Proposition 2.1 — consumer's best response $\hat a(z)$, $\hat b(\gamma)$ and the closed forms of $H_m$, $H_v$ ($H_m$ corrected beyond $A_{\max}$)

In the demand-response model with effort boxes $A=\prod_i[0,\mu_iA_{\max}]$ and $B=[\varepsilon,1]^d$, costs $c_1,c_2$ and Hamiltonians $H_m,H_v$ defined by the infima (2.9), the consumer's best response to the payment rates $(z,\gamma)$ is $\hat a_i(z)=\mu_i(z^-\wedge A_{\max})$ and $\hat b_j(\gamma)=(1\wedge(\lambda_j\gamma^-)^{-1/2})\vee\varepsilon$. Precisely:

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 $\gamma$,

The closed forms of the consumer's Hamiltonian are what makes the producer's problem a deterministic scalar optimisation in the payment rates.

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; it equals the printed one exactly when $z^-\le A_{\max}$. The page's index range "$j=1,\dots,N$" for $\hat b$ is read as $j=1,\dots,d$. $\hat b_j(\gamma)$ is $1$ when $\lambda_j\gamma^-\le1$, the paper's reading of $0^{-1/2}=+\infty$.