The 3D twin of this simulator tracks the relative unit price of healthcare vs manufacturing output. This 2D companion goes to the root of Baumol's original 1967 "unbalanced growth" argument instead: it derives how the labor force itself must reallocate between sectors to keep real output growing at the same rate everywhere, with a fixed total workforce.
Output/worker: A_m(t)=e^(g_m·t), A_h(t)=e^(g_h·t)
Real demand for each sector's OUTPUT grows at the
same rate n (population/income growth) — n cancels
out of every share below, so it isn't even a slider:
Q_i(t) = Q_i(0)·e^(n·t)
Workers needed = output demanded / output per worker:
L_i(t) = Q_i(t) / A_i(t)
Nominal jobs & GDP share of healthcare (wages equal
across sectors under full labor mobility, so this is
also its GDP share):
s_h(t) = L_h(t) / [L_h(t)+L_m(t)]
= 1 / [ 1 + ((1−s_h(0))/s_h(0))·e^((g_h−g_m)·t) ]
Real quantity share (physical output, not spending):
q_h(t) = Q_h(t) / [Q_h(t)+Q_m(t)] = s_h(0) ← CONSTANT for all t
That last line is the heart of the "disease": because Q_m and Q_h are assumed to grow at the same real rate, the real quantity share never moves — yet the nominal jobs/spending share climbs without bound whenever gm > gh, because it takes an ever-larger share of the (fixed) workforce to keep producing healthcare's flat real share once each healthcare worker's output stops growing. Pushed far enough, sh(t) → 100%: the entire economy would eventually work in the stagnant sector — Baumol's actual headline prediction, distinct from (and complementary to) the relative-price result shown in the 3D twin.
- Workforce grid — each of the 120 squares is one unit of the (fixed) national labor force; as t advances, squares flip from manufacturing (blue) to healthcare (green) in a fixed hiring order, visualizing sh(t) directly.
- Chart — the rising curve is nominal share sh(t); the flat dashed line is the real quantity share, which by construction never moves — the gap between them is the cost disease.
- gm / gh sliders — the productivity-growth gap (gm−gh) is the only thing that determines how fast sh(t) climbs; set them equal and the workforce split freezes at sh(0), exactly as the formula predicts.
- sh(0) slider — the economy's starting healthcare employment/GDP share.
Real-world relevance: US health-sector employment share has risen for decades even though the physical volume of care per capita hasn't grown anywhere near as fast — this model is the standard textbook explanation (Baumol & Bowen 1966; Baumol 1967) for why that keeps happening without any inefficiency anywhere in the system.