AI x Billable Hours Model

An economic model of how AI affects billable-hours law firms. Four lines of work (premium, commodity-elite, mid-market, small-client) run through hours compression, a make-or-buy decision by clients, endogenous pricing under competition, and a pass-through parameter. Outputs are long-run percentage changes in price, book, revenue, hours by tier, leverage, and profit per partner.

All parameter values are illustrative, not calibrated to data.

Quick start

pip install -r requirements.txt
python scripts/generate_tables.py     # writes tables/*.csv and tables/*.md
python scripts/generate_charts.py     # writes charts/*.svg (light and dark) and charts/*.png
python -m pytest tests

from mmbillhr import run_all, sweep, grid2d
run_all()[["price", "revenue", "assoc_hours", "ppp"]]
sweep("delta", [0, .2, .4, .6, .8])
grid2d("delta", [0, .4, .8], "beta", [0, .5, 1], "ppp", "Premium")

Any segment or global parameter can be overridden by keyword: run_all(delta=0.6, g=6).

Model specification

Each line of work is solved independently. A baseline matter uses 1 partner hour and l0 associate hours; baseline price is P0 = r_S + r_J*l0.

Step 1 — Hours compression. phi_i = (1 - a_i) + a_i/g for tier i. Blended phi_bar = (phi_S + l0*phi_J)/(1 + l0). In-house compression phi_c = 1 - inhouse_adoption*(1 - phi_bar): in-house legal captures that share of the firms' hours savings (1 = the same, 0 = none).

Step 2 — Make-or-buy. Matters indexed by complexity/stakes s ∈ [0,1] with density Beta(s_a, s_b) and value weight v(s) = 1 + nu*s. Costs in units of in-house cost per baseline hour; psi = P/P0.

• Outside: mu * psi
• In-house, pre-AI: 1 + kappa*s + rho*s
• In-house, post-AI: phi_c*(1 + kappa*(1 - delta(s))*s) + rho*s, with delta(s) = clip(delta*(1 + omega*(1 - 2s)), 0, 1). delta is the leveling at mid-complexity (s = 0.5); omega sets how fast it fades with complexity. At omega = 1 leveling reaches zero on the hardest work (s = 1); at omega = 0 it is the same at every level. Whether AI can close the expertise gap on the most specialised work is what omega encodes.

Probability a matter is outsourced: p = 1/(1 + exp(-(C_in - mu*psi)/tau)). Share of clients able to insource: eta0 pre-AI, eta1 = min(1, eta0*phi_c^(-alpha_F)) post. Book B(psi) = (1-eta)*W + eta*∫ f v p ds; retention R(psi) = B_post(psi)/B_pre(1).

Step 3 — Pricing. Baseline insourcing elasticity e0 = -dlnB_pre/dlnpsi at 1. Costs are calibrated so baseline prices are an equilibrium: c0 = P0*(1 - 1/(theta + e0)); non-associate cost o = c0 - w_J*l0 scales with partner hours; post-AI cost c1 = w_J*l0*phi_J + o*phi_S. Full-pass-through price solves the symmetric Nash condition (psi*P0 - c1)/(psi*P0) * (theta + e_R(psi)) = 1. Actual price psi = psi_full^beta. Implied lambda solves r_S*phi_S^(1-lambda) + r_J*l0*phi_J^(1-lambda) = psi*P0.

Step 4 — Outcomes. Q = R(psi)*psi^(-eps); revenue Q*psi; associate hours Q*phi_J; partner hours Q*phi_S; profit pool Q*(psi*P0 - w_J*l0*phi_J) vs P0 - w_J*l0; profit per partner = pool change / max(1, partner hours). Sustainability check: a single deviating firm picks x ≤ psi to maximise (x*P0 - c1)*R(x)*(x/psi)^(-theta); the table reports its profit gain and required partner hours.

Firm mix. firm_mix() combines lines by revenue share; profit is weighted by share × baseline margin, hours by baseline hours per revenue dollar.

Parameters

	Meaning
a_J, a_S	share of associate / partner tasks exposed to AI
g	speed-up on exposed tasks (global)
l0	baseline leverage
r_S, r_J, w_J	partner rate, associate rate, associate cost
eps	market-level demand elasticity
theta	firm-level (cross-firm) elasticity; sets baseline margin 1/(theta+e0)
beta	pass-through of cost savings to price
mu	outside rate / in-house cost per hour
kappa	in-house expertise penalty slope
rho	insurance/reputation value of outside counsel
delta	expertise leveling at mid-complexity
omega	how fast leveling fades with complexity (1 = gone at the top, 0 = flat)
inhouse_adoption	share of firms' AI hours savings that in-house legal also captures
eta0	share of clients able to insource pre-AI
s_a, s_b	Beta shape of matter distribution over s
tau, nu, alpha_F	choice softness, value weight slope, fixed-cost elasticity (global)

Defaults are in mmbillhr/params.py.

Known limitations

• Static, long-run only; no adoption path or dynamics.
• beta is reduced-form; not derived from a capacity-constrained pricing game.
• No stable equilibrium at low theta (roughly < 1.2); reported as n/a.
• Leverage and AI exposure do not vary with s within a line of work.
• No training pipeline, firm entry/exit, or AI tool costs.

Layout

mmbillhr/params.py   parameter dataclasses and defaults
mmbillhr/model.py    SegmentModel, Results, firm_mix
mmbillhr/grid.py     run_all, sweep, grid2d
scripts/generate_tables.py
scripts/generate_charts.py
tables/             generated CSV + Markdown (see tables/README.md)
charts/             generated SVG (light/dark) + PNG
tests/