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 :