Joint Optimization of Human Headcount and Stochastic AI Resource Capacity
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摘要
This paper develops a joint optimization framework for allocating a fixed monthly budget between human headcount and generative-AI token capacity under cognitive frictions and stochastic spending. Using a Cobb–Douglas production core augmented with an exponential friction term, the model yields a closed-form optimal headcount (N*) and a transcendental condition for per-capita token intensity (T*), with a bordered-Hessian condition ensuring a local maximum. Comparative statics show that higher per-engineer token volatility raises T* and lowers N*, implying a substitution at the margin even though inputs are production complements. Skewness (γ1) acts as a pure deadweight tax on the budget, shrinking available resources without altering substitution dynamics. The direction of substitution depends on whether dispersion is invariant to mean or scales with it, offering an empirically testable regime change. These insights are framed within a chance-constrained stochastic budgeting approach and central-limit/Cornish–Fisher corrections [page::1].
