A Compartmental Differential-Equation Model of the Imperial Examination and Social-Class Mobility in Ming China

The imperial examination (keju) is conventionally credited with keeping the late-imperial Chinese elite open to talent, yet that openness is usually measured as a rate for a period or a cohort rather than modelled as a process. This paper builds a compartmental ordinary differential equation (ODE) model in which two household stocks — commoner households C and elite households E, the latter defined as households with a degree or an office-holding within three generations — interact through an examination gate whose selection function π(t)=C/(C+ρE) is exactly the quantity historians report as the share of newly enrolled jinshi coming from strictly commoner families. The model is calibrated on a dataset assembled for this study from published scholarship and public databases: the per-examination jinshi intake for 88 examinations (24,588 graduates) and the per-examination origin composition of 57 examinations (15,528 usable observations). Two parameters are pinned to documented historical anchors (elite decay δ=0.034 yr−1 from the finding that close to 87% of jinshi lineages did not survive three generations; the initial commoner stock from the 1393 census of 10,652,870 households); the remainder are estimated by weighted least squares, with class-transfer terms that are conservative by construction. The fitted model reproduces the observed trajectory with a weighted R^2 of 0.746 and a mean absolute percentage error of 7.6%. A specification in which the elite stock grows without limit fits markedly worse (R^2=0.609), and the two specifications are separated on fit quality; the implied elite stock of the retained specification grows 14- to 20-fold over the dynasty (14.5-fold at the fitted point; the level of the stock and the elite advantage ρ are identified only jointly), against an independently documented 16.7-fold rise in the licentiate population that played no part in the fitting. Sensitivity analysis, computed as partial rank correlation coefficients, ranks elite decay first (PRCC +0.97), the degree quota second (−0.95) and the entry barrier facing commoners a distant third (−0.09, marginal at p=0.049). The jinshi quota is almost inert: doubling it moves the endpoint commoner share by about 0.2 percentage points, because the metropolitan channel is far narrower than the elite stock it feeds.