Deadlines and investment depth in production hierarchies

We study how a deadline affects the optimal allocation of scarce installation effort across layers of a production hierarchy. Heterogeneity in productivity, costs, and propagation delays generates a depth-horizon return kernel that satisfies the single-crossing property. Under convex installation technologies, this property orders pure choices; under arbitrary layer-specific concave technologies, it shifts the unique allocation toward deeper layers as the horizon expands. Isoelastic technologies yield a closed-form likelihood-ratio ordering. The results hold with finite marginal productivity at zero and arbitrary layer-specific depreciation rates. Thus, curvature determines whether adjustment is discrete or diversified, but not its direction.