Investment and financing in incomplete markets

Abstract

I propose a model of production with incomplete financial markets, in which a firm can act as a financial innovator by issuing claims against its stock. In this environment, market value maximization may be against the firm’s shareholders’ interests. I propose instead a new measure of adjusted value, which is the sum between the market value and the shareholders’ surplus from their trades in the stock market. If a firm maximizes its adjusted value, then its financial policy is relevant (that is, Modigliani–Miller theorem does not hold), equilibrium outcomes are stable to shareholders’ renegotiation, and endogenously incomplete markets can arise at the equilibrium. If the firm is competitive, the adjusted value coincides with the objective proposed by Grossman and Hart (Econometrica 47(2):293–329, 1979). In a competitive market with no production-specific uninsurable risk (that is, spanning property holds), the adjusted value coincides with the market value.

Appendix

Lemma 8.1

The economy $\mathcal {E}_{\mathcal {P}}$ has an equilibrium for every $\mathcal {P\in }\widehat{\mathcal {F}}$.

Proof

Note first that, using part (iv) of Definition 4.1 and formula (3), the budget constraints (4) can be written, equivalently, as:

$$\begin{aligned}&c_{0}^{i}+q(b^{i}-\delta ^i b^f)+p(r^{i}-\delta ^i \mathbf {1}_N)+v(\theta ^{i}-\delta ^i (1+\theta ^f))=\omega _{0}^{i}+\delta ^{i} y_{0}\nonumber \\&c^{i\mathbf{1}}=\omega ^{i\mathbf{1}}+\delta ^i y^\mathbf{1}+A(b^{i}-\delta ^i b^f)+X(r^{i}-\delta ^i \mathbf {1}_N)+\theta ^f \cdot D(\theta ^{i}-\delta ^i) \nonumber \\&\quad c^{i}\ge 0,\quad \left( b^{i},r^{i},\theta ^{i}\right) \ge -L. \end{aligned}$$

(16)

Therefore, if the firm chooses policy $\mathcal {P}=\left( y,D,X,b^{f},\theta ^{f}\right)$, having an initial endowment of $\delta$ shares is equivalent, in terms of generating the same consumption streams, to receiving an additional endowment of goods $\delta y$ and trading in an economy in which all the $J+N+1$ securities are in zero net supply. Therefore, the economy $\mathcal {E}_{\mathcal {P}}$ is equivalent to a standard stock-exchange economy, $\mathcal {E}_{\mathcal {P}}^{0}$, in which

• 1. consumers’ endowments of goods are $\left( \omega ^{i}+\delta ^{i}y\right) _{i\in I}$,

• 2. asset structure is given by $\left( A,X,\theta ^f\cdot D\right) ,$

• 3. there is no initial endowment of assets,

• 4. consumers face “personalized” short-sale bounds, $L^{i,\mathcal {P} }\in \mathbb {R}^{J+N+1}$, given by:

  • (a) $L_{j}+\delta ^{i}b^{f}$ for every exogenously given security $A^j$,

  • (b) $L_{n}+\delta ^{i}$ for every firm-issued security $X^n,$

  • (c) $L_{D}+\delta ^{i}$ for the security $\theta ^f \cdot D$.

Since $y\in \widehat{Y}$, every consumer’s endowment of goods in $\mathcal {E} _{\mathcal {P}}^{0}$ is strictly positive.

The proof of the existence of an equilibrium for $\mathcal {E}_{\mathcal {P} }^{0}$ is similar to standard general equilibrium existence proofs. The reader is referred to Debreu (1962) and, especially, Geanakoplos and Polemarchakis (1986) for the details.¹⁹ An important step in the proof is the construction of an appropriate convex and compact price space. For this specification of the model, the set can be defined as follows.

Let $\lambda _{0}$ be the price of date-0 consumption, expressed in some arbitrary unit of account, and let $\pi$ be the price vector for the $J+N+1$ assets (expressed in the same units). Let

$$\begin{aligned} Q:=\left\{ \left( \lambda _{0},\pi \right) \in \mathbb {R} _{+}\times \mathbb {R}^{J+N+1}\mid \exists \lambda \in \mathbb {R}_{+}^{S} \text { s.t. }\pi =\lambda \left( A,X,D\right) \right\} . \end{aligned}$$

If $(\lambda _0, \pi )$ are equilibrium prices for $\mathcal E^0_{{\mathcal {P}}}$, then $(\lambda _0, \pi )\in Q.$ Clearly, Q is a convex and closed cone. If Q does not contain a full line then there exists a hyperplane $H\subseteq \mathbb {R}^{J+N+2}$ (of dimension $J+N+1)$ such that $0\ne \left( \lambda _{0},\pi \right) \in Q$ if and only if $\alpha \left( \lambda _{0},\pi \right) \in Q\cap H$ for some $\alpha >0,$ and $Q^0:=Q\cap H$ is compact. If Q contains a full line, take H to be half the unit sphere in $\mathbb {R}^{J+N+2},$ centered at origin and let $Q^{0}:=Q\cap H$ be the price space. Then $Q^{0}$ is a convex and compact set (or an acyclic absolute neighborhood retract if H is the half sphere) and the technique used by Geanakoplos and Polemarchakis (1986) can be applied here. $\square$

Let $\widetilde{\prod }^{0}\left( \mathcal {P}\right)$ be the set of normalized (to lie in $Q^0$) date-0 consumption and asset equilibrium prices of $\mathcal {E}_{\mathcal {P}}^{0},$ so that $\widetilde{\prod }^{0}:\widehat{\mathcal {F}}\rightrightarrows Q^{0}$. Then $\widetilde{\prod }\left( \mathcal {P}\right) =\left\{ \frac{\pi }{\lambda _{0}} \mid \left( \lambda _{0},\pi \right) \in \widetilde{ \prod }^{0}\left( \mathcal {P}\right) \right\}$.

Lemma 8.2

The equilibrium price correspondence $\widetilde{\prod }:\widehat{ \mathcal {F}}\rightrightarrows {\mathbb {R}}^{J+N+1}$ is upper hemi-continuous, with compact values.

Proof

It is enough to show that $\widetilde{\prod }^{0}:\widehat{ \mathcal {F}}\rightrightarrows Q^{0}$ has closed graph. Since $Q^{0}$ is compact, this implies that $\widetilde{\prod }^{0}$ is upper hemi-continuous with compact values. On the other hand, since the utility functions of the agents are assumed to be strictly increasing in date-0 consumption, for every ${\mathcal {P}} \in \widehat{{\mathcal {F}}}$ and every $(\lambda _0, \pi )\in \widetilde{\prod }^{0}({\mathcal {P}})$, $\lambda _0>0$. Thus, if $\widetilde{\prod }^{0}$ is upper hemi-continuous and compact-valued, ${\mathcal {P}} \mapsto \widetilde{\prod }({\mathcal {P}})$ must be upper hemi-continuous and compact-valued as well.

To prove that $\widetilde{\prod }^{0}:\widehat{ \mathcal {F}}\rightrightarrows Q^{0}$ has closed graph, notice first that the equilibrium portfolios are bounded, due to the short sale constraints. Let K be a cube in $\mathbb {R}^{J+N+1},$ large enough so that it contains all the portfolio bounds. Consider the truncated portfolio demands $Z_{K}^{i}:\widehat{\mathcal {F}}\times Q^{0}\rightrightarrows K.$ Then $Z_{K}^{i}$ has non-empty, convex and compact values, and it is upper hemi-continuous at every $\left( \mathcal {P} ,\pi \right) \in \widehat{\mathcal {F}}\times Q^{0}$ with $\lambda _{0}\left( \omega _{0}+\delta ^{i}y_{0}\right) +\pi L^{i,\mathcal {P}}\ne 0.$

To overcome the possible discontinuity of the demand at points $\left( \mathcal {P},\pi \right) \in \widehat{\mathcal {F}}\times Q^{0}$ for which $\lambda _{0}\left( \omega _{0}+\delta ^{i}y_{0}\right) +\pi L^{i,\mathcal {P} }=0,$ it is enough to construct a smoothed demand correspondence, $\widehat{Z_{K}^{i}},$ and a quasi-equilibrium as in Debreu (1962). It can be shown that every quasi-equilibrium of $\mathcal {E}_{\mathcal {P}}^{0}$ is an equilibrium, and that the smoothed demand correspondence is upper hemi-continuous everywhere.

The closed graph property of $\widetilde{\prod }^{0}$ follows now immediately from the upper hemi-continuity of the smoothed aggregate demand. $\square$

Proof of Theorem 6.1

For a given $\mu \in \mathcal {M}$, let $\varGamma _{\mu }$ be the normal-form, two-player game defined as follows:

• The strategy set of each player is

  $$\begin{aligned} \widehat{\mathcal {F}}^{^{\prime }}=\left\{ \mathcal {P=}\left( y,D,X,b^{f},\theta ^{f}\right) \in \widehat{\mathcal {F}}\mid \left( b^{f}, \mathbf {0}_{N},0\right) \in K\right\} , \end{aligned}$$

  where K is a cube in ${\mathbb {R}}^{J+N+1}$ which is large enough to contain all the portfolio bounds.

• The first player’s payoff function is

  $$\begin{aligned} \varPhi _{\mu }^{1}\left( \mathcal {P}_{1},\mathcal {P}_{2}\right) :=\int _{ \mathcal {M}}\mathcal {V}_{\mathcal {P}_{2}}^{C}\left( \mathcal {P}_{1}\right) \mathrm{d}\mu \left( \varPi \right) . \end{aligned}$$

• The second player’s payoff function is

  $$\begin{aligned} \varPhi _{\mu }^{2}\left( \mathcal {P}_{1},\mathcal {P}_{2}\right) : =-\left\| \mathcal {P}_{1}-\mathcal {P}_{2}\right\| , \end{aligned}$$

  where $\left\| \cdot \right\|$ is the Euclidean norm on $\mathbb {R} ^{2S+SN+J+2}$ ($\mathcal {P}_{1}$ and $\mathcal {P}_{2}$ are viewed as $\left( 2S+SN+J+2\right)$-dimensional vectors here).

It is easy to see that $\mathcal {P}$ is an equilibrium production-financial plan consistent with the belief $\mu$ if and only if $\left( \mathcal {P,P} \right)$ is a Nash equilibrium of the game $\varGamma _{\mu }$. Therefore, it is enough to prove that there exists $\mu \in {\mathcal {M}}$ such that $\varGamma _\mu$ has a Nash equilibrium. The proof will proceed in two steps.

Step 1: The strategy space $\widehat{\mathcal {F}} ^{^{\prime }}$ is compact.

I prove first that $\widehat{Y}$ is compact. Since $\widehat{Y}$ is a closed subset of $\mathbb {R}^{S+1},$ it is enough to prove that it is bounded. Suppose it is not. Then, there exists a sequence $\left( y^{n}\right) _{n}\subseteq \widehat{Y}$ such that $\left\| y^{n}\right\| >n,$$\forall n\ge 1.$ Convexity of $\widehat{Y}$, together with $0\in \widehat{Y}$, implies that:

$$\begin{aligned} \frac{1}{\left\| y^{n}\right\| }y^{n}+\left( 1-\frac{1}{\left\| y^{n}\right\| }\right) 0\in \widehat{Y},\quad \forall n\ge 1. \end{aligned}$$

Since $\left\| \frac{1}{\left\| y^{n}\right\| }y^{n}\right\| =1,$ it can assumed, without loss of generality, that $\frac{1}{\left\| y^{n}\right\| } y^{n}\rightarrow {\hat{y}}\in \mathbb {R}^{S+1},$ with $\left\| {\hat{y}}\right\| =1.$$\widehat{Y}$ closed implies then that ${\hat{y}}\in \widehat{Y}.$

On the other hand, $y^{n}\in \widehat{Y}\Longrightarrow y^{n}_s\ge -\min _{i}\frac{\omega _{s}^i}{\delta ^{i}}+\varepsilon$ for every $s=0,1,\ldots ,S$ and therefore

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\left\| y^{n}\right\| }y^{n}\ge -\lim _{n\rightarrow \infty }\frac{\left( -\min _{i}\frac{\omega _{s}^i}{\delta ^{i}}+\varepsilon \right) _{s=0,\ldots ,S} }{\left\| y^{n}\right\| }={\mathbf {0}}_{S+1}. \end{aligned}$$

Since $Y \cap {\mathbb {R}}^{S+1}_+=\{{\mathbf {0}}_{S+1}\}$, the above inequality implies that ${\hat{y}}={\mathbf {0}}_{S+1},$ which contradicts $\left\| {\hat{y}}\right\| =1.$ Compactness of $\widehat{\mathcal {F}}^{^{\prime }}$ follows now immediately from compactness of $\widehat{Y}$ and K, and the assumption that $y\longmapsto \mathcal {K}\left( y\right)$ is upper hemi-continuous with compact values.

Step 2: The game $\varGamma _{\mu }$ has a Nash equilibrium, for some $\mu \in {\mathcal {M}}$.

I will show that the family of games $\left( \varGamma _{\mu }\right) _{\mu \in \mathcal {M}}$ induces a “game with endogenous sharing rules” which satisfies all the hypotheses of the main theorem in Simon and Zame (1990).

Define the payoff correspondences $Q^{1},Q^{2}:\widehat{\mathcal {F}}^{^{\prime }}\times \widehat{\mathcal {F}} ^{^{\prime }}\rightrightarrows \mathbb {R}$ as follows:

$$\begin{aligned} Q^{1}\left( \mathcal {P}_{1},\mathcal {P}_{2}\right):= & {} \left\{ \int _{\mathcal {M} }\mathcal {V}^C_{\mathcal {P}_{2}}\left( \mathcal {P}_{1}\right) \mathrm{d}\mu \left( \varPi \right) \mid \mu \text {{ = rational belief} } \right\} , \\ Q^{2}\left( \mathcal {P}_{1},\mathcal {P}_{2}\right):= & {} -\left\| \mathcal {P}_{1}- \mathcal {P}_{2}\right\| . \end{aligned}$$

The game satisfies the hypotheses of the main theorem in Simon and Zame (1990) if: (a) the strategy sets are compact metric spaces, and (b) correspondences $Q^{1}$ and $Q^{2}$ are upper hemi-continuous with compact and convex values. The above conditions are indeed satisfied for the following reasons:

• a. The strategy space $\widehat{\mathcal {F}}^{^{\prime }}$ can be organized as a metric space with the distance induced by the Euclidian metric of $\mathbb {R}^{2S+SN+J+2}.$ According to step 1, $\widehat{\mathcal {F}}^{^{\prime }}$ is also compact.

• b. $Q^{2}$ is a continuous function and thus upper hemi-continuous as a correspondence. Clearly, it has compact and convex values. Upper hemi-continuity of $Q^{1}$ (as well as compactness of its values) follows from the upper hemi-continuity and compactness of the values of $\widetilde{\prod }$, together with the continuity of the optimal consumption as a function of prices and endowments. Convexity of values follows from the linearity of the integral with respect to ${\mu }.$

Therefore, there exists ${\mu }$ such that the game $\varGamma _{{\mu }}$ has a Nash equilibrium. $\square$