which is almost always a binding one due to the
fact that the firm only occasionally places large positive orders and
on most days orders nothing. My work on an improved algorithm to solve
this problem lead me to develop and implement
a very promising new solution method
I call ``parametric policy iteration'' that I will describe below
in more detail.
such that if the
firm's current inventory q is less than s(p,x) it is optimal for
it to order an amount
qo(p,q,x)=S(p,x)-q to bring its inventory
level up to the desired level S(p,x). If
q < s(p,x) then
it is optimal not to order, i.e.
qo(p,q,x)=0. These functions depend
on the current wholesale price p and a vector x of other
variables that affect the firm's beliefs about future sales and
costs including ``macro shocks'' (e.g. industrial production indices
and indices indicating overall steel prices, etc.) and current interest
rates (which affects the firm's interest opportunity costs for holding
steel). We also observed that when the fixed costs of placing an order
K is zero we had
S(p,x)=s(p,x), otherwise whenever
s(p,x) > 0 we
found that S(p,x) is strictly greater than s(p,x). However when
we presented our work at a seminar at CREST in Paris, France, our
discussant, Guy Laroque (an expert in inventory theory and commodity
pricing) claimed that our numerical results could not be right and
that the optimal trading strategy would not be of the generalized
(S,s) form. However in a subsequent theoretical paper,
``The
(S,s) Rule
is an Optimal Trading Strategy in a Class of Commodity
Price Speculation Problems'',
we were able to prove that the generalized (S,s) rule suggested
by our computational results is in fact an optimal trading strategy
in a fairly wide class of speculative trading
dynamic inventory problems. We proved the result by
providing a sufficient condition under which the value function of the
firm's dynamic programming problem has the property of K-concavity (a function f(q) is K-concave if
for all
satisfying
we
have:
![$ f(q+z) - K \le f(q) + z[ f(q) - f(q-b)]/b$](img5.gif){kind=small}
).
Following original work by Herbert Scarf (who was the first to define
the analogous property of K-convexity and use it to prove the
optimality of the (S,s) rule in the simpler context of minimizing costs
of holding inventory in a problem without any Markovian cost shocks)
we proved that if the value function V(p,q,x) is K-concave in q
for any fixed values of (p,x) then the generalized (S,s) rule is an
optimal trading strategy for a commodity price speculator. Our new
contribution was a fairly general sufficient condition for the
K-concavity property to hold: we call it the no expected loss
condition. A strong form of
this condition is that the retail price on day t always exceeds the
expected discounted procurement price on day t+1,
,
where
is the speculator's
discount factor. This is an extremely natural and intuitive condition
that we would expect to be automatically satisfied by profit maximizing
speculators: it states that no speculator would sell a unit of
inventory for an amount less than the expected discounted cost of
replacing that unit on the next trading day.
We believe this represents an important new theoretical contribution
that links formerly separate liteatures on the theory of speculation and
optimal inventory theory. This paper was very well received when
we presented it last year at the MIT Operations Research seminar and
one of the editors of the journal Operations Research invited
us to submit it for consideration for publication. We are currently
extending the paper to allow endogenous determination of the retail
price prt in which case we believe we can derive the no expected
loss condition as a consequence of expected profit maximization and
prove our results for an even wider class of problems than has
ever been considered in the literature. We expect to be submitting the
paper to Operations Research shortly.
governing the vector Markov process
where pt is the wholesale price of a particular steel
product and xt is a vector of macro varaiables such as industrial
production indices, unemployment rates, interest rates, etc. that are correlated
with and useful for forecasting future steel prices), 2) John H's
beliefs about the demand for steel at his New England plant (which we
represent by a conditional probability distribution
which gives the conditional
probability distribution of retail sales qrt on trading day tconditional on the retail sales price prt, the wholesale price ptand the other macro variables, xt). Our objective is to find values
of the unknown parameters
characterizing
the key objects determining John H's trading strategy for steel by
finding values for which the implied optimal trading strategy
(computed from our dynamic programming model) ``best fits'' the observed
sequence of purchases, sales, retail and wholesale prices, and
inventory levels for each of the 2200+ steel products that John H's
firm trades. Note that the conditional distribution H constitutes a
stochastic downward sloping demand function for steel, in the sense
that the conditional expectation of demand for steel,
is a
downward sloping function of the retail price prt that John H sets,
reflecting his local market power due to informational problems and
search costs and the relatively high transportation costs for steel.
This demand is complicated by having a mass point at 0, reflecting the
fact that on given business day there is a positive probability of zero
retail sales. The straightforward way to approach the
problem is to estimate the unknown parameters 
by the method
of maximum likelihood. However this presents several major problems.
First, deriving the likelihood function is quite complicated due to
the fact that the stochastic process for purchases, sales and
inventory levels has discrete and continuous components. For example
as noted above, on any given day there is positive probability that
purchases and sales will be zero, but also positive probability of
a stockout, i.e. of receiving a retail demand that would exceed current
stock on hand, resulting in inventory level of qt+1=0 on the
next trading day. The second problem is that the likelihood function
governing the full data,
(where
![$q^s_t=\min[q_t+q^o_t,q^r_t]$](img15.gif){kind=small}
is actual retail sales, the smaller of
retail demand qrt and stock on hand after orders are made,
qt+qot) depends on the solution to the full dynamic programming
problem. Thus it is not possible to express the likelihood function in
closed form, but rather a ``nested fixed point'' algorithm similar to
the algorithm I pioneered in my earlier work described in the appendix
must be employed. The nested fixed point is the contraction fixed
point representation of the infinite horizon dynamic prorgramming
problem. Our computational advances allow us to solve this problem
rapidly, but it still must be solved many hundreds of times for the
various trial
values of
that are encountered in the search
for the the maximum likelihood estimates
.
where
is the optimal order threshold corresponding
to a dynamic program with parameter
described
previously. While I have been able to provide sufficient
conditions under which such ``ultra high dimensional'' numerical
integration is both possible and can be done at essentially
uni-dimensional rates when the ``effective dimension'' of the
problem is low (this work is forthcoming
in a recently accepted Econometrica
paper with computer scientists Joseph Traub and Henryk Wozniakowski),
it is hard to verify analytically that our likelihood statisfies the
regularity conditions in our forthcoming Econometrica paper.
While George Hall and I plan to try to implement direct maximum
likelihood estimation using these ``low discrepancy''
quasi monte carlo methods, we discovered an alternative much simpler
estimation method that we call ``simulated minimum distance
estimation'' (SMD). We have verified that this new estimation procedure not
only is computationally feasible, but it also works (both theoretically
in terms of
its asymptotic properties, and in practice in
monte carlo studies of the actual
numerical performance of the estimator, where we generated artificial
data from the model and verified that the SMD estimator converges to
values very close to our choices of the ``true'' parameter values
.
This work is described in our paper,
``Econometric Methods for Endogenously Sampled Time Series: The Case of
Commodity Price Speculation in the Steel Market''.
.
The idea is that once we solve the DP problem and
uncover the optimal dynamic trading strategy for any trial value of the
unknown parameters ,
we can simulate the model and censor and
endogenously sample the simulated data in exactly the same way as the
actual data are censored and endogenously sampled, i.e. we throw away
the simulated wholesale prices of steel on the days that the simulated
trading strategy does not make purchases of steel. Thus some of the
simulated moments will also be biased and inconsistent estimates of the
corresponding ergodic moments of the process, but the key point is that
both the actual sample moments and the simulated moments will be biased
in symmetrical ways so that an estimator that is formed by choosing
values
that minimizes a quadratic form in the difference
between the simulated and sample momemnts can be shown to be
consistent and asymptotically normally distributed. The estimator is
similar in many respects to the ``method of simulated momemts'' (MSM)
estimator pioneered by the recent Nobel prize winner Daniel
McFadden (my thesis advisor at MIT), and the asymptotic properties
of the SMD estimator is similar to the MSM estimator (although the
former is derived in a time series context and the latter in a
cross sectional context). In particular, the SMD estimator can be
shown to be consistent and asymptotically normal using only
a single time series realization of the trading process to form
simulated moments.
The cost to doing only one simulation of the time series of
trading behavior
(as opposed to averaging the simulated moments
from S independent time series realizations) is that the covariance
matrix for the
parameters is multiplied by a factor
(1+1/S). Thus, when S=1 the penalty to doing only a single
simulation is to double the variances of the parameter estimates
compared to a situation where 
which would essentially be
equivalent to doing exact numerical integration to form the moments
implied by the DP solution.
- 1.
- The level of search costs,
- 2.
- The market maker's per unit transactions cost, c
- 3.
- The most efficient middle man's transaction cost, kl
,
the three regimes
depend on the values of kl and c and can be characterized as
follows:
- 1.
- If c is sufficiently large relative to kl, it is not
profitable for a market maker to enter, so middle men handle all trade.
- 2.
- If kl is sufficiently large relative to c, the market
maker is able to drive all middle men out of business, so
the market maker handles all trade.
- 3.
- For intermediate values of kl and c middle men and market
makers coexist. The division of trade depends on the relative values
of c and kl.
- 1.
- Estimate the DP model using all available waves of the
Health and Retirement Survey (HRS). (The Rust-Phelan model was
estimated on the older and now out of date Retirement History Survey
based on the 1905-1912 birth cohorts whowere followed over the
decade 1969 to 1979).
- 2.
- Relax the assumption that consumption equals savings
in the Rust-Phelan model and allow individuals to accumulate and
decumulate assets in an optimal fashion.
- 3.
- Expand the treatment of Social Security to include the
Social Security Disability Insurance program (SSDI) and the Social
Security Supplemental Security Income program (SSI).
Appendix: Research Summary Prepared for Yale University in 1996
Nearly all of the research that I have done after receiving my PhD in Economics at MIT in 1983 has focused on developing econometric and computational methods for understanding and predicting human decision-making over time and under uncertainty, an area I refer to as empirical models of stochastic decision processes (SDP's). Although mathematicians and statisticians have provided us with an elegant but abstract normative theory of optimal sequential decision making under uncertainty (known as statistical decision theory or stochastic control), my interest has been on the positive application of this theory to develop improved empirical models of human decision making. I have been able to solve a number of difficult problems associated with the practical implementation of this abstract theory, resulting in the development of a set of computational and econometric tools for building, estimating, and simulating fairly detailed and realistic empirical models of a wide range of economic phenomena. A considerable body of empirical research conducted by myself and others over the last decade has demonstrated that the SDP framework leads to improved understanding and more accurate quantitative predictions of individual behavior. My primary contributions have been in the following three areas:
1. development of econometric methods for structural estimation of the primitives of an SDP, namely, the individual's underlying preferences and beliefs. These estimation methods are used in conjunction with diagnostic goodness of fit tests to help us judge whether SDP models provide accurate descriptions of human behavior.
2. development of faster algorithms to allow us to compute approximate solutions to increasingly detailed and realistic dynamic choice models. The speed and accuracy of these algorithms are critical to our ability to do empirical work in this framework due to the fact SDP model is repeatedly re-solved many times for varying parameter values in a subroutine of an outer maximum likelihood estimation algorithm.
3. successful empirical applications of SDP models with particular focus on dynamic models of retirement behavior. The estimated retirement model generates accurate predictions of individual retirement behavior and provides simple explanations of several puzzling features of retirement behavior in the U.S. that previous models had been unable to explain.
Section 1 of this summary begins with a general discussion of the problems that motivated my research on SDP's, with special focus on the use of this framework to improve our understanding of retirement behavior. Section 2 summarizes some of my recent work in computational economics, including a new algorithm that succeeds in breaking the ``curse of dimensionality'' of computing approximate solutions to an important sub-class of SDP problems. Section 3 describes joint research with John Miller of Carnegie Mellon and Richard Palmer of Duke University on the use of artificial intelligence methods and automata trading in double auction markets. Section 4 concludes with an outline of some of my future research plans.
1. Empirical Models of Human Decision Making using Stochastic Decision Processes
Initially most economic theories
and behavioral models (including the classical
general equilibrium model of Arrow and Debreu)
abstracted from a realistic treatment of time and
uncertainty. However research by economists, mathematicians,
and statisticians in the 1950's and 1960's lead to
the development of the theory of dynamic programming, (DP),
a very powerful tool for
building an immense variety of models of
optimal behavior that explicitly incorporates time and
uncertainty. The
``behavior'' implied by a DP model is given by an
optimal decision rule
, i.e. a function
specifying the individual's optimal decision dt in state stat time t (the precise sense in which this rule is optimal will
be explained shortly). Starting in the late
1960's and early 1970's the theory of dynamic programming
lead to a virtual revolution in economic theory, providing
a major impetus for the development
of the theory of rational expectations and
dynamic game theory -- areas that now constitute the
core of modern economic theory.
Before we could assess
the empirical validity of these new dynamic
economic theories, we needed to develope a completely
different econometric methodology, which I refer
to as dynamic structural econometrics.
Traditional ``reduced-form'' statistical
and econometric methods (including regression
analysis) can be viewed as a way of summarizing the stochastic
process governing the observable variables,
where
st is the state and dt is the action taken by
a decision maker at time t. Traditional
estimation methods posit a specific functional form for this stochastic
process and estimate its unknown parameters. Dynamic structural
econometric methods treat
as a controlled
stochastic process whose law of motion cannot be
specified a priori but must be derived
from the solution to the DP problem. The object of inference
in dynamic structural econometrics is not
the stochastic process
, but rather the
individual's underlying
preferences and beliefs. In a DP problem
an individual's preferences are represented by a utility
function
u(st,dt) specifying the psychological or monetary
reward received by an individual or firm who is in state
st and takes action dt at time t, and also by a
subjective discount factor
at which
the individual trades off current versus future
utility. Beliefs are
specified by a transition probability
p(st+1|st,dt)representing the individual's subjective beliefs about next
period's state given the current state and action.
Given any particular specification for beliefs and
preferences,
,
the method of dynamic programming can be used to
derive the optimal decision rule and the
controlled stochastic process
implied by
as the solution to the following optimization problem
where V is the value function representing the maximum expected disounted utility of an individual in state s at time t=0 and
denotes mathematical expectation over the stochastic
process
induced by the law of motion
p and the decision rule .
Dynamic structural econometrics can be viewed
as solving the following inverse problem:
given a stochastic process
governing the states and decisions of a
decision maker, is there a set
beliefs and preferences
for which
coincides with the solution to a DP problem,
and if so, can we infer
from observations of
?
In practice we solve this inverse or ``revealed preference'' problem by
parameterizing
in terms of a vector of unknown
parameters, so that different values of index
different sets of beliefs and preferences. Let
denote the preferences and
beliefs corresponding to the vector , and let
and 
denote the corresponding optimal
decision rule and value function. The goal of dynamic structural
estimation is to compute an estimate
so that
the implied optimal decision rule
``best
fits'' our observations of
in a sense I will make precise below.
One of the first contributors to this new econometric
methodology was Thomas Sargent in his paper on
dynamic labor demand in the 1978 Journal of Political
Economy. This work and subsequent joint work by Sargent
and Lars Hansen in
the early 1980's yielded a well-developed estimation theory for the
class of linear-quadratic DP problems, i.e. problems
where the individual's
utility
is assumed to be a quadratic function of their
state s and action d and the individual's beliefs
are specified by
linear stochastic difference equations. Subsequent work by
Hansen and Kenneth Singleton in 1982 developed a method for structural
estimation of a much wide class of models, namely DP problems
whose optimal decision rules satisfy a stochastic
Euler equation, a functional equation characterizing the first
order necessary condition for the optimal decision rule
. Since the derivation of this first order
condition requires differentiation with respect to the decision
variable d, the
method only applies to continuous decision processes
(CDP's), i.e. SDP's where the individual has
a continuum of possible actions in each state s.
In particular, these methods cannot be used for structural
estimation of discrete decision processes (DDP's),
i.e. SDP's where the individual has only a finite number of possible
actions in each state s. An example of a DDP is
an optimal search or stopping problem where there
are two possible actions: dt=0 (reject current job
offer and continue searching), and
dt=1 (accept current job offer and stop searching).
Motivated by prior work by my thesis adviser Dan McFadden
on structural estimation of static discrete choice models,
in 1984 I developed a nested fixed point maximum likelihood
estimation algorithm (NFXP) that could be used to estimate preferences
and beliefs of a wide class of DDP problems. The
first empirical application of this
approach was subsequently published in Econometrica in 1987.
I established the asymptotic statistical properties of
of the NFXP algorithm and
the associated maximum likelihood estimator of (i.e.
consistency and asymptotic normality) in my paper
in the 1988 SIAM Journal on Control and Optimization.
The idea behind the NFXP algorithm
is conceptually quite simple: one uses fast algorithms
that repeatedly re-solve the DP problem (1) for different
trial parameter values until the predictions of the DP model
``best fit'' the data in the sense of maximizing the likelihood
of the observed states and decisions of a sample of individuals.
The NFXP approach was initially treated with a
great deal of skepticism in the economics
profession since it was widely believed that
the nested numerical solution of the inner DP problem would be
computationally intractable and round-off and
approximation error in the numerical solution
would lead to instabilities
in the outer maximum likelihood algorithm. However I was
able to develop a fast, numerically stable DP solution
algorithm that enabled me to estimate small to medium sized
problems on personal computers, and large scale problems on
supercomputers. The implementation of the NFXP algorithm
is complicated by the fact that my estimation
framework requires the individual's state s to be partitioned into two
components,
where x is observed by the
individual and the econometrician and 
is observed
only by the individual. This decomposition reflects the practical
reality that no data set will ever be able to completely measure
the full state of an individual. It is also motivated by
the fact that the optimal decision rule
is a deterministic function of
, which implies
that in principle we could perfectly predict an individual's
behavior if we knew
and were able to observe
in addition to x. Of course, no theory
is sufficiently powerful to be able to perfectly
predict an individual's behavior, and the
assumption of fully observed state leads to a statistically degenerate
econometric model (i.e. there may be no value of that
can perfectly predict an individual's actual behavior). I was able to
avoid this degeneracy problem by integrating out the unobserved state
variable to obtain a conditional choice
probability
:
where
denotes the indicator function (1 if
and 0 otherwise), and
denotes
the conditional density of the unobserved state variable
given the
value x of the observed state variable. The choice probability
implies that there is a positive probability (perhaps
very small) that the individual will choose any decision
d from the set D(s) of feasible decisions in state s.
While this approach solves
the problem of statistical degeneracy and yields a well-defined
likelihood function 
given by:
it creates additional computational problems since the unobserved state variable
will typically be a vector of continuous
state variables. This leads to difficult numerical integration problems
in order to compute the value function
and the conditional choice probability
.
Borrowing from the previous work of my thesis
adviser Dan McFadden, I showed that if
is a multivariate
extreme value distribution, and if the individual's
utility function takes the form
where
is a vector with as many components
as there are alternatives in the individual's choice set D(s), then
one obtains a very simple closed-form
expression for the choice probabilities
:
where the function
is the unique solution to the functional
equation:
![\begin{displaymath}v_\theta(x,d) = u_\theta(x,d) + \beta \int \log\left[ \sum_{d...
...exp\{ v_\theta(x',d')\}\right] p_\theta(dx'\vert x,d). \eqno(6)\end{displaymath}](img52.gif)
Equation (5) is a dynamic generalization of the classical multinomial logit model that includes the well known static logit model as a special case when
.
The function
can be regarded as an intertemporal
generalization
of the static utility function
of
the static logit model and is related to the
value function
given in equation (1) via the
identity:
![\begin{displaymath}V_\theta(x,\epsilon) = \max_{d \in D(s)} \left[ v_\theta(x,d) + \epsilon(d)\right].
\eqno(7) \end{displaymath}](img57.gif){kind=small}
A similar approach to estimation has been adopted in independent contributions by other econometricians including Gotz and McCall's 1979 paper on exit decisions of air force pilots, Miller's 1984 work on occupational choice, Pakes's 1987 paper on patent renewal, and Wolpin's 1984 paper on fertility decision of Malayasian households. With the exception of Miller's method (which is restricted to the relatively narrow class of multi-armed bandit problems), the approach adopted by these other authors does not easily extend to non-binary decision problems whereas the methodology I developed applies to DDP problems with an arbitrary finite number of alternatives. Due in part to the relative simplicity of my estimation framework and the computational advantages of the closed-form multinomial logit expressions for the choice probabilities, nearly all applied work on estimation of non-binary DDP problems to date has adopted my estimation framework, estimating the structural parameters using versions of my NFXP algorithm. A recent example is Donna Boswell's 1994 use of the NFXP algorithm to estimate an SDP model of the effect of employer health care policies on worker absenteeism.
Having developed the econometric tools for estimating discrete dynamic programming models, my subsequent research turned to the more interesting economic question: do human decision makers actually behave according to the theory of dynamic programming? My first empirical application of the method was the paper ``Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher'' which appeared in the 1987 Econometrica. Zurcher was the head of maintenance at the Madison Metro Bus Company. One of Zurcher's regular duties was to decide which buses in his fleet should have their engines replaced with a new or overhauled engine, and which buses should have their current engines left in place with only routine engine maintenance being performed. I constructed a DP model that predicted the optimal time to replace an engine on a bus. The decision variable is binary: dt=0(keep the current engine), dt=1 (replace the current engine). The observed state variable xt was the cumulative mileage on the bus since last engine replacement. The optimal replacement policy was chosen to minimize the expected discounted costs of operating the bus. I found that the DP model did a remarkably good job of predicting Zurcher's replacement decisions, and was able to provide simple explanations for several puzzling features in the data. One of puzzles was the fact that Zurcher replaced the engines in a group of newer GMC buses an average of 57,000 miles earlier than the engines on older GMC buses despite the fact that the cost of replacment engines for the new GMC buses were 25% higher. The DP model ``explained'' this fact by imputing a faster rate of deterioration to the newer GMC engines. Follow-up discussions with Zurcher revealed that this was indeed the case and was precisely the reason why engines on the newer buses were replaced more frequently. The DP model also yielded predictions of the ``replacement demand function'', i.e. the number of annual replacement bus engines installed as a function of the cost of a replacement engine. I experimented with traditional econometric techniques for estimating this demand function including regressing the annual number of engine replacements on the cost of replacement engines. I found that these simpler econometric methods lead to inaccurate estimates of the replacement demand function for two reasons: 1) I had less than 10 years of time series data on this group of GMC buses so the regressions had very few degrees of freedom, 2) there had been little or no variation in the real cost of replacement engines for various models of GMC buses. When I ignored model year differences and pooled the older GMC buses with the newer GMC buses (whose engines are 25% more expensive) the regression model predicted an upward sloping replacement demand function.
The Zurcher paper succeeded in convincing many skeptics that the NFXP algorithm is computationally tractable and that at least for certain well-structured problems the dynamic programming framework can be quite successful in predicting the behavior of the ordinary ``man on the street''.1 In recognition of this contribution to our understanding of dynamic decision making, the Econometric Society awarded me the Ragnar Frisch Medal in 1992 (for leading empirical paper published in Econometrica in the preceding 5 years).
After I finished my initial work on the Zurcher paper in 1986, I became interested in using the DP framework to model retirement behavior. The challenges involved in developing an accurate and realistic empirical models that could capture the complex range of observed retirement patterns required much more of my research time than I initially anticipated, consuming nearly 50% of my available research time over the last 8 years. I decided to concentrate my research in this area for several reasons. Retirement is an important national and international policy issue in view increasing longevity, the continued trend towards early retirement, the impending wave of retirements of aging baby boomers, and the shaky financial condition of the Social Security trust funds. It seemed clear that the SDP framework could lead to improved economic models of retirement behavior and that these models could be of practical use to help governments and firms design improved Social Security systems and pension policies. However estimation and testing of these models is extremely data-intensive, requiring thousands of detailed longitudinal observations of individual retirement trajectories. Fortunately, an excellent panel data set had already been collected by the Social Security Administration -- the Retirement History Survey (RHS) -- representing more than 500 meagbytes of information on income, assets, labor supply, health and family status, and Social Security benefits for 11,000 men and women aged 58-63 interviewed in biennial surveys from 1969 to 1979.
Although the data-processing and computational challenges involved in estimating a comprehensive unified SDP model of retirement behavior were quite daunting, I believed the large up-front investment required was justified in view of several compelling advantages of using the SDP framework to model retirement behavior. From a policy perspective, the most important advantage of the SDP framework is that it allows us accurately model the impact of the complicated details of private pensions and Social Security rules. These rules lead to uncertain time, state, and decision-dependent benefit streams that depend on factors such as the person's age, earnings history, health, and labor force participation decisions. It turned out that accurate modelling the Social Security rules within the SDP framework has enabled me to solve a number of empirical puzzles about retirement behavior that previous models have been unable to explain. One of these puzzles is the large peaks in retirements at ages 62 and 65 -- the ages of eligibility for early and normal Social Security retirement benefits. There are several reasons why previous models have failed to explain these puzzles, the most obvious being their unrealistic treatment of time and uncertainty and their failure to model the pattern of incomplete insurance markets facing various individuals. The limitations create further problems in the models' ability to accurately capture the incentive effects of Social Security and pensions. For example, most of the existing retirement models assume that individuals pre-commit to a fixed retirement age at some arbitrary ``planning date'' such as age 55 and treat retirement as an absorbing state (i.e. one cannot ``un-retire'' after retiring). These models also assume that individuals have perfect ability to borrow against future income and pension benefits, and have either ignored uncertainty about future health status and earnings potential or assumed that all individuals have access to fairly priced health and disability insurance. I realized that between the NFXP algorithm and the RHS data, I had the technology and data necessary to estimate a SDP model that could circumvent all of these limitations. The result was a model that does a much better job of capturing the constraints created by incomplete markets and the uncertainties older individuals face as they approach retirement.
The details involved in carrying out this task turned
out to be much more challenging and time consuming than
I had originally anticipated. However I believe the results
obtained from this research, summarized in
my recent paper co-authored with Chris Phelan ``How Social Security
and Medicare Affect Retirement Behavior in a World of
Incomplete Markets'', justify the time and effort that
I devoted to it. In my opinion it
represents my best empirical work
to date, and I hope it will ultimately be regarded as a significant
contribution to the literature on retirement
behavior. Our DP model succeeds in
providing relatively simple explanations
for many of the puzzling aspects of retirement behavior discussed
above. The DP model permits a fairly realistic treatment
of the sequential nature of the retirement
process and individual subjective
uncertainties about future mortality, marital status, health status,
employment status, income levels, and health expenditures.
The model places no restrictions on labor
supply paths (i.e. retirement is not assumed to be
an absorbing state), and separates the labor
supply decision from the ``retirement decision'', which we
identify with the decision to apply for Social Security benefits.
The DP model delivers a rich set of predictions about the dynamics of
retirement behavior, and comparisons of actual vs. predicted
behavior show that the DP model
is able to account for wide variety of phenomena observed in the data,
including the pronounced peaks in the distribution of retirement
ages at 62 and 65 (the ages of early and normal eligibility for
Social Security benefits, respectively). The peak at 62 is a result of
borrowing constraints that prevent individuals with
relatively little tangible net worth from retiring prior to the
age of first eligibility for early retirement benefits.
The peak at age 65 is a result of incomplete
annuities markets and the fact that Social
Security benefit formula is actuarially
unfair for retirements after age 65.
However this only accounts for part of the large peak in
retirements that occur precisely at age 65. The remainder
of the peak is explained by another form of
market incompleteness - incomplete health insurance - and the
fact that Medicare insurance is only available to
individuals over 65 who have applied for Social
Security retirement benefits.
Although the overall effect of Social
Security is to create strong disincentives to continued labor
force participation, it creates strong incentives for
certain individuals to remain employed up until their
birthday. We identify a significant fraction
of ``health insurance constrained'' individuals who have
no form of retiree health insurance other than Medicare, and
who can only obtain fairly priced private health insurance
via their employer's group health plan.
The combination of significant individual
risk aversion and a long tailed (Pareto) distribution of
health care expenditures implies that these individuals
impute a significant
``security value'' to remaining employed
until they are eligible for Medicare coverage at 65. Overall,
our findings suggest that the so-called ``age 65 retirement
puzzle'' and several other puzzling aspects of retirement
behavior can be viewed as artifacts of particular details
of the Social Security rules whose incentive effects can be
quite strong for lower income individuals and those
who do not have access
to fairly priced loans, annuities, and health insurance.
In order to obtain these findings, I had to
overcome a number of difficult
computational, econometric, data and measurement problems.
The solutions to many of these problems represent independent
contributions in their own right. The computational
challenge was to develop an efficient algorithm to
solve the DP model sufficiently rapidly to enable us
to estimate on a limited computer budget.
The DP model that we ultimately estimated contains
7 state variables and
2 control variables, and is to our knowledge the largest and most
realistic model of its kind that has ever been estimated
in this literature. I developed a highly efficient
solution algorithm that was able to solve the DP
problem in less than .5 cpu seconds on a Cray
supercomputer. A paper describing this algorithm,
``A Dynamic Programming Model of Retirement Behavior''
won 3rd prize in the
1989 IBM Supercomputing Competition.
The computational problems turned out to be minor in comparison to
the econometric problems of obtaining
accurate estimates of individuals' beliefs,
. Individual subjective
beliefs are obviously rather slipperly objects to estimate,
so strong identifying assumptions had to be imposed.
In this case I imposed the assumption of rational expectations,
i.e. that subjective beliefs about mortality, earnings,
etc. correspond to the objective probability distributions of
realized outcomes for mortality, income, etc.
for individuals in a homogenous population.
However even given the assumption of rational expectations and
the relatively large number of observations in the RHS data set
it is still impossible to estimate p directly.
The reason is that p still contains far too many unknowns than
we have data to estimate. To see this note that
even given our relatively coarse discretization of the state and
decision variables, (with x assuming 14,400 possible
values, d assuming 6 possible values in a total
of 23 possible biennial time periods from age 56 to 102)
the array p of conditional
probability distributions representing
an individual's beliefs contains
a total of 1.9 million probability distributions
over the 14,400 possible values of xt+1 -- amounting to over
unknown probability values to be estimated!
It is clearly impossible to reliably
estimate all these probabilities from my cleaned RHS sample of
approximately 7500 person/year transitions. I solved this
problem by developing
a method for decomposing p into a product of ``sub-transition''
densities for individual components of xt and imposing
certain ``exclusion restrictions'' on the conditioning
elements entering the sub-transition densities.
This decomposition enabled me to obtain accurate estimates of the overall
transition probability p using the limited number of
available observations. Examples of the sub-transition
densities include separate models of mortality, marital status,
health status and wage earnings of husband and spouse.
I developed a unique method for embedding Social Security
rules within the p matrix via a set ``Social Security transition
matrices'' transforming wage earnings exclusive of
Social Security benefits to wage earnings inclusive of Social Security.
Once all of the separate sub-transition densities
had been estimated, it was a relatively simple
matter to build the massive overall array of beliefs p by simple
matrix multliplication. However the large number of elements in
p necessitated development of a specialized highly efficient algorithm
to rapidly re-construct p avoiding ``unecessary'' multplications
implied by the exclusion restrictions.
The result was a computationally and econometrically tractable algorithm
capable of solving the the DP problem extremely rapidly (in less than
.5 cpu seconds on a Cray supercomputer and less than 30 cpu seconds
on a fast workstation).
Solving the data and measurement problems turned out to be by far the most time consuming aspect of my research on retirement behavior. Creating a consistently defined set of observations on variables such health status, labor supply, earnings, and consumption from successive waves of the RHS survey presented many challenges. An example of these problems is determining how to deal with internal inconsistencies in respondents' reports about labor supply (e.g. a respondent who reports that he is working at the 1969 interview but in the 1971 interview reports having been continuously unemployed since the previous interview). Determining how to create good measurements of subjective variables such as health status created additional difficulties, especially since the RHS did not use a consistent set of questions on health status throughout all survey waves. Other state variables required imputations to determine individuals' opportunity sets. An example is the health insurance state variable, which indicates whether an individual is constrained in their ability to purchase private health insurance at a fair price. As discussed above, this turned out to be one of the key variables to understanding the incentive effects of Social Security and Medicare. Since the RHS did not directly ask respondents about whether they had access to fairly priced private health insurance, I had to infer this information from their responses to other survey questions. However the problem of obtaining accurate measurements of consumption expenditures ct was the most severe, and ultimately forced me to simplify the specification of the DP model to circumvent it. I experimented with several alternative methods for constructing measures of consumption from the information in the RHS. For example I attempted to solve for ct from the budget equation wt+1=wt+yt-ct where wt+1 and wt are the observed net worth and yt is the imputed income flow to the household between successive survey waves. Measurement errors in wt+1 and wtto a lesser extent imputation errors involved in constructing yt lead to implausibly erratic measurements of ct, including a disturbingly high incidence of negative measurements of ct. Even though the original version of my algorithm included ct as one of the decision variables in the DP model, I found that the large measurement errors in ct interfered with the ability of the DP model to fit the other variables in the data such as labor supply and income that I was able to measure much more accurately. As a result all of the versions of the DP model I have estimated to date make the simplifying assumption that ct=yt. Although this assumption has been criticized by certain economists, I believe it is a highly defensible assumption for my sample of predominantly blue collar workers in the RHS. These individuals have very little net worth outside of their housing equity, and rely nearly exclusively on their pension and Social Security benefits to finance retirement consumption. There is very little evidence in the RHS that individuals spend down their net worth as they age, and indeed the distribution of changes in net worth wt+1-wt has a large spike at 0, indicating that ct=yt is in fact a very good approximation to individuals' behavior in the RHS. Indeed if we were to treat all fluctuations in net worth as arising from random measurement error, statistical test would be unable to reject the hypothesis that ct=yt. Furthermore, Deaton's 1991 Econometrica article demonstrated that it is optimal to set ct=yt if there are borrowing constraints and income is highly serially correlated. Since observed income is in fact highly serially correlated the fact that ct=yt appears to be satisfied in the data may be a confirmation that these individuals are behaving optimally in the face of borrowing constraints, exactly in accordance with Deaton's theoretical results. Borrowing constraints are certainly a fact of life for most older individuals: in fact it is actually illegal to borrow against one's future Social Security benefits.
My painstaking analysis of the RHS data, reported in my 1990 paper ``Behavior of Male Workers at the End of the Life-Cycle: An Empirical Analysis of States and Controls'', has contributed to my reputation for serious concern for survey sampling and data measurement issues. In 1990 I was appointed to as a consultant to assist in the design of the National Institute on Aging's new Health and Retirement Survey (HRS), a successor the RHS designed to obtain much better information on health, working conditions, and pension plan characteristics than was available in the RHS, and in 1993 I was appointed to the Panel on Retirement Income Modelling at the Committee on National Statistics of the National Academy of Sciences. I have recently submitted a 3 year research proposal to the National Institute on Aging to fund further research into retirement behavior using the HRS data. The HRS will enable me to overcome several shortcomings of the DP model that stemmed from limitations in the RHS data. In my opinion the most important shortcoming was the exclusion of 40% of the sample who expected to receive private pension benefits. These individuals were excluded due to the fact that the RHS has very sketchy information on the benefit provisions of the more than 600,000 different types of pension plans available in the U.S. in the 1970's. The HRS collects much more detailed information on the benefit structures of an individual's main pension programs, and this information will allow me to re-estimate a DP model with an integrated treatment of pensions and Social Security. The HRS will also allow me to create improved measures of health status, and to model disability and unemployment as additional exit routes from labor force participation. While I will be devoting substantial effort to devising new ways to obtain improved measurements of consumption and net worth, I am not optimistic that the HRS data will enable me to obtain significantly better measurements of these items than I could obtain using the RHS. Our inability to obtain good measurements of consumption is bad news for those interested in modelling consumption/savings decisions since even the most basic ``stylized facts'' about trends in consumption, savings, and net worth at the end of the life-cycle are still subject of considerable professional dispute. Fortunately I do not believe that the problem of obtaining accurate measurements of consumption will seriously impinge on my ability to model the retirement decision, at least for the vast majority of individuals who have failed to accumulate substantial net worth as they approach retirement. This is certainly true for the vast majority of individuals in the HRS, where 88% of all older full-time workers have accumulated level net worth that is less than 5 times their current annual income, and the mean ratio of net worth to current income for full time workers is only 3.8. My future research will continue to concentrate on the analysis of variables such as labor supply that we can measure relatively accurately, and will resort to reasonable simplifying assumptions (such as ct=yt) for variables such as consumption that are dominated by measurement error.
In summary,
my overall assessment of my research on empirical models of SDP's
is that
the bus replacement and retirement applications are unqualified
successes that demonstrate that the NFXP algorithm is a practical
econometric tool and that the behavior of the ``man on the street''
can be well-approximated by an optimal decision rule to a
dynamic programming problem.
However a cynic could argue that any behavior
can always be ``rationalized'' as optimal for some choice
of preferences and beliefs. Indeed in my recent paper
``Do People Behave According to Bellman's Principle of
Optimality?'' I show that in a dynamic context the hypothesis of
optimization per se has no empirical content if we
are unwilling to place any a priori restrictions on individual's
preferences or beliefs. In formal terms, I proved that
given an abitrary function
, we can always find a
set of preferences and beliefs
for which
is an optimal decision rule. However it is important
to emphasize that there is no guarantee
that the preferences and beliefs that
rationalize an arbitrary behavior pattern
will
be regarded as ``reasonable''.
This implies that in order to judge whether an
SDP model provides a ``successful'' explanation of an individual's
behavior we need to consider whether it does so via a simple,
parsimonious, and intuitively plausible specification of preferences
and beliefs. Although the issue of what
constitutes a ``reasonable'' specification of preferences
and beliefs is clearly subjective, most economists
are capable of identifying an artificially
concocted theory when they see it. A more practical test of
the validity of an SDP model is how well it forecasts behavior,
especially in out-of-sample predictive tests and in quasi-experimental
settings where agents' beliefs or preferences can be altered in
controllable and predictable ways. However I must
ackowledge that the hypothesis of expected utility
maximization underlying the SDP framework has been rather
convincingly rejected by a number of fairly robust
experimental tests (such as the famous ``Allais paradox''), so
SDP models can at best be regarded as idealized approximations of
human decision making processes. My view is that we should judge
SDP models the same way we would judge any other idealized,
approximate model: are these models of practical value in the sense of
improving our ability to predict and understand human behavior?
The answer to this question is almost certainly yes at least
in terms of large and rapidly growing
number of empirical applications of
the SDP framework to problems ranging from business
investment decisions (Das, 1992) to women's contraceptive
decisions (Montgomery, 1988, Ahn, 1994). We also have a number of
very concrete examples
demonstrating that SDP models do in fact yield much more accurate out-of-sample
predictions than alternative models, especially with respect
to policy changes.
On a personal level, I have found my research in this area to be very rewarding even though doing empirical work in the SDP framework involves many difficulties and is quite labor intensive and time consuming. I have been fortunate to receive research support from agencies such as the National Science Foundation, the National Institute on Aging, the Sloan Foundation, IBM and the Bradley Foundation, and professional recognition of my research contributions including my 1988 Alfred Sloan Fellowship, an invitation to present a symposium paper on estimation of SDP's at the 1990 Sixth World Congress of the Econometric Society, the 1991 Lief Johansen Lectures at the University of Oslo, the 1992 Ragnar Frisch Medal, an invited lecture on ``Pensions and the Labor Market'' at the Tinbergen Institute in Rotterdam in 1993, an invited chapter for the forthcoming volume 4 of the Handbook of Econometrics, and my 1993 election as a Fellow in the Econometric Society and my appoitment to the Committee on National Statistics of the National Academy of Sciences.
2. Research in Computational Economics
I have had a long interest in computation arising out of my original development of the nested fixed point algorithm for estimating discrete dynamic programming models. In 1992 I was asked to co-edit the Handbook of Computational Economics with Hans Amman of the University of Amsterdam and David Kendrick of the University of Texas. I have written a chapter for this volume entitled ``Numerical Dynamic Programming in Economics''. The chapter summarizes the current state of the art in numerical methods for solving various types of dynamic programming problems.
One of the most frustrating aspects of computing numerical solutions to DP problems is the computational burden involved in solving increasingly realistic models. Although there is a well developed literature on numerical methods for solving DP problems, the usefulness of these methods is limited by a problem that Richard Bellman termed the curse of dimensionality: the amount of computer time required to solve a DP problem increases exponentially fast in any relevant measure of the ``size'' or realism of the DP problem. The curse of dimensionality sets very severe limits on the level of realism and detail that we can expect to incorporate in a DP model. Even though the speed of computer hardware is increasing at an exponential rate, the curse of dimensionality implies that size of problems we can solve with this faster hardware grows at a much less dramatic rate: in practice the size of problems that are solvable seems to grow only linearly as hardware improves. Solving DP models at the level of detail that we encounter in everyday life -- such as choosing which items to buy from the thousands of possibilities at a grocery store or choosing which of the tens to hundreds of possible commuting routes to get from home to work -- is far beyond the reach of current hardware and software. Since humans are able to do these tasks virtually effortlessly, the computational problems involved in solving these problems in the SDP framework cast doubt on whether humans really are behaving ``as if'' they were solving a dynamic programming problem. Put differently, if the world's best mathematicians and economists are only able to solve relatively ``simple'' DP problems (even with the assistance of very powerful supercomputers), can we seriously expect DP models to do a good job of predicting the behavior of the ordinary ``man on the street'' who faces each day vastly more complicated decision problems?
In a recent paper ``Using Randomization to Break
the Curse of Dimensionality'' I present a simple
monte carlo algorithm that succeeds in breaking the
curse of dimensionality of solving DDP problem, i.e.
dynamic programming problems where the decision
maker has a finite number of possible alternative actions
in each state. The cpu time required to solve a DDP
problem using this algorithm increases only as polynomial
rather than exponential function of the size of the
problem using other solution methods.
This result is of considerable theoretical
significance since most computer scientists have previously
believed that this class of DP problems are subject to an
inherent
curse of dimensionality that no deterministic or
randomized algorithm could circumvent
regardless of how cleverly it is designed.
The practical significance
of the result is that it offers the possibility that we
can solve much larger and realistic DDP problems than
we previously thought was possible. The algorithm will have
immediate application to estimation of SDP models, and
will be especially useful in relaxing the strong assumption
that the unobservable state variables in the
DDP problem are an IID extreme value process. The
new algorithm will make it feasible to estimate DDP models with
serially correlated unobservables with more general
non-extreme value marginal distributions. The result is
also significant from a behavioral perspective, since
it leaves open the logical possibility that humans really do
use approximately optimal strategies to solve the
highly complex decision problems that they face on
a daily basis.
It also raises the interesting
question whether we really use ``algorithms'' that involve
implicit randomization, or whether our processing
capabilities are more of a result of effective harnessing of massive parallism
in our uncounted trillions of neural circuits.
3. Research on Artificial Intelligence and Automata Trading in Double Auction Markets
An outgrowth of my interest in computational methods for solving DP problems is my interest in heuristic methods for solving difficult SDP problems. The heuristic methods I am interested in include ``learning algorithms'' and methods from the artificial intelligence literature. When one looks closely at most practical decision making situations one realizes that it is unrealistic to assume that individuals have well-defined probabilistic beliefs about uncertain aspects of their environment -- at least it seems unrealistic that most people would have highly detailed subjective probability assessments of the huge range of possible outcomes that could happen to them.2 This is especially true in strategic environments where individuals need to make probability assessments about the strategies used by their opponents. A classical example of such an environment is the double auction market, where bidding activity of buyers and sellers leads to endogenous formation of trading prices for commodities (a classic example of a double auction market are the commodity trading pits at the Chicago Board of Trade). The complexities involved in inferring whether certain bids and asks are results of private information possessed by other traders or whether their bids primarily reflect predictable information about their trading strategies poses an extremely difficult learning problem. It is natural to try to model the learning problem in a Bayesian fashion, and treat the observed price trajectories in these markets as Bayesian Nash equilibrium outcomes of a dynamic game of incomplete information. Unfortunately this game is far too complex to be solved either analytically or numerically using the fastest available supercomputers. Another problem with the game-theoretic approach is that it depends criticially on the common knowledge assumption that each trader knows that each of his opponents are rational and knows the exact form of his opponents' equilibrium trading strategies. As discussed above, this is a highly dubious assumption in practice. The common knowledge assumption has also been widely criticized since it presumes a high degree of implicit coordination amongst the traders, begging the question of how decentralized coordination is achieved in the first place. Thus, economists have been able very little about one of the most fundamental economic problems: how are prices determined and why does supply equal demand?
In order to obtain new insights into this problem, I got involved in a project at the Santa Fe Institute in the fall of 1989 which lead to the sponsorship of the 1990 Santa Fe Double Auction Tournament, co-organized with John Miller and Richard Palmer (a physicist at Duke Univeristy). Our computerized tournament attracted a collection of 30 heterogenous computer programs playing the role of buyers and sellers vying for a pool of $10,000 prize money (from my Alfred Sloan Fellowship award). The idea was to have contestants encode their trading intuition or favoriate artificial intelligence procedure into a computerized trading program. The computerized strategies competed over a broad range of trading environments to provide a rigorous test of their relative trading ability, where ability can be measured unambiguously in terms of the profits each program earned (we structured the tournament so that each program had the same potential trading surplus with probability 1). Since we had access to the code, we could essentially ``observe'' the underlying trading strategies, something that is impossible to do in actual or even experimental double auction markets using human subjects. We found that the collection of computerized traders behaved very similar to human subjects in double auction experiments, with price trajectories converging to competitive equilibrium and allocations that were nearly 100% efficient. Surprisingly we found that a very simple ``wait in the background'' trading strategy emerged as the winner of the tournament, beating out many more complex algorithms using statistically based predictions of future transaction prices, explicit optimizing principles, and sophisticated learning principles. The strategy required remarkably little information beyond its private token values, current time, and the current bid and ask. Whether or not good human traders really behave according to a few simple decision rules is the subject of ongoing investigations, but the results do seem to confirm Nobel laureate Friederik von Hayek's conjecture about the ``economy of information needed to take the right action in a competitive market''. This project resulted in a publication in the 1994 Journal of Economic Dynamics and Control and an article in a book I co-edited with Daniel Friedman of University of California Santa Cruz, The Double Auction Market: Institutions, Theories and Evidence. The National Science Foundation funded subsequent joint research with Vernon Smith of the University of Arizona that lead to the creation of the Arizona Token Exchange which allowed us to use the computerized strategies we obtained from the 1990 Santa Fe tournament as computerized opponents to human traders trading worldwide via the Internet. Our goal was to see if humans can learn to exploit or at least form best responses to fixed collections of computerized strategies in the double auction environment under repeated play. We gathered observations on over 10,000 DA trading periods from over 90 subjects, providing the largest dataset on time-series observations of strategy-formation and trading behavior ever collected. We are currently analyzing this voluminous data (several hundred megabytes worth of information on tens of thousands of separate double auctions) and expect to have a paper ready for submission to a journal by the end of this year. We expect analysis of this data to yield insights into the way humans learn effective trading strategies, and to improve understanding of the forces driving markets to competitive equilibria. We also expect to formulate improved computerized trading strategies and more realistic computerized models of working DA markets.
4. Future Research Plans
In rough terms the first decade of my research career focused on the development of dynamic programming as a general tool for understanding human economic behavior, with special focus on its application to understanding retirement behavior. The next decade of my research career will focus on systematic application of this tool for prediction and design of improved policies by governments and firms, with special focus on firm pension plans and government Social Security policy as it affects retirement behavior.
The primary advantage of the dynamic structural approach to econometric forecasting and policy-making is that it allows us to to make much more detailed and accurate predictions of the effects of hypothetical policy changes than is possible using traditional reduced-form econometric methods. I intend to use the SDP framework to design more efficient policies in a number of areas such as improved pension plans (to help private firms encourage the retirement of less productive older workers while retaining and rewarding the most productive), improved government Social insurance schemes (to direct health insurance disability and retirement benefits to individuals who truly need these benefits at minimum cost to taxpayers and with minimum distortionary impacts on savings and labor supply decisions), and regulatory policies for nuclear power plants (helping public utilities strike an appropriate balance between public safety and cost-efficient power generation).
The DP approach to predicting the impacts of policy changes is
computationally intensive but conceptually straightforward.
Let 
denote a given policy: for example in the case of
Social Security could be represented by a vector
where 
is the age of early
retirement, 
is the age of normal retirement, 
is
the Medicare co-insurance rate, and so forth. Let denote
a vector of parameters characterizing individuals' preferences and
beliefs as discussed in section 1.
We assume that individuals know their own values of
but that this vector is unknown to the econometrician
and must be estimated. Thus,
represents the utility
(i.e. psychological well-being) of an
individual whose preferences are given by the vector
, who is facing a given policy
, and who is in state st and takes action dt at time
t. Similarly, the conditional probability density
represents this individual's
beliefs about his next period state st+1 conditional
on their current state and action (st,dt). Note that policy
can affect a person's current welfare (for example by
the level of Social Security retirement benefit currently received)
and also one's beliefs about the future (e.g. one's expectations
of future Social Security benefits).
Given panel data on a sample of individuals'
states and actions
of
individuals
followed over time periods
, under an existing policy
, we estimate
by finding the value
such that the
decision rule
best fits
the observed data
using the NFXP algorithm described
in section 1. Given
we can then
predict individuals' behavior under an alternative hypothetical
policy 
by re-solving the DP problem under the new policy
resulting in a new optimal decision rule
. A more systematic and
ambitious method of policy analysis is the method of
``computational mechanism design'' that I have been developing
with my colleague Chris Phelan. This approach use
the revelation principle to simplify the process
of searching over the
space of all possible policies to maximize a certain objective
function subject to certain individual rationality
and incentive constraints, where the latter constraints
arise from the presence of private information. For example,
in our paper ``U.S. Social Security Policy: A Dynamic Analysis of
Incentives and Self-Selection'' we used the computational
mechanism design approach to find new Social Security policies
that minimize the cost of providing retiree benefits to taxpayers
subject to the individual rationality constraints that retirees
should not be any worse off under the new policy 
than
under the existing Social Security policy , and subject
to the incentive constraints that individuals should not want
to lie about their true health status (e.g. to claim they are
disabled in order to gain early retirement benefits).
We are currently using this method in a paper entitled
``Measuring the Inefficiency of the U.S. Social Security System''
which we presented at a recent conference on ``The Future
of the Welfare State'' in Ebeltoft, Denmark in 1994.
John Rust
2001-04-10