Economic Model & Calibration

Here the economic model of Rust (1987) is documented and two ways of calibrating its parameters are introduced. The first one is the nested fixed point algorithm (NFXP) initially suggested by Rust (1987) and the second one is mathematical programming with equilibrium constraints (MPEC) based on Su and Judd (2012). This builds the theoretic background for the estimation and simulation modules of ruspy.

The Economic Model

The model is set up as an infinite horizon regenerative optimal stopping problem. It considers the dynamic decisions by a maintenance manager, Harold Zurcher, for a fleet of buses. As the buses are all identical and the decisions are assumed to be independent across buses, there are no indications of the bus in the following notation. Harold Zurcher makes repeated decisions \(a\) about their maintenance in order to maximize his expected total discounted utility with respect to the expected mileage usage of the bus. Each month \(t\), a bus arrives at the bus depot in state \(s_t = (x_t, \epsilon_t(a_t))\) containing the mileage since last engine replacement \(x_t\) and other signs of wear and tear plus decision specific information \(\epsilon_t(a_t)\). Harold Zurcher is faced with the decision to either conduct a complete engine replacement \((a_t = 1)\) or to perform basic maintenance work \((a_t = 0)\). The cost of maintenance \(c(x_t, \theta_1)\) increases with the mileage state, while the cost of replacement \(RC\) remains constant. Notationwise \(\theta_1\) captures the structural parameters shaping the maintenance cost function. In the case of an engine replacement, the mileage state is reset to zero.

The immediate utility of each action in month \(t\) is assumed to be additively separable and given by:

\[\begin{split}\begin{align} u(a_t, x_t, \theta_1, RC) + \epsilon_t(a_t) \quad \text{with} \quad u(a_t, x_t, \theta_1, RC) = \begin{cases} -RC - c(0, \theta_1) & a_t = 1 \\ -c(x_t, \theta_1) & a_t = 0. \end{cases} \end{align}\end{split}\]

The objective of Harold Zurcher is to choose a strategy \(\pi\) of all strategies \(\Pi\) to maximize the utility over infinite horizon and therefore the current value in period \(t\) and state \(s_t\) is given by:

\[\begin{align} \tilde{v}^{\pi}(s_t) \equiv \max_{\pi\in\Pi} E^\pi\left[\sum^{\infty}_{i = t} \delta^{i - t} u(a_t, x_t, \theta_1, RC)) + \epsilon_t(a_t) \right]. \end{align}\]

The discount factor \(\delta\) weighs the utilities over all periods and therefore captures the preference of utility in current and future time periods. As the model assumes stationary utility, as well as stationary transition probabilities the future looks the same, whether the agent is at time \(t\) in state \(s\) or at any other time. Therefore the optimal decision in every period can be captures by the Bellman equation:

\[\begin{equation} v_\theta(x_t, \epsilon_t) = \max_{a_t \in \{0,1\}} \biggl[u(x_t, a_t, \theta_1, RC) + \epsilon_t(a_t) + \delta EV_\theta(x_t, \epsilon_t, a_t)\biggr], \end{equation}\]


\[\begin{equation} EV_\theta(x_t, \epsilon_t, a_t) = \int \int v_\theta(\gamma, \eta) p(d\gamma, d\eta | x_t, \epsilon_t, a_t, \theta_2, \theta_3) \end{equation}\]

and \(\theta\) captures the parametrization of the model given by \(\{\delta, RC, \theta_1, \theta_2, \theta_3 \}\). Thus Harold Zurcher makes his decision in light of uncertainty about next month’s state realization captured by the their conditional distribution \(p(x_{t+1}, \epsilon_{t+1} | x_t, \epsilon_t, a_t, \theta_2, \theta_3)\).

Rust (1987) imposes conditional independence between the probability densities of the observable and unobservable state variables, i.e.

\[\begin{equation} p(x_{t+1}, \epsilon_{t+1}| x_t, a_t, \epsilon_t, \theta_2, \theta_3) = p(x_{t+1}| x_t, a_t, \theta_3) p(\epsilon_{t+1}|\epsilon_t, \theta_2) \end{equation}\]

and furthermore assumes that the unobservables \(\epsilon_t(a_t)\) are independent and identically distributed according to an extreme value distribution with mean zero and scale parameter one, i.e.:

\[\begin{equation} p(\epsilon_{t+1}| \theta_2) = \exp\{-\epsilon_{t+1} + \theta_2\} \exp\{-\exp\{-\epsilon_{t+1} + \theta_2 \}\} \end{equation}\]

where \(\theta_2 = 0.577216\), i.e. the Euler-Mascheroni constant.

Rust (1988) shows that these two assumptions, together with the additive separability between the observed and unobserved state variables in the immediate utilities, imply that \(EV_\theta\) is a function independent of \(\epsilon_t\) and the unique fixed point of a contraction mapping on the reduced space of all state action pairs \((x,a)\). Furthermore, the regenerative property of the process yields for all states \(x\), that the expected value of replacement corresponds to the expected value of maintenance in state \(0\), i.e. \(EV_\theta(x, 1) = EV_\theta(0, 0)\). Thus \(EV_\theta\) is the unique fixed point on the observed mileage state \(x\) only. Therefore in the following \(EV_\theta(x)\) refers to \(EV_\theta(x, 0)\). The contraction mapping is then given by:

\[\begin{equation} EV_\theta(x) = \sum_{x' \in X} p(x'|x, \theta_3) \log \sum_{a \in \{0, 1\}} \exp( u(x' , a, \theta_1, RC) + \delta EV_\theta(x')) \end{equation}\]

This gives rise to the shorthand notation of the above formula:

\[\begin{equation} EV_\theta(x) = T_\theta(EV_\theta(x)) \end{equation}\]

In addition, the conditional choice probabilities \(P(a| x, \theta)\) have a closed-form solution given by the multinomial logit formula (McFadden, 1973):

\[\begin{equation} P(a|x, \theta) = \frac{\exp(u(a, x, RC, \theta_1) + \delta EV_\theta((a-1) \cdot x))}{\sum_{i \in \{0, 1\}} \exp(u(i, x, RC, \theta_1) + \delta EV_\theta((i - 1)x))} \end{equation}\]

These closed form solutions allow to estimate the structural parameters driving Zurcher’s decisions. Given the data \(\{a_0, ....a_T, x_0, ..., x_T\}\) for a single bus, one can form the likelihood function \(l^f(a_1, ..., a_T, x_1, ...., x_T | a_0, x_0, \theta)\) and estimate the parameter vector \(\theta\) by maximum likelihood. Rust (1988) proofs that this function has due to the conditional independence assumption a simple form:

\[\begin{equation} l^f(a_1, ..., a_T, x_1, ...., x_T | a_0, x_0, \theta) = \prod_{t=1}^T P(a_t|x_t, \theta) p(x_t| x_{t-1}, a_{t-1}, \theta_3) \end{equation}\]

Therefore the estimation can be split into two separate partial likelihood functions, given by:

\[\begin{equation} l^1(a_1, ..., a_T, x_1, ...., x_T | a_0, x_0, \theta_3) = \prod_{t=1}^T p(x_t| x_{t-1}, a_{t-1}, \theta_3) \end{equation}\]


\[\begin{equation} l^2(a_1, ..., a_T, x_1, ...., x_T | \theta) = \prod_{t=1}^T P(a_t|x_t, \theta) \end{equation}\]

Nested Fixed Point Algorithm

The calibration strategy employed by Rust (1987) involves handing the logarithm of the above \(l^f(a_1, ..., a_T, x_1, ...., x_T | a_0, x_0, \theta)\) to an unconstrained optimization algorithm. Rust originally suggests a polyalgorithm of the BHHH and the BFGS for this purpose. This optimizer fixes a guess of the structural parameter vector \(\hat\theta\) for which the unique fixed point of the economic model is found. Through this the conditional choice probabilities \(P(a|x, \hat\theta)\) are obtained which in turn are used to evaluate the log likelihood function. On the basis of this, the optimization algorithm comes up with a new guess for the structural parameters and the procedure starts over until a certain convergence criteria is met.

The algorithm consequently corresponds to solving the following optimization problem in an outer loop:

\[\begin{equation} \max_{\theta} \; log \; l^f(a_1, ..., a_T, x_1, ...., x_T | a_0, x_0, \theta) \end{equation}\]

while finding the unique fixed point of \(EV_\theta(x) = T_\theta(EV_\theta(x))\) in an inner loop for a given parameter guess produced in the outer loop.

Mathematical Programming with Equilibrium Constraints

The approach developed by Su and Judd (2012) casts this unconstrained nested problem into a constrained optimization problem. For this they plug the conditional choice probabilities \(P(a|x, \theta)\) into the likelihood function \(l^f(.)\):

\[\begin{split}\begin{equation} \begin{split} l^f_{aug}(. | a_0, x_0, \theta, EV) = & \prod_{t=1}^T \frac{ \exp(u(a, x, RC, \theta_1) + \delta EV((a-1) \cdot x))}{ \sum_{a \in \{0, 1\}} \exp(u(a, x, RC, \theta_1) + \delta EV((a - 1)x))} \\ \\ & \times p(x_t| x_{t-1}, a_{t-1}, \theta_3). \end{split} \end{equation}\end{split}\]

They coin the term augmented likelihood function for \(l^f_{aug}\). The particular feature now is that the likelihood depends explicitly on both the structural parameter vector \(\theta\) as well as the choice of \(EV\). In order to ensure that guesses of both vectors are consistent in the spirit of the economic model, the contraction mapping of the expected value function is imposed as a constraint to the augmented likelihood function. Consequently, the calibration problem boils down to a constrained optimization looking like the following:

\[\begin{split}\begin{equation} \max_{(\theta, EV)} \; log \; l^f_{aug}(a_1, ..., a_T, x_1, ...., x_T | a_0, x_0, \theta, EV) \\ \text{subject to } \; EV = T(EV, \theta). \end{equation}\end{split}\]

The constraints are generally nonlinear functions which restricts the use of optimization algorithms. An non-exhaustive list of optimizers that can handle the above problem are the commercial KNITRO (see Byrd et al. (2006)), as well as the open source IPOPT (see Wächter and Biegler (2006)) and the SLSQP (see Kraft (1994)) provided by NLOPT.

The Implied Demand Function

Rust (1987) shortly describes a way to uncover an implied demand function of engine replacement from his model and its estimated parameters. Theoretically, for Harold Zurcher the random annual implied demand function takes the following form:

\[\begin{equation*} \tilde{d}(RC) = \sum_{t=1}^{12} \sum_{m=1}^{M} \tilde{a}^m_t \end{equation*}\]

where \(\tilde{a}^m_t\) is the replacement decision for a certain bus \(m\) in a certain month \(t\) derived from the process {\(a^m_t, x^m_t\)}.

For convenience I will drop the index for the bus in the following. Its probability distribution is therefore the result of the process described by \(P(a_t|x_t; \theta)p(x_t|x_{t-1}, a_{t-1}; \theta_3)\). For simplification Rust actually derives the expected demand function \(d(RC)=E[\tilde{d}(RC)]\). Assuming that \(\pi\) is the long-run stationary distribution of the process {\(a_t, x_t\)} and that the observed initial state {\(a_0, x_0\)} is in the long run equilibrium, \(\pi\) can be described by the following functional equation:

\[\begin{equation} \pi(x, a; \theta) = \int_{y} \int_{j} P(a|x; \theta)p_3(x|y, j, \theta_3) \pi(dy, dj; \theta). \end{equation}\]

Further assuming that the processes of {\(a_t, x_t\)} are independent across buses the annual expected implied demand function boils down to:

\[\begin{equation} d(RC) = 12 M \int_{0}^{\infty} \pi(dx, 1; \theta). \end{equation}\]

Given some estimated parameters \(\hat\theta\) from calibrating the Rust Model and parametrically varying \(RC\) results in different estimates of \(P(a_t|x_t; \theta)p(x_t|x_{t-1}, a_{t-1}; \theta_3)\) which in turn affects the probability distribution \(\pi\) which changes the implied demand. In the representation above it is clearly assumed that both the mileage state \(x\) and the replacement decision \(a\) are continuous. The replacement decision is actually discrete, though, and the mileage state has to be discretized again which in the end results in a sum representation of the function \(d(RC)\) that is taken to calculate the expected annual demand.

This demand function can be calculated in the ruspy package for a given parametrization of the model. A description how to do this can be found in Demand Function Calculation.