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 maximization as its objective function. Under well known conditions, every competitive equilibrium is Paretoefficient 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 condition, 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 attainable 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