Lemma 1. For any TP b ∈ B$_{1}$ with any budget Γ$_{b}$ ≤ 1 , the slope curve S ( Z$_{b}$, Γ$_{b}$ , b ) is piecewise constant and decreasing in Z$_{b}$ for all Z$_{b}$ ≥ ¯ h (Γ$_{b}$ , b ) . Further, for any given Γ$_{b}$ and any scalar δ$_{b}$ ≤ Γ$_{b}$ , we have For any given Z$_{b}$ , let T ( Z$_{b}$, Γ$_{b}$ , b ) denote the slope (with respect to the pico resource) of the function ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) at any Γ$_{b}$ : Γ$_{b}$ ≥ h ( Z$_{b}$, b ) . In particular, Then, for any Z$_{b}$ ≤ 1 , the slope curve T ( Z$_{b}$, Γ$_{b}$ , b ) is piecewise constant and decreasing in Γ$_{b}$ for all Γ$_{b}$ ≥ h ( Z$_{b}$, b ) . Further, for any z$_{b}$ ≤ Z$_{b}$ , we have that Proof of Proposition 4 : We will first prove the claim for b$_{1}$ = b$_{2}$ and then consider the case b$_{1}$ = b$_{2}$ . Consider the LHS of (15) and parse the difference as The RHS of (15) can be parsed analogously. Then, it suffices to show that and that We first consider (33). Letting ˜ Γ = Γ + ˜ δ, ˜ Γ = Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ , ˘ Γ = ˜ Γ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ = Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ , S ( Z$_{b}$, Γ$_{b}$ , b ) ≤ S ( Z$_{b}$, Γ$_{b}$ − δ$_{b}$, b ) , ∀ Z$_{b}$ ≥ ¯ h (Γ$_{b}$ − δ$_{b}$, b ) (29) T ( Z$_{b}$, Γ$_{b}$ , b ) = lim δ → 0$_{+}$ ˆ O ( Z$_{b}$, Γ$_{b}$ + δ, b ) − ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) δ (30) T ( Z$_{b}$, Γ$_{b}$ , b ) ≤ T ( Z$_{b}$ − z$_{b}$, Γ$_{b}$ , b ) , ∀ Γ$_{b}$ ≥ h ( Z$_{b}$ − z$_{b}$, b ) . (31) ˆ O (Γ , Γ ) − ˆ O (Γ + δ, Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) = ˆ O (Γ , Γ ) − ˆ O (Γ , Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) ︸ ︷︷ ︸ + ˆ O (Γ , Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) − ˆ O (Γ + δ, Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) ︸ ︷︷ ︸ ˆ O (Γ , Γ ) − ˆ O (Γ , Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) ≤ ˆ O (Γ + ˜ δ, Γ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ ) − ˆ O (Γ + ˜ δ, Γ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) (32) ˆ O (Γ , Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) − ˆ O (Γ + δ, Γ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) ≤ ˆ O (Γ + ˜ δ, Γ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) − ˆ O (Γ + ˜ δ + δ, Γ + ˜ δ$_{b}$$_{2}$ e$_{b}$$_{2}$ + δ$_{b}$$_{1}$ e$_{b}$$_{1}$ ) (33)