Theorem 1, p. 14 — the unregulated aggregation model attains the benchmark welfare, dispatch and wholesale prices

Consider the networked market under Assumptions 1 and 2, and assume the benchmark dispatch (9) is feasible (Assumption 3). Let $(h^*_B,z^*_B,y^*_B)$ be the optimal solution of (9) and $\mathcal W_B^*$ its welfare. Then:

1. the aggregation dispatch (21) has an optimal solution;
2. every optimal solution $(h_A,x_A,d_A,y_A)$ of (21) has welfare

and the same dispatch as the benchmark: $x_A-d_A=z^*_B$, $y_A=y^*_B$, $h_A=h^*_B$; 3. the split $x^*=[z^*_B]^+$, $d^*=[-z^*_B]^+$ (with $h^*_B$, $y^*_B$) is an optimal solution of (21); 4. the wholesale prices coincide: a vector $\lambda$ is a vector of optimal Lagrange multipliers of (5a) at an optimal solution of (21) if and only if it is one at an optimal solution of (9).

Hence a monopolistic, profit-seeking aggregator using two-part pricing causes no loss of market efficiency: welfare, dispatch and locational prices are those of direct participation.

Formalization Note Theorem 1 names no assumptions; it is stated under Assumptions 1–3, which the paper's proof uses through Propositions 1 and 3 (Assumption 4 follows from Assumption 3 via the split). Problem (21) determines only the net sales $x-d$, not $x$ and $d$ separately; "the optimal $x^*,d^*,y^*$ … are the same as those solving (9)" is stated as items 2 and 3. Problem (21) is written in the separate variables $x$, $d$ with the constraints (18) as printed.