# Stage 3 Derivation: Optimal Rotation in a Research Program This note replaces the maintained horizon-push argument with a solved allocation problem. The benchmark is deliberately small: a fixed-capacity research program, an explicit task density, and locally constant lag annuities. Its role is to prove that horizon-first and then reliability is an optimal path, not an assumed path. ## 1. A task density with depletion and complementarity Shift the initial frontier to `(h,f)=(0,0)` and consider a compact rectangle. There are three value measures: - horizon-only tasks have threshold density `u0-u1 h`; - reliability-only tasks have constant threshold density `v0`; - joint tasks have constant density `v1` over the rectangle. All parameters are positive and `u0-u1 Hbar>0`. With sharp service thresholds, the gross value of tasks served at frontier `(h,f)` is exactly `W(h,f) = u0 h - (u1/2) h^2 + v0 f + v1 h f`. (DR1) The boundary values and cross-boundary statistic are `B_H(h,f)=u0-u1 h+v1 f`, `B_F(h,f)=v0+v1 h`, `C(h,f)=v1>0`. (DR2) Thus horizon value depletes along a horizon push, while the same push unlocks joint tasks and raises the reliability boundary value. The polynomial is not an arbitrary reduced form: it is the exact integral of the stated task density. Smooth task-success kernels approximate the sharp-threshold benchmark on compact subsets. ## 2. The research-program problem Let `a_i=(1-exp(-r tau_i))/r` be the local annuity of direction `i`, with `a_F>a_H` whenever `tau_F>tau_H`. The firm has `Q` units of research capacity. Research input `s in [0,Q]` is the program clock. It chooses measurable shares `(x_H,x_F)` satisfying `x_H+x_F=1`, `x_i>=0`, and the frontier evolves according to `h'=x_H`, `f'=x_F`. One unit of direction-`i` research creates capitalized private value `a_i B_i` and costs `c_i`. The program therefore solves `max integral_0^Q [(a_H B_H-c_H)x_H + (a_F B_F-c_F)x_F] ds`. (DR3) This is the exact committed-cohort/local-annuity benchmark: each marginal task cohort keeps its creation-state boundary value during its imitation window. In the full model, Theorem 3's variation terms perturb these returns. Strict policy regions below are robust when those perturbations are smaller than the displayed switching margin. The fixed-capacity restriction is substantive and useful. It makes `s` cumulative research input, so the theorem first signs the switching *position*. If capacity is a constant `xbar` per unit of calendar time, the switching time is position divided by `xbar`. If capacity responds to imitation, the calendar-time sign needs an additional elasticity condition and is not inferred from the position sign alone. ## 3. Why the optimal path is ordered Consider two paths around a rectangle of width `dh` and height `df`, starting at the same state and ending at the same state. Horizon then reliability yields `HF = a_H [W(h+dh,f)-W(h,f)]` ` + a_F [W(h+dh,f+df)-W(h+dh,f)]`. Reliability then horizon yields the analogous `FH`. Directional research costs cancel. Using (DR1), `HF-FH=(a_F-a_H)v1 dh df>0`. (DR4) Equivalently, the payoff one-form `(a_H B_H-c_H) dh + (a_F B_F-c_F) df` has positive curl `(a_F-a_H)C`. Every local reliability-then-horizon inversion can therefore be improved by swapping its order. Repeated adjacent interchange sorts any optimal path: all horizon research occurs before all reliability research. This is the key step the old theorem lacked. The horizon push is an equilibrium implication of complementarity and the annuity gap. ## 4. Endpoint choice and switch surface At a current state `(h,f)` with `R` units of program capacity left, suppose the ordered path uses `y` additional units of horizon research and `R-y` reliability units. Its continuation value, apart from accumulated past payoff, is `J(y;h,f,R)` ` = a_H[(u0-u1 h+v1 f)y-(u1/2)y^2]-c_H y` ` +(R-y)[a_F{v0+v1(h+y)}-c_F]`. (DR5) Define `D = a_H u1 + 2 a_F v1 > 0`, (DR6) and the switching index `S(h,f,R)` ` = a_H B_H(h,f)-c_H - {a_F B_F(h,f)-c_F} + a_F v1 R`. (DR7) Then `J_y=S-Dy`, `J_yy=-D<0`, so the unique amount of horizon research remaining is `y*(h,f,R)=projection_[0,R] {S(h,f,R)/D}`. (DR8) The first two terms in `S` compare current net project values. The last term is the dynamic option value of horizon research: one horizon increment raises the value of each of the `R` remaining potential reliability increments by `a_F v1`. The policy has three regions: - if `S<=0`, use all remaining capacity on reliability; - if `0=DR`, use all remaining capacity on horizon. Along horizon investment, `h` rises and `R` falls at equal rates. Hence `dS/ds = -a_H u1 - 2a_F v1 = -D<0`. (DR9) Along reliability investment, `f` rises and `R` falls, so `dS/ds = (a_H-a_F)v1<0`. (DR10) Consequently the switch surface can be crossed at most once and there is no reversal within the program. If initially `0 0`, (DR12) provided the horizon boundary remains positive, and `dh*/da_F = (v1 Q-v0-2v1 h*)/D`. (DR13) Thus a shorter horizon lag lowers `a_H` and reduces the amount of horizon research before rotation. If `v0>=v1 Q`, a longer reliability lag raises `a_F` and also brings rotation forward. Since `da_i/dtau_i=exp(-r tau_i)>0`, these signs translate directly to the lag comparative statics. If fixed calendar capacity is `xbar`, then `T*=h*/xbar` and position and time have the same signs. With endogenous capacity, `d ln T*/d tau_H = d ln h*/d tau_H - d ln xbar/d tau_H`; (DR14) the time sign requires the capacity elasticity to be smaller than the position elasticity. This is the distinction omitted by the old maintained-path theorem. ## 6. Task creation Let creation of new horizon-only tasks add `g h` to the horizon boundary. Net horizon depletion becomes `u1-g`. The switch index then changes along horizon research at rate `dS_g/ds = -{a_H(u1-g)+2a_Fv1}`. (DR15) The exact single-crossing condition in this benchmark is `g < u1 + 2(a_F/a_H)v1`. (DR16) Task creation can replenish horizon value faster than the original task stock depletes and rotation can still occur because horizon research both raises the reliability boundary and exhausts remaining research capacity. If (DR16) fails, the switch index need not decline, and an initially horizon-directed program need not rotate. This converts the old qualitative task-creation caveat into a growth-rate condition. ## 7. Scope and robustness The theorem is an optimizing finite research program, not a claim that the economy stays forever in the reliability phase. Repeated programs can generate cycles when reliability investment replenishes the horizon boundary, as the manuscript's later remark suggests. Stage 4 must separately solve strategic interaction between labs. Three perturbations preserve the result locally: 1. a nonconstant positive cross statistic preserves the ordering result whenever `(a_F-a_H)C` stays positive; 2. small annuity-remainder terms preserve a strict switch by continuity; 3. discounting research completion dates preserves a unique switch among ordered paths, with the remaining-program option `R` replaced by a discounted annuity of remaining research dates. Global ordering additionally requires the complementarity gain to dominate the timing gain from moving a currently more valuable project earlier. The result intentionally signs calendar time only under fixed capacity. This is not a weakness: it removes the false inference that a lower switching state automatically implies an earlier date under endogenous convex research effort. ## 8. Completion-date discounting Let research payoffs at program input `s` receive the additional factor `exp(-rho s)`, with `rho>0`, and write `m_i=a_i B_i-c_i`. For two adjacent intervals of length `ds`, a second-order expansion gives `value(HF)-value(FH)` ` = exp(-rho s){(a_F-a_H)C + rho(m_H-m_F)}ds^2 + o(ds^2)`. (DR17) Thus the sufficient global ordering condition is `(a_F-a_H)C(y)+rho{m_H(y)-m_F(y)}>0` (DR18) throughout the compact program region. It says that the complementarity-annuity gain from doing horizon first must dominate the timing gain from accelerating a currently more valuable reliability project. In the polynomial benchmark, conditional on this order, a switch after `z` units has `J_rho(z)=integral_0^z exp(-rho s)m_H(s,0)ds` ` +integral_z^Q exp(-rho s)m_F(z,0)ds`. The first-order condition is `m_H(z,0)-m_F(z,0)+a_F v1{1-exp[-rho(Q-z)]}/rho=0`. (DR19) Its left side has derivative `-a_H u1-a_F v1-a_F v1 exp[-rho(Q-z)]<0`, so the switch is unique when endpoint signs bracket zero. As `rho->0`, the discounted remaining-program annuity converges to `Q-z`, and (DR19) becomes the baseline switch condition. This extension makes explicit which part of the result uses the research- input clock and which part survives calendar discounting.