Supplement to ``Martingale properties of self-enforcing debt''

We present some complementary results to Bidian and Bejan (2012). Part 1 provides necessary and sufficient transversality conditions for an agent's optimization problem. They are extensions to stochastic environments of the conditions given by Kocherlakota (1992), or alternatively, extensions to nonzero debt constraints of the corresponding conditions in Forno2003}. Part \ref{ap:B} presents an elementary proof of the characterization of NTT debt limits (Theorem 3.5 in the main paper) for the case when debt constraints bind in bounded time, that requires no martingale techniques or boundedness assumptions on the discounted debt limits. Part \ref{ap:C} complements results in Section 5.1 (in the main paper), showing that all the equilibria that can sustain bubbles under an interdiction to trade can be achieved from fixed, zero initial wealth for the agents. Thus endogeneity of debt limits causes multiplicity of not only asset prices (through bubbles), but also of real equilibrium allocations.rational bubbles, transversality conditions, endogenous debt limits, not-too-tight constraints, self-enforcing debt, limited enforcement

Martingale properties of self-enforcing debt

Not-too-tight (NTT) debt limits are endogenous restrictions on debt that prevent agents from defaulting and opting for a specified continuation utility, while allowing for maximal credit expansion (Alvarez and Jermann, 2000). For an agent facing some fixed prices for the Arrow securities, we prove that discounted NTT debt limits must differ by a martingale. Discounted debt limits are submartingales/martingales under an interdiction to trade/borrow, and can be supermartingales under a temporary interdiction to trade. With high interest rates and borrowing limited by the agent's ability to repay debt out of his future endowments, nonpositive NTT debt limits are unique. With low interest rates, bubbles limited by the size of the total martingale components in debt limits can be sustained in equilibrium. Bubbles arise in response to debt limits more restrictive (at the prevailing interest rates) than the total amount of self-enforcing debt allowed by the underlying enforcement limitations

Ownership Structure and Efficiency in Large Economies

We analyze the limit behavior of sequences of oligopolistic equilibria in which firms follow objectives consistent with their shareholders' interests. We show that the efficiency of the limit allocation depends on how firms' shares are distributed across consumers, and provide a characterization of the class of ownership structures that lead to Walrasian equilibrium allocations in the limit

Limited enforcement, bubbles and trading in incomplete markets

Rational bubbles are believed to be fragile and unable to explain the trading frenzy associated to price run-ups. With limited enforcement of credit contracts and endogenous debt limits designed to prevent default and allow for maximal credit expansion, a large class of bubbles can be introduced in asset prices by appropriately tightening agents' debt limits. By not affecting consumption, these bubbles are ideally suited to explain a variety of asset pricing puzzles. They can generate large increases in trade volume until they crash. Nonpositivity of debt limits restricts the potential for bubble injections to assets in zero supply or to equilibria with an infinite present value of aggregate endowment. Such equilibria are common in economies with limited enforcement, where interest rates are low to induce debt repayment (Bidian and Bejan 2012)

Supplement to ``Martingale properties of self-enforcing debt''

We present some complementary results to Bidian and Bejan (2012). Part 1 provides necessary and sufficient transversality conditions for an agent's optimization problem. They are extensions to stochastic environments of the conditions given by Kocherlakota (1992), or alternatively, extensions to nonzero debt constraints of the corresponding conditions in Forno2003}. Part \ref{ap:B} presents an elementary proof of the characterization of NTT debt limits (Theorem 3.5 in the main paper) for the case when debt constraints bind in bounded time, that requires no martingale techniques or boundedness assumptions on the discounted debt limits. Part \ref{ap:C} complements results in Section 5.1 (in the main paper), showing that all the equilibria that can sustain bubbles under an interdiction to trade can be achieved from fixed, zero initial wealth for the agents. Thus endogeneity of debt limits causes multiplicity of not only asset prices (through bubbles), but also of real equilibrium allocations

Ownership Structure and Efficiency in Large Economies

We analyze the limit behavior of sequences of oligopolistic equilibria in which firms follow objectives consistent with their shareholders' interests. We show that the efficiency of the limit allocation depends on how firms' shares are distributed across consumers, and provide a characterization of the class of ownership structures that lead to Walrasian equilibrium allocations in the limit

Ownership Structure and Efficiency in Large Economies

We analyze the limit behavior of sequences of oligopolistic equilibria in which firms follow objectives consistent with their shareholders' interests. We show that the efficiency of the limit allocation depends on how firms' shares are distributed across consumers, and provide a characterization of the class of ownership structures that lead to Walrasian equilibrium allocations in the limit.the objective of the firm, oligopolistic competition, ownership structure, efficiency

Martingale properties of self-enforcing debt

Not-too-tight (NTT) debt limits are endogenous restrictions on debt that prevent agents from defaulting and opting for a specified continuation utility, while allowing for maximal credit expansion (Alvarez and Jermann, 2000). For an agent facing some fixed prices for the Arrow securities, we prove that discounted NTT debt limits must differ by a martingale. Discounted debt limits are submartingales/martingales under an interdiction to trade/borrow, and can be supermartingales under a temporary interdiction to trade. With high interest rates and borrowing limited by the agent's ability to repay debt out of his future endowments, nonpositive NTT debt limits are unique. With low interest rates, bubbles limited by the size of the total martingale components in debt limits can be sustained in equilibrium. Bubbles arise in response to debt limits more restrictive (at the prevailing interest rates) than the total amount of self-enforcing debt allowed by the underlying enforcement limitations.rational bubbles, endogenous debt limits, not-too-tight constraints, self-enforcing debt, limited enforcement