complex.

Around the same time that he was developing his path-breaking insights regarding bounded rationality (Simon 1955, 1957), Herbert Simon demonstrated that a particular class of dynamic programming problems obeys certainty equivalence, which enables the dynamic component to be separated from the uncertainties. Simon’s method was extended to the multivariate case by Theil (1957). An influential application of certainty equivalent programming methods to production planning can be found in the work of Holt and colleagues (1960). More recently, it has been shown that certainty equivalence properties emerge even when there is uncertainty about model specification (see, e.g., Hansen and Sargent 2007: 21–37). Specifically, in a dynamic programming problem, at every moment in time the decision maker selects a value of the control variable to maximize the discounted present value of the utility (or other objective function), subject to the evolution of the state variables, which describe the underlying economic environment as a function of past control variables as well as exogenous variables and random shocks (normally assumed to be the source of the uncertainty in the problem). Simon showed that if the single-period utility is a quadratic function of the state and the control, and the current state evolves as a linear function of the past states and the random shock’s value, then the optimal control law is found by assuming that future state variables will take their expected values with certainty. In other words, one can solve such a ‘linear-quadratic’ problem as if there is no uncertainty, using the expected values of all uncertain variables in place of probability density functions, without worrying about the possible interactions between the uncertain environment and optimal dynamic decisions. The existence of risk or uncertainty reduces the maximized value of the dynamic utility function, but does not change the optimal choices of the control variables.

Certainty equivalence therefore separates dynamic decision-making problems into two separate parts – ‘optimization’ and ‘forecasting’ – which highlights the formation of expectations and beliefs, and how they are translated into behaviour.

This insight has spawned two related and very substantial but disparate literatures. Because complicated optimization problems can be approximated by related problems for which certainty equivalence holds – what Simon characterized in his 1978 Nobel address as ‘an approximating, satisficing simplification’ – then, as Simon suggests, one might plausibly imagine a bounded-rational decision maker acting as if they are solving a dynamic optimization problem that obeys certainty equivalence (Simon 1992: 359). He states:

Hence the assumption of quadratic costs reduces the original problem to one that is readily solved. Of course the solution, though it provides optimal decisions for the simplified world of our assumptions, provides, at best, satisfactory solutions for the real-world decision problem that the quadratic function approximates. In principle, unattainable optimization is sacrificed for in practice, attainable satisfaction. If human decision makers are as rational as their limited computational capabilities and their incomplete information permit them to be, then there will be a close relation between normative and descriptive decision theory. (Simon 1992: 351)

But as a descriptive matter or as the basis for normative or policy analysis, indeterminacy still exists: how are expectations formed? For this reason, Simon argued that uncertainty requires an ‘elaboration of the model of the decision process’, which would ‘incorporate the notions of bounded rationality: the need to search for decision alternatives, the replacement of optimization by targets and satisficing goals, and mechanisms of learning and adaptation’ (Simon 1992: 360, 366). Alternatively, one can follow the logic of certainty equivalence to the rational expectations hypothesis pioneered by Simon’s colleague and co-author at Carnegie Mellon, John Muth, and simply assume that ‘expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory’ (Muth 1961: 318). Certainty equivalence is widely acknowledged to be crucial to the development of the theory of rational expectations (see, e.g., Lucas and Sargent 1981: xiv–xvi).

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