Lemma B.2, p. 28 — for max_{A∈[0,Ā]} y·u(A/y, d) − C(A): A*(y) increasing, A*(y)/y decreasing, V*(y) concave, dV*/dy decreasing in d

Assume Assumptions 1–2 (with the mission's standing pins). For $y > 0$ and $d \ge 0$ consider

the consumer's problem when she spreads total attention $A$ evenly over "$y$ platforms" of addictiveness $d$. Let $A^*(y,d)$ be a maximizer and $V^*(y,d)$ the maximized value. Then for every $d \ge 0$:

1. $A^*(y,d)$ is (weakly) increasing in $y$ on $(0,\infty)$;
2. $A^*(y,d)/y$ is (weakly) decreasing in $y$ on $(0,\infty)$;
3. $V^*(\cdot,d)$ is concave on $(0,\infty)$;

and 4. $\partial V^*/\partial y$ is decreasing in $d$, in integrated form: for all $0 < y_1 < y_2$, the map

is (weakly) decreasing on $[0,\infty)$.

At $y = K$ the problem defines $A_K(d)$, and $V^*(K,d) - V^*(K-1,d)$ is the consumer's gain from joining the $K$-th platform; the lemma is the engine of Step 1 of Proposition 2.

Formalization Note. The page says "$V^*(y)$ is strictly concave in $y$". That is false: on the paper's Figure 3 instance $u(a,d) = 1 - e^{-(a-d)}$, $C(a) = a/2$ with a large cap, $V^*(y,d) = y\,(1/2 - (d + \ln 2)/2)$ is linear in $y$. The proof shows concavity, which is all that is used downstream, so concavity is stated. The derivative claim is stated as a decreasing difference because the paper does not establish differentiability of $V^*$. The maximizer is a parameter Astar assumed to maximize (A.1) for every $y > 0$, $d \ge 0$ (it is unique by strict concavity, and exists when $\overline{A} < \infty$ by compactness).