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---

### Model setup and agents
- Time: continuous time t.
- Commodities and skills: discrete sets j ∈ J and s ∈ S. Commodity j = 0 is the numeraire.
- Agents:
  - Financiers: mass ε, own firms, no labor endowments, consume the numeraire; financiers set r_t = r̄.
  - Regular households (superscript h): mass 1, endowed with skill vector n^h_t = ∫_{s∈S} n^h_{s t}, consume bundles c^h_t = ∫_{j∈J} c^h_{j t}, maximize lifetime utility subject to flow budget constraints and asset restriction a^h_t ∈ R. Prices p_t = ∫_{j∈J} p_{j t} (p_{0 t} = 1); wages w_t = ∫_{s∈S} w_{s t}; household wages w^h_t = w_t · n^h_t. The term a^h_t ∈ R captures potential constraints.
- AI replication:
  - AI can replicate labor inputs in subset A ⊂ S.
  - For s ∈ A: total input ℓ_{s t} = (∫_h n^h_{s t} dh) + q_{s t}.
  - For s ∉ A: ℓ_{s t} = ∫_h n^h_{s t} dh.
  - q_{s t} denotes units of AI-generated output, indistinguishable from worker output.

### AI production, computing resources, and feasibility
- Producing q_{s t} uses computing resources; model text: “To produce AI-generated output, the AI firm uses 1ùψ_{s t} units of computing resources, where ψ_{s t} denotes the efficiency of algorithms reproducing input s.”
- Computing resources x_t are produced one-to-one from the numeraire.
- Feasibility constraint for computing resources:
  - ∑_{s∈A} ψ_{s t} q_{s t} ≤ x_t.
- Goods production: y produced using labor (or AI) ℓ with technology F(y, ℓ) ≤ 0, constant returns to scale, operated competitively.
- Resource feasibility:
  - ∫_h c^h_{0 t} dh + x_t ≤ y_{0 t}
  - ∫_h c^h_{j t} dh ≤ y_{j t} for j ≠ 0.

### Equilibrium concept and AI firm profits
- Equilibrium given AI firm choices {q_{s t}, x_t}: prices {r_t, p_t, w_t}, consumption plans {c^h_t, c^0_t}, asset positions {a^h_t, a_t}, production plans y, ℓ such that consumers maximize, competitive firms maximize profits, and markets clear.
- Equilibrium AI firm profits at time t:
  - π_t = ∑_{s∈A} π_{w s t} − ψ_{s t} ∫ q_{s t} (as given in the source notation).

### Objectives of socially-oriented AI firms
- Firm objective V includes three components:
  - PDV of profits (profit-maximization motive).
  - Welfare term: PDV of ∫_h μ_h u^h dh, where μ_h are household-specific welfare weights; average welfare weight μ = ∫_h μ_h dh; normalized weights g_h = μ_h/μ − 1, with ∫_h g_h dh = 0.
  - Conservative / labor-market disruption term: weight λ times penalties for percent declines in labor income of household h relative to status quo ¯w_h; penalties apply to households with w^h_t < ¯w_h and are computed as the percent decline in labor income.
- Interpretation:
  - Profit motive tends to increase output and markups.
  - Efficiency motive (μ > 0) pushes prices toward marginal cost (P = MC).
  - Distributional motive (μ and g_h) can either restrict or expand AI depending on whether AI competes more with poor or rich households.
  - Conservative motive (λ) always reduces quantities to minimize labor-market disruptions for exposed households.

### First-order effect of a marginal deviation in AI quantities
- Variation δV from a deviation δ q_{s t}:
  - δV = ∫_t e^{−ρ t} { π term − μ ∫ (Σ_{s∈A} (w_{s t} − ψ_{s t}) δ q_{s t}) + μ ∫_h g_h Σ_s n^h_{s t} δ w_{s t} dh − λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) Σ_s n^h_{s t} δ w_{s t} dh } dt.
  - Definitions preserved: μ = ∫_h μ_h dh, g_h = μ_h/μ − 1, assumption 1 > μ so firm retains profit incentive.

### Proposition 1 — Modified Lerner Rule (quasi-linear case)
- In interior equilibria of the quasi-linear case, the socially minded firm produces q_{s t} until first-order conditions in the source hold; elasticity notation ε_{s s' t} ≡ −∂ ln q_{s t} / ∂ ln w_{s' t}, q_{s' t} = 0 for s' ∉ A.
- Special cases and compact intuition:
  - Profit-maximizing firm (μ = λ = 0): Lerner’s Rule L_{s t} = 1 / ε_{s t}.
  - Socially minded firm: Modified Lerner rule L_{s t} = M / ε_{s t}, where
    - M ≡ 1 − μ + μ ∫_h g_h n^h_{s t} q_{s t} dh + λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s t} q_{s t} dh.
  - Components of M:
    - "1": profit term.
    - "−μ": pushes toward lower markups (higher quantities).
    - μ ∫_h g_h n^h_{s t} q_{s t} dh: ambiguous sign (positive if AI harms poorer households → higher markups; negative if AI displaces richer workers → lower markups).
    - λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s t} q_{s t} dh: always positive, reduces quantities to limit disruption concentrated on low-wage households.

### Characterizations of firm objective types
- Autilitarian AI firm (μ > 0, g_h = 0, λ = 0):
  - L_{s t} = (1 − μ)/ε_{s t} — prices below profit-maximizing level, closer to marginal-cost pricing.
- Welfarist firm (μ > 0, g_h ≠ 0, λ = 0):
  - L_{s t} = (1 − μ + μ ∫_h g_h n^h_{s t} q_{s t} dh)/ε_{s t}.
- Conservative firm (μ = 0, λ > 0):
  - L_{s t} = (1 + λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s t} q_{s t} dh)/ε_{s t} — exceeds profit-maximizing level.
- General case with cross effects ε_{s s' t} ≠ 0: formula accounts for price effects on all workers and revenue from other AI products; μ = λ = 0 recovers standard multi-product Lerner formula.

### Key mechanisms and directional implications
- Distributional and stability motives can raise markups (restrict quantities) when AI disproportionately harms poorer workers.
- If AI mainly displaces higher-income workers, socially minded firms may lower prices and expand AI usage.
- Conservative motive λ always induces quantity restriction to reduce labor-market disruption; effect stronger when AI competes against low-wage households.
- Cross-product effects produce multi-product pricing interactions as in standard models.

### Firm tradeoff example and parameter choices
- Example statement from source: "the firm is willing to reduce profits by $ 1 if it raises wages for the average displaced worker by $ 2."
- Welfare weights g_h taken from Lockwood and Weinzierl(2016) used at reported income percentiles and interpolated.
- λ is rescaled by the average wage and set to λ = 0.5 in relevant scenarios.
- Variable costs include computational resources and residual costs; ψ_{s t} can vary across jobs.

### Equilibrium markups and AI deployment plans (empirical scenarios)
- Horizons reported: impact (t = 0), short run (t = 5 years), long run (t = 100 years).
- Benchmark profit-maximizing firm:
  - At t = 0, markups are around 32% and constant across occupations.
  - In the long run the AI firm becomes sole supplier of skills and charges a markup of =50% across the board (as presented in the source).
- Utilitarian firm:
  - Charges about 15% at t = 0.
  - Converges to 20% in the long run.
- Welfarist firm:
  - Distributional concerns cause a 0.1 percentage point higher markup at the bottom relative to utilitarian.
  - Lowers markups at the top by a full percentage point (from 15% to 14%) relative to utilitarian.
- Conservative firm:
  - Charges prices above profit-maximizing level to minimize labor-market impact; higher markups at the bottom; stability concerns vanish in long run.
- Multi-objective firm:
  - At impact charges a 33% markup on AIs competing against low-paying workers and a 15% markup on AIs substituting high-paying occupations.
  - In the long run converges to a common 20% markup.

### Decomposition of motives driving markups and quantities
- Profit motives push for high markups, especially for high-paying jobs with less worker competition.
- Aggregate efficiency considerations uniformly lower markups to improve access.
- Distributional considerations have negligible impact at the bottom and call for lower markups at the top.
- Wage stability concerns increase prices and curb quantities, especially for AIs replacing low-wage jobs.
- Equilibrium quantities across scenarios:
  - AI-produced quantities range from one to four times baseline pre-AI production levels.
  - Utilitarian firm generates the maximum increase in AI usage.
  - Conservative and profit-maximizing firms restrict quantities the most.

### Why distributional considerations play a small role (empirical reasons)
- Two forces reduce the role of distributional motives:
  1. Strength of distributional motives vanishes as AI use deepens (e.g., AI output for utilitarian and welfarist firms is already 4 times the baseline supplied by workers).
  2. Distributional effects of changing wages in a given occupation are small, especially at the bottom.
- Average Pareto weight for occupation s computed as: Average Pareto Weight_s = ∑_h g_h ̄n_h_s / ̄n_s (as in source).
- Empirical findings reported exactly:
  - All occupations in the bottom half of the pay distribution have positive average weights, but these weights are small and close to zero.
  - "Increasing income in jobs at the bottom carries a tiny distributional gain of 1 cent for every dollar."
  - Increasing income in jobs at the top carries a distributional penalty of 12 cents for every dollar.
  - Example: Cashiers — average employee household at the 42nd percentile; employees as cashiers span incomes up to the 70th percentile.
  - Example: Economists — average economist located at the 90th percentile.

### Sensitivity to much stronger redistributive preferences
- If welfare weights are ten times those estimated (̃g_h = 10 × g_h):
  - For bottom occupations: stronger distributional considerations call for one percentage point higher markups relative to utilitarian.
  - Even with 10× weights, incentive to increase aggregate efficiency remains dominant; welfarist firm still expands AI quantity to broaden access.
  - For top occupations: welfarist firm with 10× distributional concerns should charge markups that are half of what a utilitarian firm would charge and produce 20% more output.
- Summary: stronger distributional concerns matter at the top but have no significant implications for AIs substituting bottom jobs.

### Pricing more productive AI
- Baseline: AIs replacing workers at 50% of their cost. Consider more productive AI with ψ_s = 0.2 ̄w_s (AI replaces workers at 20% of their cost).
- At t = 0 for ψ_s = 0.2:
  - Markups are slightly higher relative to baseline.
  - Quantities are an order of magnitude larger.
  - More productive AI reduces worker competition; firm captures larger market share and can charge higher markups.
  - Distributional and stability motives are weaker: conservative firm outcomes close to profit-maximizer; welfarist and multi-objective outcomes close to utilitarian.
- Conclusion: as AI becomes more productive and less costly, distributional and stability considerations become less pressing; a socially minded firm with sufficiently productive AI behaves essentially as a utilitarian one.

### Social planner vs socially minded firm
- Planner objective: maximize social welfare and maintain labor market stability, giving weight μ to financiers equal to the average household.
- Planner supplies AI until the planner first-order condition reproduced in the source holds (equation in the text).
- Planner trades off reduction in aggregate efficiency from lowering output with distributional and stability gains.
- Implied optimal planner markups at impact (t = 0):
  - Conservative social planner with μ = λ (as multi-objective firm) would set an optimal markup at the bottom of 7% and at the top of less than 1% on impact.
  - Over time, as distributional and stability concerns subside, the planner would impose no markups/taxes and implement a competitive outcome.
- Key result: planner allocation features lower AI prices than a responsible-AI firm with same social objectives; socially responsible firms remain constrained by private profit incentives.

### Applicability to occupations with high AI replacement risk
- Using Eloundou et al.(2023) task exposure data:
  - Occupations at risk do not stand out in distributional or stability considerations.
  - Panel A: highly exposed occupations have average Pareto weights near zero.
  - Panel B: average cost of disruptions among employees does not systematically vary with AI exposure.
  - Reason: occupations at risk are spread throughout the income distribution, not concentrated at bottom or top.
- Conclusion: earlier conclusions apply equally to the subset of occupations more at risk of substitution by AIs.

### Extensions: taxes, new goods, and competition (A.1–A.3)
- A.1 Taxes and the safety net:
  - After-tax labor income of household h: After-tax labor income_h_t = Tπ(w_h_t), Tπ'(·) = marginal tax rate; firm stability term depends on reductions in Tπ(w_h_t) relative to Tπ( w̄_h ).
  - Proposition 3 (qualitative): In the quasi-linear case with government taxes, the socially-responsible firm produces q_{s t} until L_{s t} satisfies equation (7) in the source.
  - Main implication: A more progressive tax system (lower Tπ'(w_h_t)) weakens the firm’s distributional and stability concerns; full redistribution (Tπ'(w_h_t) = 0) eliminates these private concerns.
- A.2 AI creating new goods and services:
  - Allow AIs to produce new goods s′ ∈ N with n_h_s′ = 0 ∀ h.
  - Proposition 4: For novel AI-produced goods, optimal pricing satisfies L_{s′ t} = (1 − μ)^(1/σ_{s′}) (equation (8) in source).
  - Implication: novel AI goods broaden access without disrupting existing labor markets; socially minded firms should price close to marginal cost.
- A.3 Competition among AI producers:
  - For skill s ∈ A with M_s > 1 identical Cournot firms:
    - Proposition 5: equilibrium price L_{s t} given by equation (9) in source; increased M_s divides the (1 − μ) term by M_s.
    - Comparative statics: more competition moves prices closer to marginal cost and makes distributional/stability concerns relatively more important for socially minded firms.

### Derivations and proofs (appendix B summary)
- Derivations use financier indifference r_t = ρ, variations δV and δS, envelope theorem and competitive zeros profits for j ≠ 0, quasi-linearity, and ∫_h g_h dh = 0 to simplify first-order conditions and derive Propositions 1–5 (sketched in source).

### Conclusion (Section 4) — principal messages
- Extended Lerner’s Rule to incorporate profits, social welfare, and labor-market disruption; derived pricing formulas showing how these objectives shape markups, deployment speed, and access.
- Main empirical findings for US occupations:
  - A firm that cares about welfare and stability equally should price near profit-maximizing level in short run, closer to marginal cost over time.
  - Gradual deployment limits short-run disruptions and balances improved access in medium run.
  - Most pro-social course for a monopolist AI firm is to refrain from exploiting market power except in very short run when stability matters most.
- Dependence on competition:
  - If AI firms face little competition, incentive to broaden access by lowering prices dominates.
  - If competition already yields near-efficient quantities, distributional and stability concerns dominate.
- Scope limitation: analysis focuses on labor market and economic efficiency, abstracting from broader societal and existential risks; conclusions apply only to well-aligned or narrow AIs without existential risks.

*Content summarized from the provided PDF chapter/section wpiea2025234-source-pdf.*

### 2.1  The Economy

### 2.1  The Economy

### Model setup and agents
- Time: continuous time t.
- Commodities and skills: discrete sets j ∈ J and s ∈ S. Commodity j = 0 is the numeraire.
- Agents:
  - Financiers: mass ε, own firms, no labor endowments, consume the numeraire. They maximize expected discounted utility with interest rate r_t (financiers set r_t = r̄).
  - Regular households (superscript h): mass 1, endowed with a vector of skills n^h_t = ∫_{s∈S} n^h_{s t}, consume commodity bundles c^h_t = ∫_{j∈J} c^h_{j t}, and maximize lifetime utility subject to flow budget constraints and asset restrictions a^h_t ∈ R. Prices: p_t = ∫_{j∈J} p_{j t} (with p_{0 t} = 1) and wages w_t = ∫_{s∈S} w_{s t}, with household wages w^h_t = w_t · n^h_t. The term a^h_t ∈ R captures potential constraints (assumed independent of prices).
- AI replication: AI can replicate labor input in subset A ⊂ S. For s ∈ A, total input ℓ_{s t} = (∫_h n^h_{s t} dh) + q_{s t}; otherwise ℓ_{s t} = ∫_h n^h_{s t} dh. q_{s t} denotes units of AI-generated output, assumed indistinguishable from worker output.

### AI production, computing resources, and feasibility
- AI computing: producing q_{s t} requires (1 − ψ_{s t}) units? (model text: “To produce AI-generated output, the AI firm uses 1ùψ_{s t} units of computing resources, where ψ_{s t} denotes the efficiency of algorithms reproducing input s.”) Computing resources x_t are produced one-to-one from the numeraire commodity.
- Feasibility constraint for computing resources:
  - ∑_{s∈A} ψ_{s t} q_{s t} ≤ x_t.
- Goods production: commodities y are produced using labor (or AI) ℓ with plans y = ∫_{j∈J} y_{j t} and ℓ = ∫_{s∈S} ℓ_{s t}, technology F(y, ℓ) ≤ 0, constant returns to scale, operated competitively.
- Resource feasibility:
  - ∫_h c^h_{0 t} dh + x_t ≤ y_{0 t}
  - ∫_h c^h_{j t} dh ≤ y_{j t} for j ≠ 0.

### Equilibrium concept
- Equilibrium given AI firm choices {q_{s t}, x_t}: prices {r_t, p_t, w_t}, consumption plans {c^h_t, c^0_t}, asset positions {a^h_t, a_t}, and production plans y, ℓ such that:
  - Consumers maximize utility subject to constraints.
  - Competitive firms operating F maximize profits taking prices as given.
  - Commodity markets and asset markets clear.
- Equilibrium profits for the AI-producing firm at time t:
  - π_t = ∑_{s∈A} π_{w s t} − ψ_{s t} ∫ q_{s t} (as in source notation).

### Objectives of socially-oriented AI firms
- AI firm objective V includes three terms:
  - PDV of profits: captures profit-maximization motives.
  - Welfare term: PDV of ∫_h μ_h u^h dh, where μ_h are household-specific welfare weights (larger weight on poorer households). The average welfare weight is μ = ∫_h μ_h dh and normalized weights g_h = μ_h/μ − 1 satisfy ∫_h g_h dh = 0.
  - Conservative / labor-market disruption term: weight λ times penalties for percent declines in labor income of household h relative to status quo ¯w_h. Penalties apply to households with w^h_t < ¯w_h and are computed as the percent decline in labor income.
- Interpretation of motives:
  - Profit motive (first term): favors higher output and markups.
  - Efficiency motive (welfare μ > 0 second term): pushes prices toward marginal cost (P = MC).
  - Distributional motive (term involving μ and g_h): can push to restrict AI if AI competes more intensely against poor households (positive contribution) or expand AI if it benefits poorer households (negative contribution).
  - Conservative motive (λ term): always positive and reduces quantities to minimize labor-market disruptions across all households exposed to AI.

### First-order effects of a marginal deviation in AI quantities
- Change in firm objective from a deviation δ q_{s t}:
  - δV = ∫_t e^{−ρ t} { π term − μ ∫ (Σ_{s∈A} (w_{s t} − ψ_{s t}) δ q_{s t}) + μ ∫_h g_h Σ_s n^h_{s t} δ w_{s t} dh − λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) Σ_s n^h_{s t} δ w_{s t} dh } dt.
  - Definitions: μ = ∫_h μ_h dh, g_h = μ_h/μ − 1. Sign of g_h represents distributional motives.
  - Assumption: 1 > μ so firm retains profit incentive.

### Proposition 1 — Modified Lerner Rule (quasi-linear case)
- In interior equilibria of the quasi-linear case, the socially minded firm produces q_{s t} until:
  - L_{s t} = [ Σ_{s'} ( (1 − μ) ∫ q_{s'0 t} w_{s'0 t} q_{s' t} w_{s' t}^{−1} μ ∫_h g_h n^h_{s'0 t} w_{s'0 t} q_{s' t} w_{s' t}^{−1} dh + λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s'0 t} w_{s'0 t} q_{s' t} w_{s' t}^{−1} dh ) ]^{1/ε_{s' s t} }
  - (Source presents the full expression in equation (2) with ε_{s s' t} ≡ −∂ ln q_{s t} / ∂ ln w_{s' t}. q_{s' t} = 0 for s' ∉ A.)
- Special cases and intuition:
  - If cross-demand elasticity is 0 for s ≠ s', own-demand elasticity ε_{s t} > 0 remains in formula.
  - Profit-maximizing firm (μ = λ = 0): Lerner’s Rule L_{s t} = 1 / ε_{s t}. (Equation (3).)
  - Socially minded firm: Modified Lerner rule L_{s t} = M / ε_{s t}, where
    - M ≡ 1 − μ + μ ∫_h g_h n^h_{s t} q_{s t} dh + λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s t} q_{s t} dh.
  - Components of M:
    - "1": profit maximization term.
    - "−μ": pushes toward lower markups and higher quantities (raises access for efficiency).
    - μ ∫_h g_h n^h_{s t} q_{s t} dh: ambiguous sign; positive when AI competes more with poor households (leads to higher markups/restriction), negative if AI displaces rich workers (leads to lower markups/expansion).
    - λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s t} q_{s t} dh: always positive, reduces quantities to minimize labor-market disruptions; stronger when AI competes against low-wage households.

### Scenarios of firm objectives (characterizations)
- Autilitarian AI firm (μ > 0, g_h = 0, λ = 0):
  - L_{s t} = (1 − μ)/ε_{s t} — prices below profit-maximizing level, closer to marginal-cost pricing.
- Welfarist firm (μ > 0, g_h ≠ 0, λ = 0):
  - L_{s t} = (1 − μ + μ ∫_h g_h n^h_{s t} q_{s t} dh)/ε_{s t}.
- Conservative firm (μ = 0, λ > 0):
  - L_{s t} = (1 + λ ∫_{h: w^h_t < ¯w_h} (1/¯w_h) n^h_{s t} q_{s t} dh)/ε_{s t}, which exceeds the profit-maximizing level.
- General case with cross effects ε_{s s' t} ≠ 0: formula accounts for price effects on all workers and revenue from other AI products; μ = λ = 0 recovers standard multi-product Lerner formula.

### Key implications and mechanisms
- Distributional and labor-market stability motives can lead socially minded firms to restrict quantities (raise markups) relative to profit-maximizing firms when AI disproportionately harms poorer workers.
- Conversely, when AI mostly displaces higher-income workers, socially minded firms may lower prices and expand AI usage.
- Conservative motive (λ) always induces quantity restriction to reduce labor-market disruption.
- With cross-product effects, standard multi-product pricing interactions appear.

*Italicized source attribution line*

### 2. The values imply that the firm is willing to give $ 1 of profits to increase

### wpiea2025234-source-pdf - 2. The values imply that the firm is willing to give $ 1 of profits to increase

### Firm objectives, welfare weights, and parameter choices
- The firm tradeoff example: "the firm is willing to reduce profits by $ 1 if it raises wages for the average displaced worker by $ 2."
- Welfare weights푔
ℎ are taken from Lockwood and Weinzierl(2016) and used directly at reported income percentiles; weights are interpolated to span the income distribution.
- In relevant scenarios, 휆 is rescaled by the average wage in the economy to ensure equal scale across terms and set to 휆=0.5.
- Variable costs include computational resources to run the AI and replicate human skills푠, plus residual costs for integration, prompting, or inspection; 휓
푠푡 can vary across jobs.

### Equilibrium markups and AI deployment plans (Section 3.1)
- Markups are reported for five firm types across three horizons: impact (푡=0), short run (푡=5 years), long run (푡=100 years).
- Benchmark profit-maximizing firm:
  - At 푡=0, markups are around 32% and constant across occupations (assumes common productivity improvement).
  - In the long run, the AI firm becomes sole supplier of skill푠 and charges a markup of 휎
푠 ùπ휎
푠1 ∫ =50% across the board.
- Utilitarian firm:
  - Charges about 15% at 푡=0.
  - Converges to 20% in the long run (lowers prices below profit-maximizing level to expand access and increase aggregate efficiency).
- Welfarist firm:
  - Prioritizes aggregate efficiency and distributional concerns.
  - Distributional concerns cause a 0.1 percentage point higher markup at the bottom relative to utilitarian.
  - Lowers markups at the top by a full percentage point (from 15% to 14%) relative to utilitarian.
- Conservative firm:
  - Balances profits against labor-market stability, charging prices above profit-maximizing level to minimize labor-market impact.
  - Higher markups at the bottom (low-wage occupations) to reduce disruptions; stability concerns vanish in long run.
- Multi-objective firm:
  - Balances profit, efficiency, redistribution, and stability.
  - Charges a 33% markup on AIs competing against low-paying workers and a 15% markup on AIs substituting high-paying occupations at impact.
  - In the long run converges to a common 20% markup.

### Decomposition of motives driving markups (Figure 4 summary)
- Profit motives push for high markups, especially for high-paying jobs with less worker competition.
- Aggregate efficiency considerations uniformly lower markups to improve access.
- Distributional considerations have negligible impact at the bottom and call for lower markups at the top.
- Wage stability concerns increase prices and curb quantities, especially for AIs replacing low-wage jobs.
- Result: multi-objective firm markups reflect the net of these motives, with profit motives dominant at the top and stability motives important at the bottom.

### Equilibrium quantities (Figure 5 summary)
- AI-produced quantities range from one to four times baseline pre-AI production levels.
- Utilitarian firm generates the maximum increase in AI usage.
- Conservative and profit-maximizing firms restrict quantities the most.

### Why distributional considerations play a small role (Section 3.2)
- Two forces:
  1. Strength of distributional motives vanishes as AI use deepens (e.g., AI output for utilitarian and welfarist firms is already 4 times the baseline supplied by workers).
  2. Distributional effects of changing wages in a given occupation are small, especially at the bottom.
- Average Pareto weight for an occupation s computed as: Average Pareto Weight 푠 = ∑
ℎ 푔
ℎ ̄푛
ℎ
푠 ̄푛
푠 .
- Empirical findings:
  - All occupations in the bottom half of the pay distribution have positive average weights, but these weights are small and close to zero.
  - "Increasing income in jobs at the bottom carries a tiny distributional gain of 1 cent for every dollar."
  - Increasing income in jobs at the top carries a distributional penalty of 12 cents for every dollar.
  - Example: Cashiers (one of the 5% lowest paying jobs) — average employee household is at the 42nd percentile; people employed as cashiers span incomes up to the 70th percentile.
  - Example: Economists (one of the 5% highest paid jobs) — average economist located at the 90th percentile.

### Sensitivity to much stronger redistributive preferences (Figure 7)
- If welfare weights are ten times those estimated (̃푔
ℎ =10×푔
ℎ):
  - For bottom occupations: stronger distributional considerations call for one percentage point higher markups relative to utilitarian.
  - Even with 10× weights, incentive to increase aggregate efficiency remains dominant; welfarist firm still expands AI quantity to broaden access.
  - For top occupations: welfarist firm with 10× distributional concerns should charge markups that are half of what a utilitarian firm would charge and produce 20% more output.
- Summary: stronger distributional concerns matter at the top but have no significant implications for AIs substituting bottom jobs.

### Pricing more productive AI (Section 3.3)
- Baseline considered AIs replacing workers at 50% of their cost. Consider more productive AI with 휓
푠 =0.2 ̄푤
푠 (AI replaces workers at 20% of their cost).
- At 푡=0 for 휓
푠 =0.2:
  - Markups are slightly higher relative to baseline.
  - Quantities are an order of magnitude larger.
  - More productive AI reduces worker competition, so firm captures larger market share and can charge higher markups.
  - Distributional and labor-market stability motives are weaker: conservative firm outcomes close to profit-maximizer; welfarist and multi-objective outcomes close to utilitarian.
- Conclusion: as AI becomes more productive and less costly, distributional and stability considerations become less pressing; a socially minded firm with sufficiently productive AI behaves essentially as a utilitarian one.

### Social planner vs socially minded firm (Section 3.4)
- Planner objective: maximize social welfare and maintain labor market stability, giving weight 휇 to financiers equal to the average household.
- Planner supplies AI until:
  1 휓
푠푡 푤
푠푡 =
(
∫
ℎ 푔
ℎ ̄푛
ℎ
푠 푞
푠푡 푒훼푡 dℎ ∏ 휆 휇 ∫
ℎ 1 ̄푤
ℎ ̄푛
ℎ
푠 푞
푠푡 푒훼푡 dℎ ) 푞
푠푡 푞
푠푡 ∏ ̄푛
푠 푒훼푡 1 휎
푠 (equation reproduced as in source).
- Planner trades off reduction in aggregate efficiency from lowering output below competitive level with distributional and stability gains.
- Implied optimal planner markups (or taxes) at impact (푡=0):
  - Conservative social planner with 휇=휆 (as multi-objective firm) would set an optimal markup at the bottom of 7% and at the top of less than 1% on impact.
  - Over time, as distributional and stability concerns subside, the planner would impose no markups/taxes and implement a competitive outcome.
- Key result: planner allocation features lower AI prices than a responsible-AI firm with same social objectives; socially responsible firms remain constrained by private profit incentives.

### Applicability to occupations with high AI replacement risk (Section 3.5)
- Using data from Eloundou et al.(2023) on share of core tasks automatable by LLM-powered systems:
  - Occupations at risk do not stand out in distributional or stability considerations.
  - Panel A: highly exposed occupations have average Pareto weights near zero.
  - Panel B: average cost of disruptions among employees does not systematically vary with AI exposure.
  - Reason: occupations at risk are spread throughout the income distribution, not concentrated at bottom or top.
- Conclusion: earlier conclusions apply equally to the subset of occupations more at risk of substitution by AIs.

### Conclusion (Section 4)
- Extended Lerner’s Rule to consider profits, social welfare, and labor-market disruption; derived pricing formulas showing how these objectives shape markups, deployment speed, and access.
- Main empirical findings for US occupations:
  - A firm that cares about welfare and stability equally should price near profit-maximizing level in short run, closer to marginal cost over time.
  - Gradual strategy limits short-run disruptions and balances improved access in medium run.
  - Most pro-social course for a monopolist AI firm is to refrain from exploiting market power except in very short run when stability matters most.
- Dependence on competition:
  - If AI firms face little competition, incentive to broaden access by lowering prices dominates.
  - If competition already yields near-efficient quantities, incentive to broaden access is less relevant; distributional and stability concerns dominate.
- Scope limitation: analysis focuses on labor market and economic efficiency, abstracting from broader societal and existential risks; conclusions apply only to well-aligned or narrow AIs without existential risks.

*Italic: Content derived from the provided PDF chapter/section.*

### References

### References

### A Theory Extensions — overview
- This section extends the baseline model to consider three main cases:
  - How taxes and the safety net affect deployment choices by socially minded AI firms (A.1).
  - The possibility that AIs create entirely new goods and services rather than merely substituting human skills (A.2).
  - Competition among multiple AI firms competing à la Cournot (A.3).

### A.1 Taxes and the safety net — key results
- Setup:
  - After-tax labor income of household h: After-tax labor income_h_t = Tπ(w_h_t), where T_t is a common transfer balancing the government budget and Tπ(·) is an increasing tax function with Tπ'(·) = marginal tax rate; Tπ'(·)=0 at some reference and 1 − Tπ' (w_h_t) > 0 gives marginal tax rate experienced by households at different income points.
  - Firm objective V modified to include stability term that depends on reductions in after-tax labor income Tπ(w_h_t) relative to status quo Tπ( w̄_h ).
- Proposition 3 (qualitative statement):
  - In the quasi-linear case with government taxes, a socially-responsible firm produces q_st until L_st satisfies equation (7) as stated in the source.
  - Main implication: A more progressive tax system (as evidenced by a lower Tπ'(w_h_t)) weakens the firm’s distributional and stability concerns.
  - Extreme case: Full redistribution (i.e., Tπ'(w_h_t)=0) eliminates distributional and stability considerations; policy implication is that public redistribution substitutes for some private self-regulation by AI firms.

### A.2 AI as creating new goods and services — key results
- Setup:
  - Allow AIs to produce new goods/services N (indexed s′ ∈ N) that humans previously could not produce (n_h_s′ = 0 ∀ s′ ∈ N).
  - Household utility specified with nested CES structure; σ_s > 1 for goods.
- Proposition 4 (pricing rule):
  - For novel AI-produced goods s′ ∈ N, the optimal pricing by a socially-minded firm satisfies a modified Lerner rule:
    - L_s′_t = (1 − μ)^(1/σ_s′)  (equation (8) in the source).
- Main implication:
  - AIs that expand the range of goods and services benefit all workers without disrupting existing labor markets, raising no distributional or stability concerns.
  - For this class of AIs, the primary responsibility of a socially minded firm is to price close to marginal cost and broaden access.

### A.3 Competition among AI producers — key results
- Setup:
  - For each skill s ∈ A, suppose M_s > 1 identical firms produce the AI and compete in quantities (Cournot).
- Proposition 5 (equilibrium price under Cournot competition):
  - Equilibrium price L_s_t satisfies the expression given in equation (9) in the source, where q_s_t is aggregate AI quantity used in s ∈ A.
  - Comparative statics:
    - Competition forces prices closer to marginal cost via division of the (1 − μ) term by M_s.
    - As competition intensifies (M_s increases), distributional and stability concerns become increasingly important relative to broadening access: competition already pushes prices toward marginal cost and aggregate efficiency, so the remaining deviation from profit-maximizing pricing by socially minded firms is driven by distributional/stability motives.
  - Contrast: A pure-profit-maximizing firm would price according to L_s_t = 1/M_s * q_s_t / q_s_t * ... (as stated in the source); more competition → price closer to marginal cost.

### B Derivations and Proofs — key derivations and logical structure
- Derivation of equation (1):
  - Start with financier indifference r_t = ρ.
  - Compute variation δV from arbitrary change in AI firm quantities; expression decomposes into:
    - Changes in profits to financiers.
    - Changes in households’ utility (marginal value of income e^(−ρ t) × μ_h).
    - Effects via labor market disruptions (function of wages).
  - Use definitions ℓ_s_t = ∫_h n_h_s_t dh for s ∈ A and market clearing for commodities j ≠ 0.
  - Competitive production with constant returns to scale → zero profits and envelope theorem yield cancellation: ∑_{j≠0} y_j_t δp_j_t − ∑_s ℓ_s_t δw_s_t = 0.
  - Quasi-linearity (c_h_j_t = y_j_t for j ≠ 0) and ∫_h g_h dh = 0 further simplify to the form of equation (1).
- Derivation of planner condition (equation (6)):
  - Define planner objective S and compute δS from perturbations in AI quantities.
  - Apply same cancellations as in the private firm case (zero profits, envelope theorem, quasi-linearity).
  - Set δS = 0 at optimum; rearrange to obtain planner first-order conditions that lead to equation (6) in the simplified economy.
- Proof sketches for Propositions:
  - Proposition 1: Interior equilibrium (q_s_t > 0) implies δV = 0; rearranging yields expression (2).
  - Proposition 2: Uses formula in Proposition 1 plus elasticity expression ε_s_t = q_s_t / (q_s_t ∏ ̄n_s e^(−α_t)) × σ_s; derives demand curve by equating MRS to price w_s.
  - Proposition 3: Follows from computing δV with taxes and transfers, substituting δT_t (rebate condition), and applying previous derivation steps to obtain the tax-augmented variant of equation (1); leads to the result summarized in A.1.
  - Proposition 4: For novel AIs s′ ∈ N with n_h_s′ = 0, compute δV and apply same steps to show optimal pricing satisfies modified Lerner rule (8).
  - Proposition 5: For Cournot competition with M_s symmetric firms:
    - Let q^{(i)}_{s t} denote firm i’s supply; compute δV^{(i)} and apply same cancellations.
    - First-order condition yields residual elasticity 1/ε^{(i)}_{s t} = −∂ ln w_s_t / ∂ ln q^{(i)}_{s t}.
    - With iso-elastic specification, 1/ε^{(i)}_{s t} = q^{(i)}_{s t} / q_s_t ∏ ̄n_s e^(−α_t) × 1/σ_s.
    - Symmetry q^{(i)}_{s t} = q_s_t / M_s leads to equation (9) and the comparative-static conclusions summarized in A.3.

*Italic: Content summarized from the source PDF wpiea2025234-source-pdf (References and Appendices A–B).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025234-source-pdf.pdf_