A job offer just arrived. It's decent — but is it good enough? What if something better shows up next week? And what if you wait too long and the bills pile up?
This is the job search problem. It's been studied by economists for 50 years. You're about to feel exactly why it's hard — and then learn the math that solves it.
Each week, one wage offer appears. Accept and you earn that wage for the rest of your career. Reject and collect $10 in unemployment benefits while you wait for the next one.
Offers follow a realistic salary distribution — about % will fall somewhere between $ and $. A few will be much lower. A few will be much higher. You won't know which until it arrives.
Your goal: maximize your lifetime income. The catch — you must decide on each offer before seeing the next one.
Now that you understand the mechanics, practice it for real. Twenty offers. One chance at each.
The last game treated every dollar equally — a $50 offer twice as good as a $25 one. But that's not how people actually experience money. The difference between $0 and $25 feels nothing like the difference between $975 and $1,000.
Before the math can work properly, we need a better model of what a dollar is actually worth.
This next game will show you what we mean. It's a clicker — your clicks per second represent your wage. Click faster and you can afford more, but you'll notice the value of each new luxury tapers off. At some point it just takes too much effort to sustain, and the extra income buys you almost nothing new.
Economists capture this effect by converting dollars into an abstract concept called utility — a measure of satisfaction rather than raw income. And because each additional dollar buys less and less satisfaction, they reach for the logarithm: u(w) = ln(w). The log function grows quickly at first and slowly thereafter, embodying exactly the diminishing returns you just felt. The jump from $1 to $10 is enormous. The jump from $1,000 to $1,010 is nearly invisible. This is why a worker at $20/hr will quit for $25 but a worker at $90/hr won't bother.
Patience isn't a personality trait here — it's a financial position. Your discount factor β answers one question: how much trouble are you in right now, and how confident are you that waiting won't make it worse?
Someone with a repo notice on their car and an empty account can't wait. Someone with three months of runway and no debt can afford to be very picky.
Higher β doesn't make you more virtuous or strategic. It means the world is giving you the luxury of time. The math treats both situations equally — it just finds the best rule given your actual position.
We've assumed: accept a job, earn that wage forever. But jobs end. Companies downsize. Contracts expire. You might be back here sooner than you think.
We model this with a separation rate α: each period there's an α chance you get laid off and return to searching. This changes what a job offer is actually worth.
Counterintuitively, higher job insecurity lowers your reservation wage. If you'll likely be searching again soon regardless, holding out for perfect costs more than it saves. The model captures this exactly.
We have all the pieces. The Bellman equation is the mathematical decision rule that ties it all together. Click through to build it one piece at a time.
Adjust the three parameters. The model solves instantly.