Proposition 2.1 — consumer's best response â(z), b̂(γ) and the closed forms of H_m, H_v (H_m corrected beyond A_max)

In the demand-response model, let $A=\prod_i[0,\mu_iA_{\max}]$, $B=[\varepsilon,1]^d$, $c_1(a)=\frac12\sum_ia_i^2/\mu_i$, $c_2(b)=\sum_j\frac{\sigma_j^2}{\lambda_j}(b_j^{-1}-1)$ and $|\sigma(b)|^2=\sum_j\sigma_j^2b_j$, and let $H_m$, $H_v$ be the Hamiltonians (2.9). Let $\hat a_i(z)=\mu_i(z^-\wedge A_{\max})$ and $\hat b_j(\gamma)=(1\wedge(\lambda_j\gamma^-)^{-1/2})\vee\varepsilon$ for $j=1,\dots,d$. Then:

1. for every $z\in\mathbb R$, $\hat a(z)\in A$ and $\hat a(z)$ minimises $a\mapsto a\cdot\mathbf 1\,z+c_1(a)$ over $A$;
2. for every $\gamma\in\mathbb R$, $\hat b(\gamma)\in B$ and $\hat b(\gamma)$ minimises $b\mapsto c_2(b)-\gamma|\sigma(b)|^2$ over $B$;
3. for every $z$, with $m:=z^-\wedge A_{\max}$,

4. for every $\gamma$, $H_v(\gamma)=-\frac12\big(c_2(\hat b(\gamma))-\gamma|\sigma(\hat b(\gamma))|^2\big)$.

The Hamiltonian is the consumer's instantaneous benefit from a payment rate $z$ on the consumption drift and $\gamma$ on its volatility; this proposition makes it explicit, and it is what turns the first-best value into a closed form.

Formalization Note The paper prints $H_m(z)=\frac12\bar\mu(z^-\wedge A_{\max})^2$, which is false when $z^->A_{\max}$ (for $N=1$, $\mu=1$, $A_{\max}=1$, $z=-10$ the minimum of $-10a+a^2/2$ over $[0,1]$ is $-9.5$, so $H_m=9.5$, not $0.5$). Item 3 is the corrected formula, equal to the printed one exactly when $z^-\le A_{\max}$. The page's index range "$j=1,\dots,N$" for $\hat b$ is read as $j=1,\dots,d$. When $\gamma^-=0$ the paper reads $(\lambda_j\gamma^-)^{-1/2}$ as $+\infty$, so $\hat b_j=1$; the Lean definition makes this case explicit.