A. Agents’ Optimization Problem

Agents act atomistically in competitive markets and choose consumption (c_t), land holdings (l_t+₁) and holdings of one-period discount bonds (b_t+1), taking as given the price of land (q_t) and the gross real interest rate (R) so as to maximize a standard intertemporal utility function:

It is critical to note that $E_t^s$ represents the expectations operator conditional on the representative agent’s beliefs formulated with the information available up to and including date t. As we explain below, these beliefs will differ in general from the rational expectations formulated with perfect information about the persistence of the financial regime, which are denoted $E_t^a$.

The agents’ budget constraint is:

Agents operate a concave neoclassical production function g(l_t) subject to a stochastic TFP shock z_t. Since land is in fixed aggregate supply, a linear production technology could also be used. We will use the curvature of g(l_t), however, to calibrate the model so that the condition that arbitrages returns across bonds and land is consistent with U.S. data on the risk-free interest rate and the value of residential land as a share of GDP (see Section III for details).

TFP shocks follow an exogenous discrete Markov process (which can be enriched to include also interest rate shocks). For these shocks, we assume that agents know their true Markov process without informational frictions. That is they know the Markov transition matrix π(z_t+1 | z_t) and the corresponding set Z of M possible realizations of z at any point in time (i.e., z_t ∈ Z = {z₁ < z₂ < .... < z_M)). Alternatively, we could assume that TFP shocks are also affected by imperfect information.

Frictions in the credit market force agents to comply with a collateral constraint that limits the value of debt (given by b_t+1/R since 1/R is the price of discount bonds) to a time-varying fraction κ_t of the market value of their land holdings:

In this constraint, κ_t is a random variable that follows a “true” Markov process characterized by a standard two-point, regime-switching process with regimes κ^h and κ^l, with κ^h > κ^l, and transition probabilities given by F^a = p^a(κ_t+1 | κ_t)⁶. The continuation transition probabilities are denoted $F_{hh}^a\equiv p^a\left({\kappa }_{t+1}={\kappa }^h|{\kappa }_t={\kappa }^h\right)$ and $F_{ll}^a\equiv p^a\left({\kappa }_{t+1}={\kappa }^l|{\kappa }_t={\kappa }^l\right)$, and the switching probabilities are $F_{hl}^a=1-F_{hh}^a$ and $F_{lh}^a=1-F_{ll}^a$. The long-run probabilities of the high- and low-leverage regimes are ${\Pi }^h=F_{lh}^a/(F_{lh}^a+F_{hl}^a)$ and ${\Pi }^l=F_{hl}^a/(F_{lh}^a+F_{hl}^a)$ respectively, and the corresponding mean durations are $1/F_{hl}^a$ and $1/F_{lh}^a$. The long-run unconditional mean, variance, and first-order autocorrelation of κ are:

As explained earlier, agents know κ^h and κ^l but do not know F^a. Hence, they make decisions based on their individual beliefs characterized by E^s which evolve over time. Using μ to denote the Lagrange multiplier on the credit constraint, the agents’ optimality conditions for bonds and land are:

With the caveat that agents use subjective beliefs to form expectations, these conditions are standard from models with credit constraints. Following Mendoza (2010), we can show that the second condition implies a forward solution for land prices in which the future stream of land dividends is discounted at the stochastic discount factors adjusted for the shadow value of the credit constraint:

Defining the return on land as $R_{t+1}^q\equiv \left(z_{t+1}{g}^{\prime }\left(l_{t+1}\right)+q_{t+1}\right)/q_t$ and the period marginal utility of consumption as λ_t+1 = βu′ (c_t+1), the excess return on land can be expressed as:

Thus, as in Mendoza (2010), the borrowing constraint enlarges the standard premium on land holdings, driven by the covariance between marginal utility and asset returns, by introducing direct and indirect effects. The direct effect is represented by the term (1 − κ_t)μ_t. The indirect effect is represented by the fact that the credit constraint hampers the agents’ ability to smooth consumption and hence reduces $co{v}_t^s\equiv ({\lambda }_{t+1},R_{t+1}^q)$. Moreover, since the expected land returns satisfy $q_t{E}_t^s[R_{t+1}^q]\equiv E_t^s\left[z_{t+1}{g}^{\prime }\left(l_{t+1}\right)+q_{t+1}\right]$, we can rewrite the forward solution for the agents’ land valuation as:

The expressions in ((10)) and ((11)) imply that the collateral constraint lowers land prices because it increases the rate of return at which future land dividends are discounted. Note also that land valuations are reduced at t not just when the constraint binds at t, which increases $E_t^s[R_{t+1}^q]$, but also if agents expect (given their beliefs F^s) that the constraint can bind at any future date, which increases $E_t^s[R_{t+1+i}^q]$ for some i > 0. Thus this expression suggests that the learning process and the collateral constraint interact in an important way. For instance, suppose the credit constraint was binding at t, pessimistic beliefs such that agents expect higher future land returns because of tight credit conditions will depress more current land prices, which will tighten more the collateral constraint.

The economy has a fixed unit supply of land, hence market clearing in the land market implies that the land holdings of the representative agent must satisfy l_t = 1 for all t, and the rest of the equilibrium conditions reduce to the following:

B. General Features of the Learning Setup

Following Cogley and Sargent (2008), our learning setup features two-point passive learning without noise, so that the belief transition probability matrix denoted by $p_t^s\left({\kappa }_{t+1}|{\kappa }_t\right)$ converges to its true value $p_t^s\left({\kappa }_{t+1}|{\kappa }_t\right)\to p^a\left({\kappa }_{t+1}|{\kappa }_t\right)$ for sufficiently large t. With this setup, agents learn about the transition probability matrix only as they observe realizations of the shocks. In particular, they learn about the true regime-switching probabilities of κ only after observing a sufficiently long set of realizations of κ^h and κ^l.⁷

This learning setup fits nicely our goal of studying a situation in which financial innovation represents an initial condition with a state κ^h but with imperfect information about the true riskiness of this new environment. Agents are ignorant about the true transition distribution of κ, since there is no data history to infer it from. Over time, if they observe a sequence of realizations of κ^h for a few periods, they build optimism by assigning a probability to the possibility of continuing in κ^h that is higher than the true value. We refer to this situation as the “optimistic phase.” Such optimism by itself is a source of vulnerability, because it is quickly reversed into a “pessimistic phase” with the opposite characteristics as the first few realizations of κ^l hit the economy. In addition, during the optimistic phase, the incentives to build precautionary savings against the risk of a shift in the ability to leverage are weaker than in the long-run RE equilibrium. This increases the agents’ risk of being caught “off-guard” (i.e., with too much debt) when the first shift to the low-leverage regime occurs.

Modeling imperfect information in this fashion implies a deviation from rational expectations. This is a key feature of our model, because it highlights the role of the short history of a new financial regime in hampering the ability of agents to correctly assess risk. This approach seems better suited for studying the role of financial innovation in causing the financial crisis, as opposed to a standard signal extraction problem with noisy signals and rational expectations.

Since κ is exogenous, we are modeling a passive learning structure from and about exogenous variables, which facilitates significantly the analysis and numerical solution of the model. In particular, it allows us to split the analysis into two parts. The first part uses Bayesian learning to generate the agents’ sequence of posterior density functions ${\left\{f\left(F^s|{\kappa }^t\right)\right\}}_{t=1}^T$. Each of these density functions (one for each date t) is a probability distribution over possible Markov transition matrices F^s. Since agents do not know the true transition matrix F^a, the density function changes with the sequence of realizations observed up to date t (i.e., { κ^t, κ^t−1,…, κ¹ } where κ^t = (κ_t, κ_t—1,…,κ₁)) and with the initial date-0 priors, as we explain below. If date T is high enough to accomodate sufficient sampling of regime switches across κ^h and κ^l, the sequence ${\left\{f\left(F^s|{\kappa }^t\right)\right\}}_{t=1}^T$ converges to a distribution with all its mass in F^a. In other words, beliefs converge to the true values, so that in the long-run the model converges to the RE equilibrium.

The second part of the analysis characterizes the model’s equilibrium by combining the sequences of posterior densities obtained in the first part, ${\left\{f\left(F^s|{\kappa }^t\right)\right\}}_{t=1}^T$, with chained solutions from a set of “conditional” optimization problems. These problems are conditional on the posterior density function of F^s that agents form with the history of realizations up to each date t. This approach takes advantage of the fact that, because of the passive learning, agents do not benefit from trying to improve their inference about the regime switching probabilities by “experimenting” using their optimization problems. As a result, the evolution of beliefs can be analyzed separately from the agents’ optimal consumption and savings plans. The remainder of this Section examines in more detail the Bayesian learning setup and the construction of the model’s equilibrium.

C. Learning and the Sequence of Beliefs

The learning framework takes as given an observed history of realizations of T periods of leverage regimes, denoted by κ^T, and a prior of F^s for date t = 0, p(F^s), and it yields a sequence of posteriors over F^s for every date t,${\left\{f\left(F^s|{\kappa }^t\right)\right\}}_{t=1}^T$.⁸ At every date t, from 0 to T, the information set of the agent includes κ^t as well as the possible values that κ can take (κ^h and κ^l).

Agents form posteriors from priors using a beta-binomial probability model. Since agents know the realization vector of κ, information is imperfect only with regard to the Markov transition matrix across κ′s, and, because κ can only take two values, this boils down to imperfect information about the continuation probabilities $F_{hh}^a$ and $F_{ll}^a$. The other two elements of the transition matrix of κ are recovered using $F_{ii}^a+F_{ij}^a=1$, where $F_{ij}^a$ denotes the true probability of switching from state i to state j.

The agents’ posteriors about $F_{hh}^s$ and $F_{ll}^s$ have Beta distributions as well, and the parameters that define them are determined by the number of regime switches observed in a particular history κ^t. As in Cogley and Sargent (2008), we assume that the priors for $F_{hh}^s$ and $F_{ll}^s$ included in p(F^s) are independent and determined by the number of transitions assumed to have been observed prior to date t = 1. More formally,

where $n_0^{ij}$ denotes the number of transitions from state i to state j assumed to have been observed prior to date 1.

The likelihood function of κ^t conditional on $F_{hh}^s$ and $F_{ll}^s$ is obtained by multiplying the densities of $F_{hh}^s$ and $F_{ll}^s$:

Multiplying the likelihood function by the priors delivers the posterior kernel:

and dividing the kernel using the normalizing constant M(κ^t) yields the posterior density:

where

The number of transitions across regimes is updated as follows:

Note that the posteriors are of the form $F_{hh}^s\propto Beta\left(n_t^{hh},n_t^{hl}\right)$ and $F_{ll}^s\propto Beta\left(n_t^{lh},n_t^{ll}\right)$. That is, the posteriors for κ^h only depend on $n_t^{hh}$ and $n_t^{hl}$ and not on the other two counters, $n_t^{lh}$ and $n_t^{ll}$, and the posteriors for κ^l only depend on $n_t^{lh}$ and $n_t^{ll}$. This is important because it implies that the posteriors of $F_{hh}^s$ change only as $n_t^{hh}$ and $n_t^{hl}$ change, and this only happens when the date-t realization is κ^h. If, for example, the economy experiences realizations κ = κ^h for several periods, agents learn only about the persistence of the high-leverage regime. They learn nothing about the persistence of the low-leverage regime. To learn about this, they need to observe realizations of κ^l. Since in a two-point, regime-switching setup persistence parameters also determine mean durations, it follows that both the persistence and the mean durations of leverage regimes can be learned only as the economy actually experiences κ^l and κ^h.⁹

We illustrate the learning dynamics of this setup by means of a simple numerical example. We choose a set of values for $F_{hh}^a$, $F_{ll}^a$, and initial priors, and then simulate the learning process for 300 quarters (75 years) using a hypothetical sequence of κ realizations produced by a stochastic simulation of the true Markov-switching process. The results are plotted in Figure 4, which shows the time paths of the conditional averages of $F_{hh}^s$ and $F_{ll}^s$ based on the beliefs formed at each date t in the horizontal axis, after observing the corresponding κ_t shown in the bottom panel. The true regime-switching probabilities are set to $F_{hh}^a=0.95$ and $F_{ll}^a=0.5$. These values are used only for illustration purposes (they are not calibrated to actual data as in the solution of the full model in Section III). In addition, the date-0 priors are set to $F_{hh}^s\sim Beta\left(0.7,0.7\right)$ and $F_{ll}^s\sim Beta\left(0.7,0.7\right)$. With these priors, the agents update their beliefs about the persistence of the high-leverage regime to around 0.78 after observing κ₁ = κ^h.

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Evolution of Beliefs

Citation: IMF Working Papers 2010, 164; 10.5089/9781455201754.001.A001

Notes: This figure plots the evolution of beliefs about $F_{hh}^s$ (top panel), $F_{ll}^s$ (middle panel), and the associated realizations of κ (lower panel). The horizontal red lines in the upper two panels mark the true values of the corresponding variables.

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The most striking result evident in Figure 4 is that financial innovation can lead to significant underestimation of risk. Specifically, the initial sequence of realizations of κ^h observed until just before the first realization of κ^l (the “optimistic phase”) generates substantial optimism. In this example, the optimistic phase covers the first 30 periods. The degree of optimism produced during this phase can be measured by the difference between the conditional expectation based on the date-t beliefs, $F_{hh}^s$, and the corresponding rational expectations value, $F_{hh}^a$ (shown in the horizontal line of the top panel). As the Figure shows, the difference grows much larger during the optimistic phase than in any of the subsequent periods. For example, even though the economy remains in κ^h from around date 40 to date 80, the magnitude of the optimism that this period generates is smaller than in the initial optimistic phase. This is because it is only after observing the first switch to κ^l that agents rule out the possibility of κ^h being an absorbent state. As a result, $F_{hh}^s$ cannot surge as high as it did during the initial optimistic phase. Notice also that the first realizations of κ^l generate a “pessimistic phase,” in which $F_{ll}^s$ is significantly higher than $F_{ll}^a$, so the period of unduly optimistic expectations is followed by a period of unduly pessimistic expectations.

Figure 4 also illustrates the fact that the beliefs about the average duration of κ^h (κ¹) are updated only when the economy is in the high- (low-) leverage state. This is evident, for example, in the initial optimistic phase (the first 30 periods), when $F_{ll}^s$ does not change at all. As explained above, for the agents to learn about the duration of the high- (low-) leverage regime, the economy needs to actually be in that regime. This feature of the learning dynamics also explains why in this example $F_{hh}^s$ converges to $F_{hh}^a$ faster than $F_{ll}^s$. Given that the low-leverage regime is visited much less frequently, it takes longer for the agents to learn about its persistence, and therefore $F_{ll}^s$ takes longer to converge to $F_{ll}^a$.

D. Recursive Competitive Equilibrium

We define the model’s competitive equilibrium in recursive form. Since in the quantitative analysis we solve the model by policy function iteration on the equilibrium conditions ((12))-((15)), we formulate the recursive equilibrium using these conditions instead of a Bellman equation (Appendix A describes the solution method in detail). The state variables in the recursive equilibrium are defined by the triple (b, z, κ).

The solution strategy works by breaking down the problem into a set of conditional optimization problems (COP) that are conditional on the beliefs agents have each period. We add time indices to the policy and pricing functions in the recursive equilibrium so as to identify the date of the beliefs that match the corresponding COP. It is critical to note that this solution strategy works because the law of iterated expectations holds with passive Bayesian learning (see Appendix B in Cogley and Sargent (2008)). This is important because, in solving each COP, agents take into account that they are in a learning environment, so they form expectations about the future, including future κ′s and the associated future beliefs, conditional on the information and beliefs they have in the current planning period. Since the law of iterated expectations holds, however, $E_t^s[E_{t+1}^s\left(x_{t+2}\right)]=E_t^s\left[x_{t+2}\right]$.

Consider first the date-1 COP. At this point agents have observed κ₁, and use it to update their beliefs. Thus, we pull f (F^s | κ¹) from the sequence of posterior density functions solved for in the previous subsection. This is the first density function in the sequence ${\left\{f\left(F^s|{\kappa }^t\right)\right\}}_{t=1}^T$, and it represents the first posterior about the distribution of F^s that agents form, given that they have observed κ₁ and given their initial priors. The solution to the date-1 COP is given by policy functions (b′₁ (b, z, κ), c₁ (b, z, κ),μ₁(b, z, κ)) and a pricing function q₁ (b, z, κ) that satisfy the equilibrium conditions ((12))-((15)) rewritten in recursive form:

where the expectations inside the integrals in ((20)) and ((21)) are of the form $E_1^s[x|f\left({\kappa }^{\prime }|{\kappa }^1,F^s\right)]\equiv {\displaystyle {\sum }_{i=l}^h\mathrm{Pr}({\kappa }^{\prime }={\kappa }^i|{\kappa }^1,F^s)}x$, for a random variable x that depends on κ′. These expectations taken by themselves are analogous to those we would calculate if we were solving a standard RE model using an Euler-equations method with an invariant Markov transition function F^s. Since agents here do not know F^a, however, the expectations in ((20)) and ((21)) also integrate over f (F^s|κ¹). The time subscripts that index the policy and pricing functions indicate the date of the beliefs use to form the expectations (which is also the date of the most recent observation of κ, which is date 1 in this case).

Recall that Equations ((20))-((23)) incorporate the market-clearing condition in the land market, which requires l = 1. Moreover, given ((20))-((21)), the pricing function q₁ (b, z, κ) satisfies the asset pricing equation ((11)).

It is critical to note that solving for date-1 policy and pricing functions means solving for a full set of optimal plans over the entire (b, z, κ) domain of the state space and conditional on f (F^s | κ¹). Thus, we are solving for the optimal plans agents “conjecture” they would make over the infinite future acting under the beliefs given by f (F^s | κ¹). This COP remains recursive, particularly in terms of forming expectations about future variables and beliefs, because the law of iterated expectations still holds. For characterizing the “actual” equilibrium dynamics to match against the data, however, the solution of the date-1 COP determines optimal plans for date 1 only.

Generalizing the date-1 problem we can define COPs for all subsequent dates t = 2,…, T using the corresponding density function f (F^s | κ^t) for each date t in the sequence of beliefs solved for earlier. This is crucial because f (F^s | κ^t) changes as time passes and each subsequent κ_t is observed, reflecting the passive Bayesian learning, which implies that the policy and pricing functions that solve each COP also change. Thus, history matters for the “full solution” of the model because assuming different histories κ^t yields different densities f (F^s | κ^t), and hence different sets of policy functions. If at any two dates t and t + j we give the agents the same values for (b, z, κ), they in general, will not choose the same bond holdings for the following period because f (F^s | κ^t) and f (F^s | κ^t+j) will differ.

The solution to the date-t COP is given by policy functions (b′_t (b, z, κ), c_t (b, z, κ), μ_t (b, z, κ)) and a pricing function q_t (b, z, κ) that satisfy the model’s equilibrium conditions: