1 宏微观区别
This is perhaps a good opportunity to point out that the "certainty equivalence" concept means one thing in microeconomics/choice under uncertainty theory, while it means something different in macroeconomics.
Microeconomics/choice under uncertainty The Certainty Equivalent of a lottery/gamble, is the amount of wealth which, if given with certainty, provides the same utility as the lottery/gamble. See for example Jehle & Reny's "Advanced Microeconomic Theory" (2011, 3d ed.), p.113.
Macroeconomics "Certainty Equivalence" is the situation in a stochastic model, where optimal decision rules prove to be identical to those that would have been derived in a deterministic framework (see for example Lungqvist & Sargent's "Recursive Macroeconomic Theory" 2004, 2nd ed, p. 113-115). Informally, this is sometimes described as "agents behave as if the stochastic processes are not stochastic", or "the decision rule is not affected by stochastic variability".
The concepts are evidently linked by the "as if there was no uncertainty" angle, but they are different in essential aspects: the micro-concept is a "buy-out" of uncertainty, providing a certain alternative towards which the agent that faces the gamble is indifferent in terms of utility, while the macro-concept emerges as a property of the solution (and only under restrictive model structures, or linear approximations thereof).
Hint: search "Certainty Equivalence Principle" "Certainty Equivalent Control" for macro-concept
2 含义
2.1 教科书定义
Lungqvist & Sargent's "Recursive Macroeconomic Theory" 2004, 2nd ed:
a) p.113: CERTAINTY EQUIVALENCE PRINCIPLE: The decision rule that solves the stochastic optimal linear regulator problem is identical with the decision rule for the corresponding nonstochastic linear optimal regulator problem.
b) p.115: The certainty equivalence principle is a special property of the optimal linear regulator problem, and comes from the quadratic objective function, the linear transition equation, and the property that future disturbances have zero mean conditional on the current state. Certainty equivalence does not characterize stochastic control problems generally.
2.2 Wikipedia 解释
An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is the certainty equivalence property:[2] that the optimal control solution in this case is the same as would be obtained in the absence of the additive disturbances. This property is applicable to all centralized systems with linear equations of evolution, quadratic cost function, and noise entering the model only additively; the quadratic assumption allows for the optimal control laws, which follow the certainty-equivalence property, to be linear functions of the observations of the controllers.
Any deviation from the above assumptions—a nonlinear state equation, a non-quadratic objective function, noise in the multiplicative parameters of the model, or decentralization of control—causes the certainty equivalence property not to hold. For example, its failure to hold for decentralized control was demonstrated in Witsenhausen's counterexample.
3 理解
Intuitively, it means that the model has such characteristics that "the best we can say" about remaining uncertainty, is that it will be zero. From general experience, we know that it won't be zero, but the information we possess does not permit us to say anything else than that it will be zero. The even deeper assumption here is that the information and knowledge we possess make all things that remain unpredictable to average to zero.
In such a case, of course the actual realized path of the model is affected by these totally unpredictable shocks, but the decisions we take are based on a rule "as if" the future shocks will be zero (it is implied that we have to take decisions prior to observe the realization of the shock).
4 数学表达
Refer: 见所附链接
