



S1: okay on focus? alright? i've discovered you guys are all much more sensitive to that than i am so, uh okay, we've been talking about uh formal structural models of the retirement process. uh, we did the Gustman and Steinmeier, model last class and today, um, guess we'll spend all the time on so-called option value models Lumsdaine, Stock and Wise, although we'll get a small head start in thinking about uh, dynamic programming models as well because part of what they do is compare their preferred model, to one that is still the more elaborate. uh... basically the the, theme of the paper, um is to compare, three models of increasing complexity, using exactly the same data. and, just as a general... point about doing research, bu- on the same data, is a really nice feature of this paper. because normally what happens when you try to teach a course like this, is that you have, researchers and models, and data, and you hope that the differences and conclusions are <LAUGH> not due to the researchers, but you're never really sure how much they're due to the models and how much they're due to the data. okay so if you have model one, estimated on data set A and model, two estimated on data set B and they get different results, um you're never really sure whether the results are due to differences in the model or due to differences in the data. and so [S2: (excuse me) ] it's a it's a good idea, and it's a remarkably_ yeah? 
S2: today we we have no... handout? 
S1: oh i'm sorry, (xx) handouts. [SU-M: thanks ] <P :04> so i- it's you know, you wouldn't think this would be an issue. i mean you stop and think about it for a minute. um when you're in the second paper in a field, not only, presenting you're pr- pr- preferred model. uh but also kind of... estimating or re-estimating earlier models on your data, it would seem to be an obvious thing to do. and it helps people like me trying to teach the stuff because now we can compare model against model, holding data constant, uh which is in some sense the right way to evaluate the model. um as you probably discovered from just, earlier rendi- you know earlier topics in the course, it's remarkably rare, that one actually sees that. um oftentimes people just go off and estimate their model on their data, and don't think about the poor, uh six-twenty-two teacher who's trying to make sense of the literature. uh so having all three models estimated on the same data, uh gives, you know, it seems like something you wouldn't need to praise or you shouldn't need to praise um <LAUGH> but it's actually unusual enough um that it that it's worth emphasizing and and saying you know yes do like they did. okay? um a lotta the paper is driven by the data so i'm actually going to start by the data, um and then talk about the models. okay the data that they have are from the personnel files of a, large firm, which they don't name that's part of their agreement with the firm, and the workers are nonmanagerial workers. okay? uh they have company payroll records which means they have excellent data on earnings, and they have the cooperation of the firm so they know the details of the pension plan, probably better than the workers do. okay? um, the details of this potenti- particular pension plan, um is that it discourages leaving before early retirement age, okay so if you leave before early retir- retirement, the vested deferred benefit that you're eligible for is pretty unattractive, okay? and then if you stay after sixty-five, the increment to your pension benefits doesn't, repay the fact that you're going to be drawing those benefits over one less year. okay? and so basically there's a big incentive to stay till fifty-five so you qualify for early retirement, and a fairly strong incentive to leave beginning at sixty-five because, the pension system just does not reward work beyond sixty-five in this particular firm. okay? uh then when we talked earlier about there being def- different definitions of retirement and here when they talk about retired they mean leave the firm. okay? the good news is they have very good data about what happens to you as long as you work for this firm, the bad news is they know nothing about you after you leave. okay? so uh_ yeah Mato 
S3: but that's a problem because it's like, when they will model_ i don't know the model but when they will model their choice of going into retirement, people who retire don't think of_ just about that but about missing opportunities.
S1: th- that's particularly important for the application that they're going to have. so the_ yeah the i mean um, well lemme, yes th- part of what the paper is about, is about retirement in normal times. okay? um and once you make a kind of_ one can have a discussion within oneself about whether, partial retirement jobs or omitting partial retirement jobs or not modeling partial retirement jobs is an important flaw in that context. um they're going to apply it to an early-out window. okay? which is, a deliberate attempt to get people who are not really ready to retire, to retire to leave... okay? and so the early retirement window is going to induce people who otherwise wouldn't be retiring, to leave early, and the question of what they do after they leave is is kind of doubly important. okay? so for the the group of workers that that that they would be studying um, normally uh the issue of what happens to them after leaving the firm is potentially important. for the group of workers who are be- whose departure is being accelerated by the early-out window, uh the issue of what they do afterwards is doubly important. and and you know what this reflects is just the fact that, uh you know we started out saying here's a model, here're all the things we would like to fix about that model, and now we're gonna watch the papers try to fix them one or two at a time. okay, so this is one that as you'll see is going to not do a very good job, uh with_ is not going to do anything really about modeling what happens after you leave the firm, um but it does a very good job with other things and, so it's a it's a, [SU-M: uh ] you know it's a step toward a larger understanding of of what an ideal model would be. okay so as i said that this particular firm offered an early-out window in nineteen eighty-two so an early-out window is not, early retirement. okay so a typical pension plan has early retirement provision. and that retir- early retirement provision is a constant feature of the pension plan. okay? so whatever age you re- at whatever year you reach age fifty-five you could pretty well count on, you being eligible for early retirement and, you know if you have the time and tenacity to work through the plan you can figure out what benefits you'd be eligible for. okay? the early-out window is a special, incentive that says we're particularly eager to get rid of workers, this year. okay? um and if you agree to retire right away we'll give you whatever the pension plan says you're entitled to, plus we'll give you a bonus. okay now sometimes that bonus will take the form of extra pension benefits, sometimes it'll take the form of just, money up front, and for this firm it took the form of money up front. okay so three to twelve months of salary if you agree to retire right away, and uh sort of high end of the bonus applying to people fifty-eight to sixty-two. okay so people who have reached early retirement, but haven't decided to retire, um despite being eligible to do so. um and so here's the strategy they're gonna fit the data each of the models to nineteen eighty data. and they're going to use these estimates for out-of-sample prediction. okay? so they're going to_ (you) estimate the parameters off of the data from nineteen eighty. and the- they're going to use that to predict response, to the window. okay? and so that's a bit like um, what one often sees in other fields where one has, a data set there where where you estimate the the parameters and then by the time the paper is ready to get published there're two more years of data, and so you ask well how well would my data_ would my model fit um these out-years of data? Mato 
S3: so so what, what this, what these uh, nineteen eighty-two (perspective) (xx) 
S1: no, no when when uh, ear- this is one of the very first early retirement windows offered. okay so wh- when this was offered, um the authors are reasonably confident and i'm reasonably confident, that it was not anticipated by the workers. okay? uh what's happened since is that at least in some firms there have been, serial o- early-out windows. so they'll announce that we're trying to get rid of workers uh, you know here's the deal are you interested, and people can sign up or not, and then six months later they come back and say well if, three months' salary wasn't enough how about four months salary. okay and in that context once they come to be expected, when somebody, rejects the offer you're not, really sure whether they're rejecting the offer as not good enough, or they're rejecting the offer because they expect something better to come down the road in a couple of weeks, or a couple of months anyway. uh but for this firm i think that this was a sufficiently unusual departure from what firms as of that date had done. um and i think it's reasonable to think of it as A unanticipated and B reasonable to think of it as likely not recurring. so a worker probably wouldn't have resisted re- rejected this offer, uh because, he or she expected a better offer to be coming next year. okay? but for, more recent data that would be a real concern. okay any questions about, that? <P :11> okay so normally we start with the model and then go into data this morning we started with the data and went to the model and the reason for that is because, the details of what's in these models uh is really driven by the strengths and weaknesses of the data. okay so for example there's a rather careful attempt at projecting future wages future pensions future social security benefits and that becau- that's because these folks really have the data to do it. okay the Gustman and Steinmeier paper that we looked at, um on Thursday for example has really primitive information, about pension entitlements and they spent a lot of effort, making rough approximations to what people are entitled to. um here these folks can just gracefully look it up. and so they <LAUGH> spent less effort and have i think much more reliable, um information. um, the different models reflect differing handle handling of uncertainty that's part of what they're trying to emphasize. um as (xx) mentioned there's a lack of data and so a lack of model uh for partial retirement. okay and so there will be a state which is, retirement that is having retired from this firm, and there's no explicit attempt to ask does that reflect, retiring from the labor force altogether or does it reflect, going across the street to this firm's rival and doing exactly what you'd done before, uh or does it reflect something, you know undetermined? um unlike much of the literature um the health problems that people often have as they get older and often lead people to retire, are not explicitly modeled. okay so there's a, sort of random term um about which one is uncertain and you can think of the health problems as being embedded there. um but unlike virtually all the other papers in this literature um there's no discu- there's no, explicit discussion of health. why is that? because all of the other papers are run off of survey data where asking people about their health is an important part of the research project. here they're basically exploiting company personnel records, where they don't really know if you're sick unless you take sick leave. okay if you have some chronic problem that motivates your retirement, uh unless you're eligible for some sort of disability benefit the firm typically wouldn't record it. okay so there's nothing in the personnel files and therefore nothing in the data set that they're using, uh which tell them about the fine details of the health status of the workers. and so you know as is always true when something isn't measured, its effect such as it is is in the random term in the error term. and so part of the, um error term part of the uncertainty in the model, deals with health problems that in other data sets might actually be measured at least in some rudimentary way, and in this one they're not. um finally, uh <LAUGH> i've never really had a chance to sit down with the authors and ask them whether the the last assumption is because it's what they really believe is the right way to go, or because it's the way they can implement but it's different. um the Gustman-Steinmeier model remember assumed perfect capital markets. indeed overly perfect capital markets if you read the model literally they let you borrow against your social security benefits. okay which you know would, get you in line for a pardon from President Clinton if you'd actually done it, okay? um, this model assumes exactly the reverse. okay, that your utility in period T depends only on your inflow of income in period T. and so one way of imagining thinking about that is i consume my income. which isn't terribly, attractive. uh another way to think about it my utility depends on my income rather than my consumption, which is certainly not what you're taught in six-oh-one or six-oh-two. uh, but it's a, basically a way of of saying, um the idea that people can borrow and lend in perfect capital markets i- is really an exaggeration. um and worse here it would complicate the estimation so it would be, an assumption that probably isn't right, and makes the analysis harder. um let's focus on something which has a little, bit of i would think a a folksy element to it, uh but makes the computations a lot uh, a lot easier. uh so what are sort of the authors actually, you know, if they had infinite computing power and a and a vast army of research assistants to implement their every whim, would they be making this assumption or the perfect capital markets assumption? i'm not quite sure. um but they're making this assumption for some combination of the tractability it provides and perhaps disbelief in perfect capital markets which is the other, kind of reigning alternative, uh would be an appropriate assumption for this situation. alright now the title says three models and you've been remarkably patient in not asking me <LAUGH> so what are the three models Charlie? um, the three models are are listed at the bottom of the slide there. uh one is a simple probit model. okay? so the probability that individual I retires in year T um, is a function of their Xs and an error term if X and beta plus that error term is greater than zero then i retire. um the Xs are kind of the usual suspects, uh meaning uh the wage social security wealth that is the present value of the benefits that i could claim if i retired today, the amount by which that present value changes if i delay my retirement by one year. my pension wealth that is the present value of the pension benefits that i would be entitled to if i retired today, and then the change in pension wealth. okay so the amount by which that present value would go up, if i retired one year later. um so, and you can really think of this as both a simplified and a reduced form version of the Gustman-Steinmeier model. okay so remember when we talked about the Gustman-Steinmeier model, um people were thinking about, you know the increment to their compensation from working one more year, and asking whethe- how that compared to the marginal utility of their lesion. okay so the Gustman-Steinmeier model is a much more complicated model but it's fundamentally in this spirit of basically looking one year ahead. okay? uh the option value model which is the model that the authors are selling, okay uh the- there was a joke that in the Carter administration that if you were an advisor, um and you knew which option you wanted the president to select um you would, always make that the option B, because (it's really) when he wasn't sure what he would do he would sort of pick option B. okay? well option B here is the one that they're trying to sell you. okay it's the option value model. it's kind of mid-range in complexity between the probit which is relatively simple, and the dynamic program and prop model which is actually, quite difficult. and the idea behind the option value model is that i retire now, if the expected utility from retiring now is greater, than the utility i can expect at any later retirement date. okay? so while the probit you can think of as comparing the utility from retiring now versus the retire- utility from, working at least one more year, or working one more year. uh the option value model, uh compares the util- expected utility from retiring now, and compares that to the maximum of the expected utilities at later retirement dates. okay so if i retire in one year what utility can i expect? if i retire in two years what utility can i expect? if i retire in three years what utility can i expect? pick the maximum of those, compare that to what i would get if i retired now, and if retiring now is at least as good as any of those other options, then i should retire now... the stochastic dynamic programming model, is, to a first blush at first blush, subtly different but actually, in terms of the complexity of the models substantially different. okay, what it says is i retire now if the expected utility from retiring now is greater than the expected value of the maximum utility, of future retirement dates. [S4: so what does that (mean) ] okay? so the difference is here we are talking about the maximum of the expected value, and here we're talking about the expected value of maximum. 
S3: but they are the same [S1: mhm ] if you have if you impose the probability function changing dates if you have erratic utility functions it means you're changing them correct? [S1: um ] or no or not? 
S4: do you mean like certainty equivalence or 
S3: yeah 
S1: it depends also on the learning process. so let let me come back to that and you'll_ i'll give you sort of a simple example of where they would be striking error. but one of the things it's going to depend on is, um, sort of how, uh h- how the error terms line up over time. whether you can learn_ whether observing today's error term tells you something important about tomorrow's error term. because then there's some gain from waiting until you can observe today's error term. w- w- we'll come back to that. and we'll actually t- talk i think more about it on Thursday as well. okay? um <P :10> alright, now this is getting to be the point where we roll up the sleeves here. uh <LAUGH> so for that the, probit model really i'm not gonna say much more about it it's it's kind of, what we've been talking about and it's really not what the authors were fundamentally, emphasizing. the option value model is their preferred model. in terms of notation, um T is my ret- is my current age, R is some potential retirement age, S is just going to be an index for age so when i'm talking about my age in general i'll index that by S. uh, <LAUGH> there are a number of interesting notational choices in this paper. uh why for example the notation A is so scrupulously avoided for age, is a bit of a puzzle but tha- that's okay. um, Y-sub-S is my earnings at age S if i'm still working for this firm. okay? and B-sub-S is my pension plus social security benefit, edge at age S, if i retire at age R. okay? so the idea that my benefit would vary with how long i work is an idea that we've talked about before, right? that if you retire before early retirement the, B-of-R is a very small number, if i work past sixty-five, B-of-R may actually decline because i'm not compensated for the loss in the number of years that i'll be able to collect benefits. it's less obvious why B would vary with S. okay? because most pension plans once you retire, um you get that benefit. okay? uh the easiest way to see why it would vary is suppose i retire before age sixty-two, i may be able to draw pension benefits from this firm immediately, but i won't be able to draw social security until s- i started sixty_ until i reach sixty-two. okay so suppose i retire suppose my current age is is fifty-two, i retire at fifty-five, uh my benefits would be just my pension benefits fifty-five until sixty-one, and then my pension benefits plus my social security benefits beginning at sixty-two. okay so, benefits would vary with my age, for a given retirement date, as well as the thing that we've been emphasizing more that my benefits will reti- vary with my retirement age. <P :04> okay and cap-V-of-T reflects my rest-of-life utility at age T if i retire at age R. okay? so i'm say fifty-two, and i'm thinking about retiring at fifty-five this would be my, utility if i retire at fifty-five. i'm thinking about retiring at fifty-six this would be my retirement at age fifty-six. V-little-T-of-T would be my utility if i quit right now... okay? so that's just the definition how is that actually defined, uh well, here's where things get interesting, okay? my utility as of age T if i were to retire at some future date R, is, composed of two pieces. one piece is, the utility i had experience while i'm working. okay so between, my current age T and the last year that i work, i get some utility associated with working, and that's discounted by this discount factor D-S-comma-T which is actually my notation not theirs. it's a combination of, some discount factor to the, you know raised to the appropriate exponent, and the conditional survival probability. okay so the assumption is if you're not there you get no utility. and so your utility is discounted not only for, t- time preference, but also for survival probabilities. uh combining those things is kind of a pain in the neck from a computational point of view but for your understanding of what's in the paper there's not much point to distinguishing between them. okay? so a one percent higher mortality risk and a one percent higher discount rate, basically work the same, in the model. (xx) okay, so here's my utility in each year that i work. that's the discounted, present value of the utility that i get in the years between my current age, and the last year that i work. okay once i retire instead of getting Y i get B. okay so this second term reflects the discounted value of my utility, from the year_ age at which i retire, to cap-S which is, sort of, the oldest age i can imagine living to. um, which i know for you guys is about forty-six but uh <SU-F LAUGH> you know for some of the rest of us it gets to be a, high double digit number let's just stop there. <SU-M LAUGH> um... and, what's assumed is that my utility in those years depends on the retirement benefits that i get in each of those years... okay so notice that there's no Cs here there's not_ we're not asking how much do you consume, at each of those ages, it's just sort of what is your income flow, in each of those ages and that reflects as i say the kind of polar opposite assumption, the nonperfect capital markets alternative. okay? uh, the the you know just for, you know your own personal reference and how to make_ when you write articles how to make the people who teach them happier, um using cap-S as the upper limit of lower-S works fine for those of us who are crazy enough to be kind of typing out slides at, one o'clock in the morning. it's a real nasty notational choice if somebody's gonna use_ try to teach the paper on the blackboard, alright <LAUGH> because little-S and big-S look very similar if you write them on the blackboard. <LAUGH> um so you know when when you do this stuff, have the the poor teacher in mind, uh have in mind that you may be teaching this paper someday and uh you know, using this and this to mean separate things is is really, you know it's a cute notational choice but it really actually makes the paper much harder to teach. okay uh, so we need now some explicit assumptions about what these two utility functions look like, and they make the simple assumption that there is um, kind of constant sort of utility, of income-while-working is income-while-working to the gamma plus an error term. okay so the error term might reflect things like how much you like your job, um how healthy you are, all that good stuff. and then your utility um, while retired is just your benefits raised to some exponent, and the factor K is meant to reflect the intuition that, if you offer me a dollar's worth of wages or a dollar's worth of retirement benefits, i would prefer the dollar's worth of retirement benefits cuz i don't have to work for those. okay? so we expect K is greater than one. okay that the utility that comes from ten thousand dollars' worth of retirement benefits, is probably greater than the utility that comes from ten thousand dollars' worth of earnings, cuz i have to work for the latter, i don't have to work for the former. okay? 
S3: so, so what, why you don- you don't use that in the discount you you could use that y- through the discount factor. 
S1: well no it's not through the discount factor so much right because this cou- this is sort of present versus future and this is just a matter of do i have to work for the money or not? 
S3: yeah but, that will be the same thing it's like because that would mean that you would more heavily discount the, that's actually very similar to saying that you, more heavily discount the future after you start you start retirement... [S1: ah ] or less heavily compared to 
S1: it's, if i put in the discount factor though um, it's going to differ not only between this segment and this segment but it's going to get heavier and heavier the older i get. right so within this segment it's going to accumulate the way discount factors do, and writing it this way it doesn't. it sort of, it's different in retirement and nonretirement but it's not different within the retirement period. that's probably the only, reason. um if you wanted this to have some micro foundations and i <LAUGH> don't want to stretch the point <SU-F LAUGH> cuz these are kind of lower-case M micro foundations. you can imagine sort of a Cobb-Douglas, function where there's leisure and income, and we let, working correspond to leisure-equal-to-one, and we let not working correspond to leisure-greater-than-one, and so K then_ so leisure to that exponent would be what we would call K. okay i'm sorry th- this is definitely lower-case M micro foundations um... i'm sort of trying to, relate something that has a kind of, folksy feel to it, um to the broader literature when i say that. okay they experiment with various forms for um these error terms, um but what they basically assume is that, first of all what'll matter is only the difference. so it doesn't bear, a whole lot of thinking about each of the components. and they make the assumption that this is a random walk. okay so th- so that the piece of my utility which i can see, and you the analyst can't, this year, is equal to last year's value plus some, fresh disturbance. okay and we'll we'll we'll talk about that a little bit at the wrap-up. um and that just, simplifies things just, whole great deals, a whole great deal and given how complicated everything else is, that simplification is is certainly welcome. okay so now we get to the heavy lifting, um G, at age T-of-R is the gain at age T for postponing retirement, to age R. and so it's the utility that i can expect if i retire at age R, the expectation is conducted at period T. okay i'm making the decision at age T. okay so that's the expectation conditional on the information that i have at age T. okay i asked how happy do i expect to be if i retire, at age_ over the whole rest of my life if i retire at age R? how happy do i expect to be if i retire now? okay?
S4: he notes though, you have a (sine N) there so do you really need an expectation T there? i mean you know that, right? 
S1: no i don't know how happy i'm gonna be right cuz i don't know what the (xx) 
S4: yeah but i'm talking about in time T, the for the second part of that.
S1: um, [S4: wouldn't you ] well it includes my error term at time T so [S4: okay ] yes at times, it it it's a little odd. [S4: okay, alright thanks ] um... R-star is the value of R that maximizes this, and, G is both at evaluated at R-star is called the option value. and basically i will postpone retiring as long as the option value is positive, i'll retire if the option value is zero or negative. <P :09> okay, notice the last line of this where d- where were we you're on your honor this time you guys failed me miserably <LAUGH> last time we tried doing this. um, so don't turn beyond this page. (xx) <SS LAUGH> alright, so the decision rule is keep working if the option value's positive retire when it's not. uh... figuring out what that thing looks like involves a lot of ugly algebra. uh which actually has if you make enough assumptions along the way, a relatively pleasant resolution. okay? so while i am, often inclined to say well it's in the paper look it up, uh here i've actually sort of, tried to show you how it, plays out and simplifies. so here's the expected value, of my lifetime utility if i retire at age R, and that's just borrowed from the, previous slide. okay? here's my expected utility if i retire today, it's just i get a stream of retirement benefits for the whole rest of my life. okay, um i'm gonna re-write that as one piece that reflects, the current age up to R... and then another piece which reflects from R to cap-S. <P :06> okay now i'm gonna subtract this second expected value from the first because that's how i defined the option value function. and the reason that i broke this up into two pieces is so i can tr- subtract the, first piece of this second term, from the first piece of the first equation. and subtract the second piece of the second equation, from the second piece of the first equation. okay? <P :05> so when i do that i get the discount factor multiplied by here, this first term is the years between, now and when i'm thinking about retiring. okay it's the years between age T and years R-minus-one. and the difference between retiring then and retiring now, is that if i retire then i have earnings in each of those years. whereas if i retire now i get retirement benefits, in each of those years. okay so this term reflects the, difference, between now and when i'm thinking of retiring. between the earnings i will get if i keep working, and the retirement benefits that i will get if i retire now instead... okay? <P :04> here's the error term which is just the difference between, this guy and this guy, more on that in a minute, and then this last_ so the difference in first terms in the two equations, are given on the first line, down here. and the difference between the second terms in the two equations, is given down here... and what that says in words or as close to words as <LAUGH> i'm gonna be able to come up with, is, from R until, i would eventually die. okay if i retire now i get a benefit based on retiring at age T, if i wait to retire at a- uh age R i get a benefit, based on that later retirement age. okay? so typically this is going to be positive if i postpone my my retirement, i get a higher value of retirement benefits, in each of those years. <P :04> okay? and again where we're headed is we wanna ask, whether this is positive or negative. remember if this is positive i don't retire, if it's_ once it becomes zero i do retire. <P :06> so now we're going to do a little notational thing we're gonna take the first summation. here and the third summation notice those are all nonstochastic. okay they're just things that, either one can observe we're going to pretend that they can observe because they can pretty well project. okay so the nonstochastic terms we're going to call little-G, this this expectation of the stochastic terms, um turns out to have a nice simple form and that comes because we've assumed, that V follows a random walk. okay? and so the statement i retire at age T, if my option value is, nonpositive <P :08> i think we had a sign error here um... yeah this should be i don't retire at age T, okay <P :13> so i keep on working if this option value is positive <P :08> okay? and that's gonna reduce to this relatively simple thing. now, given how, kinda hard we've worked to get here <P :05> it may not be quite obvious, how big a simplification this is. okay, so suppose you know survival probabilities you can look that up in a, actuarial table, and suppose you picked values of beta_ okay it's the survival probabilities and the discount rate that determine this thing that we've been calling D, gamma K and the variance of the error. okay so suppose i knew those four things, okay. then i could calculate this probability. okay? the computer i- would, grunt and groan, the lights in the building would probably dim, okay but there'd be a lot of, computing that goes beyond y- you know (that) right because we're calculating present values of, appropriately discounted differences between two earning streams, okay and and getting those right requires a fair amount of effort. okay but conceptually, this right-hand side, is a computable function of four simple parameters. okay? Mato 
S3: what's K-T of (xx) [S1: yeah? ] i get it now 
S1: okay <P :07> okay s- so so this, nasty thing is a computable function of, three parameters... okay, and then when i ask what's the probability that it exceeds the random term that is going to involve, how big the variance of the error term is. okay you have if V were normal that's just a, kind of you read that off a standard normal table. <LAUGH> once you've done everything else. okay? and so if i knew the four key parameters i could calculate the probability that each individual would retire, at age T. okay? well of course i don't know the parameters <LAUGH> that's why i'm in the business. right? if i knew the parameters they wouldn't, pay me to tell them the parameters. okay so our job is to try to estimate the parameters, but basically, that just runs this process in reverse. it says, calculate the probability for each individual and therefore the likelihood for the sample, for several parameters, see how that likelihood changes as i change the parameters, and use some, maximization routine, to find the values of those parameters that will maximize this likelihood. okay so there's kind of two steps in any of these problems one is just seeing, where does the likelihood come from, and being able to at least imagine writing it down if one doesn't actually write it down. and then there's the question of, how do you numerically maximize the thing? and the answer is you, go to somebody in the computer lab who's good at that stuff and, there's some routine that will do it. okay, so i mean the computational issues of how to maximize this thing are are are not trivial but they're not labor economics. um... now a warning, this is the probability that i retire at age T if i have a whole panel of data, then i've got to combine statements like this for different Ts. and that's actually you know going to be, again heavier lifting still. okay but the point is they can estimate and in fact they do estimate the parameters off of a single cross-section. okay they ask, in nineteen eighty at whatever age each individual is, do they choose to retire in that year? okay so they basically have one observation per person and they ask what happens if they include more, and find out it doesn't much matter. okay that's in a separate (paper.) basically they can estimate the things that drive their retirement model off of one cross-section. and, as these things go, this is computationally not mathematic. i mean, you know... if you f- i mean if you guys are motivated, this would be you know, well within the range of what you guys can do. whether you <LAUGH> want to or not you know is sort of would rather see the sun occasionally, um is a different matter but but, as computational problems go this is actually not that bad. okay in fact i'll predict that some of you will do things that are at least this complicated and probably more so by the time you're out of here. if you're taking Willis's course you probably are, <LAUGH> doing things that are more complicated. okay...? let's see why did i say don't turn the page? <P :05> ah okay up up up_ [S5: (i'm) sorry ] don't turn the page don't turn the page. now <P :04> we said that... the comparison between this model and the kind of the Gustman-Steinmeier approach or or the, simple probits more generally, is that, those models look one year ahead... okay so they say what happens if i retire now, what happens if i retire next year? and if retiring, if i'm not gonna be any better off by working one more year, i retire now. okay, so it's a comparison of T versus T-plus-one. my current age versus next year. the option value as we saw looks at T versus T-plus-one, T-plus-two, and so on... and so the question is when would those decisions differ? 
S3: like if you have like (a rash) uh, adaptive expectation they should be the same. 
S1: uhh no. 
S3: or very similar, wasn't it just 
S1: no not n- well i'll i'll give you at least two examples where they're different... okay can you guys think of any? without turning the page 
<P :04> 
S5: wait where they would be the same? to 
S1: where they'd be different. so wh- why would i... if i look at just this year versus next year, okay, decide, say to retire, but if i look at this year versus a bunch of future years, decide not to retire. 
S4: well maybe there's some information that like you know like you know is coming some, renegotiation or 
S1: well okay gimme a_ well there's a particular [S4: with ] not even a renegotiation but a particular, fo- 
S6: the normal retirement age and the early retirement age, [S1: ] 
S1: yeah [S6: can be the same ] so suppose your, suppose the early retirement age is fifty-five. okay? and you're fifty-two. okay? the probit model says well what happens if i retire at fifty-two, versus what happens if i retire at fifty-three? okay in terms of the financial incentives the answer is gonna be not much. in either case i'll have to take a vested deferred pension. in either case they're gonna be stealing from me. <LAUGH> and so the reward for working, to fifty-three rather than fifty-two is just not very large. okay so if my health isn't good or or you know, my spouse is nagging me to start taking vacations or or i don't like my boss um, or it's just too cold to get up and go into work on days like this, um i might decide to retire at fifty-two. okay? suppose i took the option value model and said well, okay i could retire now, i can retire, next year, that wouldn't be any better, i can retire when i'm fifty-four... that wouldn't be any better. but if i wait till i'm fifty-five now i get the early retirement, provision, and i'm a lot better off. okay so the gain for postponing retirement from fifty-two to fifty-three might be negative, from fifty-two to fifty-four might be negative, but retiring the g- the gain from going fr- from fifty-two to fifty-five might be strongly positive. okay so so the the, easiest example e- e- the easiest in quotes example i can think of, um is the decision before early retirement, to keep on slugging through to early retirement, will in many cases be a multi-year horizon, decision. a decision that you wouldn't get right if you just looked one year ahead. okay? <P :04> not unlike graduate school i might add right? you think well okay i finished my second year what's the reward for completing my third year. okay, but if i complete my third year i have the option of actually finishing, and going on to get a fun job. if i like getting up at eight-thirty in the morning and, <SS LAUGH> teaching your graduate students. okay? so you guys are pretty well you know if you're still here you've probably got the option value idea at least intuitively right in the back of your heads. right the uh you know, the reward from staying this extra year, probably doesn't repay <LAUGH> the rather extreme cost not only in effort but in terms of foregone earnings, that you're suffering as a result. the me- the reason you're around is because you realize that if you hang around <LAUGH> a couple more years, and wear down the faculty member of your choice he or she will sign your dissertation, and then you know you can go out into the the wider world and have a higher earnings and a much more fun job than you could get, either this year or next year. okay? so that's the guts of the option value intuition. okay there are decisions that, why where postponing one year doesn't make me a whole lot better off, but postponing, two three four five six years might make me a lot better off. [SU-M: mm ] i said two three <LAUGH> 
SU-M: you kept going though you didn't stop. <SS LAUGH> 
S1: well i meant it to include things not just graduate school alright? [SU-M: ah no ] <SS LAUGH> we could be talking about a PhD in other fields would that make you feel better? 
SU-M: no, no no 
S1: like like history <SS LAUGH> okay, alright the second thing that i had in mind um, is specific to this particular, application, which is the early-out window. okay so suppose i'm, fifty-five now i'm eligible for early retirement, and they offer me, nine months' salary to leave. well if i think about retiring this year versus next year, <LAUGH> it's pretty clear that i wanna retire this year, right? the gain from me turning that down would to be work one more year and, if they were offering me nine months' pay to leave then i'm working for three months' pay if i stay. so a one-year-ahead horizon would say, if they're giving you six or nine months' salary you're out the door. okay, but if you were fifty-five and you were thinking about working to sixty-five, now you've gotta say well retire now versus retire next year that's a no-brainer i should retire now. but retire now versus retire at sixty-five well now they're giving me nine months' of pay and i'm giving up ten years that i otherwise would have wanted to work for this firm. not so clear i should take the cake. mkay? so i'm thinking about early retirement window if you had a one-year horizon you would almost surely take it. okay? but if you had a longer run horizon, and you were thinking of retiring soon anyway, you would take it. but if the kind of optimum retirement date after this year was something ten years in the future, giving you six months' pay might well not be enough to tip the decision. okay? so a probit person, will almost always take an early retirement window. an option value person, might or might not depending on what otherwise their optimal retirement age is gonna be. <P :07> okay 
S6: so from the firm's perspective why would they offer higher bonuses to people who are closer to sixty-five? 
S1: uh probably because they desperately wanna get those people out. i mean i- in the optimal design of these things is actually quite complicated um, it depends <P :06> well it depends on a lot of things right? it depe- the the, i mean in general to induce somebody to retire, you would think the, younger they are the more you have to offer them. so if you were indifferent about getting people out at different ages you'd have higher bonuses for younger people. mkay but then there's a question about should you be indifferent about getting rid of people at different ages? and then there's the question of how will the age discrimination law look at what you do, if you offer different bonuses. and and here it's really kind of bizarre right because it's not entirely clear whether we're offering a big bonus to old workers, is treating them nicely, or nudging them out the door. there will be attorneys that will argue either position, no matter what you do. okay, um, so it's a combination of at what age would you like people to be gone um, what are the legal constraints and how do they, restrict you what you might otherwise be d- doing. um just as an aside, um, it turns out that in the real world when these early-out windows are offered, um only about a third of the workers take them. okay (xx) before and so the firm, is really offering the window to a whole bunch of people, and then taking its chances about who actually takes them up. and it's not uncommon to find that some of the people who take up the windows are people who are <LAUGH> extraordinarily employable also, and who you really would not like to lose. and they say thank you for the window and then, you know if there is nothing in their contract preventing it they go to work across the street for your competitor. you know which is all_ the worst of all worlds is to stimulate some of your, best workers, to consider their options, and their options end up including working for your competitor. um, but on the other hand you know if you really need to get rid of workers if you're a firm, which has, lived off the notion that we provide a career not a one-year-at-a-time employment contract and now you find it's really optimal to wanna get rid of some of the people that you induced to believe they could stay as long as they wanted to, but then you've gotta change, their wanting to. and the way to do that or at least one way to do that is with the early-out window. okay <P :10> okay uh the next_ now you can turn the page and we just basically said what, is on page there. <P :05> okay stochas- stochastic dynamic programming models, um we're gonna do this once over lightly, um in part because it's really, not the preferred model um unless you're gonna be stocking while you're selling and in part because you'll get another look at this in in John Russ's paper on Thursday. so, the option value compares, the expected value of retiring now, against the maximum of the expected values of retiring later. okay? and then dynamic programming compares the expected value of retiring now against, the expected maximum... of retiring later... but in general this second, the one that, dynamic programming focuses on, is going to be bigger, than the one that the option value, focuses on. um, lemme give you a simple example. suppose that uh the gain from retiring is zero at all ages. okay? so i'm just, as indifferent i'm just totally, you know, nudge me either way and i'll retire or not at any of these ages. okay so basically it's a, total coin flip for me at any age. okay? it's the option value model then would say i'm indifferent about retiring. okay i have a certain level of utility now, i'll have that, same level of utility if i p- average if i postpone a year, i'll have that same level of utility on average if i postpone it two years. uh, it really doesn't matter. (xx) programming model on the other hand would sort of put its arm around you and say well now think about this a little bit more carefully. okay? and the bottom line is gonna be in this situation you should keep working. okay and th- uh why is that? well, suppose you get a bad, omega T-plus-one. okay so suppose you you keep working for a year, uh either good things or bad things might happen, right? and on average they should be, work out to zero. if it works out bad, well then you lose, then you leave so, you suffer a loss, okay okay so suppose your health turns out not to be very good that year, and you kind of slug your way through it. um or suppose your boss turn- you know you get a new boss and that boss turns out to be a, raging imbecile. um well then you kind of you know suffer the year, and then you leave. okay? but suppose you get a good shock, from working another year. well you not only get, that exp- that, good shock, but on average, you can expect good things to happen for the next several years as well. remember the random-walk process. okay...? and so, what that cl- stochastic dynamic programming model basically emphasizes is, that by not retiring, you have the option of sticking around, if you earn something good in the meantime. okay that you'll know more about the future years, next year, than you do at this time. and there's some value to that additional information that you'll gain... the option value model basically ignores that. okay and the stochastic dynamic programming model says in a year you'll know more about the situation than you do, and so if you were just indifferent about leaving or not, stick around, because if the information is bad you can still leave, but if the information is good you will definitely wanna stick with it you'll wish you had stuck with it, stuck with it. okay but we'll talk more about that classic model Th- um, on Thursday. i'm sort of, working up my energy level to, take this on th- that's gonna be a hard topic. okay so, when you actually estimate the option value model um what sort of things do you get? um, you get gamma, less than one which i guess is what you'd probably guess. um you get K greater than one, although not estimated very precisely. (xx) you get kind of remarkably sensible discount factor alright that says basically an interest rate of ten percent. um and one one of the um, underappreciated, crises uh of people who try to estimate structural models is that oftentimes the discount factors that you get when you estimate them freely are just awful. uh and you get discount rates of eighty-six percent. <LAUGH> boy can i write you a loan um you know or or zero uh which doesn't make a whole lot of sense and so, papers which actually estimate a discount factor freely and come out with sensible results, are are surprisingly, uncommon and i think at least i don't have a sense of why, papers in general, often seem to have that problem um, and, you know this one kind of lands on its feet. um and finally there's the, variance of the error term okay which in some sense reflects the relative importance of the unmeasured nonfinancial factors in people's retirement decision. and there's an estimate there and it's kind of, you know is that big or small i don't know it's kinda hard to tell. hard to evaluate. but it's estimated reasonably precisely. okay so one question you can ask is okay i've got nineteen eighty data, i know as of nineteen eighty, how much people have earned, i know what they can expect to earn if they continue to work for this firm more or less, i know what their retirement benefits will be, um if they retire now or if they retire at some point in the future, um, armed with all of that information, how well can i predict who retires in the next year? okay and i have three models for doing that, and the bottom line turns out to be that all three models do about equally well. 
S3: so, how does this, comply with the critical focus because it said that when you this i don't know this seems like very, problematic for me from the Lucas point of view, in which he said that when you change the the policy like when you change the policy so people change their, expectations so we should not expect the uh rationally not expect that, anything would happen. 
S1: okay well well so far we're still talking about the, kind of normal policy. okay so nineteen eighty, was a normal year not the year of the early retirement window. okay so i think that's, where you more have the Lucas is um what happens once they offer the window and how does that change people's expectations. okay? the story line that they're going to want you to accept i believe, um which i think is probably okay in this application and maybe not as okay in other applications, is that this was really seen where the window was really seen as a once and for all, experience. and so, it really didn't, change people's expectations a- although i'll talk about a subtle way in which it might have even if it were a one off. l- let me come back to that i- i'll i won't answer it fully but i i'll at least, touch on that issue, um the way in which that would where you_ i wouldn't have called it the Lucas critique but actually you'll you'll see it in in the next slide. okay? so for predicting who retires in nineteen eighty which is normal times all of the models do about equally well. um but for predicting the response to the window, um the two more sophisticated models do much better. okay? and why is that? well we kind of, let the cat out of the bag a couple slides ago right? we said that, if you're a probit person and they offer you six months' pay to leave now, and you think about do i wanna leave now or work another year, then you wanna leave now. right? but if you're a, option value or a stochastic dynamic programming person you say well if i leave now i won't be able to work till sixty-five like i was planning to do. so i'll compare that six months' of pay, against, not just working one more year where it clearly wouldn't make sense to do so but, whole bunches of future years. and if i think about it that way it's a much less attractive option. okay so basically the option value and the stochastic dynamic programming models get right the idea that the window is not very attractive to people who were planning on staying a good while longer. and the probit model doesn't get that and so the probit model basically predicts that everybody will wanna leave. okay? and i'm slightly exaggerating. but basically it will overpredict how many people'll want to leave. and so what i think that the the paper clearly does, is it shows that the kinds of sophistication that are in the option value or the stochastic dynamic program models, are really quite important uh for problems where the incentives become, sharply nonlinear at one point in time as they do, obviously with an early-out window. i mean to be honest i'm surprised given the early retirement, accrual spike um that the probit does as well as it does. okay that is i would have thought that the probit would have had problems uh predicting that people would retire the year before_ two years before early retirement. uh whereas the option value and the S-D-P would tend to get that right too. um so i'm actually surprised the probit does as well as it does. um, but but that's what they find. <P :06> okay now let's kind of look back on on what we've done here... um, first of all uh to come back to the point that we made earlier, um utility is depending only on current income. okay, and, the more i think about this the, the more it bothers me. um, i mean, for most problems the idea that, people consume more or less their income, is kind of depressingly true. for an ama- i mean you know an awful lot of people arrive at retirement with no assets other than their house. okay? so the idea that that people more or less consume their income so that their consumption is equal to their income so their utility of consumption is equal to the utility of their income, doesn't seem like that bad an assumption. but it's a little, sort of the the rhetoric is not just right when you're embedding that assumption in a model where people are thinking eight years in the future about their retirement, but they don't think one year in the future about, planning their consumption and asset accumulation. um, that sounds really weird. okay that so if you tell me that people are grasshoppers they, you know, eat their income each year, that actually doesn't bother me. but to say well people are that way but, they look eight years in the future about deciding, how much they're going to consume. or when they're going in to retire i mean. that seems a little inconsistent. okay so the people who i know who... consume their income are not people who i imagine look ten years in the future to decide their, labor force, plans. and so there's a little tension between um this assumption which would be i think appropriate for people who are not very forward-looking, and the guts of the model which actually relies on forward-looking behavior to give it its distinctive character... um i mentioned before that little-K makes sense if one thinks of utility as a Cobb-Douglas function of leisure and consumption, um, but only if you assume that those who will retire don't work elsewhere. okay and so this gets to the point where, um i mean if you read the paper quickly, you might not notice that the issue of partial retirement was totally dodged. okay there's the utility if i work here there's the utility if i retire the utility if i retire depends on my benefits that all seems sensible. okay but it's only when you sort of, put this paper next to the papers like Gustman and Steinmeier, where partial retirement is front and center, that you realize the partial retirement issue, is really missing. okay and so in thinking about whether i should leave this firm, i should be thinking not only about my current pay with this firm versus my retirement benefits from this firm, but my current pay from this firm versus my retirement benefits from this firm and what i ever i could earn if i go across the street. okay and it it's kind of that [S3: so what ] which is just not modeled. 
S3: but would that would mean would actually, retire from that firm even earlier... or not? 
S1: holding constant the values of the parameters that's exactly right. [S3: okay ] okay so if if we sort of have all of the parameters taking their true value, and then we said, how would someone who has the option of working elsewhere, behave differently? the answer is yes they would be likely to retire earlier. but of course if we've got the model wrong we get the parameter estimates wrong. [S3: so ] so something else in the model is going to want to capture the fact that these people are actually retiring. 
S3: so but can you something is like when they_ to what extent like i could expect that K should be, less not not more? 
S1: um i'm not sure which parameter it would bias. i mean o- one of the, problems with these nonlinear models is that as soon as you notice something wrong, you're kind of, you don't have the intuition that you have in a linear model, about which parameter it would likely bias. um, mkay just to wrap up cuz we're running out of time, uh, the zero mean innovations to the error term. okay? um this is the idea that the marginal utility of leisure increases with age. what's kind of strikingly missing from the paper is, age or time as an independent determiner of retirement. okay so we don't have health in there we let health be in the error term. but the error term's a zero mean. taken literally that means that if the error term reflects health i'm not getting sicker as i get older. and while we would like that to be true we kinda suspect it's not. okay so there's nothing in the model that's capturing the intuition that as people get older, on average it gets harder for them to show up for work... um this gets now to Mato's point. even if i think that the window offer is a once-in-a-lifetime opportunity, okay which is actually probably how these workers saw it, there's still the issue of how does that policy change by the firm change the expectation of the workers. and i think you could reasonably expect that the window offers would even if it did nothing else even if it didn't lead workers to expect another window offer in a year, it probably should revise their forecasts about their earnings. i mean a booming vibrant firm does not say hey guys, wouldn't you like to retire? alright a firm that says, hey guys wouldn't you like to retire is probably a firm that's having trouble. if the firm is in some sense that desperate is it likely to be giving ten percent wage increases to the people that stay next year? right after having, asked me politely would i like to leave in exchange for a small, subsidy, this year, are they likely to offer me the big bucks as a raise next year if i stay? well probably not. okay so the model assumes that this firm is a stable and stodgy enough large employer that if you know your earnings at fifty-five you can pretty well guess your earnings at sixty-five. and whatever the reliability of that assumption in normal times, you would think that, people might change their expectations about, future retirement. in some sense what would get you to retire in response to the window, is a combination of the dreck financial incentive which is modeled, and the implicit bad news about your career prospects at this firm, which are not modeled, okay? so suppose somebody didn't respond to the financial incentive of the window offer at all, but said hey, this likely means that, my remaining years at this firm are not going to be as <LAUGH> golden as i imagine they would be. okay they might dec- decide to retire as much for that reason, as for the naked financial incentive given by the window, but the model sort of puts all the weight on the financial aspects of the window and none on the, there's an implication here about what, wages and life at the firm are likely to be like if i stay. Ben? 
S6: but if you're comparing_ if you're just trying to compare the option value to the probit, and the probit said that too many people le- th- this would, this would mean that pe- more people would leave than would otherwise here, (xx) my so 
S1: right so should so you right i you're saying wo- does this bias the comparison between the probit and the option value the answer's probably not. and so i guess my claim is more, you know, how would you like to, crank a little more on the option value model and my my my sense is that, this is a dimension at least that you wanna think about. um but you know look i gotta say every sort of complained about various aspects of the paper. the idea of using the window as basically external validation, is an idea that that happens way too infrequently. okay i mean a- actually Thursday's econometrics seminar if you have time is is about, this set of issues broadly. all of our, you know statistical tests are based on the presumption that i specify the model, have my research assistant, estimate the model once on a short-term contract, and here are the results. okay? what people in fact do is they sort of run the model, they notice something isn't working they sort of think why is it not working they change the model they change the you know what variables are included or, you know so so basically there's interaction between the model and the data. which you know i- in some sense is not entirely unwholesome, but it certainly means that when somebody after sixteen tries, ends up with T-ratio of two um, you've gotta think that the possibility of this happening if the true parameter is zero is really more than point-oh-five. okay if they get a T-ratio of two after sixteen tries, it's not the same as the T-ratio in a textbook, uh and what people in other social sciences often do is they reserve part of the data, and then having sort of fiddled with the model the way we all do, estimate the quote final model on fresh data. okay and see at least does it predict outside the sample or is it ability its ability to predict inside the sample just because i've, sort of fiddled with it to line up with every quirk in the data that i happen to have? so the idea of using a completely different experience as external validation for the parameter estimates um, you know is a neat idea you see it more in other social sciences than you do in economics and it would be nice to see applications where it's possible to do it here. okay? <P :05> i'm still trying to figure out when we're going to meet on Tuesday i need to do that i need to find a time to talk to you about your paper, um i need to give you guys a problem set. uh the Royal Shakespeare Company is here Tuesday twice on Wednesday and Thursday night i'm, <LAUGH> it's gonna be a coup- funny couple of days. um, but starting on Friday i come back up for air, uh so, you know you'll be back in (Monday) so w- we'll do the Russ optio- the Russ c- econometrical paper, in class on Thursday. 
S3: (xx) (you know) next week we we don't have (time) 
S1: let's try for Friday and then with the option to cancel how's that? i i would feel better if i 
S3: (xx) Friday like i don't have class (xx) so 
S1: oh Friday doesn't work okay 
S3: (xx) so it's like the best, time (to meet) 
S1: fine um
S3: because i the trouble is i ca- i can't, now (xx) come to the point uh be on uh s- be stuck with the empirical question. 
S1: okay well uh, let's see what would would work Monday <P :10> (xx) oh okay well well w- w- what works for you on Monday? 
S3: on Monday it's from one to two thirty or, or <P :04> or after four.
S1: how about two o'clock, would that two two-thirty does that work for you? 
S3: yeah that's fine.
S6: did you leave entire bag (xx)
SU-F: it was i- it was on purpose, i had to go to the_ see (xx) (i keep forgetting to) (xx) 
S1: okay i'm sorry to (xx) 
S3: that's not the problem because, i have done at least some examples (xx.) 
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