Rational Expectations Under Knightian Uncertainty — Part 3
How to represent model-consistent expectations in a model with Knightian uncertainty
This is the third post in my series that explains the main components of Roman Frydman’s and my new theoretical approach to opening macroeconomic models to unforeseeable structural change and representing rational expectations in the presence of the Knightian uncertainty arising from such change.
We present this approach, which we call the Knight-Muth hypothesis (KMH), in the paper titled Rational Expectations of Inflation Undergoing Unforeseeable Change.
In the first post, I discussed our motivation for developing KMH: our argument that economists must open their models to unforeseeable structural changes to represent the expectations of rational individuals in real-world markets.
In the second post, I explained how we open a New Keynesian Phillips curve (NKPC) relation for inflation to unforeseeable structural change and how that implies Knightian uncertainty about future inflation.
In this post, I focus on how we provide a new representation of model-consistent expectations in a model where rational individuals face Knightian uncertainty arising from unforeseeable change.
Rationality and Model-Consistent Expectations
The assumption that individuals are rational plays an important role in most economic models. A key component of this is that economists represent individuals’ (or typically a representative agent’s) decision-making as if the individual maximizes subjective utility. Another component is that economists represent their utility-maximizing decisions as based on rational expectations of future outcomes.
But what does it mean that expectations are rational?
A general definition that I find useful is that individuals’ expectations are rational if they are based on a reasonable understanding of how the economy works.
Individuals in the real world obviously have different understandings of how the economy works. It is impossible, and not useful, for an economist to attempt to realistically formalize all these different understandings and expectation formations in economic models.
Instead, the relevant question is how an economist can represent expectations in economic models to acknowledge that individuals are rational.
John Muth (1961) provided an answer to this question when he proposed the rational expectations hypothesis (REH). Muth argued that economists should acknowledge that individuals are rational by representing their expectations as being consistent with the predictions of the economist’s own model. As he put it (p. 316):
I should like to suggest that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory.
Muth’s important insight was that when building an economic model, an economist formalizes a hypothesis about how (a specific part of) the economy works. Conditional on this hypothesis, the economist can only acknowledge that individuals are rational by representing their expectations as consistent with the model’s predictions of future outcomes. Thereby, the economist’s model imposes consistency between its hypothesis about how the economy works and its representation of individuals’ expectations.
Muth implemented this general idea in a simple Cobweb model with constant parameters by representing individuals’ expectations with the model’s conditional expectation of future outcomes. He referred to it as the rational expectations hypothesis (REH). It formalizes Muth’s conjecture that:
expectations of firms (or, more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the prediction of the theory (or the “objective” probability distribution of outcomes). (Muth, 1961, p. 316)
Thus, REH effectively assumes that rational individuals (or at least the market in aggregate) have perfect probabilistic foresight.
In the 1970s, Robert Lucas’ Nobel prize-winning papers persuaded economists to embrace REH. As Roman and I write in our paper (pp. 8-9):
In the 1970s, Robert Lucas advanced compelling arguments that persuaded most macroeconomists to rely on REH to represent participants’ expectations. As he recounted in his Nobel lecture, intertemporal models that do not conform to Muth’s hypothesis suffer from a “glaring” (Lucas, 1995, p. 255) inconsistency between “actual equilibrium prices and (…) the price expectations that the theory imputed to individual agents” (p. 254). Indeed, as Lucas emphasized, specifying participants’ expectations as model-consistent is an essential condition of coherent macroeconomic model building.
Lucas argued that representing expectations as model-consistent was the only sensible way to build coherent macroeconomic models. Here is the full quote from Lucas’ Nobel lecture:
Some microeconomic analyses treated these prices as known; others imputed adaptive forecasting rules to maximizing firms and households. However it was done, though, the “church supper” models assembled from such individual components implied behavior of actual equilibrium prices and incomes that bore no relation to, and were in general grossly inconsistent with, the price expectations that the theory imputed to agents.
As intertemporal elements and expectations came to play an increasingly explicit and important role, this modeling inconsistency became more and more glaring. John Muth’s (1961) “Rational Expectations and the Theory of Price Movements” focused on this inconsistency, and showed how it could be removed by taking into account the influences of prices, including future prices, on quantities and simultaneously the effects of quantities on equilibrium prices. The principle of rational expectations he proposed thus forces the modeler towards a market equilibrium point of view, although it took some time before a style of thinking that recognized this fact had a major effect on macroeconomic modeling. (Lucas, 1995, pp. 254-255, emphasis in the original)
As Lucas pointed out, before the “rational expectations revolution,” economic models relied on simple rules, such as adaptive expectations, that were inconsistent with a model’s implied prices and incomes. Thus, they led to a “glaring inconsistency” in the form of obvious forecast errors, which are inconsistent with the idea that individuals are rational. REH solved this problem by ensuring consistency between a model’s hypothesis about how the economy works and individuals’ expectations.
Similar to REH, KMH acknowledges that a model-consistent representation of expectations is the only way to acknowledge market participants’ rationality.
KMH’s representation of model-consistent expectations, however, deviates from REH’s because it is implemented in a model with Knightian uncertainty arising from unforeseeable change. Because this change renders the model’s conditional expectation of future outcomes inherently unknowable ex ante, it cannot represent individuals’ expectations as it would in an REH model.
Before turning to how KMH represents expectations as model-consistent, let me emphasize that both KMH’s and REH’s representations of expectations should be interpreted as abstract representations. Muth (1961, p. 316) called REH a “purely descriptive hypothesis” and emphasized that it
does not assert that the scratch work of entrepreneurs resembles the system of equations in any way” (p. 317, emphasis in the original).
In other words, neither KMH nor REH seeks to provide a realistic formalization of the process by which rational individuals actually form, or should form, expectations.
Instead, they are model representations of expectations that are “rational” conditional on the model’s assumptions about how the economy works. They hypothesize that a model-consistent representation of expectations can adequately capture the key features of the expectations of rational individuals in real-world markets (or markets in the aggregate).
This requires, however, that the economist’s model adequately characterizes how the economy works.
As I wrote about in the first post in this series, Roman Frydman and I argue that REH does not characterize the expectations of rational individuals in real-world markets because REH models assume that the future is a probabilistic replica of the past. Thereby, they abstract from an important feature of how the economy works that has crucial implications for rational expectations: that individuals in real-world markets cannot have perfect probabilistic foresight, because they face Knightian uncertainty arising from unforeseeable change in the economy’s structure.
Let’s now turn to how KMH represen t expectations as model-consistent when the model’s conditional expectation of future outcomes is inherently unknowable. First, let’s consider how our formalization of unforeseeable change in the NKPC changes the model’s predictions about future inflation.
Unforeseeable Change in the New Keynesian Phillips Curve (NKPC)
Recall from the previous post in this series that we consider the NKPC given by:
for t=1,2,…,T, where 𝜋ₜ denotes inflation, 𝑥ₜ denotes an exogenous output gap, 𝐹ₜ(𝜋ₜ₊₁) denotes the market’s expectation of the next period’s inflation, 0<𝛽<1 is a discount factor, and 𝜅ₜ is a parameter that represents the pass-through of the output gap to inflation at time t.
For simplicity, we assume that the output gap follows a mean-zero stationary first-order autoregressive process with constant parameters:
where |𝜌|<1 and 𝜖ₜ is an i.i.d. normally distributed shock.
Moreover, we assume that the pass-through parameter 𝜅ₜ intermittently undergoes structural shifts between nonrepetitive values, which we formalize by:
and so on.
Finally, we assume that parameter values in each of the nonrepetitive regimes lie within a positive interval:
As I discussed in the previous post, this formalization of structural change implies that at any point in time t, the values of the future pass-through parameters are inherently unknowable. Because the future parameters differ from the past ones, there is simply no way they can be estimated based on the information available at time t.
This renders REH inapplicable because the model’s conditional expectation of future inflation depends on the inherently unknowable future pass-through parameters.
So what are the model’s predictions at time t of the next period’s inflation?
By iterating the NKPC in (1) forward one period, the model represents inflation at time t+1, 𝜋ₜ₊₁, as a linear function of the output gap, 𝑥ₜ₊₁, with pass-through parameter 𝜅ₜ₊₁ and the link to 𝑥ₜ determined by (2), and the market’s expectation at time t+1 of inflation at t+2, discounted by 𝛽.
Although the future pass-through parameter, 𝜅ₜ₊₁, is inherently unknowable at time t, (4) restricts it to lie within the positive interval 𝐼. Thus, the model predicts at time t that inflation at t+1 will be characterized by (1) iterated forward one period with 𝜅ₜ₊₁ lying within the positive interval 𝐼. The model predicts that inflation further into the future will be characterized by a similar structure with all future pass-through parameters, 𝜅ₜ₊ᵢ for i=1,2,…, lying within the interval.
We next derive the representation of the market’s inflation expectation that is consistent with these predictions.
Model-Consistent Expectations Under Knightian Uncertainty
We represent the market’s expectation of next period’s inflation with the conditional expectation of (1) and (2) iterated forward, but with the unknown future pass-through parameters, 𝜅ₜ₊ᵢ for i=1,2,…, replaced by a subjective parameter. We denote this subjective parameter by 𝜙ₜ, and we assume it lies within the same interval as the actual pass-through parameter at all times, i.e. 𝜙ₜ∈𝐼 for all t.
(Here, I use a slightly different notation for the subjective parameter than in the paper due to Substack’s poor support for mathematical expressions and notation.)
We denote the conditional expectation, given the subjective parameter 𝜙ₜ, by
where 𝜋ₜ₊ᵢ(𝜙ₜ) for i=1,2,… is given by
Note that here we only iterate forward over the horizon i=1,2,…, while keeping the subjective parameter 𝜙ₜ and the time index t fixed. Note also that we have here made the simplifying assumption of replacing the future pass-through parameters, 𝜅ₜ₊ᵢ, by the fixed subjective parameter, 𝜙ₜ, for all horizons i=1,2,…. As I will discuss below, this assumption simplifies the derivations and notation, but we show that it is not required.
By iterating the conditional expectation in (5) forward using (6), and using the law of iterated expectation (which applies because the subjective parameter is fixed), we show that we can derive an explicit expression for the model-consistent expectation given 𝜙ₜ. The first step of this forward iteration is given by:
Note here that because 𝜙ₜ is a fixed parameter, it “jumps” out of the conditional expectations. Moreover, the conditional expectation of future output gaps is determined by the constant-parameter AR(1) process in (2).
In Proposition 1, we show that by continuing this forward iteration, the model-consistent representation of the market’s expectation has a unique expression, given by:
This is just the expected discounted sum of future output gaps, but where the unknown future pass-through parameters are replaced by the subjective parameter 𝜙ₜ.
This illustrates the main difference between KMH and REH: KMH represents the market’s inflation expectations in terms of subjective parameters that generally differ from the actual future pass-through parameters in (1). As we write (p. 11):
because KMH recognizes that participants face Knightian uncertainty, and thus cannot have perfect probabilistic foresight, it represents their model-consistent inflation expectations in terms of exogenous subjective parameters that generally differ from the future [pass-through] parameters of the inflation process.
KMH, however, maintains REH’s feature that the functional form and relevant information of its model-consistent expectations are completely determined by (1). As we write in the paper (p. 11):
The functional form and relevant information of KMH’s model-consistent representation of participants’ inflation expectations are completely determined by the model’s specification of the inflation and output gap processes in (1) and (2). Thereby, KMH maintains Muth’s assertion that “[t]he way expectations are formed depends specifically on the structure of the relevant system describing the economy (Muth, 1961, p. 316).
Importantly, we show that the simultaneous specification of the NKPC in (1) and the model-consistent inflation expectation in (7) imposes consistency between these expectations and the resulting expression for inflation at time t+1. This is the Knightian uncertainty equivalent of Lucas’ argument that coherent model-building imposes consistency between the representation of expectations and actual future outcomes.
To see this, first insert the model-consistent representation of the market’s inflation expectation in (7) into the NKPC in (1) to derive the model’s reduced-form expression for inflation at t:
Iterating (8) one period forward shows that inflation at t+1 is given by:
Viewed from time t, however, both the pass-through parameter 𝜅ₜ₊₁ and the subjective parameter 𝜙ₜ₊₁ are unknown, but restricted to lie within the interval 𝐼. Thus, we can derive the set of conditional expectations of 𝜋ₜ₊₁ for all possible values of 𝜅ₜ₊₁ and 𝜙ₜ₊₁ within the interval. This set constitutes the model’s time-t prediction of inflation at t+1, and it formalizes the model’s Knightian uncertainty about future inflation.
In Proposition 2, we derive this set and show that the representation of the market’s inflation expectation lies within this set. Thus, it is consistent with the model’s implied prediction about future inflation.
Note that KMH nests REH as the special case where the Knightian uncertainty is reduced to probabilistic risk. For example, if the pass-through parameter is assumed to be constant instead of undergoing unforeseeable change, the Knightian uncertainty about future inflation is reduced to probabilistic risk. Consequently, KMH’s model-consistent representation of expectations in (6) is reduced to REH.
The Exogeneity of the Subjective Parameters
Within the model, the subjective parameter 𝜙ₜ represents the market’s subjective assessment of the pass-through parameter at time t+1, as well as its subjective assessment at time t of its subjective assessment at time t+1 of the pass-through parameter at t+2, and similar higher-order subjective assessments.
As I briefly mentioned above, we make the simplifying assumption that this subjective assessment can be represented as the same for all forecasting horizons. Thus, above we replaced all future pass-through parameters 𝜅ₜ₊ᵢ for i=1,2,… by the same fixed subjective parameter 𝜙ₜ to derive the model-consistent inflation expectation at time t. This simplifies the notation and derivations a lot.
In the appendix, however, we show that a model-consistent representation of expectations can be derived for any arbitrary sequence of subjective parameters as long as they all lie within the interval 𝐼. Thus, we can address questions such as what happens to inflation if market participants assess that the pass-through will shift k periods into the future?
Over time, we assume that the subjective parameter undergoes nonrepetitive, and thus unforeseeable, shifts. We formalize these shifts similarly to our formalization of the shifts in the pass-through parameter in (3), so I don’t include the equations here. This formalizes the idea that, over time, rational participants revise how they map relevant information into inflation expectations in nonrepetitive, and thus unforeseeable, ways.
The most important thing to note about the subjective parameters, however, is that KMH leaves their values, and shifts over time, exogenously determined. In other words, we do not specify a mechanism that determines their values endogenously. Nor do we specify endogenously when and how they shift in response to and anticipation of the shifts in the pass-through parameter. In the paper, we write (p. 11):
By leaving the subjective parameters exogenous and only bound to lie within the [actual pass-through] parameters’ interval, KMH acknowledges that an economist cannot know exactly how participants form […] their inflation expectations, when the inflation process undergoes unforeseeable change. Moreover, by allowing the subjective parameters to shift in nonrepetitive ways, KMH acknowledges that neither an economist nor market participants can know in advance how and when participants will revise their inflation expectations in anticipation and response to […] unforeseeable changes in the inflation process.
This illustrates a crucial implication of formalizing the inflation process in (1) to undergo unforeseeable structural changes: that we cannot specify how the subjective parameters are determined endogenously. This is because the unforeseeable change in the pass-through parameter implies that there is no objective way to assess its future values.
KMH does, however, hypothesize that there exists a sequence of subjective parameters, {𝜙ᵢ} for i=1,2,…, such that rational participants’ expectations of next period’s inflation can be represented by the model-consistent expectation in (6).
Because KMH allows these subjective parameters to differ from the model’s future pass-through parameters, it predicts that rational individuals’ inflation expectations generally differ from actual ex post inflation outcomes—even when they have access to full information.
This is just one of KMH’s key predictions about rational expectations under Knightian uncertainty that differ from REH’s under probabilistic risk.
In the next and final post of this series, I will elaborate on the differences between KMH’s and REH’s representations of rational expectations under Knightian uncertainty and probabilistic risk, respectively. In particular, I will explain how KMH reconciles market participants’ rationality with key characteristics of survey-based forecasts of inflation and other key macroeconomic variables.
References
Frydman, Roman, and Morten Nyboe Tabor (2025), “Rational Expectations of Inflation Undergoing Unforeseeable Change,” INET Center on Knightian Uncertainty Working Paper 02. Link.
Lucas (1995), “Monetary Neutrality,” Nobel Prize Lecture. Link.
Muth (1961), “Rational Expectations and the Theory of Price Movements,” Econometrica, Vol. 29, No. 3. Link.