Price of Intelligence:
How Should Socially-minded Firms Price
and Deploy AI?
Nils H. Lehr, Pascual Restrepo
WP/25/234
IMF Working Papers describe research in
progress by the author(s) and are published to
elicit comments and to encourage debate.
The views expressed in IMF Working Papers are
those of the author(s) and do not necessarily
represent the views of the IMF, its Executive Board,
or IMF management.
2025
NOV
INTERNATIONAL MONETARY FUND 2
© 2025 International Monetary Fund WP/25/234
IMF Working Paper
Western Hemisphere Department
Price of Intelligence: How Should Socially-minded Firms Price and Deploy AI?
Prepared by Nils H. Lehr, Pascual Restrepo
Authorized for distribution by Esteban Vesperoni
November 2025
IMF Working Papers describe research in progress by the author(s) and are published to elicit
comments and to encourage debate. The views expressed in IMF Working Papers are those of the
author(s) and do not necessarily represent the views of the IMF, its Executive Board, or IMF management.
ABSTRACT: Leading AI firms claim to prioritize social welfare. How should firms with a social mandate price
and deploy AI? We derive pricing formulas that depart from profit maximization by incorporating incentives to
improve welfare and reduce labor disruptions. Using US data, we evaluate several scenarios. A welfarist firm
that values both profit and welfare should price closer to marginal cost, as efficiency gains outweigh
distributional concerns. A conservative firm focused on labor-market stability should price above the profit-
maximizing level in the short run, especially when its AI may displace low-income workers. Overall, socially
minded firms face a trade-off between expanding access to AI and the resulting loss in profits and labor
market risks.
RECOMMENDED CITATION: Lehr, Nils H., Pascual Restrepo. 2025. “Price of Intelligence: How Should
Socially-minded Firms Price and Deploy AI?”, IMF Working Paper No. 25/234
JEL Classification Numbers: O30, J24, J30
Keywords: Artificial intelligence; automation; corporate social responsibility
Author’s E-Mail Address: nlehr@, @
The Price of Intelligence: How Should Socially-minded
Firms Price and Deploy AI?∗
Nils H. Lehr
International Monetary Fund
Pascual Restrepo†
Yale University
October 24, 2025
Abstract
Leading AI firms claim to prioritize social welfare. How should firms with a social man-
date price and deploy AI? We derive pricing formulas that depart from profit maximization
by incorporating incentives to improve welfare and reduce labor disruptions. Using US data,
we evaluate several scenarios. A welfarist firm that values both profit and welfare should price
closer to marginal cost, as efficiency gains outweigh distributional concerns. A conservative
firm focused on labor-market stability should price above the profit-maximizing level in the
short run, especially when its AI may displace low-income workers. Overall, socially minded
firms face a trade-off between expanding access to AI and the resulting loss in profits and labor
market risks.
JEL-Codes: O30, J24, J30
Keywords: Artificial intelligence, automation, corporate social responsibility
∗The views expressed in this paper are our own and do not necessarily reflect those of the IMF, its Executive Board,
or its Management. Restrepo is part of the Anthropic Economic Advisory Council.
†Corresponding author.
E-mail addresses: @ (P. Restrepo), nlehr@ (N. H. Lehr).
1 Introduction
Artificial Intelligence (AI) promises to transform the economy, raising new questions about how
firms should price, deploy, and manage this technology. Leading AI firms present themselves as
socially responsible entities. They claim a dual mandate: to generate profits for shareholders while
enhancing social welfare and mitigating risks. OpenAI adopted a capped-profit model. Investors
earn returns up to a fixed multiple, after which the organization prioritizes its mission to “benefit all
of humanity” by “building safe and beneficial AGI and helping create broadly distributed benefits.”1
Anthropic declares to “make decisions that maximize positive outcomes for humanity in the long
run”.2 Both companies claim to have been conservative in deploying more advanced models and
capabilities, aiming to manage societal risks—such as economic displacement—while giving the
labor market time to adjust.
How should firms with a social mandate price and deploy AI? Is a commitment to maximizing
shareholder returns the best way to promote welfare? Should they expand access by pricing below
profit-maximizing levels? Or should they deploy AI slowly to mitigate labor market risks?
This paper addresses these questions by providing optimal-price formulas for socially minded
AI firms. The formulas extend Lerner’s Rule, which says that profit-maximizing firms should set
𝑃 − 𝑀𝐶
𝑃
=
1
𝜀
,
with 𝜀 the demand elasticity. Optimal pricing is given by a Modified Lerner Rule
𝑃 − 𝑀𝐶
𝑃
=
M
𝜀
,
where M summarizes the motives of a socially minded firm. We derive the formulas in a gen-
eral equilibrium environment where a tech firm has a monopoly over an AI capable of replicating
human skills. The deployment of this AI reduces production costs but disrupts labor markets for
workers with skills that are substitutable. The AI firm prioritizes profits, broader social welfare,
1See
2
1
and minimizing labor market disruptions.
The optimal deployment strategy balances four distinct considerations: (1) Profit motives push
towards M = 1, as in the traditional Lerner Rule; (2) Aggregate efficiency considerations push
towards M = 0, or marginal-cost pricing. This achieves the level of AI production and access
that maximizes the size of the pie; (3) These aggregate benefits are weighed against distributional
considerations, which capture who benefits the most from AI. These can be positive or negative,
depending on whether AI substitutes for high- or low-income workers; and (4) the incentive to
minimize labor market disruptions pushes for higher values of M that can exceed one in the short
run but not in the long run. This motive calls for a gradual deployment path, with the firm acting
conservatively. This is because the cost of disrupting the labor market is higher in the short run,
while workers adjust. The resulting formula highlight a tension between expanding access to AI (to
maximize aggregate efficiency) and the resulting short-term loss in profits and labor market risks.
We then present an exploration of the formulas, using US data. We compute the optimal de-
ployment path and prices of an AI capable of replacing human labor in each of 525 detailed jobs.
For each job, we imagine our tech firm develops an AI capable of replacing labor at 50% of the cost
and ask how a socially minded firm should price and deploy such AI.
We report optimal plans for firms that value welfare and minimizing disruptions to varying
degrees. Awelfarist firm that values profits and welfare should price closer to marginal cost. This is
because for all jobs considered, efficiency gains outweigh distributional concerns by a wide margin,
since losses do not concentrate among low-income workers. On the other hand, a conservative
firm focused solely on balancing profits with labor-market stability should price above the profit-
maximizing level in the short run. A firm that values welfare and stability equally should price
close to the profit-maximizing level in the short run and closer to marginal cost in the long run.
We conclude that the most pro-social course of action for AI firms with considerable market
power is to refrain from exploiting it. A socially minded AI firm should price closer to marginal
cost in an effort to broaden access, with the only possible exception being the very short run, when
stability concerns are most significant. This conclusion contradicts recommendations to tax AI
2
and automation technologies to mitigate their adverse effects on the labor market. What these
recommendations miss is that AI firms can have considerable market power. If we worry that an AI
can have sizable impacts on prices and wages and is controlled by a small number of firms, we must
accept the possibility that these firmswield considerable market power andwould limit output to bid
up prices. This exercise of market power already protects workers from the substituting effects of
AI at the expense of consumers. Further increasing the price of AI through taxes or self-regulation
would have an adverse first-order impact on consumers, whose access to AI is already limited, with
only modest protective benefits for workers.
We conclude the paper with extensions that explore the robustness of this conclusion. For
example, we demonstrate that the incentive for socially minded AI firms to price closer to marginal
cost becomes stronger when a progressive tax system is in place, providing some redistribution and
insurance for workers, or when its AI does not substitute for workers but instead creates value by
introducing new products. Conversely, we demonstrate that distributional and stability concerns
become more significant when there is increased competition among AI suppliers.
Literature This paper contributes to the long-standing debate on the social responsibilities of
firms. Following Friedman (1970), the traditional view holds that a firm’s sole obligation is to
maximize shareholder value. Leading AI companies explicitly reject this view by adopting mission
statements that emphasize societal welfare, long-term human outcomes, and labor stability. This
paper explores how such objectives should alter their pricing strategies.
This paper also contributes to the growing literature on optimal policy responses to AI and
automation. A strand examines the optimal taxation of automation technologies, motivated either
by distributional concerns (Guerreiro, Rebelo and Teles, 2021; Donald, 2022; Thuemmel, 2023;
Costinot and Werning, 2022; Lehr and Restrepo, 2024; Bond and Kremens, 2025) or efficiency
considerations (Acemoglu, Manera and Restrepo, 2020; Beraja and Zorzi, 2024). Our work relates
to this literature in that socially responsible AI firms partially internalize distributional concerns
by curbing the scale of AI deployment—much like how a tax on automation can reduce its use
3
and mitigate inequality. However, a key distinction is that, in the models studied in the literature,
it is always optimal to tax technologies that worsen inequality (assuming the set of fiscal tools is
limited). This result relies on the assumption of an efficient baseline economy, where the cost of
a small tax is second-order, while the distributional gains are first-order. In contrast, our setting
begins with an inefficient allocation due to market power, resulting in insufficient AI production.
In this context, expanding the use of AI yields first-order efficiency gains, which must be balanced
against concerns regarding distributional and labor market stability.
A third related literature study the optimal deployment of AI accounting for existential risks
(Jones, 2024, 2025) and social risks that can be learned over time or via testing (Acemoglu and
Lensman, 2024; Guerreiro, Rebelo and Teles, 2023). Our work abstracts from these risks and
focuses exclusively on the question of how firms should deploy well-aligned or narrow AIs that
carry no existential risks.
Finally, our paper contributes to a growing empirical literature exploring how AI could disrupt
labor markets by measuring the capabilities of AI (Webb, 2020; Brynjolfsson, Mitchell and Rock,
2018; Felten, Raj and Seamans, 2021, 2023; Eloundou et al., 2023; Handa et al., 2025) and studying
the deployment of AI and Large Language Models in specific contexts (Peng et al., 2023; Brynjolf-
sson, Li and Raymond, 2025; Noy and Zhang, 2023). These papers show that AI can substitute for
human labor in various domains at a fraction of the cost and with minimal input from expert human
workers. We use some of the estimates from these papers in our numerical exploration.
2 Model of labor-replacing AI
This section outlines a general model of how AI affects wages, prices, and households’ welfare. We
focus on AI technology capable of replicating human skills or inputs in some areas of the economy.
Examples include the use of AI systems to automate tasks such as radiology, copywriting, journal-
ism, customer service, and driving. These are domains where AI systems can be trained to replicate
human input. We assume that the technology is sufficiently advanced to operate autonomously and
4
without requiring input fromworkers. In our discussion section, we extend our theory to account for
the possibility that AI is used for novel applications beyond replicating human input. Derivations
and proofs are presented in Appendix B.
The Economy
The economy flows in continuous time 𝑡. There is a discrete set of commodities 𝑗 ∈ J and skills
or labor inputs 𝑠 ∈ S. Commodity 𝑗 = 0 serves as the numeraire.
The economy is populated by a mass 𝜖 of financiers and a mass 1 of regular households (iden-
tified with the superscript ℎ). Financiers own firms and no labor endowments. They consume the
numeraire good and make consumption and saving decisions to maximize
𝑢 ≡
∫ ∞
0
𝑒−𝜌𝑡 𝑐0𝑡 𝑑𝑡 st: ¤𝑎𝑡 = 𝑟𝑡𝑎𝑡 + 𝜋𝑡 − 𝑐0𝑡 .
Regular household ℎ is endowed with a vector of skills or labor inputs 𝑛ℎ𝑡 = (𝑛ℎ𝑠𝑡)𝑠∈S that can
change over time. They consume commodity bundles 𝑐ℎ𝑡 = (𝑐ℎ𝑗𝑡) 𝑗∈J and maximize
𝑢ℎ ≡
∫ ∞
0
𝑒−𝜌𝑡 𝑢(𝑐ℎ𝑡 ) 𝑑𝑡 st: ¤𝑎
ℎ
𝑡 = 𝑟𝑡𝑎
ℎ
𝑡 + 𝑤𝑡 · 𝑛
ℎ
𝑡 − 𝑝𝑡 · 𝑐
ℎ
𝑡 and 𝑎
ℎ
𝑡 ∈ R .
Here 𝑝𝑡 = (𝑝 𝑗 𝑡) 𝑗∈J is the price of commodities at time 𝑡 (with 𝑝0𝑡 = 1) and 𝑤𝑡 = (𝑤𝑠𝑡)𝑠∈G are
wages, with household wages given by𝑤ℎ𝑡 = 𝑤𝑡 ·𝑛ℎ𝑡 . The term 𝑎ℎ𝑡 ∈ R captures potential constraints,
assumed independent of prices.
AI can replicate labor input in a subset A of S. The quantity of 𝑠 input is
ℓ𝑠𝑡 =
∫
ℎ
𝑛ℎ𝑠𝑡 𝑑ℎ + 𝑞𝑠𝑡 for 𝑠 ∈ A∫
ℎ
𝑛ℎ𝑠𝑡 𝑑ℎ otherwise.
𝑞𝑠𝑡 represents units of AI-generated output, assumed to be indistinguishable from that of workers.
To produce AI-generated output, the AI firm uses 1/𝜓𝑠𝑡 units of computing resources, where
𝜓𝑠𝑡 denotes the efficiency of algorithms reproducing input 𝑠. Computing resources, denoted as 𝑥𝑡 ,
5
are produced at a one-to-one rate from the numeraire commodity. Feasibility requires
∑
𝑠∈A
𝜓𝑠𝑡 𝑞𝑠𝑡 ≤ 𝑥𝑡 ,
so that the consumption of computational resources by AI does not exceed supply.
Commodities 𝑦 are produced using labor (or AI) ℓ. Plans 𝑦 = (𝑦 𝑗 𝑡) 𝑗∈J ,𝑡 and ℓ = (ℓ𝑠𝑡)𝑠∈S,𝑡 with
𝐹 (𝑦, ℓ) ≤ 0
can be produced. 𝐹 has constant returns to scale and is operated competitively. Feasibility requires
𝑐𝜔 +
∫
ℎ
𝑐ℎ0𝑡 𝑑ℎ + 𝑥𝑡 ≤ 𝑦0𝑡 and
∫
ℎ
𝑐ℎ𝑗𝑡 𝑑ℎ ≤ 𝑦 𝑗 𝑡 otherwise
so that consumption of commodities does not exceed production.
Equilibrium: we are interested in an equilibrium where the AI company sets a feasible choice of
𝑞𝑠𝑡 and 𝑥𝑡 anticipating the effects of its actions on prices, profits, and the economy.
Given the choices of 𝑞𝑠𝑡 and 𝑥𝑡 , the equilibrium is defined in a standard way. It is given by a set of
prices {𝑟𝑡 , 𝑝𝑡 , 𝑤𝑡}, consumption plans {𝑐ℎ𝑡 , 𝑐0𝑡}, asset positions {𝑎ℎ𝑡 , 𝑎𝑡}, and production plans 𝑦, ℓ
such that consumers maximize utility subject to their flow-budget constraint and asset restrictions,
competitive firms maximize profits from operating 𝐹 taking prices as given, commodity markets
clear, and the asset market clears. Equilibrium profits for the AI-producing firm at time 𝑡 are
𝜋𝑡 =
∑
𝑠∈A
(𝑤𝑠𝑡 − 𝜓𝑠𝑡) 𝑞𝑠𝑡
To derive our formulas, we do not need to solve for the equilibrium explicitly. It suffices that
financiers set the interest rate 𝑟𝑡 = ℎ𝑜 and determine the discount factor used by firms.
The objective of socially-oriented AI firms: The AI firm operates under three objectives: profit
maximization, social welfare, and minimizing labor market disruptions. Its objective function is
𝑉 = PDV 𝜋𝑡 +
∫
ℎ
𝜇ℎ 𝑢ℎ 𝑑ℎ + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
PDV
𝑤ℎ𝑡
�̄�ℎ
𝑑ℎ.
The first term captures profit maximization motives.
6
The second term captures welfare considerations in a reduced-formway. Here 𝜇ℎ is the value the
firm attaches to increasing the income of household ℎ. The 𝜇ℎ’s differ across households, reflecting
distributional considerations. As in standard welfare functions, the firm attaches greater weight to
poor households than richer ones. Investor welfare is already accounted for in profits, so we do not
include it again to avoid double-counting.
The third term captures the objective of minimizing labor-market disruptions created by AI,
with a weight of 𝜆. The AI firm penalizes labor-market losses incurred by exposed households,
computed as the percent decline in labor income of household ℎ relative to its initial status quo of
�̄�ℎ. These penalties represent various considerations. Firms may adopt the principle that reducing
people’s wages below their status quo level is undesirable, either because people are particularly
averse to wage losses or because the firm adopts a conservative stance when judging its labor-
market impact that regards these deviations as unfair (as in Corden, 1974). Penalties may also
capture strategic considerations, with the firm minimizing disruptions to reduce discontent. In our
formulation, the firm penalizes all wage losses, without accounting for indirect benefits via reduced
product prices. AI firmsmay attach greater weight to wage losses because people are more sensitive
or responsive to their labor-market outcomes, either because these are more salient (benefits from
reduced product prices are “out of sight; out of mind”) or because they derive status from their high
To summarize, AI firms want to avoid major shifts in the way labor markets operate, with
the status and wages of different jobs falling in ways that may be perceived as unfair or arbitrary by
workers.
To simplify the exposition, we derive formulas assuming a quasi-linear aggregator of the form
𝑢(𝑐ℎ𝑡 ) = 𝑐
ℎ
0𝑡 +
∑
𝑗
𝑢 𝑗 (𝑐ℎ𝑗𝑡).
We also assume the equilibrium is such that all households ℎ consume 𝑐ℎ0𝑡 > 0 at all times.
3In our formulation, the AI firm penalizes a reduction in wages in percent terms, so that a reduction in wages of
$10,000 receives a higher penalty if experienced by low-income workers.
7
To understand firm incentives, consider how a deviation in plans {𝛿𝑞𝑠𝑡} affects its objective:
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
(1)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡 + 𝜇
∫
ℎ
𝑔ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
𝜇 =
∫
ℎ
𝜇ℎ 𝑑ℎ is the average welfare weight across households and 𝑔ℎ = 𝜇ℎ/𝜇−1 are the normalized
weights. By construction,
∫
ℎ
𝑔ℎ 𝑑ℎ = 0 and the sign of 𝑔ℎ represents distributional motives.
The first term in the right of (1) represents profit motives. We assume 1 > 𝜇 so that the firm has
an incentive to maximize profits.
The second term represents efficiency motives. Because the firm cares about welfare, it has an
incentive to produce efficiently, increasing quantities until prices equal marginal cost, 𝑃 = 𝑀𝐶.
The third term represents distributional motives. These call for restricting the quantity of AI if
it competes against poor households. This motive is weighed against efficiency considerations.
The last term represents conservative motives. These receive a weight 𝜆 and capture the value of
minimizing the labor-market disruptions from AI. These are different from standard distributional
motives as the AI firm is concerned about disrupting the labor market of rich and poor households,
all of whom experience wage pressures due to the deployment of AI. We assume all households are
exposed to AI, in the sense that 𝑛ℎ𝑠𝑡 > 0 for at least some 𝑠 ∈ A.
The firm optimally balances these motives to ensure 𝛿𝑉 = 0. This implies:
Proposition 1. In interior equilibria of the quasi-linear case, the socially minded firm produces
𝑞𝑠𝑡 until
L𝑠𝑡 =
∑
𝑠′
(
(1 − 𝜇)
𝑞𝑠′𝑡 𝑤𝑠′𝑡
𝑞𝑠𝑡 𝑤𝑠𝑡
+ 𝜇
∫
ℎ
𝑔ℎ
𝑛ℎ
𝑠′𝑡 𝑤𝑠′𝑡
𝑞𝑠𝑡 𝑤𝑠𝑡
𝑑ℎ + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
𝑛ℎ
𝑠′𝑡 𝑤𝑠′𝑡
𝑞𝑠𝑡 𝑤𝑠𝑡
𝑑ℎ
) 1
𝜀𝑠′𝑠𝑡
(2)
where 𝜀𝑠𝑠′𝑡 ≡ − 𝜕 ln 𝑞𝑠𝑡𝜕𝑙𝑛𝑤𝑠′𝑡 is the cross demand elasticity between 𝑠 and 𝑠
′ and 𝑞𝑠′𝑡 = 0 for 𝑠′ ∉ A.
8
To develop intuition, assume the cross-demand elasticity is 0 for 𝑠 ≠ 𝑠′. The own demand
elasticity (𝜀𝑠𝑡 for 𝑠 = 𝑠′) is strictly positive and always remains in the formula. Let’s also consider
first a profit-maximizing firm, by setting 𝜇 = 𝜆 = 0. The formula says that a profit-maximizing AI
firm should restrict quantities until its Lerner index L ≡ (𝑃 − 𝑀𝐶)/𝑃 satisfies Lerner’s Rule
L𝑠𝑡 =
1
𝜀𝑠𝑡
, (3)
where 𝜀𝑠𝑡 ≥ 0 is the (negative) of the demand elasticity of 𝑠 at time 𝑡.
For a socially minded firm, optimal pricing satisfies a Modified Lerner Rule
L𝑠𝑡 =
M
𝜀𝑠𝑡
,
where
M ≡ 1 − 𝜇 + 𝜇
∫
ℎ
𝑔ℎ
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ.
The adjustment term differs from 1 and summarizes the different firm considerations:
• The “1” is the usual profit maximization term.
• The “−𝜇” pushes towards lower markups and higher quantities. This term reflects the firm’s
desire to increase access toAI, thereby raising aggregate efficiency at the expense of investors.
• The term “𝜇
∫
ℎ
𝑔ℎ
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ” has ambiguous sign. It is positive when AI competes more
intensely against poor households. In this case, AI deepens existing inequalities, causing a
socially minded firm to restrict its use by charging higher prices. The term can be negative
if AI competes more intensely against rich households. In this case, the use of AI reduces
underlying inequalities, causing socially minded firms to lower prices and increase quantities.
• The term “𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ” is always positive and reduces quantities of AI produced.
This captures the AI firm’s incentive to minimize labor market disruptions. This incentive to
curb the use of AI is stronger when it competes against poor segments of the labor market,
since a reduction in wages of a given amount is more costly in proportional terms for low-
wage households.
9
The formula serves to illustrate several scenarios. For a utilitarian AI firm that cares about
profits and aggregate efficiency but has no distributional or conservative inclinations (𝜇 > 0, 𝑔ℎ =
0, 𝜆 = 0), optimal prices satisfy
L𝑠𝑡 = (1 − 𝜇)
1
𝜀𝑠𝑡
.
These prices are below the profit-maximizing level and closer to marginal-cost pricing. For a wel-
farist AI firm that cares about profits, welfare, and distributional issues, but has no distributional
or conservative inclinations (𝜇 > 0, 𝑔ℎ ≠ 0, 𝜆 = 0), optimal prices satisfy
L𝑠𝑡 =
(
1 − 𝜇 + 𝜇
∫
ℎ
𝑔ℎ
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ
) 1
𝜀𝑠𝑡
.
For a conservative AI firm that cares about minimizing labor market disruptions but not about
welfare per se (𝜇 = 0, 𝜆 > 0), optimal prices satisfy
L𝑠𝑡 =
(
1 + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ
) 1
𝜀𝑠𝑡
,
which exceed the profit-maximizing level.
In the general case with cross effects (𝜀𝑠𝑠′𝑡 ≠ 0 for 𝑠 ≠ 𝑠′), the formula accounts for equilibrium
price effects on all workers and revenue from other AI products. For example, when 𝜇 = 𝜆 = 0,
we recover the standard multi-product Lerner formula, which takes into account how increasing the
quantity supplied of one good affects demand for other AI products sold by the firm.
A Tractable Example of Equilibrium with Socially-Minded Firms
In general, the equilibrium of the model is given by (i) a choice of quantities and prices by the AI
firm that satisfy the Modified Lerner’s rule, (ii) a vector of commodity prices, and (iii) production
and consumption plans that maximize households’ utility and firms’ profits (for firms producing
commodities 𝑦). The characterization of the equilibrium is generally complicated, as the residual
demand for AI depends on how skills are combined into goods, the demand for these goods by
households, and the supply of skills.
10
In this sub-section, we characterize the full equilibrium of the model in an example economy
with the following features:
(a) Each commodity is produced linearly using a commodity-specific skill, with the skill asso-
ciated with the numeraire commodity not in A.
(b) The utility function is 𝑢𝑠 (𝑐𝑠) = 𝛾1/𝜎𝑠𝑠
𝑐
1−1/𝜎𝑠
𝑠
1−1/𝜎𝑠 , with 𝜎𝑠 > 1, so that the demand for each
commodity has a constant elasticity 𝜎𝑠.
(c) Households reallocate labor away from disrupted skills at a rate 𝛼 > 0. This implies
𝑛ℎ𝑠𝑡 = �̄�
ℎ
𝑠 𝑒
−𝛼 𝑡 and 𝑛𝑠𝑡 = �̄�𝑠 𝑒−𝛼 𝑡 for 𝑠 ∈ A.
Here, {�̄�ℎ𝑠 } and �̄�𝑠 denote pre-AI quantities of labor input in skill 𝑠.
(d) AI is productive enough to justify deployment and ensure an interior equilibrium. This im-
plies
1 − 𝜇
∫
ℎ
𝑔ℎ
�̄�ℎ𝑠
�̄�𝑠
𝑑ℎ − 𝜆
∫
ℎ
1
�̄�ℎ
�̄�ℎ𝑠
�̄�𝑠
𝑑ℎ >
𝜓𝑠𝑡
�̄�𝑠
,
where 𝜓𝑠𝑡 is the marginal cost of the AI firm and �̄�𝑠 = 𝛾𝑠 �̄�
−1/𝜎𝑠
𝑠 the pre-AI price of skill 𝑠.
In this economy, the quantity and price of AI for each skill are determined independently.
Proposition 2. In an economy where (a)–(d) hold, equilibrium prices and quantities of skills inA,
are uniquely determined by two equations. The supply curve, obtained by rearranging (2):
1 −
𝜓𝑠𝑡
𝑤𝑠𝑡
=
(
1 − 𝜇 + 𝜇
∫
ℎ
𝑔ℎ
�̄�ℎ𝑠
𝑞𝑠𝑡
𝑒−𝛼 𝑡 𝑑ℎ + 𝜆
∫
ℎ
1
�̄�ℎ
�̄�ℎ𝑠
𝑞𝑠𝑡
𝑒−𝛼 𝑡 𝑑ℎ
) 𝑞𝑠𝑡
𝑞𝑠𝑡 + �̄�𝑠 𝑒−𝛼 𝑡
1
𝜎𝑠
(4)
and the demand curve, obtained from consumer demand:
𝑤𝑠𝑡 =𝛾𝑠 (𝑞𝑠𝑡 + �̄�𝑠 𝑒−𝛼 𝑡)−1/𝜎𝑠 (5)
The proposition provides formulas for computing the full deployment path of an AI that substi-
tutes for skill 𝑠. The supply and demand curve pin down quantities, prices, and markups charged in
equilibrium by socially minded AI firms. Figure 1 depicts the supply and demand curves, assuming
11
the distributional motive is positive. Condition (d) ensures the curves intersect at a unique 𝑞𝑠𝑡 > 0.
Figure 1: Equilibrium Supply and Demand for AI
Notes: The figure shows the demand and supply curves for an AI that substitutes for skill 𝑠. The supply curve and
equilibrium points are shown for a profit-maximizing firm, a utilitarian firm, awelfarist firm (assuming the distributional
motive is positive), and a conservative firm.
The supply curve for a profit-maximizing firm is upward sloping: as the quantity of AI produced
increases, the residual demand curve becomes more inelastic, leading to higher markups. The
utilitarian firm supply curve is shifted to the right, reflecting incentives to charge lower markups to
increase access and aggregate efficiency. The supply curves of welfarist and conservative firms are
shifted upward, reflecting incentives to restrict quantities and mitigate the harmful distributional or
labor-market impacts of AI.
The figure also shows that the distributional motives of a welfare-maximizing firm or the sta-
bility motives of a conservative firm vanish as quantities increase. This is why the supply curves
of a welfarist and utilitarian firm convergee and the supply curves of a conservative and a profit-
maximizing firm converge. This force can be so strong as to render the supply curve of these firms
12
downward sloping—a distinct possibility shown in the Figure. From equation (4), this occurs if
𝜇
∫
ℎ
𝑔ℎ
�̄�ℎ𝑠
�̄�𝑠
𝑑ℎ + 𝜆
∫
ℎ
1
�̄�ℎ
�̄�ℎ𝑠
�̄�𝑠
𝑑ℎ > 1 − 𝜇,
so that distributional and labor-market stability concerns are dominant.
To understand why distributional and labor-market stability concerns vanish, return to equation
(1), describing the effects of changes in quantities produced on the objective of the AI firm. The
firm balances three objectives: profits, aggregate efficiency, and distributional and stability con-
siderations. The equation indicates that profit and efficiency motives are directly proportional to
the quantity of AI used. Increasing the quantity of AI by 1% leads to a larger profit and efficiency
increase when the AI is widely used. However, distributional and stability concerns do not scale
with quantities. Increasing quantities produced by 1% reduces wages of exposed groups by at most
(1/𝜎𝑠) × 1%—an effect that remains bounded as the use of AI deepens. For this reason, socially
minded firms prioritize efficiency and profit motives as the use of AI becomes
The formulas in the proposition also highlight two new economic mechanisms introduced by
labor reallocation. First, the formulas show that distributional and labor-market stability motives
vanish over time as workers reallocate. This force calls for a gradual and backloaded deployment
plan, where AI firms first curb quantities and set higher prices to shield exposed workers from
disruptions and give them time to adjust. Over time, firms lower prices and expand quantities, as
workers slowly reallocate away from exposed skills or sectors of the labor market.
Second, the reallocation of labor away from exposed skills eases competition, making the resid-
ual demand faced by the AI firm more inelastic over time. This allows firms to set higher markups
in the long run, leading to a more front-loaded deployment plan.
The net effect of these forces over time on markups and pricing is ambiguous. For a pure
profit-maximizing firm, the second effect is the only one present, and we would expect markups to
increase over time as the AI firm becomes the sole supplier of skills in A. For a conservative firm,
4The same logic is explored in Costinot and Werning (2022). Their paper derives formulas for optimal taxes that
balance aggregate efficiency with distributional considerations. As here, the cost of distorting trade or the use of
automation technology scales with quantities, which calls for smaller taxes on trade and technology as globalization
deepens and the use of automation technology becomes widespread.
13
the second effect might dominate, leading to markups that decrease in time.
Extensions
Appendix A outlines three extensions that illustrate how the model’s main mechanisms adapt under
richer environments.
In our first extension, we show that introducing a progressive tax system attenuates the rele-
vance of firms’ distributional and stability concerns. With after-tax income entering both workers’
utility and the firm’s social objective, the optimal AI deployment condition now depends on the
marginal tax schedule. A lower “keep rate” (., steeper progressivity) reduces the effective weight
placed on distributional motives. In the limiting case of full redistribution—when marginal taxes
eliminate income dispersion—distributional and stability terms vanish, and firms behave as if they
were purely efficiency-oriented. This highlights an important interaction between public policy
and self-regulation: the stronger the fiscal safety net, the less need for firms to internalize inequal-
ity concerns themselves.
Our second extension considers the case when AI systems develop entirely novel products
rather than substituting for human labor, in which case they expand the consumption set without
displacing workers. The modified Lerner rule implies that socially minded firms optimally price
these AIs close to marginal cost, focusing primarily on broad access rather than redistribution.
Because such innovation benefits all households and leaves existing wages unaffected, distributional
and stability terms drop out—the social motive operates purely through affordability.
Finally, we consider an extension where multiple AI firms to compete à la Cournot drives
equilibrium prices closer to marginal cost and expands quantities. As competition intensifies, effi-
ciency motives are largely exhausted, leaving distributional and stability considerations as the main
sources of deviation from profit-maximizing behavior. In other words, when markets are already
competitive, self-regulation becomes less about expanding access and more about moderating po-
tential adverse downstream effects.
14
3 Scenarios for AI Transitions with Socially-Minded Firms
We now turn to a numerical exploration of our formulas. We focus on the example economy in
Proposition 2 and ask the hypothetical question: “Imagine a firm develops an AI capable of repli-
cating skill 𝑠 at a fixed fraction of its current cost. How should socially minded firms deploy and
price this technology?”
The formulas in the proposition demonstrate how to calculate the optimal deployment path for
any such AI. By focusing on these hypothetical AIs, we avoid the more challenging question of
determining which specific skills are most likely to be automated in the near term.
For our application, we focus on a firm that operates in the US economy and map skills to 525
detailed occupations from the 2017–2021 American Community Survey. For each occupation, we
compute the optimal deployment plan of an AI capable of replacing labor inputs in said occupation.
For the model parameters, we set a reallocation rate 𝛼 = 4% per year, in line with estimates
from our previous work (Lehr and Restrepo, 2024). We also set 𝜎𝑠 = 3, which is a commonly
used value for the elasticity of substitution between differentiated goods, as the ones produced by
different skills in our model (see, for example, Broda and Weinstein, 2006).5
The data inputs needed for our calculations and appearing in the formulas from Proposition 2
are computed as follows:
• We let ℎ denote the set of people at different percentiles of the US income distribution, as-
sumed to have the same relative weight 𝑔ℎ.6
• We take �̄�𝑠 as the average hourly wage across occupations from 2017–2021 ACS. For each
percentile, we then measure �̄�ℎ𝑠 as their income from occupation 𝑠 and define
�̄�ℎ𝑠 =
�̄�ℎ𝑠
�̄�𝑠
,
5A related object is the elasticity of substitution between college and non-college labor, with estimates ranging from
(as in Katz and Murphy, 1992) to 4 (as in Bils, Kaymak and Wu, 2024). For broad occupations, Burstein, Morales
and Vogel (2019) estimate an elasticity of substitution of . We use a larger value since our occupational groups are
finer.
6In defining these percentiles, we sort individuals based on household income per person. This is computed as total
household income divided by the number of adults plus a half times the number of children. This approach accounts
for intra-household income sharing, assigning children a weight of times that of an adult.
15
as the effective hours worked by households from the ℎth percentile in occupation 𝑠.
• We let �̄�ℎ denote the average labor income of people in percentile ℎ.
• We compute total labor input in 𝑠 as �̄�𝑠 =
∑
ℎ �̄�
ℎ
𝑠 and calibrate 𝛾𝑠 to match �̄�𝑠 = 𝛾𝑠 �̄�
−1/𝜎𝑠
𝑠 .
Finally, we assume 𝜓𝑠 = .5 �̄�𝑠, so that AI can replicate human labor at 50% the The
rationale for this choice is as follows. In our model, an AI substituting for skill 𝑠 and sold at a
standard markup 𝜎𝑠 / (𝜎𝑠 − 1) above marginal cost raises output per worker from 1 to
1 +
𝑞𝑠𝑡
�̄�𝑠
=
(𝜓𝑠𝑡
�̄�𝑠
𝜎𝑠
𝜎𝑠 − 1
)−𝜎𝑠
= .
This -fold increase in output per worker matches the upper end of available empirical estimates.
For example, Noy and Zhang (2023) estimate a twofold increase in output per worker in writing
tasks and Brynjolfsson, Li and Raymond (2025) estimate a increase in customer service.
In the analysis, we contrast the optimal deployment plans of various firms. We consider: (a) a
pure profit maximizer (𝜇 = 𝑔ℎ = 𝜆 = 0); (b) a utilitarian firm (𝜇 > 0, 𝑔ℎ = 𝜆 = 0); (c) a welfarist
firm (𝜇 > 0, 𝑔ℎ ≠ 0, 𝜆 = 0); (d) a conservative firm (𝜇 = 𝑔ℎ = 0, 𝜆 > 0); and (e) a multi-objective
firm (𝜇 > 0, 𝑔ℎ ≠ 0, 𝜆 > 0).
In the relevant scenarios, we set 𝜇 = and use the welfare weights 𝑔ℎ reported in Lockwood
and Weinzierl (2016), inferred from the progressivity of the US tax system. This assumes that the
welfare weights of the AI firm align with those that the US political system places on households at
different percentiles of the income distribution. Our value for 𝜇 implies the firm is willing to trade
1 dollar of profit for 2 dollars of value for the economy as a whole. The values for welfare weight
𝑔ℎ are shown in Figure 2. The values imply that the firm is willing to give $ 1 of profits to increase
incomes by $ at the bottom of the income distribution and $ 3 at the top.
Finally, in the relevant scenarios, we rescaled 𝜆 by the average wage in the economy to ensure
that all terms have an equal scale and set 𝜆 = . Thus, the firm is willing to reduce profits by $ 1
7Variable costs include the computational resources needed to run the AI and effectively replicate human input in
skills 𝑠, plus any residual costs associated with integration, prompting, or inspection of the AI output. Replicating
human input can require multiple calls to these models, explaining why 𝜓𝑠𝑡 can vary across jobs. The variable compu-
tational and energy costs of using AI are significant and have increased as AI companies train larger and more complex
models with higher inference costs.
16
Figure 2: Welfare Weights Across the Income Distribution
Notes: The figure reports welfare weights 𝑔ℎ by income percentile. These are obtained as the welfare weights that
rationalize the progressiveness of the US tax system, and are based on Figure 1 in Lockwood and Weinzierl (2016).
We directly use the reported weights at specific income percentiles and interpolate to span the income distribution.
if it raises wages for the average displaced worker by $ 2. These scenarios are meant to clarify how
AI firms may act if they pursue a broader set of social objectives; of course, we do not know what
is in the minds or hearts of their CEOs or how they will weigh different considerations in practice.
Equilibrium markups and AI deployment plans
Figure 3 reports equilibrium markups (𝑤𝑠𝑡 − 𝜓𝑠𝑡)/𝜓𝑠𝑡 for firms with different objectives at three
time horizons. Panel A shows markups on impact (𝑡 = 0), Panel B for the short run (𝑡 = 5 years),
and Panel C for the long run (𝑡 = 100 years). The figures sort the 525 detailed occupations by their
average base wage �̄�𝑠 in the horizontal axis. The movement along the curves shows how markups
vary across occupations hypothetically replaced by AI as we move from low-pay to high-pay roles.
As a benchmark, consider a pure profit-maximizing firm, in black. Markups for this firm at
𝑡 = 0 are around 32% and constant, since we assume a common productivity improvement across
all skills. As expected from our discussion of 2, markups rise at longer time horizons, reflecting
reduced competition from workers as they reallocate to other jobs. In the long run, the AI firm
becomes the sole supplier of skill 𝑠 and charges a markup of 𝜎𝑠/(𝜎𝑠 − 1) = 50% across the board.
The utilitarian firm, in orange, charges lower markups than the profit-maximizing firm, about
17
Figure 3: Equilibrium Markups in the Short and Long Run
Notes: Panel A reports equilibrium markups on impact (𝑡 = 0) for AIs capable of automating different occupations
(ranked bywage in the horizontal axis). The curves are smoothed by binning occupations into 50 quantiles and reporting
the average within each bin. Each panel shows five curves, one for each type of firm. Panels B and C report the same
curves after 5 and 100 years.
15% at 𝑡 = 0 and converging to 20% in the long run as it has an incentive to lower prices below the
profit-maximizing level to expand access and increase aggregate efficiency.
The welfarist firm, in dashed green, prioritizes both aggregate efficiency and distributional con-
cerns. The latter have a tiny impact on equilibriummarkups at the bottom. Relative to the utilitarian
firm, distributional concerns call for a percentage point higher markup at the bottom, thereby
redistributing resources towards low-income households in low-paying jobs. Distributional consid-
erations have a modest impact on markups at the top, lowering them (relative to the utilitarian firm)
by a full percentage point (from 15% to 14%). Thus, distributional considerations play a small role.
From the viewpoint of a welfarist firm, the concern of maximizing access dominates and leads to
markups that are less than half of what a profit-maximizing firm would charge.
The conservative firm, in solid purple, balances profits against labor-market stability. This firm
ends up charging prices above the profit-maximizing level to minimize its labor market impact. This
18
concern is particularly pronounced for AIs that automate low-wage occupations, as these generate
more substantial labor-market disruptions. For this reason, equilibrium markups are higher at the
bottom, for AIs that automate low-paying occupations. In the long run, stability concerns vanish
and the firm stops behaving conservatively to focus entirely on profit maximization.
Finally, the dashed blue line presents markups for a multi-objective firm, which balances profit,
efficiency, redistribution, and stability concerns. This firm charges a 33% markup on AIs com-
peting against low-paying workers and a 15% markup on AIs that substitute labor in high-paying
occupations. In the long run, distributional and stability considerations fade as workers reallocate,
and the AI firm converges to a common 20% markup, balancing broader access with its profits.
Figure 4: Decomposition of Motives Driving Markups Charged by Multi-Objective Firm
Notes: The figure decomposes equilibriummarkups charged by a multi-objective firm on impact (𝑡 = 0) for AIs capable
of automating different occupations (in the horizontal axis). The solid blue line depicts the equilibrium markup. The
black dotted line represents the contribution of profit-maximizing motives. The orange dotted line adds the contribution
of aggregate efficiency considerations. The green line takes into account distributional considerations. The gap between
this and the solid blue reflects labor-market stability considerations. The curves are smoothed by binning occupations
into 50 quantiles and reporting the average within each bin. Each panel shows five curves, one for each type of firm.
Panels B and C report the same curves after 5 and 100 years.
Figure 4 decomposes the role of each motive for the multi-objective firm. The dashed black
line represents the contribution of profit motives, which push for high markups, especially for high-
paying jobs facing less competition fromworkers. The orange line incorporates aggregate efficiency
19
considerations, which call for uniformly lower markups to improve access. The green line accounts
for distributional considerations, which have no impact at the bottom, and calls for lower markups
at the top. Finally, the blue line takes into account wage stability concerns, which call for curbing
quantities and raising prices for AIs, especially those that replace low-wage jobs.
Figure 5 complements the results by reporting equilibrium quantities. We plot the increase in
quantities relative to their baseline levels before AI implementation. AI-produced quantities range
from one to four times the baseline level. The utilitarian firm generates the maximum increase in
AI usage, while the conservative and profit-maximizing firms restrict quantities the most.
Figure 5: Equilibrium Quantities in the Short and Long Run
Notes: Panel A reports equilibrium quantities on impact (𝑡 = 0) for AIs capable of automating different occupations
(in the horizontal axis) as the percent deviation from pre-AI production levels. The curves are smoothed by binning
occupations into 50 quantiles and reporting the average within each bin. Each panel shows five curves, one for each
type of firm. Panels B and C report the same curves after 5 and 100 years.
Why do distributional considerations play such a small role?
Why do distributional considerations play such a small role, especially for low-pay jobs? Two forces
explain this finding. First, and as discussed in Proposition 2, the strength of distributional motives
20
vanishes as AI use deepens. As shown in Figure 5, AI output for the utilitarian and welfarist firms
is already 4 times that supplied by workers at baseline. This pushes the firm to prioritize aggregate
efficiency over distributional considerations. Second, the distributional effects of changing wages
in a given occupation are not as large as onemay have thought, especially at the bottom. To illustrate
this point, let’s compute the distributional gains of increasing income in occupation 𝑠, given by the
normalized Pareto weights of the average employee:
Average Pareto Weight𝑠 =
∑
ℎ
𝑔ℎ
�̄�ℎ𝑠
�̄�𝑠
.
Panel A in Figure 6 reports these average weights for the 525 detailed occupations in our data, while
Panel B plots the average income percentile of people employed in each occupation.
Figure 6: Welfare Weights Across Occupations
Notes: Panel A plots the distributional gains of increasing income by $1 across occupations. These are computed as∑
ℎ 𝑔ℎ
�̄�ℎ𝑠
�̄�𝑠
, where the normalized Pareto weights 𝑔ℎ are from Lockwood andWeinzierl (2016). Panel B plots the average
income percentile of workers within an occupation.
All occupations at the bottom half of the pay distribution have positive average weights, show-
ing that increasing income in these jobs has a positive distributional benefit. However, the average
weights are small and close to zero, suggesting these benefits are small in practice. Increasing in-
come in jobs at the bottom carries a tiny distributional gain of 1 cent for every dollar. To understand
why, consider Cashiers—one of the 5% lowest paying jobs in the US. Despite its low pay, people
21
working as cashiers come from households with a wide range of incomes, spanning from the very
bottom to the 70th percentile. On average, people employed as cashiers come from households at
the 42nd percentile of the income distribution. This lack of segmentation at the bottom implies that
protecting cashiers and other low-wage jobs benefits a wide range of households, not just the very
poor. This effect is further compounded by the fact that many of the poorest households earn no
labor income at all, and are therefore not exposed to the substituting effects from AI.
On the other hand, increasing income in jobs at the top carries a more sizable distributional
penalty of 12 cents for every dollar, reflecting the higher degree of income segmentation at the
top. Consider Economists, one of the 5% highest paid jobs. People in this field typically hail
from households at the upper end of the income distribution, with the average economist located at
the 90th percentile. This asymmetry explains why distributional concerns matter very little at the
bottom but have a more appreciable (though still modest) effect for AI pricing at the top.
Figure 7: Markups and Quantities on Impact for Stronger Redistributive Preferences
Notes: This figure reports optimal markups and quantities on impact (𝑡 = 0) for stronger redistributive preferences than
baseline, �̃�ℎ = 10× 𝑔ℎ. Panel A reports optimal markups and Panel B the associated quantities for the automated skill.
Would distributional considerations for jobs at the bottom matter if the AI firm were even more
progressive? Imagine a firm whose welfare weights 𝑔ℎ are ten times those estimated by Lockwood
andWeinzierl (2016) in Figure 2. This hypothetical firm is in effect ten times more progressive than
the US political system. The resulting equilibrium markups are shown in Figure 7. The stronger
22
distributional considerations call for one percentage point higher markups on AIs that substitute
for bottom occupations, relative to what a utilitarian firm would do. However, the incentive to
increase aggregate efficiency remains dominant, and it is still optimal for a welfarist firm to expand
the quantity of AI produced to broaden access, despite its potential adverse distributional effects at
the lower end.
Stronger distributional considerations do make a difference for jobs at the top. A welfarist
firm with ten times stronger distributional concerns should charge markups that are half of what
a utilitarian firm would charge and produce 20% more output. This is because AIs that substitute
for jobs at the top redistribute from workers at the very top (who tend to hold highly paid jobs)
towards the rest of the population (who are not exposed to these top jobs)—an extremely valuable
proposition from the firm’s viewpoint.
In summary, stronger distributional concerns lead to a more aggressive deployment of AI at the
top, but have no significant implications for AIs that substitute for jobs at the bottom.
Should more productive AI be priced differently?
Our baseline results considered the optimal deployment of AIs capable of replacing workers at 50%
of their cost. Suppose that 𝜓𝑠 = .2 �̄�𝑠, so that AI can replace workers at 20% of their cost. How
should these more productive AIs be priced and deployed?
Figure 8 reports equilibrium prices and quantities for such AIs across occupations at 𝑡 = 0.
Relative to our baseline, markups are slightly higher, while quantities are an order of magnitude
larger. This is because a more productive AI firm experiences less competition from workers and
captures a greater share of the market, allowing it to charge higher markups.
More importantly, the figure shows that both distributional and labor-market stability motives
are weaker when pricing more productive AIs. This can be seen from the fact that outcomes for
conservative firms are close to those of a pure profit maximizer, and outcomes for the welfarist and
multi-objective firms are close to the utilitarian one. As discussed in Proposition 2, this is because
profit and efficiency motives scale with the quantity of AI used, while distributional and stability
23
Figure 8: Markups and Quantities on Impact for Highly Productive AI
Notes: This figure reports optimal markups and quantities on impact (𝑡 = 0) for more productivity AI than baseline,
𝜓𝑠 = . Panel A reports optimal markups and Panel B the associated quantities for the automated skill.
motives do not.
We conclude that as firms developmore productive and less costly AIs, distributional or stability
considerations become less pressing. A socially minded firm with a sufficiently productive AI
should behave essentially as a utilitarian one, balancing access and profits only.
What would a planner do?
To conclude our empirical exploration, we contrast the optimal deployment plan pursued by a so-
cially minded firm with that of a social planner that can control the supply of AIs but has no other
tools. Suppose the objective of the planner is to maximize social welfare and maintain labor market
stability, giving a weight 𝜇 to the utility of financiers—the same given to the average household.
The same derivations we did for an AI firm above imply that the planner supplies AI until
1 −
𝜓𝑠𝑡
𝑤𝑠𝑡
=
( ∫
ℎ
𝑔ℎ
�̄�ℎ𝑠
𝑞𝑠𝑡
𝑒−𝛼 𝑡 𝑑ℎ +
𝜆
𝜇
∫
ℎ
1
�̄�ℎ
�̄�ℎ𝑠
𝑞𝑠𝑡
𝑒−𝛼 𝑡 𝑑ℎ
) 𝑞𝑠𝑡
𝑞𝑠𝑡 + �̄�𝑠 𝑒−𝛼 𝑡
1
𝜎𝑠
(6)
This equation states that the planner trades off the reduction in aggregate efficiency from reducing
output below its competitive level (the left side) with the distributional and stability gains this creates
(the right side).
24
Figure 9 depicts the optimal deployment plan that a social planner controlling the supply of AI
would choose. To ease the comparison with our previous findings, we report the implied markup
that would decentralize the planner’s allocation. These can also be interpreted as the optimal tax
that such a planner would levy on AIs substituting for jobs at the bottom and top of the income
distribution, respectively. The figure also reports themarkups that a sociallymindedmulti-objective
firm would charge (using the baseline values for 𝜇 and 𝜆 from above).
Figure 9: Implied Markups or AI Taxes that Decentralizing the Planner’s Allocation
Notes: The figure reports the optimal markup that a social planner would set to balance aggregate efficiency with
distributional and stability concerns. Panel A reports markups on impact (𝑡 = 0). Panels B and C report the same
curves after 5 and 100 years. For comparison, the figure also depicts the deployment path followed by a multi-objective
firm with 𝜇 = 𝜆 = .
A conservative social planner that cares about welfare and stability equally (𝜇 = 𝜆 as our multi-
objective firm), would set an optimal markup on AIs at the bottom of 7% and at the top of less
than 1% on impact. Over time, as distributional and stability concerns subside, the conservative
social planner would impose no markups or taxes on AI and would implement a competitive out-
come, whereas a responsible-AI firm would continue to charge a positive markup, as it balances
broadening access with its private profit incentives.
25
Themain result here is that the planner allocation features lowerAI prices thanwhat a responsible-
AI firm with the same social objectives would charge. Despite its social inclinations, socially re-
sponsible firms remain constrained by their private profit motives in howmuch they can lower prices
to broaden access.
Do our conclusions apply to occupations with high AI replacement risk?
Our results characterize the optimal deployment path for AIs capable of substituting for labor across
various occupations, from cashiers to economists, without taking a stance on which jobs could be
automated first. Do our conclusions apply to occupations with the highest risk of automation by
AI, as identified in existing prospective analyses?
Figure 10: Distributional and Stability Considerations for Occupations at Risk of Replacement
Notes: Panel A plots the redistributive motive, 𝜇
∑
ℎ 𝑔ℎ
�̄�ℎ𝑠
�̄�𝑠
, against the share of tasks automatable by AI following
Eloundou et al. (2023). Panel B plots the non-disruption motive, 𝜆
∑
ℎ
1
�̄�ℎ
�̄�ℎ𝑠
�̄�𝑠
against the same AI measure.
We answer this question using data from Eloundou et al. (2023) on the share of core tasks by
occupation that could be automated with LLM-powered systems. Figure 10 shows that occupations
at risk (in the horizontal axis) do not stand out in their distributional or stability considerations.
Panel A shows that highly exposed occupations have average Pareto weights near zero. Panel B
shows that the average cost of disruptions among employees does not systematically vary with AI
26
exposure either. This is because prospective studies, such as Eloundou et al. (2023), suggest that
the set of occupations at risk is spread throughout the income distribution and does not concentrate
at either the bottom or the top. In summary, the conclusions drawn above for the entire universe of
jobs apply equally well to the subset of occupations more at risk of being substituted by AIs.
4 Conclusion
How should socially minded firms deploy and price their AIs? This paper provides a framework
to address this question by extending Lerner’s Rule to incorporate a broader set of objectives: gen-
erating profits, promoting social welfare, and minimizing labor-market disruptions. The resulting
pricing formulas clarify how these objectives shape markups, deployment speed, and access to AI.
Applying our framework to US data across hundreds of occupations, we find that a firm that
cares about welfare and stability equally should price near the profit-maximizing level in the short
run, but closer to marginal cost over time. This gradual strategy limits short-run disruptions and
balances them with improved access to the technology in the medium run.
Our conclusion from this exercise is that the most pro-social course of action for a monopolist
AI firm is to refrain from exploiting its market power, except in the very short run, when stability
considerations are the most pressing. This conclusion is particularly relevant for AIs that produce
new goods and services without disrupting existing labor markets, and when the government is
already engaged in effective redistribution.
This conclusion also depends on the baseline level of competition in AI markets. Whether the
incentive to prioritize access dominates depends on how constrained supply was by the exercise of
market power to begin with. If AI firms face little competition, the incentive to broaden access by
lowering prices dominates the actions of a socially responsible firm. Instead, if competition among
AI firms already results in quantities that are close to efficient, the incentive to broaden access loses
relevance, while distributional and stability concerns become dominant.
Because our analysis focuses solely on labor market and economic efficiency considerations,
27
abstracting from broader societal and existential risks, our conclusions apply only to well-aligned
or narrow AIs without existential risks.
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30
A Theory Extensions
This section explores theoretical extensions. First, we discuss how taxes and the safety net affect
the deployment of AI by socially minded firms. Second, we discuss the possibility that some AIs
may not replicate human skill but could eventually acquire new capabilities that allow these systems
to produce entirely new goods and services without devaluing existing human skills. Finally, we
discuss the case when multiple AI firms are competing a la Cournot. The proofs for this extension
are in the appendix.
Taxes and the safety net
We now extend the model to account for the tax system and the safety net. Assume the after-tax
labor income of household ℎ is
After-tax labor incomeℎ𝑡 ≡ T (𝑤
ℎ
𝑡 ) + 𝑇𝑡 ,
where 𝑇𝑡 is a common transfer that balances the government budget and T (.) is an increasing tax
function, withT (0) = 0 and 1−T ′(𝑤ℎ𝑡 ) > 0 giving themarginal tax rate experienced by households
at different points of the income distribution.
The AI firm’s objective function is now
𝑉 = PDV 𝜋𝑡 +
∫
ℎ
𝜇ℎ 𝑢ℎ 𝑑ℎ + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
PDV
T (𝑤ℎ𝑡 )
T (�̄�ℎ)
𝑑ℎ,
where we assume that the stability term depends on how actions by the AI firm reduce after-tax
labor income T (𝑤ℎ𝑡 ) relative to its status quo level T (�̄�ℎ).
Proposition 3. In the quasi-linear case with government taxes, a socially-responsible firm produces
𝑞𝑠𝑡 until
L𝑠𝑡 =
(
1 − 𝜇 + 𝜇
∫
ℎ
𝑔ℎ T ′(𝑤ℎ𝑡 )
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ + 𝜆
∫
ℎ
T ′(𝑤ℎ𝑡 )
T (�̄�ℎ)
𝑛ℎ𝑠𝑡
𝑞𝑠𝑡
𝑑ℎ
) 1
𝜀𝑠𝑡
(7)
The proposition shows that a more progressive tax system (as evidenced by a lower keep rate
31
T ′(𝑤ℎ𝑡 )) weakens the firm’s distributional and stability concerns. In the extreme case of full redis-
tribution (., T ′(𝑤ℎ𝑡 ) = 0), distributional and stability considerations vanish. This highlights an
important interplay between public policy and self-regulation. The more we redistribute via the tax
system, the less an AI firm should worry about its downstream distributional effects and the more
it should prioritize broadening access.
AI as creating new goods and services
Our formulas assume that AIs substitute for human labor in existing jobs, a natural application since
these systems are trained on human-generated data to mimic us. Yet some argue that large models,
when trained on vast datasets, can develop novel capabilities and produce goods and services that
surpass anything humans have created so far.
To account for this possibility, assume the firm also develops AIs that create new goods and
services, such as new proteins that it can sell or license to medical laboratories. Assume also that
household utility is given by
𝑢ℎ (𝑐) = 𝑐ℎ0𝑡 +
∑
𝑠∈S
𝛾
1/𝜎𝑠
𝑠
𝑐
1−1/𝜎𝑠
𝑠
1 − 1/𝜎𝑠
+
∑
𝑠′∈N
𝛾
1/𝜎𝑠′
𝑠
𝑐
1−1/𝜎𝑠′
𝑠′
1 − 1/𝜎𝑠′
,
for 𝜎𝑠 > 1. The set N represents new goods and services produced by AIs, indexed by 𝑠′. We let
𝑛ℎ
𝑠′ = 0 for all 𝑠
′ ∈ N , indicating that humans were not able to produce these novel goods and
services.
Proposition 4. In an economy where (a)–(d) hold, the optimal pricing of novel AIs 𝑠′ ∈ N satisfies
a modified Lerner rule
L𝑠′𝑡 =
(
1 − 𝜇
) 1
𝜎𝑠′
. (8)
AIs that expand the range of goods and services benefit all workers without disrupting existing
labor markets, and thus raise no concerns regarding distribution or stability. For this class of AIs,
the main responsibility of a socially minded firm is to price close to marginal cost and broaden
32
access.
Competition among AI producers
We extend the baseline model in Section to incorporate competition among AI companies. For
each 𝑠 ∈ A, suppose 𝑀𝑠 > 1 identical firms produce the AI and compete in quantities à la Cournot.
Proposition 5. In an economy where (a)–(d) hold and 𝑀𝑠 symmetric companies compete in quan-
tities, the equilibrium price of AI satisfies
L𝑠𝑡 =
(1 − 𝜇
𝑀𝑠
+ 𝜇
∫
ℎ
𝑔ℎ
�̄�ℎ𝑠
𝑞𝑠𝑡
𝑒−𝛼 𝑡 𝑑ℎ + 𝜆
∫
ℎ
1
�̄�ℎ
�̄�ℎ𝑠
𝑞𝑠𝑡
𝑒−𝛼 𝑡 𝑑ℎ
) 𝑞𝑠𝑡
𝑞𝑠𝑡 + �̄�𝑠 𝑒−𝛼 𝑡
1
𝜎𝑠
, (9)
where 𝑞𝑠𝑡 is the aggregate quantity of AI used in 𝑠 ∈ A.
As usual, competition forces firms to set prices closer to their marginal cost and expand quan-
tities. This is evidenced by the fact that, in (9), the term ”1 − 𝜇” is divided by 𝑀𝑠𝑡 .
The proposition also shows that distributional and stability concerns become increasingly rele-
vant as competition between AI suppliers intensifies. To see this, note that a pure-profit-maximizing
firm would price according to
L𝑠𝑡 =
1
𝑀𝑠𝑡
𝑞𝑠𝑡
𝑞𝑠𝑡 + �̄�𝑠 𝑒−𝛼 𝑡
1
𝜎𝑠
The more competition this firm faces, the closer it would price to marginal cost. Consider now a
socially minded firm. As competition intensifies, distributional and stability considerations become
the sole forces causing the firm to deviate from profit-maximizing pricing. Broadening access is
no longer a first-order concern.
The reason why this happens is that competition pushes firms to produce closer to the efficient
level of AI (from an aggregate efficiency point of view). Starting from this level, the aggregate
efficiency gains from further expanding access are limited, since firms are already pricing close to
their marginal cost. Instead, the incentive to curb quantities to limit adverse distributional effects
or maintain stability remains active and becomes the dominant force guiding firms’ actions.
33
B Derivations and Proofs
This appendix derives equation (1) and also the planner solution in (6). It then provides proofs for
the extensions.
Derivation of (1): First, note that 𝑟𝑡 = 𝜌, since financiers must be indifferent between consuming
or saving.
Following an arbitrary change in quantities by the AI firm, we get
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+
∫
ℎ
𝜇ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
In this expression:
• The first line indicates the changes in profits that flow to financiers.
• The second line gives the change in households’ utility. By assumption, 𝑐ℎ0𝑡 > 0, which im-
plies that the marginal value of income in period 𝑡 is 𝑒−𝜌𝑡 . This is then multiplied by 𝜇ℎ,
capturing the social value of increasing utility for household ℎ, and the change in household
net income resulting from the perturbation. Note that while households adjust their consump-
tion and savings decisions in response to price changes, these changes are second-order due
to the envelope theorem. This is why only the change in income resulting from price changes
is reflected. Note also that commodity 𝑗 = 0 is the numeraire, and so its price is fixed at one.
• The third line gives the effects via labor market disruptions, which are assumed to be a func-
tion of wages. Note that this applies to all households, as AI necessarily reduces nominal
wages for all households with 𝑛ℎ𝑠𝑡 > 0.
34
We can rewrite the above expression as
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
+ 𝜇
(∑
𝑠∈A
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
∫
ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
)
+ 𝜇
∫
ℎ
𝑔ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Using ℓ𝑠𝑡 =
∫
ℎ
𝑛ℎ𝑠𝑡 𝑑ℎ + 𝑞𝑠𝑡 for 𝑠 ∈ A and ℓ𝑠𝑡 =
∫
ℎ
𝑛ℎ𝑠𝑡 𝑑ℎ for 𝑠 ∉ A), plus market clearing for
commodities 𝑗 ≠ 0, this simplifies to:
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
+ 𝜇
( ∑
𝑠
ℓ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑦 𝑗 𝑡 𝛿𝑝 𝑗 𝑡
)
+ 𝜇
∫
ℎ
𝑔ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Because the production of commodities is competitive and features constant returns to scale, firms
make zero profits, and the envelope theorem (applied to their profits) implies
∑
𝑗≠0
𝑦 𝑗 𝑡 𝛿𝑝 𝑗 𝑡 −
∑
𝑠
ℓ𝑠𝑡 𝛿𝑤𝑠𝑡 = 0.
That is, the second line in the equation for 𝛿𝑉 is zero. To conclude, note that 𝑐ℎ
𝑗𝑡
= 𝑦 𝑗 𝑡 for 𝑗 ≠ 0 in
the third line because of quasi-linearity. The term
∑
𝑗≠0 𝑐
ℎ
𝑗𝑡
𝛿𝑝 𝑗 𝑡 is then common to all households
and cancels because
∫
ℎ
𝑔ℎ 𝑑ℎ = 0. This simplification yields (1).
35
Derivation of (6): Let
𝑆 = 𝜇 𝑢 +
∫
ℎ
𝜇ℎ 𝑢ℎ 𝑑ℎ + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
PDV
𝑤ℎ𝑡
�̄�ℎ
𝑑ℎ
be the planners’ objective. Consider a small perturbation in the quantity of AI produced. This
affects the equilibrium value of 𝑆 as follows:
𝛿𝑆 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
𝜇
∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+
∫
ℎ
𝜇ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
We can rewrite the above expression as
𝛿𝑆 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
𝜇
(∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+
∫
ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
)
+ 𝜇
∫
ℎ
𝑔ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Using ℓ𝑠𝑡 =
∫
ℎ
𝑛ℎ𝑠𝑡 𝑑ℎ + 𝑞𝑠𝑡 for 𝑠 ∈ A and ℓ𝑠𝑡 =
∫
ℎ
𝑛ℎ𝑠𝑡 𝑑ℎ for 𝑠 ∉ A), plus market clearing for
commodities 𝑗 ≠ 0, the first line simplifies to:
𝛿𝑆 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
𝜇
(∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡 +
∑
𝑠
ℓ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑦 𝑗 𝑡 𝛿𝑝 𝑗 𝑡
)
+ 𝜇
∫
ℎ
𝑔ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Because the production of commodities is competitive and features constant returns to scale, firms
36
make zero profits, and the envelope theorem (applied to their profits) implies
∑
𝑗≠0
𝑦 𝑗 𝑡 𝛿𝑝 𝑗 𝑡 −
∑
𝑠
ℓ𝑠𝑡 𝛿𝑤𝑠𝑡 = 0.
Moreover, 𝑐ℎ
𝑗𝑡
= 𝑦 𝑗 𝑡 for 𝑗 ≠ 0 because of quasi-linearity. The term
∑
𝑗≠0 𝑐
ℎ
𝑗𝑡
𝛿𝑝 𝑗 𝑡 is then common
to all households and cancels because
∫
ℎ
𝑔ℎ 𝑑ℎ = 0. Making both replacements in the equation for
𝛿𝑆 yields the planner’s variant of (1):
𝛿𝑆 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡 + 𝜇
∫
ℎ
𝑔ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
At an optimum, the planner sets quantities so that 𝛿𝑆 = 0. This implies that for all 𝑡 and 𝑠 ∈ A,
0 = 𝜇
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡 + 𝜇
∫
ℎ
𝑔ℎ
∑
𝑠′
𝑛ℎ𝑠′𝑡 𝛿𝑤𝑠′𝑡 𝑑ℎ + 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠′
𝑛ℎ𝑠′𝑡 𝛿𝑤𝑠′𝑡 𝑑ℎ,
which in the simplified economy considered in Proposition 2 can be written as in equation (6).
Proof of Proposition 1: In an interior equilibrium where 𝑞𝑠𝑡 > 0, any deviation in 𝑞𝑠𝑡 must yield
𝛿𝑉 = 0. Setting 𝛿𝑉 = 0 in (1) and rearranging yields (2).
Proof of Proposition 2: Equation (4) follows from the formula in Proposition 1, using the fact
that in this economy, the elasticity of demand for AI (accounting for worker production) exceeds
𝜎𝑠 and is given by
𝜀𝑠𝑡 =
𝑞𝑠𝑡 + �̄�𝑠𝑒−𝛼𝑡
𝑞𝑠𝑡
𝜎𝑠 .
The demand curve in (5) is derived by equating the marginal rate of substitution for commodity 𝑠
(relative to the numeraire) to its price 𝑤𝑠.
Note that in this economy, there are no complementarities across jobs. As a result, 𝑤ℎ𝑡 < �̄�
ℎ for
all households with 𝑛ℎ𝑠𝑡 > 0 for some 𝑠 ∈ A. This is why the cost of disruption sums over all ℎ.
37
Proof of Proposition 3: Following an arbitrary change in quantities by the AI firm, we get
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+
∫
ℎ
𝜇ℎ
(
T ′(𝑤ℎ𝑡 )
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 + 𝛿𝑇𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
T ′(𝑤ℎ𝑡 )
T (�̄�ℎ)
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Using the fact that
𝛿𝑇𝑡 =
∫
ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 − T
′(𝑤ℎ𝑡 )
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡
)
𝑑ℎ,
which follows from the requirement that tax revenue is rebated to households, the above expression
for 𝛿𝑉 can be rewritten as
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
+ 𝜇
(∑
𝑠∈A
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
∫
ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
)
+ 𝜇
∫
ℎ
𝑔ℎ
(
T ′(𝑤ℎ𝑡 )
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
T ′(𝑤ℎ𝑡 )
T (�̄�ℎ)
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
From here on, we follow the same steps from the derivation of equation (1) to obtain the variant:
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
+ 𝜇
∫
ℎ
𝑔ℎ T ′(𝑤ℎ𝑡 )
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
T ′(𝑤ℎ𝑡 )
T (�̄�ℎ)
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
From here on, we proceed as in the proof of Proposition 2.
38
Proof of Proposition 4: Following an arbitrary change in quantities for AIs in N or A, we get
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{ ∑
𝑠∈N∪A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+
∫
ℎ
𝜇ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Following the same steps as in the derivation of equation (1), we can write this as
𝛿𝑉 =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈N∪A
(
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
)
+ 𝜇
∑
𝑠∈N∪A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞𝑠𝑡
+ 𝜇
∫
ℎ
𝑔ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
At an optimum, the firm sets quantities so that 𝛿𝑉 = 0. In an economy with (a)–(d), this implies
that for every 𝑠′ ∈ N and time 𝑡,
0 = (1 − 𝜇)
(
𝑞𝑠′𝑡 𝛿𝑤𝑠′𝑡 +
(
𝑤𝑠′𝑡 − 𝜓𝑠′𝑡
)
𝛿𝑞𝑠′𝑡
)
+ 𝜇
(
𝑤𝑠′𝑡 − 𝜓𝑠′𝑡
)
𝛿𝑞𝑠′𝑡 ,
where we used the fact that 𝑛ℎ
𝑠′𝑡 = 0 and the fact that changing 𝑞𝑠′𝑡 does not affect wages for other
skills 𝑠 ≠ 𝑠′. This expression can then be rearranged into (8).
Proof of Proposition 5: Let 𝑞 (𝑖)𝑠𝑡 be the quantity supplied by one of the AI firms in 𝑠 ∈ A, where
𝑖 = 1, 2, . . . , 𝑀𝑠. Denote by 𝑞
(−𝑖)
𝑠𝑡 the quantity supplied by its competitors. A perturbation in 𝑞
(𝑖)
𝑠𝑡
changes the firm objective (𝑉 (𝑖)) by
𝛿𝑉 (𝑖) =
∫ 𝑡
0
𝑒−𝜌𝑡
{∑
𝑠∈A
(
𝑞
(𝑖)
𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞
(𝑖)
𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
𝑞
(−𝑖)
𝑠𝑡 𝛿𝑤𝑠𝑡
+
∫
ℎ
𝜇ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
39
This expression assumes the AI firm values the utility of owners of other AI firms at a rate 𝜇, which
is the same as the average household.
Rearranging terms, this can be expressed as
𝛿𝑉 (𝑖) =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞
(𝑖)
𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞
(𝑖)
𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞
(𝑖)
𝑠𝑡
+ 𝜇
(∑
𝑠∈A
𝑞𝑠𝑡 𝛿𝑤𝑠𝑡 +
∫
ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
)
+ 𝜇
∫
ℎ
𝑔ℎ
(∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 −
∑
𝑗≠0
𝑐ℎ𝑗𝑡 𝛿𝑝 𝑗 𝑡
)
𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
Following the same steps as in the derivation of (1), we have that the second line is zero and the
effect of prices on the third line also averages to zero. This yields the variant of (1):
𝛿𝑉 (𝑖) =
∫ 𝑡
0
𝑒−𝜌𝑡
{
(1 − 𝜇)
∑
𝑠∈A
(
𝑞
(𝑖)
𝑠𝑡 𝛿𝑤𝑠𝑡 +
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞
(𝑖)
𝑠𝑡
)
+ 𝜇
∑
𝑠∈A
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
𝛿𝑞
(𝑖)
𝑠𝑡
+ 𝜇
∫
ℎ
𝑔ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
+ 𝜆
∫
ℎ:𝑤ℎ𝑡 <�̄�ℎ
1
�̄�ℎ
∑
𝑠
𝑛ℎ𝑠𝑡 𝛿𝑤𝑠𝑡 𝑑ℎ
}
𝑑𝑡.
At an optimum, the firm sets quantities so that 𝛿𝑉 (𝑖) = 0. In an economy with (a)–(d), this
implies that for every 𝑠 ∈ A and time 𝑡,
0 = (1 − 𝜇)
(
𝑞
(𝑖)
𝑠𝑡
𝛿𝑤𝑠𝑡
𝛿𝑞
(𝑖)
𝑠𝑡
+
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
) )
+ 𝜇
(
𝑤𝑠𝑡 − 𝜓𝑠𝑡
)
+ 𝜇
∫
ℎ
𝑔ℎ 𝑛ℎ𝑠𝑡
𝛿𝑤𝑠𝑡
𝛿𝑞
(𝑖)
𝑠𝑡
𝑑ℎ + 𝜆
∫
ℎ
1
�̄�ℎ
𝑛ℎ𝑠𝑡
𝛿𝑤𝑠𝑡
𝛿𝑞
(𝑖)
𝑠𝑡
𝑑ℎ.
This can be written as
1 −
𝜓𝑠𝑡
𝑤𝑠𝑡
=
(
1 − 𝜇 + 𝜇
∫
ℎ
𝑔ℎ
𝑛ℎ𝑠𝑡
𝑞
(𝑖)
𝑠𝑡
𝑑ℎ + 𝜆
∫
ℎ
1
�̄�ℎ
𝑛ℎ𝑠𝑡
𝑞
(𝑖)
𝑠𝑡
𝑑ℎ
)
1
𝜀
(𝑖)
𝑠𝑡
,
where 1
𝜀
(𝑖)
𝑠𝑡
= − 𝜕 ln𝑤𝑠𝑡
𝜕 ln 𝑞 (𝑖)𝑠𝑡
is the residual elasticity of demand faced by the firm. The iso-elastic
40
specification in Section implies
1
𝜀
(𝑖)
𝑠𝑡
=
𝑞
(𝑖)
𝑠𝑡
𝑞𝑠𝑡 + �̄�𝑠𝑒−𝛼𝑡
1
𝜎𝑠
.
Plugging this expression above and using the fact that 𝑞 (𝑖)𝑠𝑡 = 𝑞𝑠𝑡/𝑀𝑠 (from symmetry), we get (9).
41
Introduction
Model of labor-replacing AI
The Economy
A Tractable Example of Equilibrium with Socially-Minded Firms
Extensions
Scenarios for AI Transitions with Socially-Minded Firms
Equilibrium markups and AI deployment plans
Why do distributional considerations play such a small role?
Should more productive AI be priced differently?
What would a planner do?
Do our conclusions apply to occupations with high AI replacement risk?
Conclusion
Theory Extensions
Taxes and the safety net
AI as creating new goods and services
Competition among AI producers
Derivations and Proofs
