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 (?,F,P) X = {x : ?→ R | x is F-measurable} h =1,...,|H| z h ∈ X i =1,2,...,|I| X i ? X,e i ∈ X i ,θ i ∈R J + ,u i : X i → R j =1,2,...,|J| Y j ? 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 ∈ ? and a single good in each state. The commodity space is R ? .Thereisafinite set of firms j ∈ J,each characterized by a production set Y j ?R ? .Thereisafinite set of consumers i ∈ I, each characterized by a consumption set X i ,anendowment e i ∈ X i , and a utility function u i : X i → R. Each agent i owns a fraction θ ij of firm j. 1 2 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM An allocation is an array (x,y)= 3 {x i } i∈I ,{y j } j∈J ′ such that x i ∈ X i for every i and y j ∈ Y j for every j. An allocation (x,y) is attainable if X i x i = X i e i + X j y j . A price system or price vector is a non-zero element p ∈ R ? .AnWalrasian equilibrium consists of an attainable allocation (x,y) and a price system such that, for every j, y j ∈ argmax{p · y j : y j ∈ Y j }, and for every i, x i ∈ argmax{u i (x i ):x i ∈ X i ,p· x i ≤ p · ? e i + X j θ ij y j ! . Note that unlike the standard model, we assume that consumers receive cash flowsineachstatedirectly. Note that shareholders unanimously want the firm to adopt profitmaxi- mization as its objective function. Under well known conditions, every competitive equilibrium is Pareto- e?cient and every Pareto-e?cient allocation is a competitive equilibrium with lump-sum transfers. 2.1.3 Securities Nowweintroduceafinite set of securities h ∈ H each represented by a vector of returns z h ∈ R ? . Securities are in zero net supply. The vector of securities prices is denoted by q ∈ R H where q h 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.Letα j (resp. α i )denote firm j’s (resp. consumer i’s) portfolio excess demand for securities. Firm j’s profitisnow p · ? y j + X h α jh z h ! ?q · α j and consumer i’s budget constraint is now p · x i + q · α i ≤ p · ? e i + X j θ ij y j + X h α ih z h ! . 2.1. ARROW-DEBREU MODEL WITH ASSETS 3 Equilibrium requires that q h = p · z h ,?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 α i + X j α j =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 a i =(x i ,α i ,β i ) ∈ A i ≡ X i ×R H ×R J a j =(y j ,α j ) ∈ A j ≡ Y j ×R H a =(a i ) i∈I × (a j ) j∈J Definition 1 An allocation a =(a i ) i∈I × (a j ) j∈J is attainable if X i∈I x i = X j∈J y j X i∈I α i + X j∈J α j =0 X i∈I α i = 1. Definition 2 An attainable allocation a =(a i ) i∈I ×(a j ) j∈J is weakly e?cient if there does not exist an attainable allocation a 0 =(a 0 i ) i∈I × (a 0 j ) j∈J such that u i (x i ) <u i (x i ) for all i. An attainable allocation a =(a i ) i∈I × (a j ) j∈J is (strongly) e?cient if there does not exist an attainable allocation a 0 = (a 0 i ) i∈I × (a 0 j ) j∈J such that u i (x i ) ≤ u i (x i ) for all i and u i (x i ) <u i (x i ) for some i. Definition 3 A Walrasian equilibrium consists of an attainable allocation a =(a i ) i∈I ×(a j ) j∈J and a price vector (p,q) ∈ X ×R H such that, for every j, a j ∈ A j maximizes the value of the firm V j = v j ? X h q h α jh = p · ? y j + X h α jh z h ! ? X h q h α jh 4 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM and, for every i, a i ∈ A i maximizes u i (x i ) subject to the budget constraint p · x i + X h α ih q h + X j β ij v j ≤ p · e i + X j θ ij V j +p · ? X h α i z h + X j β ij ? y j + X h α jh z h !! . Theorem 4 Let (a,p,q) ∈ X×R H ∈ A×X×R H be a Walrasian equilibrium and let (α 0 j ) j∈J be an arbitrary allocation of portfolios for firms. Then there exists a Walrasian equilibrium (a 0 ,p,q) such that a 0 =(a 0 i ) i∈I × (a 0 j ) j∈J a 0 i =(x i ,α 0 i ,β 0 i ),?i a 0 j =(y j ,α 0 j ),?j. Note also that, by the previous argument, V j = 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 α j has no e?ect on the value of the firm (or shareholder’s welfare); the other says that the choice of α j 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: Y j = {ˉy j },?j ∈ J. We start by assuming that firms do not trade in securities α j =0.Thereare no Arrow securities, so that consumption bundles can only be achieved by trading securities. x i = e i + X j θ ij y j + X h α ij z h + X j β ij y j . 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 u i (x i ) s.t. P j β ij v j + q · α i ≤ P j θ ij v j ; and markets for shares and securities clear: X i α i =0 and X i β i =(1,...,1). Now change α j =0to ?α j , change v j to ?v j = v j + q · α j ,andchangeα i to ?α i = α i ? P j β ij ?α j . 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 i ) i∈I ×(a j ) j∈J ∈ A and a price vector (q,v) ∈R H ×R J such that, for every j, a j ∈ A j maximizes the value of the firm V j = v j ? X h q h α jh =max i ( μ i · ? y j + X h α jh z h !) ? X h q h α jh and, for every i, a i ∈ A i maximizes u i (x i ) subject to the budget constraint X h α ih q h + X j β ij v j ≤ X j θ ij V j , where x i = e i + X h α ih z h + X j β ij ? y j + X h α jh z h ! . Theorem 6 Let (a,q,v) ∈ A×R H ×R J be an equilibrium with incomplete markets and let (α 0 j ) j∈J be an arbitrary allocation of portfolios for firms. Then there exists an equilibrium with incomplete markets (a 0 ,q,v 0 ) such that a 0 =(a 0 i ) i∈I × (a 0 j ) j∈J a 0 i =(x i ,α 0 i ,β 0 i ),?i a 0 j =(y j ,α 0 j ),?j. 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 =1with a single feasible production plan y(ω) > 0, and a single security with payo?s z(ω)=1. Limited liability raises the possibility of default and risky debt. Let ?z(α j ,ω) denote the return to risky debt and ?y(α j ,ω) the return to equity in a firm with risky debt. Then ?z(α j ,ω)= ? z(ω) if y(ω)+α j z(ω) ≥ 0 y(ω)/(?α j2 ) if y(ω)+α j z(ω) < 0. and ?y(α j ,ω)= ? y(ω)+α j z(ω) if y(ω)+α j z(ω) ≥ 0 0 if y(ω)+α j z(ω) < 0. If there are complete markets, the value of the risky debt is ?q = p · ?z(α j ) and the value of equity is ?v = p · ?y(α j ). The value of the firm to the original shareholders is ? V =?v +?qα j = p · ?y(α j )+α j p · ?z(α j ) = 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 di?erent levels of risky debt. The number of securities may be great enough to span the entire commodity space R ? . For example, suppose y(ω)=ω and choose α ω j = ?ω +1for ω =1,...,|?|. 2.3. DEFAULT 7 Then ?y 3 α |?| j ′ pays one unit if ω = |?| and nothing otherwise, that is, it is an Arrow security for the state ω = |?|. A portfolio consisting of one unit of ?y 3 α |?|?1 j ′ and minus two units of ?y 3 α |?| j ′ will yield one unit in state ω = |?| ? 1 and nothing otherwise, that is, it is an Arrow security for the state 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 =(a i ) i∈I × (a j ) ∈ A and a price vector (q,v) ∈ R H ×R such that a j ∈ A j maximizes the value of the firm V j = v j ?qα j =max i {μ i · (y j (α j )+α j ?z(α j ))}?qα j and, for every i, a i ∈ A i maximizes u i (x i ) subject to the budget constraint α i q + β i v ≤ θ i V, where x i = e i + α i ?z(α j )+β i (y j (α j )+α j ?z(α j )). 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 y j and α j has two e?ects, on the value of the firm V j and on the risk sharing that can be achieved by 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 typeofconsumercanholdsharesinaversionofthefirm 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 di?cult (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. 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