What if people value their work mattering?

[Tim H]

I, for one, do not look forward to sharing the apparent fate of mathematicians.

Contrary to my flippant response this morning on X, though, it is actually non-trivial for an economist (like me, also) to conceptualize people intrinsically valuing whether their work has impact.

Enjoyment of an activity is easy to put into utility.^[1] But there is no standard way to formalize the notion that enjoyment of doing math research is dimmed by knowing “that AI will have usually gotten there first.”

So I wrote down a toy model in which people intrinsically value their work mattering. Could it be that transformative AI even makes people worse off on net (despite everything going well safety-wise, etc.)? Spoiler: Yes, if the amount one’s work matters is a complement of consumption in utility, then even unimaginable riches may be unable to compensate for the loss of impactful work.

The main idea is to assume the marginal product of one’s labor directly enters utility. That’s how I propose to formalize the notion that people value how much their work matters (as opposed to just enjoying the activity “for its own sake”). It’s an imperfect proxy, to be sure,^[2] but it provides a way to operationalize the relationship of “mattering” to the standard quantities in a macroeconomic model.

Being the first to resolve a major math question corresponds to productive labor, contributing to the production of new math in a way that privately rediscovering something AI has already revealed does not.

---

Suppose capital may be used for either fully-automated production, , or human-complementing production, : . That is, suppose total production is

where is the productivity of full automation.^[3] The marginal product of labor, our measure of how much human work “matters,” then depends on the human-complementing capital-labor ratio:^[4] .

Dividing both sides of the aggregate production function by yields per-capita output in terms of the respective capital-labor ratios: . And optimal capital allocation by the market implies:

That is, for a given total capital level, an increase of above the threshold level diminishes because it decreases , the capital allocated to complementing human labor.^[5] Above that level, an increase in increases total production but decreases .^[6]

For simplicity, utility is a function of only and per-capita consumption, .^[7] Nothing need be assumed about the functional form of utility to make the first observation about the comparative statics of different levels. As first rises above the threshold (at which some capital begins being allocated to full automation), utility drops regardless of its specific form. That is because there is a first order effect on but only a second-order effect on .^[8]

What happens beyond that initial dip depends on the specific utility function. Let’s assume constant elasticity of substitution :

If and are substitutes (), utility will ultimately go to infinity as increases, after the initial dip. This is analogous to the typical scenario in which tech. progress harms workers initially but benefits everyone eventually.

If and are complements (), however, utility ultimately goes to zero as approaches infinity. And, to be clear, this is not about only a small minority (like mathematicians) being made miserable. This is a model in which everyone has an equal share in both the material benefits and the gut-wrenching loss of meaningful work.

Interestingly, though, when the relative weight of consumption in utility, , is high enough, there is an interior optimum at which the marginal benefits of increased consumption are exactly balanced out by the marginal harm to “mattering.” And utility at that optimum may be substantially higher than with no automation. Here is what the relationship looks like for one set of parameters:

That’s for a case in which consumption and “mattering” are complements (), but they are far from perfect complements (Leontief preferences, corresponding to ) in which greater consumption would not compensate at all for a loss of meaning.^[9] As goes to infinity, greater consumption partially compensates for the loss of human work mattering, but not enough to prevent overall utility from dropping indefinitely below the no-automation level (). And this is despite full redistribution of all output.

(I have also worked out some results for endogenous capital, with capital accumulation over time, but I will save those for another post, if there’s interest. More could also be said about grounding the parameters in empirical work, implications for policy debates, and so on, but the intention of this initial post is just to establish a theoretical possibility.)

Happily, AI is complementing my labor, at this point. Claude Fable 5 helped with working out algebra and creating the chart, as well as thinking through the interpretation.

1. ^

   For example, Anton Korinek and Megan Juelfs think through various non-pecuniary aspects of long-term loss of employment in their 2022 “Preparing for the (non-existent?) future of work.”

2. ^

   A full philosophical treatment of “mattering” is far beyond the scope here, but suffice it to say that I acknowledge forms of mattering outside of market work and even “production” more generally.

3. ^

   A often denotes human-involved TFP, but here that is set to one here, WLOG.

4. ^

   That is, take the derivative with respect to L.

5. ^

   That is, is an externality not factored into capital allocation. And the idea is that wage subsidies cannot fix the issue because such “earnings” do not reflect the marginal product assumed to quantify meaning here.

6. ^

   Specifically, and .

7. ^

   Assume workers receive not only wage earnings but an equal share of all production. Labor is assumed perfectly inelastic (perhaps due to a future four-hour workweek law).

8. ^

   This is due to the “envelope theorem.” It only depends on utility being smooth and strictly increasing in .

9. ^

   The closed form utility as a function of , for :

[localdeity]

Try flipping the status quo. If AIs were already solving all the problems in your particular field, would you advocate for banning them from the field so that people would need you to solve those problems for them (presumably more expensively, presumably making some problems’ solutions too expensive to be solved at all) and so that you would feel like your work mattered? How does the idea of such advocacy strike you?

[Tim H]

In that scenario, there wouldn’t be “folks who have spent decades at work on certain problems, in part based on a reasonable belief that they have a comparative advantage in their area” (as I put it in a different reply). But, right, such advocacy would strike me as difficult to justify.

If it were just one field’s problems being solved, I wouldn’t even consider such advocacy. My concern is about there being no interesting problems left for humans to solve whatsoever.

The scenario I worry about: my grandchildren are born into a world in which there are no interesting problems to solve at all, one in which there is effectively a genie who can both answer any question and make anything physically possible to make. If they are still born with a need to do meaningful work, that scenario seems cruel, and advocacy to change that status quo strikes me as reasonable—all the more so, advocacy to prevent that scenario if bringing it about has costs and/​or risks. (Some may propose to instead modify my grandchildren somehow to remove their desire to contribute meaningfully with their work. My instinct is to oppose that but I haven’t given it much thought.)

[skinks_basking]

What is it that AI takes from mathematicians, when it does all the math? The paid position of mathematician disappears.

What is stopping them from doing it is the absence of opportunity to dedicate one’s life to its study.

[Tim H]

We could continue paying mathematicians if we want. But what we (likely eventually) can’t do is restore their ability to make new discoveries with their minds, to receive credit for that and know their efforts contributed to knowledge. There appear to be folks who have spent decades at work on certain problems, in part based on a reasonable belief that they have a comparative advantage in their area. And within a short time, the situation is radically changed and they are transformed from an aspiring Mozart to an aspiring Salieri, only able to appreciate and perhaps help explain to others the accomplishments of AI.

[Canaletto]

perhaps help explain to others the accomplishments of AI

But at this too they would be outmatched shortly. What remains?

Integrity, trustworthiness, knowledge of their intent, commitment to some ideals.

AIs having capabilities and using them to achieve goals you know and agree with are not the same thing. Yes, this AI can persuade me that I understand this math fact, but did it do it in a valid way?

Alignment is key to this.

[_rpd]

The paid position of mathematician disappears.

at the moment, at doesn’t do even that. mathematicians are still the meaning makers with respect to mathematics. the LLM can craft a path from A to B, but cannot say “this is not babble. this is meaningful, both now, and in the future of this language game we call mathematics which has such mysterious use in describing the world in which we live.”

[skinks_basking]

So you are moved up a level of abstraction. How much time does that give you? :(((

[_rpd]

math autoresearch does appear to be of its own special kind. still requiring “research taste” (the forecasting of profitable areas of proof space exploration) without “experiments” beyond actual proof construction and lean formalization. AI experiment design needs to be much more sophisticated to extract actual knowledge. makes one sympathetic to the world model advocates.