CHAPTER 2. THE MODIGLIANI-MILLER THEOREM holding shares and risky debt. One solution to this problem: if the firms cash stream can be spanned by other firms'cash streams, the contribution to risk sharing is redundant and only the value of the firm matters. See Bell Journal Symposium(Ekern and Wilson(1974), Leland(1974), Radner (1974). Another solution: if there is a large number of identical firms, each type of consumer can hold shares in a version of the firm that uniquely optimizes his needs for risk sharing. See Hart(1979). When these are not available, for example, because the number of firms is finite, the theory of the firm becomes very difficult(see for example, Dreze(1974),Grossman and Hart(1979)). Perhaps for this reason, much fo the theory of general equilibrium with incomplete markets has been deve\,e and Gale(1988)or ed for pure exchange models. For the valuation problem in general, see alle the Allen and Gale(1994 ). For an analysis of the Modigliani-Miller Theorem with default in a partial equilibrium setting, see Stiglitz(1969) and Hellwig (1981) 2.4 Bibliography Allen, F and D. Gale, (1988)."Optimal Security Design, " Review of finan cial Studies 1. 229-263 (1992)."Arbitrage, Short Sales, and Financial Innovation"Economet rica59,1041-68 (1994). Financial Innovation and Risk: Sharing. Cambridge, MA: MIT Arrow, K(1964)."The Role of Securities in the Optimal Allocation Risk-Bearing, "Review of Economic Studies 31, 91-96 Arrow, K. and G. Debreu(1954)."Existence of equilibrium for a com- petitive economy, Econometrica 22, 265-290 A Dammon, R and R Green(1987). Tax Arbitrage and the Existence of quilibrium Prices for Financial Assets, " Journal of Finance 42, 1143-66 uffie, J. D and W. Shafer(1985)."Equilibrium in Incomplete Markets I-A Basic Model of Generic Existence, Journal of Mathematical Economics 14,285-300 (1986)."Equilibrium in Incomplete Markets: II; Generic Existence in Stochastic Economies, "Journal of Mathematical Economics 15, 199-216 Dreze,J(ed )(1974). Allocation under Uncertainty: Equilibrium and Optimality; proceedings from a workshop sponsored by the International8 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM holding shares and risky debt. One solution to this problem: if the firm’s cash stream can be spanned by other firms’ cash streams, the contribution to risk sharing is redundant and only the value of the firm matters. See Bell Journal Symposium (Ekern and Wilson (1974), Leland (1974), Radner (1974)). Another solution: if there is a large number of identical firms, each type of consumer can hold shares in a version of the firm that uniquely optimizes his needs for risk sharing. See Hart (1979). When these are not available, for example, because the number of firms is finite, the theory of the firm becomes very difficult (see for example, Dreze (1974), Grossman and Hart (1979)). Perhaps for this reason, much fo the theory of general equilibrium with incomplete markets has been developed for pure exchange models. For the valuation problem in general, see Allen and Gale (1988) or the Allen and Gale (1994). For an analysis of the Modigliani-Miller Theorem with default in a partial equilibrium setting, see Stiglitz (1969) and Hellwig (1981). 2.4 Bibliography Allen, F. and D. Gale, (1988). “Optimal Security Design,” Review of Finan￾cial Studies 1, 229-263. – (1992). “Arbitrage, Short Sales, and Financial Innovation” Economet￾rica 59, 1041-68. – (1994). Financial Innovation and Risk Sharing. Cambridge, MA: MIT Press. Arrow, K. (1964). “The Role of Securities in the Optimal Allocation of Risk-Bearing,” Review of Economic Studies 31, 91-96. Arrow, K. and G. Debreu (1954). “Existence of equilibrium for a com￾petitive economy,” Econometrica 22, 265-290. Dammon, R. and R. Green (1987). “Tax Arbitrage and the Existence of Equilibrium Prices for Financial Assets,” Journal of Finance 42, 1143-66. Duffie, J. D. and W. Shafer (1985). “Equilibrium in Incomplete Markets: I—A Basic Model of Generic Existence,” Journal of Mathematical Economics 14, 285-300. – (1986). “Equilibrium in Incomplete Markets: II; Generic Existence in Stochastic Economies,” Journal of Mathematical Economics 15, 199-216. Dreze, J. (ed.) (1974). Allocation under Uncertainty: Equilibrium and Optimality; proceedings from a workshop sponsored by the International