content_image/324432.jpg content_image/324433.jpg content_image/324434.jpg content_image/324435.jpg content_image/324436.jpg content_image/324437.jpg
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 

<img src='content_image/324432.jpg'>

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, 

<img src='content_image/324433.jpg'>

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 

<img src='content_image/324434.jpg'>

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 

<img src='content_image/324435.jpg'>

The RHS of (15) can be parsed analogously. Then, it suffices to show that 

<img src='content_image/324436.jpg'>

and that 

<img src='content_image/324437.jpg'>

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}$ , 

