<p>Check out <a href="https://en.wikipedia.org/wiki/Sunk_cost" rel="noreferrer">sunk cost</a>. Although this might seem completely counterintuitive, the investment you have made should be <strong>irrelevant to your decision</strong> to stay or quit, if this investment has already been made and cannot be recovered.</p>

<p>What matters is whether the benefits you are yet to incur outweigh the costs (that you are yet to incur). So the decision should be a <strong>forward-looking</strong> one.</p>

<p>The above reference mentions that in the real-world, people often do not behave in this way. According to sunk cost theory, this is irrational, but this is only one theory.</p>

CC BY-SA 3.0
2017-09-12T14:55:29.650Z
Some argue taking into consideration sunk costs in decision-making might be reasonable after all. [This paper](http://www.mit.edu/~rdoody/TheSunkCostFallacy.pdf) states that "_It’s reasonable to
want to maintain plausible deniability about having suffered diachronic misfortune. Sometimes, honoring sunk costs is the only way to do this. It’s reasonable to want to maintain plausible deniability because having this desire is instrumental in successful cooperation, and successful cooperation is essential to our success as social creatures_"
luchonacho
CC BY-SA 3.0
2017-09-12T19:31:11.113Z
"Sunk cost" has always sounded a lot like "lottery logic" to me. A lotto ticket is so cheap, and the reward is so great, even if the odds are worse than astronomical, what's a few bucks? Sure I've spent literally thousands on lotto tickets this year alone and never won, but the cost for that next ticket is negligible compared to the reward. And it doesn't make sense to factor the odds of winning into the potential costs, as I'm playing on the odds that I'll win with just one more ticket.
talrnu
CC BY-SA 3.0
2017-09-13T00:03:34.533Z
@talrnu The problem with your reasoning isn't that you failed to consider sunk costs. You have 2 problems. First is gambler's fallacy: "I'll win eventually if I keep playing." Second, you failed to consider your incredibly low odds of winning. Try weighing the cost of the average number of tickets required to win 80% of the time vs. the payout. I haven't done the math, but I bet it's astronomical. Occasional lottery play is okay if you consider it money spent on fun that you will not recover; assuming it will eventually be a source of income is just bad reasoning.
jpmc26
CC BY-SA 3.0
2017-09-13T00:27:23.830Z
@talrnu For example, taking [the Mega Millions Jackpot](https://www.khanacademy.org/math/precalculus/prob-comb/prob-combinatorics-precalc/v/mega-millions-jackpot-probability), you only have a [43..4%](https://www.google.com/search?q=1-%281-1%2F175711536%29^100000000) chance of winning after buying 100 million tickets at 1 dollar each. The payout is currently only sitting at 76 million dollars.
jpmc26
CC BY-SA 3.0
2017-09-13T02:25:46.287Z
@jpm26 Yes, many lottery players are delusional/irrational and succumb to the gambler's fallacy. But the players I refer to are those who decide to buy a single ticket while they're already at the store, not because "I should keep playing the lotto until I win", but because "Maybe I'll win this time." Such a decision accounts only for the cost and potential reward of a single ticket. If the cost of one ticket is negligible, and you're not planning to play more (even though you probably will), then without the perspective of sunk costs you'd feel crazy not to play.
talrnu
CC BY-SA 3.0
2017-09-13T08:58:35.217Z
Lottery tickets are not as straight forward as simple statistics tell you. You have to consider the utility of a single dollar versus the utility of the lottery winnings multiplied by the odds. The utility of a million dollars may or may not be a million times the utility of a dollar. If your basic needs are being covered, the utility of a dollar by itself, is negligible. In other words, in many cases the marginal utility of +/- 1$ = 0. The utility of a million dollars, however...
Stian
CC BY-SA 3.0
2017-09-12T08:43:05.903Z
2017-09-12T08:48:06.390Z
2017-09-12T08:48:06.390Z
M3RS
<p><a href="http://www.e-elgar.com/shop/handbook-of-choice-modelling" rel="nofollow noreferrer">This handbook</a> provides a broad overview of different modelling approaches to <strong>choice theory</strong>. Here, I give just one example: discrete choice modelling.</p>

<p>Section 2 of <a href="http://ftp.iza.org/dp9957.pdf" rel="nofollow noreferrer">this paper</a> offers an example of a dynamic discrete choice model about educational choices. The decision tree is:</p>

<p><a href="https://i.sstatic.net/QZF3f.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QZF3f.png" alt="enter image description here"></a></p>

<p>Individuals compare the return of each option, based on their ability and other observable variables.</p>

<p>The discrete choice literature is massive, and includes anything from modelling labour participation (e.g. <a href="http://www.sole-jole.org/14404.pdf" rel="nofollow noreferrer">here</a> or <a href="http://www.sciencedirect.com/science/article/pii/0304407694016941" rel="nofollow noreferrer">here</a>) to marriage/divorce (e.g. <a href="https://www.jstor.org/stable/3663512" rel="nofollow noreferrer">here</a>) to corruption (e.g. <a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2214810" rel="nofollow noreferrer">here</a>). See the handbook above for more references.</p>

CC BY-SA 3.0
2017-09-12T14:50:15.700Z
GED permits attending college.
Joshua
CC BY-SA 3.0
2017-09-12T14:57:42.810Z
@Joshua You should tell that to the authors, not me :)
luchonacho
CC BY-SA 3.0
2017-09-12T09:21:30.747Z
2017-09-13T10:22:17.000Z
2017-09-13T10:22:17.000Z
luchonacho
<p>I'm reminded of the law of diminishing returns. 
<a href="https://en.wikipedia.org/wiki/Diminishing_returns" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Diminishing_returns</a>
The idea is the more you put into production, the less you get out of it. I originally stumbled upon this theory while looking for a mathematical model to model the increased demand for addictive substances, but noted the harmonic series <a href="https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)</a> also modeled it, where the second chocolate bar, cigarette, or dose of heroin is only half as good as the first, the third even less, the fourth, etc.</p>

<p>But more salient to what you are looking for, investment into graduate school, in terms of time and money, naturally comes with greater cost, compounded by that diminished return on investment lost by decreased life time earnings in terms of raw years. This could be applied to the "worth of skill" in terms of enjoyment of a highly skilled musical instrument player vs. a skilled musical instrument player, as the number of pieces of music and number of people you can impress first opens very wide, then the limitations become more precise, and the scarcity of musical challenges increases so that a finer and finer list of musical rivals and musical challenges exists. With respect to both a musician and a graduate student, or a marathon runner, in addition to losing the time invested, there's also competition. </p>

<p>Rising in a given devotion forces you into the category of competitive X, such as competitive gambler, competitive skier, or competing against other graduates for the same job. This manifests either as literal reduction of wages, being over qualified or having to compete with a dozen other Ph.D students or a hundred scientists seeking the same grant money. That may be counter intuitive at first, but because the demand for extreme specialization decreases with the increased level of specialization, the supply of rewards for that increased specialization also decreases. With gambling or cheating gamblers, the risk continues to increase and the skill of rivals increases to to financially dangerous levels, as notoriety results in an endless sequence of people actively trying to bankrupt or arrest the gambler. Bank robbery theory is the same way, as criminals often talk about "just one more job", assessing their risk vs. rewards and concluding (usually after arrest) that they should have quit while they were ahead. </p>

<p>Opportunity costs are extreme for some of these models too, as a skilled musician, for example, may impress at 3000-5000 hours of practice, but need 7000-10,000 hours of practice for the average person to appreciate any difference, and 20,000 hours of practice for a subtle improvement beyond that. With a marathon runner, the more you practice, the older you get, and the closer to "peak" you reach, and the more frequent the opportunities for injury, such as the recent Usain Bolt run. Boxing has a similar diminishing returns effect as injuries compound and performance decreases - that was a central theme of Rocky. Football Salaries are exceptionally high with a similar peak and injury calculated into the risk assessment. </p>

<p>Diminishing returns are most poignant though with the college graduate. You are assumed to have X number of years working, and every year in school is a year where you not only are not earning optimal wages, you are also building debt that will compound. Since work experience also increases wages with time, and since student loans have interest rates, there is a point in which a person's life time earnings actually start to drop lower. At one school I was attending, the tuition total was around $50,000/year, and to complete everything through graduate school was going to total half a million dollars and take a total of 10 years. As starting age was a constant and the growth of returns depended on the time from graduation to the time of retirement minus the growth of loans, the model for staying didn't pan out, and quitting was more fiscally sound. </p>

CC BY-SA 3.0
2017-09-12T14:15:17.790Z
2017-09-12T14:15:17.790Z
Tristian
<p>Seeing Ubiquitous's answer, I want to offer a concrete model I learned originally from an Exercise in Jianjun Miao's "Economic Dynamics in Discrete Time", which looked at <strong>McCall's (1970) job search model.</strong> It asked to describe the model as an optimal stopping problem.</p>

<hr>

<p>Consider an unemployed worker searching for a job, where each period he is unemployed, the worker draws one wage offer from a cumulative distribution function $F(w)$, where $F(0) = 0, F(B &lt; \infty) = 1$. The worker may then reject the offer and receive compensation $C \geq 0$ or accept the wage $w$, which for now we will assume is received forever (no firing or quitting yet). You cannot accept past offers. The worker is risk neutral., so $\mathbb{E_0} \sum^\infty_{t=0} \beta^t y_t$, where $y_t = w$ if you accepted and $c$ if no offer has been accepted yet.</p>

<p>You can then express the question of when to accept what wage as a dynamic programming problem:</p>

<p>$$V(w) = \max_{\text{accept, reject}} \{\frac{w}{1- \beta}, \quad C+ \beta \int^\beta_0 V(\theta) dF(\theta)\}$$</p>

<p>denote the integral part as $Q$, the value function for an unemployed person receiving $w$ as an offer. So then there is some reservation wage where you accept if you get offered the wage or above.</p>

<p>That is, $\exists \ \bar{w}$ such that you accept when $w &gt; \bar{w}$. Since you accept when:</p>

<p>$$\frac{w}{1 - \beta} \geq C + \beta Q$$ </p>

<p>We have $\bar{w} = (C + \beta Q)(1 - \beta))$</p>

<hr>

<p>Now when you take this original model, what happens if we allow for quitting? No worker will want to quit since the solution is time consistent. There is no reason to quit the job if you found it optimal to accept a job in a previous period. So the model as is can't account for quitting.</p>

<p>Some ways you can tweak the model though to change the optimal reservation wage over time is to change $C$ to be time varying. Maybe if you have more security over other periods, you might quit to look for a better job that will take you. You could also change $F(w)$ to be time varying. Maybe over time the wages available to someone of your skills rise. In Joanovic's matching model, "tenure" makes your wages increase over time, so that could affect the decision to quit. Maybe your agent isn't risk neutral and their risk preferences change over time.</p>

<hr>

<p>McCall's model is a nice place to start if you want to tweak with different variables to try and model different reasons for why workers quit, and you can imagine many different extensions have built on it in the literature.</p>

CC BY-SA 3.0
2017-09-14T04:14:21.607Z
Thanks! This is indeed a good starting point. I assume that changing $c$ (instead of $C$ I presume) is equivalent to changing unemployment compensation here or an (exogenous) outside option in a more general problem.
emeryville
CC BY-SA 3.0
2017-09-14T04:22:15.803Z
Oh, I did not mean to make $c$ lowercase or whatever. If I wanted to change $C$ in the problem, the interesting way to do it would to make it an exogenous state variable as you have said, I think.
Kitsune Cavalry
CC BY-SA 3.0
2017-09-14T02:28:54.543Z
2017-09-14T04:22:47.970Z
2017-09-14T04:22:47.970Z
Kitsune Cavalry
<p>A relevant literature seems to be that on <em>optimal stopping problems</em>. There is a fairly technical Wikipedia article <a href="https://en.wikipedia.org/wiki/Optimal_stopping" rel="noreferrer">here</a> and <a href="https://www.math.ucla.edu/~tom/Stopping/sr1.pdf" rel="noreferrer">here's a book chapter</a>.  </p>

<p>In economics, such models have been used to think about <a href="http://www.sciencedirect.com/science/article/pii/S0022053111000329" rel="noreferrer">how sellers learn about selling opportunities</a>, <a href="https://www.jstor.org/stable/1556717" rel="noreferrer">investment in R&amp;D</a>, and <a href="https://www.jstor.org/stable/2297410" rel="noreferrer">optimal labor search strategies</a>.</p>

CC BY-SA 3.0
2017-09-12T10:14:35.843Z
2017-09-12T13:54:47.543Z
2017-09-12T13:54:47.543Z
Ubiquitous
<p>Is there an <strong><em>economic theory of quitting</em></strong> an activity? A theory that weighs the investment costs put into something and the opportunity costs of pursuing it. </p>

<p>I am aware of <a href="http://rady.ucsd.edu/docs/faculty/GneezyTradeOff.pdf" rel="nofollow noreferrer">Fershtman and Gneezy Quitting in Tournaments</a> paper that investigates the decision of quitting in the middle of a race or of a tournament. But, I am looking for a more general model of quitting something such as playing an instrument, a sport or even a job. Dropping-out from (grad)-school may obey the same investment-opportunity cost trade-off.</p>

<p>Update: I am looking for some explicit references modeling the exiting or quitting process, even if it concerns some decisions in particular without any generalization.</p>

<p>Edit: <a href="https://economics.meta.stackexchange.com/questions/1762/should-we-explain-accepted-answers">Here is why I picked @Ubiquitous' answer</a>.</p>

CC BY-SA 3.0
2017-09-12T06:15:46.697Z
How is this different from the standard labour market analysis of when accepting a job offer? (reservation wages and all that) Is it not just the other side of the same coin? Not sure if this needs a new theorisation.
luchonacho
CC BY-SA 3.0
2017-09-12T06:39:45.130Z
Is that simple? Are we contributing to this SE simply because its above our "reservation wage"? We would have gone otherwise?
emeryville
CC BY-SA 3.0
2017-09-12T07:51:35.763Z
I am not sure a general theory that can explain participation in SE, education, race, hobbies, employment, investment, and etc is available. Maybe a very abstract rational choice theory? There are however pretty decent theories about job-to-job or job-to-unemployment/inactivity transitions, or theories of investment or firm entry/exit, or discrete choice models of education. All together these theories show that there are important idiosyncrasies. A unified choice theory might miss important aspects of the choices we make (or more complex decision making based on culture, addiction, etc).
luchonacho
CC BY-SA 3.0
2017-09-12T08:33:37.487Z
I would be pleased to upvote an answer with some explicit references modeling the exiting or quitting process, even if it concerns some decisions in particular without any generalization.
emeryville
CC BY-SA 3.0
2017-09-12T08:55:17.863Z
One simple model of quitting grad school 1) ignore sunk costs; 2) discount lifetime increase in wage due to grad degree to the present; 3) add tuition fee yet to be paid + opportunity cost of not having a job while studying (in terms of wage foregone). If 2) bigger than 3) stay in grad school.
M3RS
CC BY-SA 3.0
2017-09-12T09:07:09.093Z
I would explore game theory, with bounded rationality + infinite games to answer this
JoaoBotelho
CC BY-SA 3.0
2017-09-12T10:55:06.740Z
@Andras It's fine to ignore sunk costs, but it could be unwise to ignore the possible effect on future job applications of having a cv / resume showing a period in which one started a course but did not complete it.  More generally, in any stay or quit choice it's worth considering how others will perceive a decision to quit and could that lead to any adverse consequences.
Adam Bailey
CC BY-SA 3.0
2017-09-12T10:56:53.313Z
@Adam Bailey: agree.
M3RS
CC BY-SA 3.0
2017-09-12T11:01:49.160Z
@AdamBailey: right. Fershtman and Gneezy highlight the social cost of quitting
emeryville
CC BY-SA 3.0
2017-09-12T12:47:19.657Z
Any chance you're writing up a PhD thesis?  I know several people (myself included) who would start every day during that period trying to rationalise quitting.  FWIW they all made it through in the end :)
Guy G
CC BY-SA 3.0
2017-09-13T04:24:26.067Z
Relevant: http://freakonomics.com/podcast/new-freakonomics-radio-podcast-the-upside-of-quitting/
Steve Bennett
CC BY-SA 3.0
2017-09-13T10:41:57.420Z
Should I stay or should I go sounds an even more catchy title. You might even get to the 10k views!
luchonacho
<microeconomics><reference-request><behavioral-economics>
Should I stay or should I quit?
emeryville
# Should I stay or should I quit?

(Asked by: username_0 on 2017-09-12)

Is there an ***economic theory of quitting*** an activity? A theory that weighs the investment costs put into something and the opportunity costs of pursuing it.

I am aware of [Fershtman and Gneezy Quitting in Tournaments](http://rady.ucsd.edu/docs/faculty/GneezyTradeOff.pdf) paper that investigates the decision of quitting in the middle of a race or of a tournament. But, I am looking for a more general model of quitting something such as playing an instrument, a sport or even a job. Dropping-out from (grad)-school may obey the same investment-opportunity cost trade-off.

Update: I am looking for some explicit references modeling the exiting or quitting process, even if it concerns some decisions in particular without any generalization.

Edit: [Here is why I picked @Ubiquitous' answer](https://economics.meta.stackexchange.com/questions/1762/should-we-explain-accepted-answers).

**Tags:** microeconomics, reference-request, behavioral-economics
> **Comments:**
> **username_1**: How is this different from the standard labour market analysis of when accepting a job offer? (reservation wages and all that) Is it not just the other side of the same coin? Not sure if this needs a new theorisation.  
> **username_0**: Is that simple? Are we contributing to this SE simply because its above our "reservation wage"? We would have gone otherwise?  
> **username_1**: I am not sure a general theory that can explain participation in SE, education, race, hobbies, employment, investment, and etc is available. Maybe a very abstract rational choice theory? There are however pretty decent theories about job-to-job or job-to-unemployment/inactivity transitions, or theories of investment or firm entry/exit, or discrete choice models of education. All together these theories show that there are important idiosyncrasies. A unified choice theory might miss important aspects of the choices we make (or more complex decision making based on culture, addiction, etc).  
> **username_0**: I would be pleased to upvote an answer with some explicit references modeling the exiting or quitting process, even if it concerns some decisions in particular without any generalization.  
> **username_2**: One simple model of quitting grad school 1) ignore sunk costs; 2) discount lifetime increase in wage due to grad degree to the present; 3) add tuition fee yet to be paid + opportunity cost of not having a job while studying (in terms of wage foregone). If 2) bigger than 3) stay in grad school.  
> **username_3**: I would explore game theory, with bounded rationality + infinite games to answer this  
> **username_4**: @Andras It's fine to ignore sunk costs, but it could be unwise to ignore the possible effect on future job applications of having a cv / resume showing a period in which one started a course but did not complete it.  More generally, in any stay or quit choice it's worth considering how others will perceive a decision to quit and could that lead to any adverse consequences.  
> **username_2**: @Adam Bailey: agree.  
> **username_0**: @AdamBailey: right. Fershtman and Gneezy highlight the social cost of quitting  
> **username_5**: Any chance you're writing up a PhD thesis?  I know several people (myself included) who would start every day during that period trying to rationalise quitting.  FWIW they all made it through in the end :)  
> **username_6**: Relevant: http://freakonomics.com/podcast/new-freakonomics-radio-podcast-the-upside-of-quitting/  
> **username_1**: Should I stay or should I go sounds an even more catchy title. You might even get to the 10k views!  

---

## 5 answers

### Answer by username_13 on 2017-09-12. Score: 13 | Accepted answer

A relevant literature seems to be that on *optimal stopping problems*. There is a fairly technical Wikipedia article [here](https://en.wikipedia.org/wiki/Optimal_stopping) and [here's a book chapter](https://www.math.ucla.edu/~tom/Stopping/sr1.pdf).

In economics, such models have been used to think about [how sellers learn about selling opportunities](http://www.sciencedirect.com/science/article/pii/S0022053111000329), [investment in R&D](https://www.jstor.org/stable/1556717), and [optimal labor search strategies](https://www.jstor.org/stable/2297410).


---

### Answer by username_2 on 2017-09-12. Score: 16

Check out [sunk cost](https://en.wikipedia.org/wiki/Sunk_cost). Although this might seem completely counterintuitive, the investment you have made should be **irrelevant to your decision** to stay or quit, if this investment has already been made and cannot be recovered.

What matters is whether the benefits you are yet to incur outweigh the costs (that you are yet to incur). So the decision should be a **forward-looking** one.

The above reference mentions that in the real-world, people often do not behave in this way. According to sunk cost theory, this is irrational, but this is only one theory.

> **Comments:**
> **username_1**: Some argue taking into consideration sunk costs in decision-making might be reasonable after all. [This paper](http://www.mit.edu/~rdoody/TheSunkCostFallacy.pdf) states that "_It’s reasonable to want to maintain plausible deniability about having suffered diachronic misfortune. Sometimes, honoring sunk costs is the only way to do this. It’s reasonable to want to maintain plausible deniability because having this desire is instrumental in successful cooperation, and successful cooperation is essential to our success as social creatures_"  
> **username_7**: "Sunk cost" has always sounded a lot like "lottery logic" to me. A lotto ticket is so cheap, and the reward is so great, even if the odds are worse than astronomical, what's a few bucks? Sure I've spent literally thousands on lotto tickets this year alone and never won, but the cost for that next ticket is negligible compared to the reward. And it doesn't make sense to factor the odds of winning into the potential costs, as I'm playing on the odds that I'll win with just one more ticket.  
> **username_8**: @talrnu The problem with your reasoning isn't that you failed to consider sunk costs. You have 2 problems. First is gambler's fallacy: "I'll win eventually if I keep playing." Second, you failed to consider your incredibly low odds of winning. Try weighing the cost of the average number of tickets required to win 80% of the time vs. the payout. I haven't done the math, but I bet it's astronomical. Occasional lottery play is okay if you consider it money spent on fun that you will not recover; assuming it will eventually be a source of income is just bad reasoning.  
> **username_8**: @talrnu For example, taking [the Mega Millions Jackpot](https://www.khanacademy.org/math/precalculus/prob-comb/prob-combinatorics-precalc/v/mega-millions-jackpot-probability), you only have a [43..4%](https://www.google.com/search?q=1-%281-1%2F175711536%29^100000000) chance of winning after buying 100 million tickets at 1 dollar each. The payout is currently only sitting at 76 million dollars.  
> **username_7**: @jpm26 Yes, many lottery players are delusional/irrational and succumb to the gambler's fallacy. But the players I refer to are those who decide to buy a single ticket while they're already at the store, not because "I should keep playing the lotto until I win", but because "Maybe I'll win this time." Such a decision accounts only for the cost and potential reward of a single ticket. If the cost of one ticket is negligible, and you're not planning to play more (even though you probably will), then without the perspective of sunk costs you'd feel crazy not to play.  
> **username_9**: Lottery tickets are not as straight forward as simple statistics tell you. You have to consider the utility of a single dollar versus the utility of the lottery winnings multiplied by the odds. The utility of a million dollars may or may not be a million times the utility of a dollar. If your basic needs are being covered, the utility of a dollar by itself, is negligible. In other words, in many cases the marginal utility of +/- 1$ = 0. The utility of a million dollars, however...  

---

### Answer by username_1 on 2017-09-12. Score: 6

[This handbook](http://www.e-elgar.com/shop/handbook-of-choice-modelling) provides a broad overview of different modelling approaches to **choice theory**. Here, I give just one example: discrete choice modelling.

Section 2 of [this paper](http://ftp.iza.org/dp9957.pdf) offers an example of a dynamic discrete choice model about educational choices. The decision tree is:

[![enter image description here](https://i.sstatic.net/QZF3f.png)](https://i.sstatic.net/QZF3f.png)

Individuals compare the return of each option, based on their ability and other observable variables.

The discrete choice literature is massive, and includes anything from modelling labour participation (e.g. [here](http://www.sole-jole.org/14404.pdf) or [here](http://www.sciencedirect.com/science/article/pii/0304407694016941)) to marriage/divorce (e.g. [here](https://www.jstor.org/stable/3663512)) to corruption (e.g. [here](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2214810)). See the handbook above for more references.

> **Comments:**
> **username_10**: GED permits attending college.  
> **username_1**: @Joshua You should tell that to the authors, not me :)  

---

### Answer by username_11 on 2017-09-12. Score: 2

I'm reminded of the law of diminishing returns.
<https://en.wikipedia.org/wiki/Diminishing_returns>
The idea is the more you put into production, the less you get out of it. I originally stumbled upon this theory while looking for a mathematical model to model the increased demand for addictive substances, but noted the harmonic series <https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)> also modeled it, where the second chocolate bar, cigarette, or dose of heroin is only half as good as the first, the third even less, the fourth, etc.

But more salient to what you are looking for, investment into graduate school, in terms of time and money, naturally comes with greater cost, compounded by that diminished return on investment lost by decreased life time earnings in terms of raw years. This could be applied to the "worth of skill" in terms of enjoyment of a highly skilled musical instrument player vs. a skilled musical instrument player, as the number of pieces of music and number of people you can impress first opens very wide, then the limitations become more precise, and the scarcity of musical challenges increases so that a finer and finer list of musical rivals and musical challenges exists. With respect to both a musician and a graduate student, or a marathon runner, in addition to losing the time invested, there's also competition.

Rising in a given devotion forces you into the category of competitive X, such as competitive gambler, competitive skier, or competing against other graduates for the same job. This manifests either as literal reduction of wages, being over qualified or having to compete with a dozen other Ph.D students or a hundred scientists seeking the same grant money. That may be counter intuitive at first, but because the demand for extreme specialization decreases with the increased level of specialization, the supply of rewards for that increased specialization also decreases. With gambling or cheating gamblers, the risk continues to increase and the skill of rivals increases to to financially dangerous levels, as notoriety results in an endless sequence of people actively trying to bankrupt or arrest the gambler. Bank robbery theory is the same way, as criminals often talk about "just one more job", assessing their risk vs. rewards and concluding (usually after arrest) that they should have quit while they were ahead.

Opportunity costs are extreme for some of these models too, as a skilled musician, for example, may impress at 3000-5000 hours of practice, but need 7000-10,000 hours of practice for the average person to appreciate any difference, and 20,000 hours of practice for a subtle improvement beyond that. With a marathon runner, the more you practice, the older you get, and the closer to "peak" you reach, and the more frequent the opportunities for injury, such as the recent Usain Bolt run. Boxing has a similar diminishing returns effect as injuries compound and performance decreases - that was a central theme of Rocky. Football Salaries are exceptionally high with a similar peak and injury calculated into the risk assessment.

Diminishing returns are most poignant though with the college graduate. You are assumed to have X number of years working, and every year in school is a year where you not only are not earning optimal wages, you are also building debt that will compound. Since work experience also increases wages with time, and since student loans have interest rates, there is a point in which a person's life time earnings actually start to drop lower. At one school I was attending, the tuition total was around $50,000/year, and to complete everything through graduate school was going to total half a million dollars and take a total of 10 years. As starting age was a constant and the growth of returns depended on the time from graduation to the time of retirement minus the growth of loans, the model for staying didn't pan out, and quitting was more fiscally sound.


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### Answer by username_12 on 2017-09-14. Score: 1

Seeing Ubiquitous's answer, I want to offer a concrete model I learned originally from an Exercise in Jianjun Miao's "Economic Dynamics in Discrete Time", which looked at **McCall's (1970) job search model.** It asked to describe the model as an optimal stopping problem.

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Consider an unemployed worker searching for a job, where each period he is unemployed, the worker draws one wage offer from a cumulative distribution function $F(w)$, where $F(0) = 0, F(B < \infty) = 1$. The worker may then reject the offer and receive compensation $C \geq 0$ or accept the wage $w$, which for now we will assume is received forever (no firing or quitting yet). You cannot accept past offers. The worker is risk neutral., so $\mathbb{E\_0} \sum^\infty\_{t=0} \beta^t y\_t$, where $y\_t = w$ if you accepted and $c$ if no offer has been accepted yet.

You can then express the question of when to accept what wage as a dynamic programming problem:

$$V(w) = \max\_{\text{accept, reject}} \{\frac{w}{1- \beta}, \quad C+ \beta \int^\beta\_0 V(\theta) dF(\theta)\}$$

denote the integral part as $Q$, the value function for an unemployed person receiving $w$ as an offer. So then there is some reservation wage where you accept if you get offered the wage or above.

That is, $\exists \ \bar{w}$ such that you accept when $w > \bar{w}$. Since you accept when:

$$\frac{w}{1 - \beta} \geq C + \beta Q$$

We have $\bar{w} = (C + \beta Q)(1 - \beta))$

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Now when you take this original model, what happens if we allow for quitting? No worker will want to quit since the solution is time consistent. There is no reason to quit the job if you found it optimal to accept a job in a previous period. So the model as is can't account for quitting.

Some ways you can tweak the model though to change the optimal reservation wage over time is to change $C$ to be time varying. Maybe if you have more security over other periods, you might quit to look for a better job that will take you. You could also change $F(w)$ to be time varying. Maybe over time the wages available to someone of your skills rise. In Joanovic's matching model, "tenure" makes your wages increase over time, so that could affect the decision to quit. Maybe your agent isn't risk neutral and their risk preferences change over time.

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McCall's model is a nice place to start if you want to tweak with different variables to try and model different reasons for why workers quit, and you can imagine many different extensions have built on it in the literature.

> **Comments:**
> **username_0**: Thanks! This is indeed a good starting point. I assume that changing $c$ (instead of $C$ I presume) is equivalent to changing unemployment compensation here or an (exogenous) outside option in a more general problem.  
> **username_12**: Oh, I did not mean to make $c$ lowercase or whatever. If I wanted to change $C$ in the problem, the interesting way to do it would to make it an exogenous state variable as you have said, I think.  

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