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Lucas Critique, Time Inconsistency, and Related Issues

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Lucas Critique, Time Inconsistency, and Related Issues Leigh Tesfatsion (Prof. of Econ, Iowa State University, Ames, IA 50011-1070) Last Updated: 1 January 2018 Course Site for Econ 502 (M.S. Macro Theory) http://www2.econ.iastate.edu/classes/econ502/tesfatsion/ Introduction to Rational Expectations: http://www2.econ.iastate.edu/tesfatsi/reintro.pdf Notes on the Lucas Critique, Time Inconsistency, and Related Issues: http://www2.econ.iastate.edu/tesfatsi/LucCrit.pdf Notes on the Lucas Critique, Time Inconsistency, and Related Issues 1 Preliminary Remarks on Government Policy Rules Consider an economy that exists over time periods t = 1; 2;:::, where period t is the interval from t to t + 1. Let It−1 denote the information set available to government at the beginning of period t (i.e., at the end of period t − 1). Let ht denote a possible policy choice vector for government in period t, where the components of ht might include tax rates, a target growth rate for the money supply, and so forth. Definition: Let I denote the collection of all possible government information sets, and let H denote the collection of all possible policy choice vectors. Then a sequence (h1; h2;:::) of government policy choice vectors ht is said to be chosen in accordance with a (determin- istic) policy rule1 if there exists some well-defined function g:I! H mapping information 1For simplicity of exposition, only deterministic policy rules are considered in this section, that is, policy rules that map each possible information state I for each period t into a single policy choice vector g(I) for period t. An alternative stochastic form would be for each possible information state I for period t to be mapped into a probability distribution PI (h) over feasible policy choice vectors h for period t. In either case, however, notice that the presumption that a government uses a single policy rule g over times t = 1; 2;::: means this government does not engage in learning; it already knows at time 1 what action(s) it will take in each possible information state in each period t ≥ 1. 1 sets into policy choice vectors such that ht = g(It−1) ; t = 1; 2;:::: (1) The policy rule g:I! H is said to be a feedback policy rule if it depends nontrivially on the information sets, i.e., if there exist information sets I∗ and I∗∗ in I such that g(I∗) is not equal to g(I∗∗). Otherwise it is said to be a fixed policy rule. If government implements a sequence (h1; h2; ···) of policy choice vectors that does not satisfy (1) for any well-defined function g:I! H, then government is said to be engaging in discretionary policy choice. Example: Optimal Policy Rule Selection via a Social Welfare Function The following example illustrates the way in which government policy choice problems were commonly represented in articles stressing microfoundations of macro prior to the ra- tional expectations debate beginning in the early nineteen seventies, and are still represented in many microfoundations of macro articles today. The framework, originally due to Nobel Laureate Jan Tinbergen (1967), will hereafter be referred to as the Tinbergen framework for government policy choice. For each t = 1;:::;T , let xt−1 be a vector describing the state of an economy at the o beginning of period t, where x0 is exogenously given in period 1. Also, let It−1 = (xt−1;I ) 2 I denote the information set available to government at the beginning of period t, where o the contents of I are clarified below. Finally, let H(It−1) ⊆ H denote the subset of policy choice vectors ht in H that are feasible for government to choose in period t, conditional on government's information set It−1. Assume the motion of the state of the economy over time is described by a system of state equations of the form xt = S(ht; vt; xt−1) ; t = 1;:::;T; (2) where vt denotes a vector of time-t stochastic shock terms and v = (v1; : : : ; vT ) in V is determined in accordance with a probability distribution function P (v) over V . For example, 2 the period-t state equation (2) might represent the period-t equations for a stochastic variant of a dynamic IS-LM model conditioned on a vector ht of government fiscal and monetary policy variables. Suppose that social welfare for the economy over periods 1 through T is measured in period 1 by a utilitarian social welfare function T X t W (h; v; x) = β Ut(ht; vt; xt−1) ; (3) t=1 where h = (h1; : : : ; hT ), x = (x0; : : : ; xT −1), Ut(·) denotes the utility function of a repre- sentative consumer in period t, and βt denotes the discount applied to this representative consumer in period t from the vantage point of the beginning of period 1. Call g:I! H a feasible policy rule for government if g(It−1) is an element of H(It−1) for each t = 1;:::;T . Suppose the problem facing government at the beginning of period 1 is to choose a feasible policy rule for generating h = (h1; : : : ; hT ) over periods 1;:::;T that maximizes expected social welfare subject to (2), conditional on the period-1 information set o o I0 = (x0;I ), where I contains the structure of the state equations (2), the form of the prob- ability distribution function P (v), and the form of the the social welfare function W (h; v; x). Formally, then, government's optimization problem in period 1 takes the following form: Maximize expected social welfare Z E [W (h; v; x)jI0] = W (h; v; x)P (dv) (4) V with respect to the choice of h = (h1; : : : ; hT ) subject to xt = S(ht; vt; xt−1) ; t = 1;:::;T ; (5) ht 2 H(It−1) ; t = 1;:::;T ; (6) o It−1 = (xt−1;I ) ; t = 1;:::;T ; (7) x0 =x ¯0 : (8) 3 By construction, the solution of this stochastic optimization problem is a feasible policy ∗ rule g :I! H that determines how a particular policy vector ht should be chosen from H(It−1) in each period t = 1;:::;T , given the dynamic state equations (5) that reveal how a choice of ht in period t affects all current and future states of the economy. Such recur- sively structured problems are often solved numerically by means of \dynamic programming" methods; see Ljungqvist and Sargent (2004). In this engineering view of government policy optimization, the economy is a machine. Once the econometrician discovers the true structure of the machine as represented by the state function S(ht; vt; xt−1) and the shock probability distribution P (v), and a social con- sensus is reached concerning the appropriate specification for the social welfare function W (h; v; x) and the feasible policy choice sets H(It−1), the government can optimally control the machine by pressing the right buttons (selecting the right policy rule g). 2 The Lucas Critique Definition: Estimated functional forms obtained for an economic model are said to be deep (or structural) if they are invariant to announced or perceived changes in government policy rules.2 In particular, when unknown functional forms are replaced by functions that are assumed known up to unknown parameter values, the estimated parameter values are said to be deep if they are invariant to announced or perceived changes in government policy rules. The Lucas Critique: Estimated functional forms obtained for macroeconomic models in the Keynesian tradition (e.g. dynamic IS-LM models) are not deep because these models do not correctly take into account the dependence of private agent behavior on perceived or 2Since the term \structural" already has many other meanings in economics and econometrics, henceforth only the term \deep" will be used. 4 anticipated government policy rules for generating current and future values for government policy variables. While such models may be useful for forecasting future states of the economy conditional on a given goverment policy rule, they are fatally flawed for the purpose of analyzing the effects on the economy of changes in government policy rules. Specifically, they can seriously mislead policy makers with regard to the effectiveness and desirability of their policy rule choices. The best known source for the Lucas Critique is Lucas (1976). In this study, Lucas criticizes government policy optimization frameworks, such as the Tinbergen framework illustrated above, for not taking into account the degree to which estimated functional forms fail to be deep. Note that the Lucas Critique does not presume that agents have rational expectations, although Lucas (1976) motivates this critique using only rational expectations contexts. Rather, the Lucas Critique points out a problem that can occur whenever private agent behavior depends to some degree on government policy rules and this dependence is not taken into account. For example, to carry out real-world policy analysis within a Tinbergen framework, an econometrician would first need to determine the form of the state function S(ht; vt; xt−1) on the basis of past observations. In traditional econometric practice (i.e. econometric practice prior to 1976), this estimation would have proceeded without any attention being paid to the particular government policy rules in effect at the time these observations were made that were used to generate the observed settings for policy variables.3 Once the form of this state function was estimated, it would then have been incorporated into the Tinbergen framework as if it were deep. That is, it would have been treated as a constant of the model as alternative choices of policy rules g:I ! H were considered for government.
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