Lucas Critique

Abstract

The ‘Lucas critique’ is a criticism of econometric policy evaluation procedures that fail to recognize that optimal decision rules of economic agents vary systematically with changes in policy. In particular, it criticizes using estimated statistical relationships from past data to forecast the effects of adopting a new policy, because the estimated regression coefficients are not invariant but will change along with agents’ decision rules in response to a new policy. A classic example of this fallacy was the erroneous inference that a regression of inflation on unemployment (the Phillips curve) represented a structural trade-off for policy to exploit.

The ‘Lucas Critique’ is a criticism of econometric policy evaluation procedures that fail to recognize the following economic logic:

[G]iven that the structure of an econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any changes in policy will systematically alter the structure of econometric models. (Lucas 1976, p. 41)

At the time of his writing, Robert E. Lucas (1976) was criticizing the prevailing approach to quantitative macroeconomic policy evaluation for ignoring this logic and, hence, as being fundamentally inconsistent with economic theory. To fully appreciate Lucas’s critique, we first consider a general theoretical argument and then turn to a particular example.

At each date t there is a vector s_t of state variables summarizing all aspects of the history that are relevant to the economy’s future evolution; for example, the vector might include the economy’s capital stock. The economy is also described by a vector x_t of government policy variables and a vector ε_t of random shocks – for example, shocks to technology or to government policy. For given specifications of the processes governing x_t and ε_t, it is common in macroeconomic theory to analyse models that yield an equilibrium law of motion in form of a difference equation,

$$ {s}_{t+1}=f\left({s}_t,{x}_t,{\varepsilon}_t\right). $$

(1)

(For many textbook examples of stochastic rational expectations models that yield such a recursive equilibrium representation; see Ljungqvist and Sargent 2004). Equation (1) is also the point of departure for the econometric policy evaluation procedures criticized by Lucas, who argued that their approach failed to recognize the optimization behaviour of economic agents that is implicit in Eq. (1). Specifically, the criticized approach proceeds as follows. First, historical data are used to estimate the equation

$$ {s}_{t+1}=F\left(\theta, {s}_t,{x}_t,{\mu}_t\right), $$

(2)

where F is specified in advance, θ is a fixed parameter vector to be estimated, and μ_t is a vector of random disturbances. Second, with the use of the estimated Eq. (2), policy evaluations are performed by comparing economic outcomes for different paths of government policy variables {x_t}. The policy choice that produces the most desirable economic outcome is deemed to be the best policy. But, as argued by Lucas, this approach violates the premises for economic theory because the parameter vector θ depends partly on agents’ decision rules that are not invariant to the conduct of government policy. That is, if the government changes its policy, the parameter θ will also change, so that the consequences of a new policy cannot be evaluated on the basis of the historical relationship in Eq. (2).

Lucas’s argument is best illustrated with an example. Consider the classic example of the so-called ‘Phillips curve’. Phillips (1958) had estimated a negative relationship between wage inflation and unemployment using British data for the period 1861–1957. Samuelson and Solow (1960) and others interpreted this and related empirical findings as evidence of a structural trade-off between an economy’s inflation rate and its unemployment rate. That is, the parameter θ in Eq. (2), estimated with historical data, was considered to be fixed and to describe how unemployment would respond to inflation outcomes associated with different monetary policies. Friedman (1968) and Phelps (1968) argued against the existence of such an exploitable trade-off because it was inconsistent with economic theory based on rational agents. To understand the fallacy of the Phillips curve and its extension – the fallacy of the econometric policy evaluation procedures criticized by Lucas – consider the monetary model of Lucas (1972). Exchange in the economy takes place in physically separated markets. Producers in a market base their output decisions on the local market-clearing price level without knowing the current economy-wide price level. The price in a market varies stochastically because there are exogenous random shocks both to the distribution of producers across markets and to the aggregate quantity of nominal money, none of which is directly observable to the agents. Hence, information on the current state of these real and monetary shocks is transmitted to agents only through the price in the market where each agent happens to be. In an equilibrium, producers in a market would like to increase their output in response to a high price driven by real but not nominal shocks. A high price due to a real shock means that the ratio of producers to consumers is low in that market and, therefore, profits on sales are high in real terms (when evaluated in terms of the economy-wide price level). But a high price in a market due to an expansion of the aggregate quantity of nominal money means that prices tend to be high in all markets and, therefore, profits on sales are high in nominal but not real terms. The inference and decision problems solved by the agents in this model are shown to give rise to a Phillips curve, as had been estimated with real-world data, but where the model’s apparent trade-off between inflation and output cannot be systematically exploited by the government in its choice of monetary policy.

To further convey the insights from this general equilibrium model of the Phillips curve, we adopt a version of Lucas’s (1976) simplified model that does not spell out all the details of the economic environment but instead postulates three equations that capture the forces at work in the fully articulated model. The economy-wide price level (in logs), p_t, is given by

$$ {p}_t={\overline{p}}_t+{m}_t, $$

(3)

where ${\overline{p}}_t$ reflects a systematic component of monetary policy that is known to all agents, and m_t reflects an i.i.d. shock to monetary policy. It is assumed that the random variable m_t is normally distributed with mean zero and variance ${\sigma}_m^2$. The price (in logs) in market i at time t, p_it, is given by

$$ {p}_{it}={p}_t+{z}_{it}, $$

(4)

where z_it is a deviation from the economy-wide price level because of shocks to the distribution of producers across markets. The real shock z_it is assumed to be a normal, i.i.d. random variable with mean zero and variance ${\sigma}_z^2$. Finally, let y_it denote the log-deviation of output from its ‘natural rate’ in market i at time t which varies with the perceived, relative price:

$$ {y}_{it}=\alpha \left[{p}_{it}-E\left({p}_t|{I}_{it}\right)\right], $$

(5)

where α > 0 reflects intertemporal substitution possibilities in supply (determined by technological factors and tastes for substituting labour over time), and E( |I_it) denotes the mathematical expectation conditioned upon information I_it available in market i at time t. The agents’ prediction problem in Eq. (5) is straightforward to solve (see, for example, Ljungqvist and Sargent 2004, ch. 5):

$$ E\left({p}_t|{I}_{it}\right)=E\left({p}_t|{p}_{it},{\overline{p}}_t\right)=\left(1-\Omega \right){p}_{it}+\Omega {\overline{p}}_t, $$

(6)

where $\Omega ={\sigma}_z^2/\left({\sigma}_m^2+{\sigma}_z^2\right).$ The substitution of Eqs. (3), (4) and (6) into Eq. (5) yields

$$ {y}_{it}=\alpha \Omega \left({m}_t+{z}_{it}\right). $$

(7)

Thus, output in market i varies with the sum of nominal and real shocks, (m_t + z_it), because producers cannot perfectly disentangle these shocks but must make inferences based on the observed price p_it. Producers’ willingness to vary output from its natural rate depends on how likely observed price variations are due to real rather than nominal shocks, as captured by the magnitude of Ω ∈ [0, 1]. Under the assumption of a large number N of markets, the real shocks, {z_it}, cancel each other out when averaged over markets, and the economy’s deviation from its natural rate of output, y_t, becomes

$$ {y}_t=\frac{1}{N}\sum_{i=1}^N{y}_{it}=\alpha \Omega {m}_t=\alpha \Omega \left({p}_t-{\overline{p}}_t\right), $$

(8)

where the last equality invokes Eq. (3) and, hence, the economy exhibits a positive relationship between unanticipated inflation and output.

If estimations were performed using data on output and inflation from the described economy, we would find a Phillips curve along which increases in inflation are associated with higher output realizations. However, any attempts by the government to exploit that relationship would fail. For example, a government that permanently increases the growth rate of the money supply to generate higher inflation in order to stimulate output will ultimately see no real effects from that change in policy. The reason for this is that, after agents have become aware of the higher underlying inflation rate in the economy, they will change their expectations when making predictions about relative price movements due to real disturbances. Formally, the change in monetary policy represents an increase in the component ${\overline{p}}_t$ and, when that systematic change becomes known to the agents, it will not affect unanticipated inflation, $\left({p}_t-{\overline{p}}_t\right)={m}_t$, so output is left unaffected in Eq. (8).

This example illustrates Lucas’s general criticism of econometric policy evaluation procedures that fail to recognize that the estimated Eq. (2) depends partly on agents’ decision rules and is therefore not invariant to changes in government policy. For a proper policy evaluation procedure, we need to revise the econometric formulation in Eq. (2) so that it becomes consistent with equilibrium outcomes as represented by Eq. (1). Recall that the latter equation is derived for given specifications of the processes governing x_t and ε_t. In particular, to analyse agents’ optimization behaviour, we need to specify the environment in which they live, including their perceptions about future government policy. As Lucas (1976, p. 40) remarked, ‘one cannot meaningfully discuss optimal decisions of agents under arbitrary sequences {x_t} of future shocks’. Instead, Lucas suggested that one proceeds by viewing government policy as a function of the state of the economy,

$$ {x}_t=G\left(\uplambda, {s}_t,{\eta}_t\right), $$

(9)

where λ is a parameter vector that characterizes government policy, and η_t is a vector of random disturbances. Then the new version of Eq. (2) becomes

$$ {s}_{t+1}=F\left(\theta \left(\uplambda \right),{s}_t,{x}_t,{\mu}_t\right), $$

(10)

and the econometric problem is that of estimating the function θ(λ). A change in government policy is viewed as a change in the parameter λ affecting the behaviour of the system in two ways: first, by altering the time series behaviour of {x_t}, and second, by leading to modification of the parameter θ governing the rest of the system, which reflects changes in agents’ decision rules in response to the new policy.

A constructive response to the Lucas critique has been the development of rational expectations econometrics. A goal of that approach has been to estimate the ‘primitives’ of dynamic rational expectations models, in the form of parameters describing tastes and technologies. If historical data can be used to obtain such estimates, the economic model can in principle be used to evaluate alternative government policies that could be without precedent, as explained by Lucas and Sargent (1981). That is, knowledge about the primitives of a model enables us to derive agents’ decision rules and equilibrium outcomes for any specified policy process. In terms of Eq. (10), this explains how the function θ(λ) could conceivably be estimated even if the historical data have been generated under a single government policy λ.

Though one of the key contributors to the methodology of rational expectations econometrics, Sargent (1984) has raised a philosophical conundrum with this approach to policy evaluation (as earlier discussed by Sargent and Wallace 1976). Suppose that the primitives of an economic model have been estimated during an estimation period in which government policy was specified to be λ, and then the estimated model is used to compare alternative policies in order to find the best future policy λ^*. But such a procedure leads to an internal contradiction under the assumption of rational expectations, because, if the procedure were in fact likely to be persuasive in having the policy recommendation actually adopted soon, it would mean that the original econometric model with it specified policy λ had been misspecified. As pointed out by Sargent (1984, p. 413): ‘A rational expectations model during the estimation period ought to reflect the procedure by which policy is thought later to be influenced, for agents are posited to be speculating about government decisions into the indefinite future.’

Given its fundamental impact on questions of economic policy both in practice and in theory, the Lucas critique figured prominently in the list of contributions when the Royal Swedish Academy of Sciences (1995) awarded Robert E. Lucas, Jr. the Nobel Prize in economics ‘for having developed and applied the hypothesis of rational expectations, and thereby having transformed macroeconomic analysis and deepened our understanding of economic policy.’

See Also

• Phillips Curve

• Rational Expectations