Compositional sources of changes in agency AI exposure

Before examining regression evidence, it is necessary to establish what drives variation in agency-level AI exposure. Our measure is constructed from fixed occupation-level scores developed by Felten et al. (Reference Felten, Raj and Seamans2021) that do not vary over time. Consequently, any observed change in agency AI exposure must arise entirely from workforce composition shifts – specifically, employment reallocation across occupations with different exposure levels – rather than from AI adoption, technological diffusion, or changes in task execution. The regression analysis therefore, relates employment pattern changes to exposure changes driven by occupational reweighting, documenting systematic workforce reallocation under institutional constraints. It does not, and cannot, identify causal effects of AI technology deployment.

The identification logic underlying this approach is made explicit by the construction of the exposure measure. We calculate agency AI exposure as a share-weighted average of occupational exposure scores:

${A_{it}}\; = \;\mathop \sum \limits_o {w_{iot}}\,{a_o},\,{w_{iot}}\; = \;{{{\rm{employmen}}{{\rm{t}}_{iot}}} \over {\mathop \sum \nolimits_{o'} {\rm{employmen}}{{\rm{t}}_{io't}}}},$

where $i$ indexes agencies, $o$ occupations, and $t$ quarters. The crucial feature is that ${a_o}$ – the occupation-specific AI exposure score from Felten et al. (Reference Felten, Raj and Seamans2021) – remains fixed over time in our data.Footnote ³ This means all changes in an agency’s AI exposure must come from staff moving between occupations, not from changes in how we measure AI susceptibility.

We can formalise this by decomposing quarterly changes in exposure:

${\rm{\Delta }}{A_{it}}\; = \;\underbrace {\mathop \sum \limits_o {a_o}\,{\rm{\Delta }}{w_{iot}}}_{{\rm{COM}}{{\rm{P}}_{it}}} + \underbrace {\mathop \sum \limits_o {w_{io,t - 1}}\,{\rm{\Delta }}{a_{ot}}}_{{\rm{TEC}}{{\rm{H}}_{it}}} + \underbrace {\mathop \sum \limits_o {\rm{\Delta }}{a_{ot}}\,{\rm{\Delta }}{w_{iot}}}_{{\rm{CROS}}{{\rm{S}}_{it}}}$

Since ${a_o}$ is constant, ${\rm{\Delta }}{a_{ot}} = 0$ , which means ${\rm{TEC}}{{\rm{H}}_{it}} = {\rm{CROS}}{{\rm{S}}_{it}} = 0$ . The entire change in exposure comes from the composition term:

${\rm{COM}}{{\rm{P}}_{it}}\; = \;\mathop \sum \limits_o {a_o}\,{\rm{\Delta }}{w_{iot}}.$

To visualise these patterns, we plot the cumulative composition changes:

${C_{it}}\; = \;\mathop \sum \limits_{\tau \le t} {\rm{COM}}{{\rm{P}}_{i\tau }},$

which tracks the net shift of employment weight toward or away from AI-exposed occupations within each agency over time.

Figure 1 shows cumulative changes for the twelve largest agencies, revealing considerable heterogeneity in workforce composition evolution. Several agencies – including the Air Force, Navy, and Veterans Affairs – show steady increases in AI-exposed employment, reflecting sustained shifts toward higher-exposure occupations. Others, like Agriculture and Interior, oscillate around zero with no clear trend, indicating temporary rather than systematic changes. Sharp discontinuities appear in some agencies – the Army’s step change in 2020 and Defense’s reversal around 2021 – likely reflecting discrete reorganisations or classification changes rather than gradual compositional shifts. Homeland Security, Health and Human Services, and Justice show sustained positive trends, whilst the Social Security Administration exhibits negative trends before partially recovering. These diverse patterns highlight how agencies with different missions, structures, and constraints respond differently to technological pressures.

Figure 1.

Cumulative composition: top 12 agencies by size. Each panel shows cumulative changes in AI exposure from occupational reweighting: ${C_{it}} = \mathop \sum \limits_{\tau \le t} \mathop \sum \limits_o {a_o}\,\Delta {w_{iot}}$ , where ${a_o}$ is the fixed occupational AIOE score and ${w_{iot}}$ is the employment share. Positive values indicate net shifts toward more AI-exposed occupations since 2019Q1. Panel-specific scales emphasise within-agency dynamics. Agencies: Air Force (AF), Agriculture (AG), Army (AR), Defense (DD), Justice (DJ), Health and Human Services (HE), Homeland Security (HS), Interior (IN), Navy (NV), Social Security Administration (SZ), Treasury (TR) and Veterans Affairs (VA).

Substitution effects: routine employment

We begin by examining how within-agency reallocation of employment across occupations with different AI exposure levels is associated with changes in routine employment shares. Because occupation-level exposure scores are fixed by construction, changes in agency-level AI exposure do not reflect technology adoption or diffusion. Instead, they summarise contemporaneous shifts in workforce composition across occupations. The empirical question addressed in this subsection is therefore not whether AI adoption causes changes in routine employment, but how compositional reweighting towards more AI-exposed occupations maps onto changes in the share of routine work within agencies.

Our approach estimates within-agency regressions that relate changes in the routine employment share to changes in agency-level AI exposure. This changes-on-changes specification is appropriate in this setting because both variables are generated by within-agency occupational reallocation, and the regression quantifies how that reallocation loads on routine versus non-routine categories under fixed institutional constraints. The baseline specification is as follows:

(1) ${\rm{\Delta }}{R_{it}} = {\beta _A}{\rm{\Delta }}{A_{it}} + {\beta _w}{\rm{\Delta ln}}\left( {{\rm{salar}}{{\rm{y}}_{it}}} \right) + {\beta _s}{\rm{\Delta ln}}\left( {{\rm{headcoun}}{{\rm{t}}_{it}}} \right) + {\lambda _t} + {\varepsilon _{it}}$

where ${R_{it}}$ is the share of routine work – comprising administrative, clerical, and blue-collar positions – and ${A_{it}}$ is the artificial intelligence exposure of agency $i$ at quarter-year $t$ . Quarter-year fixed effects ${\lambda _t}$ control for period-specific shocks common across agencies, and standard errors are clustered at the agency level.

The results in Table 1 document a consistent negative association between AI exposure changes and routine employment shares. Across all specifications, higher within-agency changes in AI exposure are associated with declines in the routine share: the coefficient on ${\rm{\Delta }}{A_{it}}$ is negative and precisely estimated at $ - 0.301$ (s.e. 0.030) in the baseline specification, attenuating to $ - 0.198$ (s.e. 0.049) as increasingly rich composition controls are added. The magnitude indicates that a one standard deviation increase in quarterly AI exposure (0.0718) is associated with a 1.42 percentage point decline in routine employment shares, with this relationship remaining robust across specifications that control for compositional shifts in age, education, and employment arrangements. A 0.10 increase in ${\rm{\Delta }}{A_{it}}$ (approximately 1.39 standard deviations) corresponds to a 1.98 percentage point decline in the routine employment share.

Table 1.

Substitution: $\Delta $ AIOE on $\Delta $ routine share

Notes: The dependent variable is the within-agency quarterly change in the routine employment share, ${\rm{\Delta }}{R_{it}}: = {R_{it}} - {R_{i,t - 1}}$ . The key regressor is the change in AI exposure, ${\rm{\Delta }}{A_{it}}$ (‘AIOE’). All specifications are estimated on first differences with a full set of quarter-year fixed effects $\left\{ {{\lambda _t}} \right\}$ ; agency time-invariant heterogeneity is removed by differencing. Core controls, included in every column, are ${\rm{\Delta ln}}\left( {{\rm{salar}}{{\rm{y}}_{it}}} \right)$ (change in average salary) and ${\rm{\Delta ln}}\left( {{\rm{headcoun}}{{\rm{t}}_{it}}} \right)$ (change in reported headcount). Column (2) augments the baseline with vectors of changes in the age and education distributions (entered as changes in category shares; one category omitted per partition). Column (3) further adds vectors of changes in tenure and appointment/scheduling type shares. Compositional-control coefficients are included in the regressions but not reported. Standard errors, reported in parentheses, are clustered at the agency level. Observations are agency-quarter differences; agencies must appear in consecutive quarters to contribute an observation. Sample size declines in column (3) reflect missingness in tenure/appointment distributions. Changes in employment shares are measured in percentage points, while log-differenced variables represent proportional changes. Significance levels: ${^{\rm{\ast}}}$ $p \lt 0.10$ , ${^{{\rm{\ast\ast}}}}$ $p \lt 0.05$ , ${^{{\rm{\ast\ast\ast}}}}$ $p \lt 0.01$ .

Complementarity effects: expert employment

We next examine how the same within-agency compositional reallocation described in the ‘Substitution effects: routine employment’ section.2 maps onto changes in expert employment shares. As before, changes in agency-level AI exposure summarise shifts in occupational employment weights rather than the adoption or diffusion of AI technologies. The empirical question here is whether reweighting towards more AI-exposed tasks is accompanied by expansion of expert roles, which comprise professional and technical positions within agencies.

Using a changes-on-changes specification that mirrors the routine employment analysis, we relate changes in the expert employment share to changes in agency-level AI exposure. The baseline specification is as follows:

(2) ${\rm{\Delta Exper}}{{\rm{t}}_{it}} = \phi {\rm{\Delta }}{A_{it}} + {\gamma _w}{\rm{\Delta ln}}\left( {{\rm{salar}}{{\rm{y}}_{it}}} \right) + {\gamma _s}{\rm{\Delta ln}}\left( {{\rm{headcoun}}{{\rm{t}}_{it}}} \right) + {\lambda _t} + {u_{it}}$

Table 2 documents a symmetric pattern for expert roles. Throughout all model specifications, increases in AI exposure exhibit positive associations with increases in the proportion of expert personnel: the coefficient estimate for ${\rm{\Delta }}{A_{it}}$ demonstrates both positive magnitude and statistical precision across columns, ranging from 0.282 (s.e. 0.027) in the baseline to 0.208 (s.e. 0.050) with full controls. Using the same standard deviation for ${\rm{\Delta }}{A_{it}}$ (0.0718), a one standard deviation increase in exposure is associated with a 1.50 percentage point rise in the expert share; a 0.10 increase in ${\rm{\Delta }}{A_{it}}$ corresponds to a 2.08 percentage point rise. These patterns are consistent with what task-based models of technological change might predict: agencies with higher AIOE exhibit employment composition changes from administrative and clerical functions toward roles requiring specialised judgment and discretionary authority. In Online Appendix B,Footnote ⁴ we redefine expert definitions using tenure, wage, and combined tenure-wage criteria. Tenure-based experts (professionals with $ \geqslant $ 10 years’ service) exhibit stronger complementarity with AI exposure, whilst wage-based definitions (top quartile earners) yield negative coefficients, as high wages also capture managerial roles less amenable to AI augmentation. The combined definition similarly produces negative estimates, confirming that the baseline results reflect genuine technical complementarity rather than wage premia or occupational composition effects.

Table 2.

Complementarity: $\Delta $ AIOE on $\Delta $ expert share

Notes: Dependent variable: ΔExpert_it (within-agency change in expert employment share). Key regressor: ΔAIOE_it. Estimation and controls as in Table 1. Column (2) adds age/education controls; column (3) adds tenure/appointment controls. Sample size declines in column (3) reflect missingness in tenure/appointment data. Significance: *p < 0.10, **p < 0.05, ***p < 0.01.

Wage compression associations

Beyond changes in employment composition, we examine how within-agency compositional adjustment towards more AI-exposed occupations is reflected in internal pay structures. Rather than treating AI exposure as a technology shock, this analysis uses changes in agency-level exposure as a summary statistic for occupational reweighting and asks how such reallocation correlates with shifts in the distribution of pay within agencies.

We measure wage compression using the ratio of median to mean salary within each agency and quarter:

(3) $\begin{align}{\alpha _{it}} = {{\rm{median\;salary}}_{it}}\over{{{\rm{mean\;salary}}_{it}}}\end{align}$

where higher values indicate more compressed pay distributions. Our specification examines how this measure responds to AI exposure:

(4) ${\rm{\Delta }}{\alpha _{it}} = \psi {\rm{\Delta }}{A_{it}} + {\theta _s}{\rm{\Delta ln}}\left( {{\rm{headcoun}}{{\rm{t}}_{it}}} \right) + {\lambda _t} + {e_{it}}$

This measure captures compositional wage compression arising from employment shifts across grades and occupations; within-grade wage dispersion is not observable in the administrative data. Table 3 shows that increases in AI exposure are associated with greater apparent compositional wage compression within agencies. Higher within-agency changes in AI exposure are associated with increases in wage compression: the coefficient on ${\rm{\Delta }}{A_{it}}$ is positive and precisely estimated at 0.193 (s.e. 0.009) in the baseline, attenuating to 0.151 (s.e. 0.046) once full composition controls are included. With ${\rm{sd}}\left( {{\rm{\Delta }}{A_{it}}} \right) = 0.0718$ , a one standard deviation increase in exposure raises the median-to-mean salary ratio by 0.0108; a 0.10 increase in ${\rm{\Delta }}{A_{it}}$ raises it by 0.0151.

Table 3.

Wage Compression: $\Delta $ AIOE on $\Delta $ wage compression

Notes: Dependent variable: Δα_it where α_it = median(salary_it)/mean(salary_it). Key regressor: ΔAIOE_it. Estimation and controls as in Table 1, except column (1) omits Δln(salary) to avoid denominator mechanics. Column (2) adds age/education; column (3) adds tenure/appointment. Significance: *p < 0.10, **p < 0.05, ***p < 0.01.

This pattern is particularly revealing given public sector institutional constraints. Unlike private firms, federal agencies cannot easily adjust individual wages in response to productivity changes, instead operating within standardised pay scales and civil service frameworks. The wage compression we observe may reflect compositional changes – employment patterns shifting toward occupations clustering at different points in the pay distribution. This aligns with earlier findings: agencies with higher AI exposure exhibit both declining routine employment (positions clustering near grade minimums) and expanding expert employment (higher pay grades). Rather than adjusting individual wages to reflect changing productivity as in private sector automation studies, government agencies operate within classification systems constraining such flexibility. Our wage compression results thus reflect institutional constraints of standardised pay scales combined with employment composition changes across occupations with different pay characteristics.

Robustness and alternative specifications

These patterns hold up well to a range of robustness checks, reported in full in Online Appendix E. Two-way clustering by agency and quarter leaves all three main coefficients significant (Table E1). A lead–lag specification finds near-zero and insignificant lead coefficients while contemporaneous effects retain the expected signs (Table E2), ruling out pre-existing trends. Substitution and complementarity results survive when AI exposure is orthogonalised relative to the routine-occupation mean (Tables E3 and E4) and when the Felten et al. (Reference Felten, Raj and Seamans2021) baseline index is replaced by the Eloundou et al. (Reference Eloundou, Manning, Mishkin and Rock2024) exposure measure or a Principal component analysis (PCA)-based composite of both indices (Tables E5 and E6).

These findings contribute to understanding how employment composition relates to occupational characteristics in institutional settings. The systematic associations between AI exposure and employment composition suggest workforce evolution in federal agencies may be more structured than assumed, exhibiting patterns correlating with occupational characteristics even within rigid institutional frameworks. Wage compression effects highlight a key distinction from private sector contexts: whilst firms can adjust individual pay to reflect changing productivity, agencies must operate within classification systems limiting such flexibility – helping explain why employment composition changes appear to be the primary adjustment mechanism in government settings.

Occupational composition within agencies is not random; agencies differ systematically in mission, regulatory environment, and institutional constraints shaping workforce structure. Our results should therefore be interpreted as documenting patterns consistent with workforce reallocation rather than causal treatment effects of AI exposure. By exploiting within-agency changes over time and fixed occupational exposure scores, the analysis isolates compositional adjustment dynamics without claiming exogenous variation in AI adoption.

These patterns are consistent with task-based models of technological change yet operate within institutional environments quite different from competitive markets. Public agencies face political oversight, budget constraints, and legitimacy concerns that may fundamentally alter workforce decisions in ways efficiency-focused models cannot capture. The wage compression suggests standard assumptions about flexible pricing may not apply in bureaucratic settings where pay scales and employment protections constrain adjustment mechanisms. This points toward theoretical frameworks explicitly accounting for political and institutional realities of government organisations, which we develop in the following sections. While AI exposure measures remain imperfect proxies for actual technological deployment, robustness checks suggest core associations are not dependent on specific measurement choices.

Formal model: actors and objectives

We now formalise the strategic interactions that will govern AI adoption within government agencies as technological integration accelerates, focusing on the distinctive institutional features that differentiate public sector implementation from private market dynamics. Our framework centres on two key actors whose competing objectives and institutional constraints will shape how agencies with different occupational compositions navigate the adoption process.

The model features an Agency Director – typically a senior civil servant or department head who serves as the primary administrative decision-maker – and a Political Overseer – representing elected officials or ministers who provide strategic direction whilst remaining sensitive to electoral consequences. The Director chooses employment levels at two organisational layers (routine workers ${N_L}$ and expert staff ${N_H}$ ) alongside AI capital stock $A$ as technological capabilities become available. Agency output $Q\left( {{N_L},{N_H},A} \right)$ emerges from the evolving interaction of human labour and artificial intelligence, whilst costs $C\left( {{N_L},{N_H},A} \right)$ reflect both personnel expenses and technology investments.

The production function specifications directly reflect how different occupational compositions will interact with AI capabilities. Agencies with higher concentrations of routine, AI-exposed occupations face different technological opportunities and constraints than those dominated by expert roles requiring human judgement. This occupational starting point becomes a crucial state variable determining how adoption can proceed and what employment transitions will follow.

What distinguishes this public sector context from private enterprise is the presence of political costs – captured through functions $P\left( \cdot \right)$ and $R\left( \cdot \right)$ below – that will arise from legislative scrutiny, interest group opposition, and public concerns about algorithmic decision-making in government. These costs have no direct parallel in commercial settings but will prove central to determining how AI adoption unfolds across different agencies and policy domains.

The agency director

The Agency Director faces a complex optimisation problem that balances institutional survival, operational effectiveness, and political acceptability. Unlike private sector executives who can focus primarily on profit maximisation, public sector administrators must navigate competing objectives that frequently conflict:

(5) ${U^D}\; = \;B\left( {{N_L} + {N_H} + \theta A} \right)\; + \;\alpha \,Q\left( {{N_L},{N_H},A} \right)\; - \;C\left( {{N_L},{N_H},A} \right)\; - \;P\left( A \right)$

The first component, $B\left( {{N_L} + {N_H} + \theta A} \right)$ , reflects budget maximisation incentives well-established in public choice theory (Niskanen, Reference Niskanen1968; Downs, Reference Downs1965). Bureaucrats possess strong incentives to maintain or expand their institutional scope, as larger agencies command greater resources, provide enhanced career opportunities, and offer increased influence within government. The parameter $\theta \geq 0$ plays a crucial role, determining whether AI systems contribute to budget justification arguments. When $\theta \gt 0$ , AI capital enhances the director’s case for resource allocation, perhaps because visible technology investment signals institutional modernisation to political overseers and budget authorities. When $\theta = 0$ , AI provides no such benefits, it is potentially because budget authorities focus exclusively on headcount metrics.

The function $B\left( \cdot \right)$ captures the political and budgetary value to the Agency Director of expanding organisational scope, rather than productive returns to scale. In the U.S. federal context, there are strong institutional reasons to expect this value to exhibit diminishing marginal returns. As agencies grow larger, incremental expansions typically attract disproportionately greater oversight, reporting requirements, and audit intensity, reducing the net marginal political benefit of further expansion. Moreover, appropriations and staffing ceilings are negotiated through incremental political processes in which marginal increases become harder to justify once the baseline scale is already large. The assumption $B'' \le 0$ therefore, reflects declining marginal appropriability of budgetary rents as organisational size expands, rather than technological scale effects, and serves as a reduced-form representation of increasing oversight and political constraint at scale.

The performance component $\alpha Q\left( {{N_L},{N_H},A} \right)$ captures the director’s concern for operational effectiveness and service delivery quality, where $\alpha \gt 0$ weights performance relative to other objectives. This reflects genuine pressures to deliver effective public services arising from political oversight, professional norms within the civil service, media scrutiny of agency performance, and intrinsic motivation among public sector leaders.

Personnel and technology costs follow $C\left( {{N_L},{N_H},A} \right)$ , but crucially, wages in the civil service are determined by standardised pay scales and collective bargaining agreements rather than marginal productivity. This creates fundamental rigidities absent from private labour markets, with implications we explore below.

Finally, $P\left( A \right)$ represents political and reputational costs unique to government contexts: legislative scrutiny over technology procurement decisions, media attention on algorithmic decision-making, potential public backlash over concerns about fairness and transparency, and compliance with government-wide AI governance frameworks. This function is assumed to be increasing and convex, reflecting that political opposition accelerates as automation scope and visibility expand.

The political overseer

Political Overseers – elected officials, political appointees, or senior ministers – face fundamentally different trade-offs from Agency Directors. Their primary concerns centre on electoral consequences, fiscal responsibility, and broader policy outcomes rather than institutional survival and bureaucratic autonomy:

(6) ${U^P}\; = \;\beta \,Q\left( {{N_L},{N_H},A} \right)\; - \;\gamma \,C\left( {{N_L},{N_H},A} \right)\; - \;R\left( A \right)\; - \;L\left( {{\rm{\Delta }}N} \right)$

The first term $\beta Q\left( {{N_L},{N_H},A} \right)$ captures electoral benefits from improved service delivery, where $\beta \gt 0$ reflects political rewards that accrue when politicians demonstrate tangible improvements in government performance. These may materialise through increased voter satisfaction, positive media coverage, or competitive advantage relative to political opponents.

The cost component $\gamma C\left( {{N_L},{N_H},A} \right)$ reflects taxpayer pressure for fiscal efficiency and legislative constraints on public expenditure, where $\gamma \gt 0$ captures sensitivity to fiscal costs that create direct political consequences for spending decisions.

The resistance function $R\left( A \right)$ represents organised opposition to automation from various interest groups: public sector unions concerned about employment security, professional associations worried about erosion of expert judgment, and civil liberties groups raising concerns about algorithmic fairness. We assume $R'\left( A \right),R''\left( A \right) \geqslant 0$ , with the convex specification reflecting that opposition intensifies at an increasing rate as automation scope expands.

The final component $L\left( {{\rm{\Delta }}N} \right)$ captures political costs associated with visible workforce changes, where ${\rm{\Delta }}N = \left| {{N_L} - N_L^0\left| + \right|{N_H} - N_H^0} \right|$ measures absolute deviation from baseline employment levels. Government employment levels carry symbolic significance about political priorities, creating costs for both increases (suggesting fiscal irresponsibility) and decreases (raising concerns about service reduction).

Institutional constraints

Three key institutional features distinguish public sector technology adoption from private markets and prove central to explaining observed patterns:

Pay-scale rigidity. Unlike private enterprises, where wages can adjust to reflect changing marginal productivities, civil service compensation follows standardised grade classifications that create significant rigidities. Personnel costs are increasing and convex in headcount:

$C\left( {{N_L},{N_H},A} \right)\; = \;{\bar w_L}N_L^{\,1 + {\eta _L}} + {\bar w_H}N_H^{\,1 + {\eta _H}} + {w_A}A,\,\,\,\,{\eta _L},{\eta _H} \ge 0,$

where the convexity parameters ${\eta _j} \geqslant 0$ capture step increases within salary grades, promotion bottlenecks as agency size grows, and the tendency for larger organisations to employ more senior personnel within each category. When ${\eta _j} = 0$ , we obtain the linear case; when ${\eta _j} \gt 0$ , grade progression effects emerge.

Employment protection. Civil service rules, union agreements, and political commitments create employment floors that limit workforce flexibility:

(7) ${N_j}\; \ge {\underline N _j},\,\,\,j \in \left\{ {L,H} \right\}$

When these constraints bind during AI adoption, agencies cannot substitute technology for labour directly but must instead pursue complementarity strategies where AI augments rather than replaces protected workers.

Procurement and adoption ceilings. Government procurement processes, security requirements, and institutional capacity constraints limit the speed and scope of AI deployment:

(8) $A\; \le \;{A_{{\rm{max}}}}\left( t \right)$

where ${A_{{\rm{max}}}}\left( t \right),$ and it grows over time as institutions develop necessary infrastructure, technical expertise, and governance frameworks to support larger-scale automation.

Production technology

The production function directly reflects our empirical findings, incorporating both substitution effects for routine tasks and complementarity effects for expert functions:

(9) $Q\left( {{N_L},{N_H},A} \right)\; = \;F\left( {{T_L}\left( {{N_L},A} \right),\,{T_H}\left( {{N_H},A} \right)} \right),\,\,\,{F_1},{F_2} \gt 0,\;{F_{11}},{F_{22}} \le 0$

where output aggregates task completion at two organisational levels.

Routine tasks. Administrative processing, data entry, and rule-based decision-making exhibit perfect substitutability between human labour and AI systems:

(10) ${T_L}\left( {{N_L},A} \right)\; = \;{\rm{min}}\left[ {{{\bar T}_L},\;{\phi _L}{N_L} + {\phi _A}A} \right],\,\,\,{\phi _L},{\phi _A} \gt 0$

This specification captures that AI can directly replace human effort in standardised functions up to the agency’s capacity requirement ${\bar T_L}$ , but some minimum human oversight remains necessary for handling exceptions and ensuring accountability.

Expert tasks. Policy analysis, legal interpretation, and complex case management exhibit complementarity rather than substitution:

(11) ${T_H}\left( {{N_H},A} \right)\; = \;{\phi _H}{N_H}\,\left( {1 + \sigma A} \right),\,\,\,{\phi _H} \gt 0,\;\sigma \gt 0$

The multiplicative form ensures that AI systems enhance rather than replace expert judgement, with $\sigma \gt 0$ capturing augmentation intensity. This reflects that whilst AI can rapidly process information and identify patterns, the interpretation and application require human insight that incorporates legal requirements, policy objectives, and contextual factors difficult to codify algorithmically.

Two-stage decision process

Decision-making occurs through a sequential process reflecting the hierarchical nature of government, whilst recognising substantial administrative discretion in implementation.

Stage 1: Political negotiation. The Political Overseer establishes the strategic framework within which the Agency Director operates, choosing constraints to maximise their utility function:

(12) $\mathop {{\rm{max}}}\limits_{{A^{{\rm{max}}}},\,{{\underline N }_L},{{\underline N }_H},\,\bar B} \;{U^P}\,\,\,\,{\rm{subject\;to\;fiscal\;and\;political\;feasibility\;constraints\;}}$

The Overseer sets the maximum AI adoption level ${A^{{\rm{max}}}} \le {A_{{\rm{max}}}}\left( t \right)$ , potentially below the technical maximum due to political risk considerations; employment floor adjustments $\left\{ {{{\underline N }_L},{{\underline N }_H}} \right\}$ that may exceed baseline levels to provide workforce protection; and overall budget allocation $\bar B$ determining the resource envelope.

Stage 2: Agency response. Given political constraints from Stage 1, the Agency Director chooses optimal employment levels and AI adoption to maximise their utility:

(13) $\mathop {{\rm{max}}}\limits_{{N_L},{N_H},A} \;{U^D}\,\,{\rm{subject\;to\;budget,\;employment\;floors,\;and\;}}A \le {A^{{\rm{max}}}}$

This sequential structure ensures that AI implementation decisions emerge from political-administrative bargaining rather than purely technocratic optimisation, whilst preserving meaningful administrative discretion in operational choices.

Comparative statics: institutional parameters and augmentation strength

This subsection summarises the directional comparative statics implied by the model with respect to key institutional parameters and the technological augmentation parameter $\sigma $ . The objective is not to derive closed-form solutions, but to clarify how equilibrium outcomes respond qualitatively to changes in (i) employment floor tightness and pay-scale rigidity, and (ii) the strength of AI–expert complementarity. The signs reported below follow directly from the Kuhn–Tucker characterisation of the Agency Director’s constrained optimisation problem and from the structure of the production technology.

Employment floor tightness. Consider first marginal increases in employment protection, captured by higher employment floors ${\underline N _j}$ for $j \in \left\{ {L,H} \right\}$ . When a given employment floor is slack at the optimum, the associated multiplier is zero, and marginal tightening has no first-order effect on optimal choices. When the constraint binds, tightening it shifts the feasible set mechanically.

In particular, if the routine employment floor binds ( ${\rm{N}}_{\rm{L}}^{\rm{*}} = {\underline N _{\rm{L}}}$ ), a marginal increase in ${\underline N _{\rm{L}}}$ raises routine employment one-for-one, $\partial {\rm{N}}_{\rm{L}}^{\rm{*}}/\partial {\underline N _{\rm{L}}} \gt 0$ . Because routine tasks are locally substitutable with AI capital, tighter routine employment protection reduces the scope for AI substitution and weakly lowers optimal AI adoption, $\partial {{\rm{A}}^{\rm{*}}}/\partial {\underline N _{\rm{L}}} \le 0$ . Through the resulting budget and cost trade-offs, tighter routine floors also weakly reduce expert employment, $\partial {\rm{N}}_{\rm{H}}^{\rm{*}}/\partial {\underline N _{\rm{L}}} \le 0$ unless offset by sufficiently strong budgetary incentives.

Analogously, if the expert employment floor binds ( ${\rm{N}}_{\rm{H}}^{\rm{*}} = {\underline N _{\rm{H}}}$ ), tightening expert protection raises expert employment mechanically while limiting the extent to which AI can be deployed for augmentation purposes. In this case, the effect on routine employment is ambiguous and depends on whether the Director reallocates resources toward routine labour or contracts overall organisational scale.

Pay-scale rigidity. Pay-scale rigidity enters through the convexity parameters ${\eta _L}$ and ${\eta _H}$ , which govern how marginal personnel costs rise with employment. An increase in ${\eta _j}$ steepens the marginal cost of expanding employment in category $j$ .

Higher routine pay rigidity ( ${{\rm{\eta }}_{\rm{L}}} \uparrow $ ) weakly reduces optimal routine employment, $\partial {\rm{N}}_{\rm{L}}^{\rm{*}}/\partial {{\rm{\eta }}_{\rm{L}}} \le 0$ , and – when substitution is feasible – raises the relative attractiveness of AI capital as a substitute input, $\partial {{\rm{A}}^{\rm{*}}}/\partial {{\rm{\eta }}_{\rm{L}}} \geqslant 0$ . The effect on expert employment is in general ambiguous, reflecting opposing forces: higher routine costs may reallocate resources toward expert–AI complementarity, but increased overall cost pressure may also contract total scale.

Higher expert pay rigidity ( ${{\rm{\eta }}_{\rm{H}}} \uparrow $ ) weakly reduces expert employment, $\partial {\rm{N}}_{\rm{H}}^{\rm{*}}/\partial {{\rm{\eta }}_{\rm{H}}} \le 0$ . Because AI capital primarily generates value through complementarity with expert labour in the production technology, steeper expert pay schedules also dampen AI adoption incentives, yielding $\partial {{\rm{A}}^{\rm{*}}}/\partial {{\rm{\eta }}_{\rm{H}}} \le 0$ when the expert margin is operative.

Augmentation strength $\bf\sigma $ . Finally, consider changes in the augmentation parameter $\sigma $ in the expert task technology ${T_H}\left( {{N_H},A} \right) = {\phi _H}{N_H}\left( {1 + \sigma A} \right)$ . This specification exhibits increasing differences in $\left( {{N_H},A} \right)$ and $\sigma $ , implying that higher $\sigma $ raises the marginal productivity of expert labour when AI is deployed and raises the marginal return to AI when expert employment is positive.

Provided the expert employment floor is slack and the AI adoption ceiling does not bind, stronger augmentation unambiguously increases expert employment, $\partial {\rm{N}}_{\rm{H}}^{\rm{*}}/\partial {\rm{\sigma }} \gt 0$ , and weakly increases AI adoption, $\partial {{\rm{A}}^{\rm{*}}}/\partial {\rm{\sigma }} \geqslant 0$ . When routine employment is adjustable, the resulting increase in AI adoption further implies weak substitution away from routine labour, $\partial {\rm{N}}_{\rm{L}}^{\rm{*}}/\partial {\rm{\sigma }} \le 0$ . If employment floors or adoption ceilings bind, these effects are locally muted, with the corresponding derivatives equal to zero.

Taken together, these comparative statics highlight how institutional parameters govern the margins along which bureaucratic adjustment occurs. Employment protection and pay rigidity shift responses away from direct substitution and toward constrained reallocation and complementarity, while stronger augmentation amplifies expert expansion where institutional constraints permit. These mechanisms distinguish bureaucratic AI adoption from market-driven adjustment and motivate the empirical focus on compositional change.