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A Discrete-Time Model of the Academic Pipeline in Mathematical Sciences with Constrained Hiring in the United States

Oluwatosin Babasola, Olayemi Adeyemi, Ron Buckmire, Daozhou Gao, Maila Hallare, Olaniyi Iyiola, Deanna Needell, Chad M. Topaz, Andrés R. Vindas-Meléndez

TL;DR

This paper develops a discrete-time, four-stock (undergraduate $U_t$, graduate $G_t$, postdoctoral $P_t$, and faculty $F_t$) compartmental model to study the mathematical sciences pipeline under vacancy-limited hiring. By anchoring $U_t$ and $G_t$ to observed degree flows via observation equations and modeling $P_t$ and $F_t$ with capacity-constrained transitions, the authors demonstrate that increases in degree inflow do not yield proportional faculty growth when hiring is turnover-limited, with postdocs absorbing excess supply. Sensitivity analyses show long-run outcomes are governed mainly by faculty exit rates and hiring capacity rather than degree production alone, signaling that vacancy-limited hiring is a central driver of persistent postdoctoral congestion. The framework provides a data-anchored, mechanistic lens for evaluating policy implications around hiring capacity and turnover in the mathematical sciences and potentially other fields with extended training pipelines.

Abstract

The field of the mathematical sciences relies on a continuous academic pipeline in which individuals progress from undergraduate study through graduate training and postdoctoral program to long term faculty employment. National statistics report trends in bachelor's, master's, and doctoral degree awards, but these data alone do not explain how individuals move through the academic system or how structural constraints shape downstream career outcomes. Persistent growth in postdoctoral appointments alongside relatively stable faculty employment indicates that degree production alone is insufficient to characterize workforce dynamics. In this study, we develop a discrete time compartmental model of the academic pipeline in the field of the mathematical sciences that links observed degree flows to latent population stocks. Undergraduate and graduate populations are reconstructed directly from nationally reported degree data, allowing postdoctoral and faculty dynamics to be examined under completion, exit, and hiring processes. Advancement to faculty positions is modeled as vacancy limited, with competition for permanent positions depending on downstream population size. Numerical simulations show that increases in degree inflow do not translate into proportional faculty growth when hiring is constrained by limited turnover. Instead, excess supply accumulates primarily at the postdoctoral stage, leading to sustained congestion and elevated competition. Sensitivity analyses indicate that long run workforce outcomes are governed mainly by faculty exit rates and hiring capacity rather than by degree production alone. These results demonstrate the central role of vacancy limited hiring in shaping academic career trajectories within the field of the mathematical sciences.

A Discrete-Time Model of the Academic Pipeline in Mathematical Sciences with Constrained Hiring in the United States

TL;DR

This paper develops a discrete-time, four-stock (undergraduate , graduate , postdoctoral , and faculty ) compartmental model to study the mathematical sciences pipeline under vacancy-limited hiring. By anchoring and to observed degree flows via observation equations and modeling and with capacity-constrained transitions, the authors demonstrate that increases in degree inflow do not yield proportional faculty growth when hiring is turnover-limited, with postdocs absorbing excess supply. Sensitivity analyses show long-run outcomes are governed mainly by faculty exit rates and hiring capacity rather than degree production alone, signaling that vacancy-limited hiring is a central driver of persistent postdoctoral congestion. The framework provides a data-anchored, mechanistic lens for evaluating policy implications around hiring capacity and turnover in the mathematical sciences and potentially other fields with extended training pipelines.

Abstract

The field of the mathematical sciences relies on a continuous academic pipeline in which individuals progress from undergraduate study through graduate training and postdoctoral program to long term faculty employment. National statistics report trends in bachelor's, master's, and doctoral degree awards, but these data alone do not explain how individuals move through the academic system or how structural constraints shape downstream career outcomes. Persistent growth in postdoctoral appointments alongside relatively stable faculty employment indicates that degree production alone is insufficient to characterize workforce dynamics. In this study, we develop a discrete time compartmental model of the academic pipeline in the field of the mathematical sciences that links observed degree flows to latent population stocks. Undergraduate and graduate populations are reconstructed directly from nationally reported degree data, allowing postdoctoral and faculty dynamics to be examined under completion, exit, and hiring processes. Advancement to faculty positions is modeled as vacancy limited, with competition for permanent positions depending on downstream population size. Numerical simulations show that increases in degree inflow do not translate into proportional faculty growth when hiring is constrained by limited turnover. Instead, excess supply accumulates primarily at the postdoctoral stage, leading to sustained congestion and elevated competition. Sensitivity analyses indicate that long run workforce outcomes are governed mainly by faculty exit rates and hiring capacity rather than by degree production alone. These results demonstrate the central role of vacancy limited hiring in shaping academic career trajectories within the field of the mathematical sciences.
Paper Structure (15 sections, 3 theorems, 25 equations, 9 figures, 1 table)

This paper contains 15 sections, 3 theorems, 25 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Model formulation and stock--flow structure
  4. Vacancy limited hiring mechanism
  5. Vacancy--limited state updates
  6. Data and model calibration
  7. Observed degree data
  8. Linking observed degrees to latent stocks
  9. Parameter calibration
  10. Consistency with observed degree trends
  11. Sensitivity analysis
  12. Model analysis
  13. Results
  14. Discussion
  15. Conclusion

Key Result

Theorem 1

Assume $U_0,G_0,P_0,F_0 \geq 0$ and $B(t)\geq 0$ for all $t\geq 0$. Suppose the coefficients satisfy and that the transition functions $p^{UG}(G_t)$ and $p^{PF}(F_t)$ are nonnegative for all $t$. Then the solution satisfies

Figures (9)

  • Figure 1: Academic pipeline flow diagram showing the four-stage stock–flow structure with external inflow, completion-driven transitions, stage-specific exits, and state--dependent progression probabilities.
  • Figure 2: Observed national degree counts (blue points ) and model-implied degree flows (red lines) for master’s and doctoral degrees. Model outputs are computed from reconstructed graduate and doctoral populations using the observation equations.
  • Figure 3: Local parameter perturbation analysis showing sensitivity of final and peak faculty and postdoctoral populations to (a) variation in competition intensity, (b) postdoctoral exit, and (c) faculty exit parameters.
  • Figure 4: Sensitivity of workforce outcomes to hiring throughput parameters, including the maximum postdoc to faculty transition rate and the faculty capacity scale.
  • Figure 5: Sensitivity of workforce outcomes to graduate progression pathways, (a) direct transitions to faculty and (b) transitions into postdoctoral positions.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 1: Positivity and feasibility
  • proof
  • Theorem 2: Boundedness under bounded inflow and positive exit
  • proof
  • Theorem 3: One-step accumulation conditions
  • proof