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a(sT aggregate endowment arbitrage arbitrary assumptions Bh(sl Bh(st bubble component budget constraints ch,zh complete markets concave concave function consumption allocation consumption set Corollary 4.7 defined denote dividend stream economy satisfying endowment is bounded Example 5.5 exists a state-price F(sT fact finite maturity finite number follows function fundamental value Furthermore given H(sT Hence household h immediate successor implies incomplete inessential borrowing limits infimum intertemporal equilibrium Lemma monetary equilibria Nh\Nh node st non-negative Note overlapping generations model Pareto optimality perturbation ph(st portfolio trading plan positive net supply possible present value product topology proof Proposition 3.1 q(st rational expectations recursive preferences result rh,ph right hand side satisfying conditions security j traded sequence simple equilibrium speculative bubble sT\st sT+1 sT|s state-price process a(s subgradient sunspot equilibria suppose supremum Theorem 4.1 Tn+i Tychonoff's theorem unique state-price process valued fiat money wealth well-defined z(sT zero net supply zh(s zh(st