《金融经济学》（英文版） Chapter 2 The Modigliani-Miller theorem

C hapter 2 p The Modigliani-Miller theorem When capital markets are perfect and compl e. corDor decisions are trivial," 2.1 Arrow-Debreu model with assets 2.1.1 Primitives (2,F,P X=f: Q-R a is F-measural h=1,,|H ah∈X i=1,2,…,| X2CX,e2∈X,1∈R,u4:X1→R j=1,2,…,|J YCX 2.1.2 Arrow-Debreu model We begin by reviewing the Arrow-Debreu model There is a finite set of states of nature w Q and a single good in each state. The commodity space is R. There is a finite set of firms j E each characterized by a production set Yi cr. There is a finite set of consumers i E I, each characterized by a consumption set Xi, an endowment ei E xi and a utility function w;: Xi- R. Each agent i owns a fraction Bi; of firm

Chapter 2 The Modigliani-Miller theorem “When capital markets are perfect and complete, corporate decisions are trivial.” 2.1 Arrow-Debreu model with assets 2.1.1 Primitives (Ω, F, P) X = {x : Ω → R | x is F-measurable} h = 1, ..., |H| zh ∈ X i = 1, 2, ..., |I| Xi ⊂ X, ei ∈ Xi, θi ∈ RJ +, ui : Xi → R j = 1, 2, ..., |J| Yj ⊂ X 2.1.2 Arrow-Debreu model We begin by reviewing the Arrow-Debreu model. There is a finite set of states of nature ω ∈ Ω and a single good in each state. The commodity space is RΩ. There is a finite set of firms j ∈ J, each characterized by a production set Yj ⊂ RΩ. There is a finite set of consumers i ∈ I, each characterized by a consumption set Xi, an endowment ei ∈ Xi, and a utility function ui : Xi → R. Each agent i owns a fraction θij of firm j. 1

CHAPTER 2. THE MODIGLIANI-MILLER THEOREM An allocation is an array (a, y)=(f=thiel, [vi)ie)such that c:E Xi for every i and y; E Y; for every j. An allocation(a, y) is attainable if eit>yj a price system or price vector is a non-zero element p E R. An Walrasian equilibrium consists of an attainable allocation(a, y) and a price system such that. for every v∈ arg max{py:v∈Y} and for every i n∈ arg max{ut(x):∈X,p≤p·(e+∑ Note that unlike the standard model. we assume that consumers receive cash Flows in each state directly Note that shareholders unanimously want the firm to adopt profit maxi mization as its objective function Under well known conditions, every competitive equilibrium is Pareto- efficient and every Pareto-efficient allocation is a competitive equilibrium with lump-sum transfers 2.1.3 Securities Now we introduce a finite set of securities h E H each represented by a vector of returns zh E R. Securities are in zero net supply. The vector of securities prices is denoted by g E r where gh is the price of security et(a, y, p)be a Walrasian equilibrium and suppose that consumers and firms are allowed to trade securities at the prices q. Let a,(resp. ai)denote j,'s(resp. consumer is) portfolio excess demand for securities. Firm j's profit is now P·y+∑an2h-q·a and consumer is budget constraint is now p+q0≤p(+∑+∑动

2 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM An allocation is an array (x, y) = ³ {xi}i∈I , {yj}j∈J ´ such that xi ∈ Xi for every i and yj ∈ Yj for every j. An allocation (x, y) is attainable if X i xi = X i ei +X j yj . A price system or price vector is a non-zero element p ∈ RΩ. An Walrasian equilibrium consists of an attainable allocation (x, y) and a price system such that, for every j, yj ∈ arg max{p · yj : yj ∈ Yj}, and for every i, xi ∈ arg max{ui(xi) : xi ∈ Xi, p · xi ≤ p · Ã ei +X j θijyj ! . Note that unlike the standard model, we assume that consumers receive cash flows in each state directly. Note that shareholders unanimously want the firm to adopt profit maxi￾mization as its objective function. Under well known conditions, every competitive equilibrium is Pareto￾efficient and every Pareto-efficient allocation is a competitive equilibrium with lump-sum transfers. 2.1.3 Securities Now we introduce a finite set of securities h ∈ H each represented by a vector of returns zh ∈ RΩ. Securities are in zero net supply. The vector of securities prices is denoted by q ∈ RH where qh is the price of security h. Let (x, y, p) be a Walrasian equilibrium and suppose that consumers and firms are allowed to trade securities at the prices q. Let αj (resp. αi) denote firm j’s (resp. consumer i’s) portfolio excess demand for securities. Firm j’s profit is now p · Ã yj +X h αjhzh ! − q · αj and consumer i’s budget constraint is now p · xi + q · αi ≤ p · Ã ei +X j θijyj +X h αihzh !

2. 1. ARROW-DEBREU MODEL WITH ASSETS Equilibrium requires that P·ah,vh∈H. Otherwise firms could increase profits without bound. But under this condi- tion, any portfolio is optimal. Thus equilibrium with securities requires only that attainability be satisfied We can do the same thing with traded equity. If equity is fairly priced, there is no reason for anyone to trade it 2.1.4 Irrelevance of capital structure a1=(x2,a2,B1)∈A≡X1×RBxR a=(9,0)∈A≡Y×R (a)ier×(a3)jeJ Definition 1 An allocation a=(oilier X(ai)iej is attainable if j∈J i∈I Definition 2 An attainable allocation a=(ai)erx(ai)ieJ is weakly efficient if there does not exist an attainable allocation a=(aier x(a')ie sUC that ui(ai)<ui(ai for all i. An attainable allocation a=(aiier x(ailieJ is(strongly) efficient if there does not exist an attainable allocation a (a')iel x(a')ieJ such that ui(i)<ui(ai) for all i and ui(ai)<ui(ai) for Definition 3 A Walrasian equilibrium consists of an attainable allocation a=(ai)iel X(ai)ieJ and a price vector(p, )E XR such that, for every j, a; E A; maximizes the value of the firm

2.1. ARROW-DEBREU MODEL WITH ASSETS 3 Equilibrium requires that qh = p · zh, ∀h ∈ H. Otherwise firms could increase profits without bound. But under this condi￾tion, any portfolio is optimal. Thus equilibrium with securities requires only that attainability be satisfied: X i αi +X j αj = 0. We can do the same thing with traded equity. If equity is fairly priced, there is no reason for anyone to trade it. 2.1.4 Irrelevance of capital structure ai = (xi, αi, βi) ∈ Ai ≡ Xi × RH × RJ aj = (yj , αj ) ∈ Aj ≡ Yj × RH a = (ai)i∈I × (aj )j∈J Definition 1 An allocation a = (ai)i∈I × (aj )j∈J is attainable if X i∈I xi = X j∈J yj X i∈I αi +X j∈J αj = 0 X i∈I αi = 1. Definition 2 An attainable allocation a = (ai)i∈I×(aj )j∈J is weakly efficient if there does not exist an attainable allocation a0 = (a0 i)i∈I × (a0 j )j∈J such that ui(xi) < ui(xi) for all i. An attainable allocation a = (ai)i∈I × (aj )j∈J is (strongly) efficient if there does not exist an attainable allocation a0 = (a0 i)i∈I × (a0 j )j∈J such that ui(xi) ≤ ui(xi) for all i and ui(xi) < ui(xi) for some i. Definition 3 A Walrasian equilibrium consists of an attainable allocation a = (ai)i∈I × (aj )j∈J and a price vector (p, q) ∈ X × RH such that, for every j, aj ∈ Aj maximizes the value of the firm Vj = vj −X h qhαjh = p · Ã yj +X h αjhzh ! −X h qhαjh

CHAPTER 2. THE MODIGLIANI-MILLER THEOREM and,for every i, a; E A: maximizes i(zi)subject to the budget constraint h ∑ P Cith+ 9+ aih2 Theorem4Let(a,p,q)∈X×RH∈AxX× r be a walrasian equilibriun and let(a)ieJ be an arbitrary allocation of portfolios for firms. Then there exists a Walrasian equilibrium(a, p, q) such that a=(,a3),v Note also that, by the previous argument, V;=V for every j There are two aspects to the Modigliani-Miller theorem: one says that the firms choice of financial strategy a; has no effect on the value of the firm(or shareholder's welfare); the other says that the choice of a, has no essential impact on equilibrium. Here we are making the second(stronger 2.2 Equilibrium with incomplete markets To simplify, and avoid some thorny issues about the objective function of the firm. we assume that production sets are singletons Y={},vj∈ We start by assurning that firms do not trade in securities a,=0. There are ies, so that consumption bundles can only be achieved by trading securities ∑+∑a+∑

4 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM and, for every i, ai ∈ Ai maximizes ui(xi) subject to the budget constraint p · xi +X h αihqh +X j βijvj ≤ p · ei +X j θijVj +p · ÃX h αizh +X j βij à yj +X h αjhzh !!. Theorem 4 Let (a, p, q) ∈ X×RH ∈ A×X×RH be a Walrasian equilibrium and let (α0 j )j∈J be an arbitrary allocation of portfolios for firms. Then there exists a Walrasian equilibrium (a0 , p, q) such that a0 = (a0 i)i∈I × (a0 j )j∈J a0 i = (xi, α0 i, β0 i), ∀i a0 j = (yj , α0 j ), ∀j. Note also that, by the previous argument, Vj = V 0 j for every j. There are two aspects to the Modigliani-Miller theorem: one says that the firm’s choice of financial strategy αj has no effect on the value of the firm (or shareholder’s welfare); the other says that the choice of αj has no essential impact on equilibrium. Here we are making the second (stronger) claim. 2.2 Equilibrium with incomplete markets To simplify, and avoid some thorny issues about the objective function of the firm, we assume that production sets are singletons: Yj = {y¯j}, ∀j ∈ J. We start by assuming that firms do not trade in securities αj = 0. There are no Arrow securities, so that consumption bundles can only be achieved by trading securities. xi = ei +X j θijyj +X h αijzh +X j βijyj

2.2. EQUILIBRIUM WITH INCOMPLETE MARKETS Since firms have no decision to make, equilibrium is achieved if consumers maximize their utility subject to the budget constraint max ∑By"+q·0≤∑6 and markets for shares and securities clear. ∑=(1, Now change a=0 to aj, change U to j=Uj+q a,, and change ai to di=ai->i Bi, ai. Checking the optimality of the consumers problem and the attainability conditions we see that the economy is still in equilibrium. Definition 5 An equilibrium with incomplete markets consists of an attain able allocation a=(a)ier×(a)ye∈ A and a price vector(q,t)∈R×R such that, for every j, a, E A, maximizes the value of the firm 1=0-∑=m{:(m+∑9)}∑ and, for every i, a; E Ai maximizes ui(i) subject to the budget constraint ∑+∑≤∑"V wnere =+∑+∑(+∑动 Theorem 6 Let(a,,vEAxRXR be an equilibrium with incomplete markets and let(a)jeJ be an arbitrary allocation of portfolios for firms. Then there exists an equilibrium with incomplete markets(a, 9, a) such that (aier x(a)ieJ

2.2. EQUILIBRIUM WITH INCOMPLETE MARKETS 5 Since firms have no decision to make, equilibrium is achieved if consumers maximize their utility subject to the budget constraint: max ui(xi) s.t. P j βijvj + q · αi ≤ P j θijvj ; and markets for shares and securities clear: X i αi = 0 and X i βi = (1, ..., 1). Now change αj = 0 to αˆj , change vj to vˆj = vj + q · αj , and change αi to αˆi = αi − P j βijαˆj . Checking the optimality of the consumers problem and the attainability conditions we see that the economy is still in equilibrium. Definition 5 An equilibrium with incomplete markets consists of an attain￾able allocation a = (ai)i∈I ×(aj )j∈J ∈ A and a price vector (q, v) ∈ RH × RJ such that, for every j, aj ∈ Aj maximizes the value of the firm Vj = vj −X h qhαjh = max i ( µi · à yj +X h αjhzh !) −X h qhαjh and, for every i, ai ∈ Ai maximizes ui(xi) subject to the budget constraint X h αihqh +X j βijvj ≤ X j θijVj , where xi = ei +X h αihzh +X j βij à yj +X h αjhzh ! . Theorem 6 Let (a, q, v) ∈ A × RH × RJ be an equilibrium with incomplete markets and let (α0 j )j∈J be an arbitrary allocation of portfolios for firms. Then there exists an equilibrium with incomplete markets (a0 , q, v0 ) such that a0 = (a0 i)i∈I × (a0 j )j∈J a0 i = (xi, α0 i, β0 i), ∀i a0 j = (yj , α0 j ), ∀j

CHAPTER 2. THE MODIGLIANI-MILLER THEOREM Note that the space of commodity bundles that can be spanned by trading uity and securities is exogenous. but only because we have assumed the firms choice of production plan is exogenous. In other words, there is no financial innovation. This assumption is crucial for the MM theorem 2.3 Default 2.3.1 Default with complete markets For simplicity we assume there is a single firm i= l with a single feasible production plan y(w)>0, and a single security with payoffs z(w)=1 Limited liability raises the possibility of default and risky debt. Let i(aj,w denote the return to risky debt and i(ai, w) the return to equity in a firm with risky debt. Then 4y,)=1((-a)i)+3)<0 and y(w)+aj2(w) if y(w)+aj2(w)20 if ya)+ <0. If there are complete markets, the value of the risky debt is q=p·2(a) and the value of equity is P·叭0j The value of the firm to the original shareholders V=+ P·(ay)+ayP·(a) So default doesn't add value to the firm Assume that there is a single type of firm consisting of a continuum of identical firms. These firms choose different levels of risky debt. The number of securities may be great enough to span the entire commodity space R For example, suppose y(w)=w and choose ai=-w+1 for w=1, ., 92

6 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM Note that the space of commodity bundles that can be spanned by trading equity and securities is exogenous, but only because we have assumed the firm’s choice of production plan is exogenous. In other words, there is no financial innovation. This assumption is crucial for the MM theorem. 2.3 Default 2.3.1 Default with complete markets For simplicity we assume there is a single firm j = 1 with a single feasible production plan y(ω) > 0, and a single security with payoffs z(ω)=1. Limited liability raises the possibility of default and risky debt. Let zˆ(αj , ω) denote the return to risky debt and yˆ(αj , ω) the return to equity in a firm with risky debt. Then zˆ(αj , ω) = ½ z(ω) if y(ω) + αjz(ω) ≥ 0 y(ω)/(−αj2) if y(ω) + αjz(ω) < 0. and yˆ(αj , ω) = ½ y(ω) + αjz(ω) if y(ω) + αjz(ω) ≥ 0 0 if y(ω) + αjz(ω) < 0. If there are complete markets, the value of the risky debt is qˆ = p · zˆ(αj ) and the value of equity is vˆ = p · yˆ(αj ). The value of the firm to the original shareholders is Vˆ = ˆv + ˆqαj = p · yˆ(αj ) + αjp · zˆ(αj ) = p · y. So default doesn’t add value to the firm. Assume that there is a single type of firm j consisting of a continuum of identical firms. These firms choose different levels of risky debt. The number of securities may be great enough to span the entire commodity space RΩ. For example, suppose y(ω) = ω and choose αω j = −ω + 1 for ω = 1, ..., |Ω|

23. DEFAULT Then i(a ) pays one unit if w=19) and nothing otherwise, that is, it i an Arrow security for the state w=Q2. a portfolio consisting of one unit of i(a a-)and minus two units of i (a,a )will yield one unit in state w=Q-1 and nothing otherwise, that is, it is an Arrow security for the state w=Q2 -1. Continuing in this way we can generate Arrow securities for each state. This is a case where capital structure is irrelevant for the individual firm, but not for the equilibrium 2.3.2 Default with incomplete markets To define an equilibrium, we assume that consumers can hold the firms debt but cannot issue debt or sell short the firms equity.(This isn't necessary but simplifies the story Definition 7 An equilibrium with incomplete markets and default consists of an attainable allocation a=(a)ie×(a)∈ A and a price vector(q,)∈ R XR such that a, E A, maximizes the value of the firm V,=U;-qa=max u;(y;(a;)+aii(ai))-qa and, for every i, a: E Ai maximizes ui(ai) subject to the budget constraint a2q+B2U≤6V wnere r=e1+a2(ay)+1((ay)+a(a3) In this case, we have to deal with the valuation problem explicitly: be- cause markets are incomplete, individuals may disagree in their valuation of urity. Only those who value it most highly will hold a positive quantity of a security or equity in equilibrium 2.3.3 Related issues With complete markets, all shareholders agree that value maximization is the right objective function for the firm. With incomplete markets, this may not be the case. The firms choice of yi and a; has two effects, on Vi and on the risk sharing that can be achieved by

2.3. DEFAULT 7 Then yˆ ³ α|Ω| j ´ pays one unit if ω = |Ω| and nothing otherwise, that is, it is an Arrow security for the state ω = |Ω|. A portfolio consisting of one unit of yˆ ³ α|Ω|−1 j ´ and minus two units of yˆ ³ α|Ω| j ´ will yield one unit in state ω = |Ω| − 1 and nothing otherwise, that is, it is an Arrow security for the state ω = |Ω| − 1. Continuing in this way we can generate Arrow securities for each state. This is a case where capital structure is irrelevant for the individual firm, but not for the equilibrium. 2.3.2 Default with incomplete markets To define an equilibrium, we assume that consumers can hold the firm’s debt but cannot issue debt or sell short the firm’s equity. (This isn’t necessary, but simplifies the story). Definition 7 An equilibrium with incomplete markets and default consists of an attainable allocation a = (ai)i∈I × (aj ) ∈ A and a price vector (q, v) ∈ RH × R such that aj ∈ Aj maximizes the value of the firm Vj = vj − qαj = max i {µi · (yj (αj ) + αjzˆ(αj ))} − qαj and, for every i, ai ∈ Ai maximizes ui(xi) subject to the budget constraint αiq + βiv ≤ θiV, where xi = ei + αizˆ(αj ) + βi (yj (αj ) + αjzˆ(αj )). In this case, we have to deal with the valuation problem explicitly: be￾cause markets are incomplete, individuals may disagree in their valuation of a security. Only those who value it most highly will hold a positive quantity of a security or equity in equilibrium. 2.3.3 Related issues With complete markets, all shareholders agree that value maximization is the right objective function for the firm. With incomplete markets, this may not be the case. The firm’s choice of yj and αj has two effects, on the value of the firm Vj and on the risk sharing that can be achieved by

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 International

8 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

2.4.BⅠ BLIOGRAPHY Economic Association. New York: Wiley. Ekern, Steinar and Robert Wilson(1974).On the Theory of the Firm n Economy with Incomplete Markets Bell Journal of Economics 5, 171-80 Grossman, S and O. Hart(1979)."A Theory of Competitive Equilibrium in Stock Market Economies, Economtrica 47, 293-329 Hart, O.(1975)."On the Optimality of Equilibrium when the Market Structure is Incomplete, Journal of Economic Theory 11, 418-43 (1979)." On Shareholder Unanimity in Large Stock Market Economies Econometrica 47. 1057-83 Hellwig, M.(1981)"Bankruptcy, Limited Liability, and the Modigliani Miller Theorem. "American Economic Review 71. 155-70 Leland, H.(1974)."Production Theory and the Stock Market, " Bell Journal of Economics 5, 125-44 Magill, M. and M. Quinzii(1996). Theory of Incomplete Markets, Volume 1. Cambridge MA: MIT Pres Radner, R(1972)."Existence of Equilibrium of Plans, Prices, and Price xpectations in a Sequence of Markets, " Econometrica 40, 289-303 (1974)."A Note on Unanimity of Stockholders'Preferences among Alternative production Plans: A Reformulation of the ekern -Wilson model"? Bell Journal of Economics 5, 181-84 Stiglitz, J (1969)"A Re-Examination of the Modigliani-Miller Theorem, American Economic review 59. 784-93

2.4. BIBLIOGRAPHY 9 Economic Association. New York: Wiley. Ekern, Steinar and Robert Wilson (1974). “On the Theory of the Firm in an Economy with Incomplete Markets Bell Journal of Economics 5, 171-80. Grossman, S. and O. Hart (1979). “A Theory of Competitive Equilibrium in Stock Market Economies,” Economtrica 47, 293-329. Hart, O. (1975). “On the Optimality of Equilibrium when the Market Structure is Incomplete,” Journal of Economic Theory 11, 418-43. – (1979). “On Shareholder Unanimity in Large Stock Market Economies,” Econometrica 47, 1057-83. Hellwig, M. (1981) “Bankruptcy, Limited Liability, and the Modigliani￾Miller Theorem,” American Economic Review 71, 155-70. Leland, H. (1974). “Production Theory and the Stock Market,” Bell Journal of Economics 5, 125-44. Magill, M. and M. Quinzii (1996). Theory of Incomplete Markets, Volume 1. Cambridge MA: MIT Press. Radner, R. (1972). “Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets,” Econometrica 40, 289-303. – (1974). “A Note on Unanimity of Stockholders’ Preferences among Alternative Production Plans: A Reformulation of the Ekern-Wilson Model” Bell Journal of Economics 5, 181-84. Stiglitz, J. (1969) “A Re-Examination of the Modigliani-Miller Theorem,” American Economic Review 59, 784-93