Both companies get a free quota allocation (their cap) but each would emit more than that under business-as-usual (BAU = 120 t/yr each). Company A's abatement cost curve is steep — cutting each extra ton gets expensive fast (an old steel mill). Company B's curve is shallow — it can cut cheaply (a biogas retrofit still has easy wins left). Left alone, each company must abate down to its own cap regardless of cost. With bilateral trading, they instead equalize marginal cost: the cheap abater (B) overshoots its own cap, abating extra tons and selling the surplus quota directly to A, who buys instead of abating expensively itself. Total required abatement across the pair stays the same — only who does the cutting changes, and total cost drops.
cost_i(a) = ½·k_i·a² (marginal cost = k_i·a)
trading: price = ΔTotal / (1/k_A + 1/k_B), a_i = price / k_i
autarky: a_i = max(0, BAU_i − cap_i) (no price, no trade)
- Abatement cost k — how steeply each ton beyond the first costs more; a low k means cheap, easy reductions remain.
- Free allocation cap — the emissions each company is permitted without buying anything.
- Trading ON/OFF — toggles whether the two companies may exchange quota directly (bilateral OTC trade) or must each hit their own cap alone (autarky).
Real-world relevance: this is the equimarginal principle behind every cap-and-trade scheme, seen here at the smallest possible scale — a single bilateral deal between two named counterparties, not an anonymous exchange-cleared price.