Which Companies Will Replace Us With AI Robots?
My Thesis Paper Is Sort Of Relevant For AI, So I’m Writing About It
There’s a lot of chatter about how many jobs AI will destroy. As always, behind these claims is a certain model of how the world works. Maybe a model that isn’t fully thought through, but it’s there. Maybe the model is just “I sell AI products and saying that they will replace human beings with decades of knowledge accumulation and practice is a banger way to move product.” Or maybe it’s something more.
When I was young and beautiful, I wrote a paper about identifying productivity distributions. It’s a mix of economic theory and econometrics. The key idea was understanding that, for many firms, the relationship between productivity improvements and the demand for “factors” (inputs to the production process, e.g., people). In fact, the relationship is fairly robust for a certain class of firms.
Here, I’ll focus on a special case of the more general set of models I discussed in the paper, but it should be enough to get the gist. The model makes predictions about what will happen if AI were to increase productivity radically, and I like to think it does so a little more carefully than the Discourse ™.
Technology Model
Technology is the magic we use to turn inputs (people, machines, land, electricity, …) into outputs (goods, services, Hot Tub Time Machine 2, …). Suppose our technology comes with an equation: Q = F(L)A, where Q represents output, L represents labor (in the form of people), and A represents productivity. We’re a firm, making a product, and so we face a demand curve, P(Q). We can sell more of our product if we charge less for it. We choose output like this:
C is our variable cost function, i.e.,
In this simplified version of the model, we’re going to show a certain relationship holds between the optimal choice of L and A, which is the question we’re asking: “Will companies demand more or less human beings if those human beings are more productive?”
Define Y = Q / A, and then because C(Q, A) only depends on Y = Q/A, we can write the profit maximization problem as choosing Y (abusing notation):
Comparative static result
Topkis’s theorem says that if the cross-partial derivative of a parameter and a choice in the objective function is positive, then the optimal choice is increasing in the parameter. In this context, the choice is Y and the parameter is A.
The derivative with respect to A is:
The derivative of that with respect to Y is:
Or:
At first, this looks inscrutable. But we can rearrange it, like I did at Michelangelo’s on State Street, Madison, Wisconsin, circa 2014 (my favorite Madison coffee place for working — highly recommended if you find yourself there).
First, remember that marginal revenue is:
And that the derivative of marginal revenue is:
So we can write the above expression as:
So, first point: suppose MR’(Q)Q + MR(Q) ≥ 0 for all Q, then we want to choose a higher Y whenever productivity is greater.
Labor demand
Okay, now suppose MR’(Q)Q + MR(Q) ≥ 0 for all Q, so that we will choose greater Y if A is larger.
When A is larger, our firm will also employ more labor because labor demand is increasing in Y. In this simplified version, this doesn’t require anything more than the production function being increasing. See the paper (of course) for more general cases.
So, if MR’(Q)Q + MR(Q) ≥ 0 for all Q, then labor demand increases as productivity increases. So, for example, AI-driven productivity improvements would lead to increased labor demand.
But, what exactly does the condition, MR’(Q)Q + MR(Q) ≥ 0, mean?
Cost-limited firms
We’ll call a firm “cost-limited” if there is no interior solution to the revenue maximization problem:
If revenue always increases with output, then the firm’s size isn’t limited by demand. This will always be the case for price-taking firms where P(Q) = P and more generally for any firm where, were output free to produce, they would do so.
Claim: If MR’(Q) Q + MR(Q) ≥ 0, then the firm is cost-limited.
Proof: Suppose otherwise, that is, that there is an interior solution to the revenue maximization problem. Then, the first order condition gives us:
And the second order condition (remember those!) gives us:
So: MR’(Q) Q + MR(Q) = MR’(Q) Q < 0, which contradicts the premise.
Therefore, there is no interior solution to the revenue maximization problem, and the firm is cost-limited if MR’(Q) Q + MR(Q) ≥ 0.
Conclusion
AI-led productivity improvements will cause cost-limited firms to increase labor demand, but not necessarily when firm size is demand-limited.
To understand why this difference matters, a demand-limited firm may produce at the revenue-maximizing level of output. Then, a shock that reduces costs will not cause the firm to want to increase production because it is already maximizing revenue. It will maintain its size and enjoy the extra profits from increased productivity by cutting the amount of inputs it purchases.
Of course, all real-world firms are, in a reductive sense, demand-limited in the sense that they can’t become literally infinite in size, but it’s about which model is more applicable.
If firm sizes are primarily limited by the cost of productive inputs, then productivity increases should generally lead to increased demand for inputs.
If firm sizes are primarily limited by the demand for their services, it might go the other way…
Thanks for reading!
Zach
Connect at: https://linkedin.com/in/zlflynn
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