Extensive form equilibrium definition

We now relax the assumption of long run profit maximization in the equilib-rium definition of chapter 3 and formally introduce the sequential model of the firm. Here, the firm’s problem is to acquire capital via stock market in period one, and then, given production capacity and a set of states of nature, each firm faces a well defined profit maximization problem in the second period, similar to the Arrow-Debreu model. The main difference to the private ownership firm introduced by Debreu [12] is that in the (EFE) model, each firm takes financial and real quantity decisions sequentially, and production sets are endogenously determined by level of capital a firm can buy by issuing stocks.

Definition 5.1 (EFE) An extensive form stock market equilibrium (¯p, ¯q) with associ-ated equilibrium allocations ((¯x, ¯z), (¯y, ˆz))for generic initial resources ω ∈ Ω, and each producer j ∈ {1, ..., n} maximizing long run profits satisfies:

(i) (¯xi, ¯zi) arg maxui(xi; zi) : xi ∈ Bzi, zi ∈ Rⁿ+ , ∀i ∈ {1, ..., m}

Chapter 3 showed that equilibrium exists generically for this economic model under the assumption of long run profits maximization and reduced form model of the firm. It therefore, remains to be shown the equivalence between the re-duced form equilibrium model of chapter 3 and the extensive form equilibrium model of the firm introduced in this chapter. This result is presented after we show that incomplete markets is a consequence of the assumption of techno-logical uncertainty. The result (5.2) has an alternative interpretation. It is some preliminary version of the Modigliani and Miller theorem suggesting the poten-tial irrelevance of the financial policies theorem [43], [14], [52], and others.

Proposition 5.1 n < S ⇐=P

jY_j|_z¯

Proof 5.2 (Proposition 5.1) Let Sj = 1 for every j ∈ {1, ..., n} , and P

jSj = S.

Then long run profit prospects π(p) > 0 imply long run capacity adjustment and market entrance until n = S. Similar for negative long run prospects, the number of firms decreases until n = S, and π(p) = 0 satisfied. This violates assumption (T ). Let S > 1 for every j ∈ {1, ..., n}, and P

jS_j = S. Then π(p) > 0 implies market entrance

and the issue of new securities such that in the limit as π(p) → 0 the number of firms increases until j → n < S by assumption (T ). Similar for π(p) < 0, firms exit the market and as π(p) → 0 the number of firms decreases until j → n < S by assumption (T ),and π(p) = 0 satisfied.

Theorem (5.2) establishes the missing gap in the existence proof in chapter 3, where existence of equilibrium was shown under assumption 3.3(P) of long run profits maximization for the reduced form model of the firm. Here, we show that every reduced form equilibrium is an extensive form equilibrium and vice versa. Hence, by the existence theorem of chapter 3, equilibria exist. The other interpretation of this result is that it suggests real allocational effects for differ-ent feasible financial policies of the firm, z(j). Consequdiffer-ently, financial policies are non-neutral. This is the simplest version of the Modigliani and Miller theo-rem of irrelevance of financial policies. This result, however, is not complete at this stage, and a full version of the Modigliani and Miller theorem needs to be studied. This involves expanding the model of the firm to a more general en-dogenous asset structure formation model, including bonds, options,and other financial assets.

Theorem 5.2 (i) If (¯p, ¯q)is an extensive form equilibrium with associated equilibrium allocations (( ¯x|_ˆz, ¯z), ( ¯y|_zˆ, ˆz))(DEFE) for generic initial resources ω ∈ Ω, then (¯p, ¯q), is a reduced form equilibrium with associated equilibrium allocations (( ¯x|ξ¯, ¯ξ), ( ¯y|ξ¯)) (DRFE) for generic initial resources ω ∈ Ω where

X^m

i=1

ξ¯_ij =X^m

i=1z¯_i(j) − ˆz_j for j = 1, ..., n (5.7)

(ii) If (¯p, ¯q)is a reduced form equilibrium with associated equilibrium allocations (( ¯x|_ξ¯, ¯ξ), ( ¯y|ξ¯))(DRFE) for generic initial resources ω ∈ Ω, then (¯p, ¯q),is an extensive form

equi-librium with associated equiequi-librium allocations (( ¯x|_zˆ, ¯z), ( ¯y|_zˆ, ˆz))(DEFE) for generic initial resources ω ∈ Ω, for any ˆz_j ≤Pm

i=1z¯_i(j)for j = 1, ..., n satisfying

X^m

i=1z¯_i(j) − ˆz_j =X^m

i=1

ξ¯_ij for j = 1, ..., n. (5.8)

Lemma 5.1 ¯xi|ξ¯is a solution of the reduced form problem

maxn

u(x; ξ)_i : x_i|_ξ¯∈ B_ξio

(5.9) if and only if, ¯x_i|ξ¯∈ B_ξi,and

∂u( ¯x_i|ξ¯)_i∩ N_Bξi( ¯x_i|ξ¯) 6= {0} (5.10) is satisfied.

Proof 5.3 (Lemma 5.1) (i) ¯x_i|_ξ¯is a solution of utility max (RFE) if and only if ¯x_i|_ξ¯∈ B_ξi and

intU_{i, ¯xi}|_ξ¯∩ B_ξi = ∅.

By the separation theorem for convex sets (appendix), there exists P = βip ∈ R₊₊^l(S+1), P 6=

0such that

H_P⁻= n

xi|ξ¯∈ R^l(S+1)₊₊ : P xi|ξ¯≤ P xi|⁰ξ¯, ∀ xi|ξ¯∈ Bξi, ∀ xi|⁰ξ¯∈ intUi, ¯xi|ξ¯

o

since ¯x_i|ξ¯∈ B_ξi,

H_P⁻=n

x_i|ξ¯∈ R^l(S+1)₊₊ : P ¯x_i|ξ¯≤ P x_i|⁰ξ¯, ∀ x_i|⁰ξ¯∈ intU_{i, ¯xi}|ξ¯

o .

By continuity of utility, intUi, ¯xi|ξ¯= U_{i, ¯xi}|ξ¯,and by continuity of the scalar product,

H_P⁺ =n

∀ x_i|⁰ξ¯∈ U_{i, ¯x|¯}

ξ : P ¯x_i|ξ¯≤ P x_i|⁰ξ¯, ∀ x_i|⁰ξ¯∈ U_{i, ¯xi}|ξ¯

o ⇔ P ∈ ∂u( ¯x_i|ξ¯)_i H_P⁻ =n

∀ x_i|ξ¯∈ B_ξi : P x_i|ξ¯≤ P ¯x_i|ξ¯,o

⇔ P ∈ N_Bξi( ¯x_i|ξ¯)_i hence, there exists p such that ∂u( ¯x_i|_ξ¯)_i∩ N_B( ¯x|_ξ¯)_i 6= {0} is satisfied.

(ii) Suppose that ¯x_i|ξ¯∈ B_ξi,and there exists P ∈ ∂u( ¯x_i|ξ¯)_i∩ N_Bξi( ¯x_i|ξ¯)_i, P 6= 0.

If ¯x_i|ξ¯ is not a solution of the (RFE) utility maximization problem, then there exists

¯

x_i|ξ¯0 ∈ intU_{i, ¯xi}|ξ¯∩ B_ξi.Since P ∈ ∂u( ¯x_i|ξ¯)_i, we have

P xi|ξ¯0 > P ¯xi|ξ¯

But since P ∈ NB_ξi( ¯x_i|_ξ¯)and ¯x_i|_ξ¯0 ∈ B_ξi,it follows that P ¯x_i|_ξ¯0 ≤ P ¯x_i|_ξ¯which contradicts that ¯x_i|ξ¯0 is preferred to ¯x_i|ξ¯.

Lemma 5.2 ¯y_j|ξ¯is a solution of

maxn

Π(p; ξ)_j : ¯y_j|_ξ¯∈ Y_j|_ξ¯o

(5.11) if and only if, ¯y_j|_ξ¯∈ Y_j|_ξ¯,and

∂Π( ¯y|ξ¯)_j ∩ Y_j|_ξ¯( ¯y_j|_ξ¯) 6= {0} (5.12) is satisfied.

Proof 5.4 (Lemma 5.2) (i) ¯yj|ξ¯ is a solution of the (RFE) profit max problem if and only if ¯y_j|_ξ¯∈ Y_j|_ξ¯and

intΠ_y¯j ∩ Y_j|_ξ¯= ∅.

By the separation theorem for convex sets (appendix), there exists p ∈ R^lS++, p 6= 0 such that

H_p⁻ =n

y_j|_ξ¯∈ Rⁿ: p y_j|_ξ¯≤ p y_j|⁰_ξ¯, ∀ y_j|_ξ¯∈ Y_j|_ξ¯, ∀ y_j|⁰_ξ¯∈ intΠ_{j ¯y|¯}

ξ

o

since ¯y_j|_ξ¯∈ Y_j|_ξ¯,

H_p⁻=n

y_j|_ξ¯∈ Rⁿ : p ¯y|_ξ¯≤ p y_j|⁰_ξ¯, ∀ y_j|⁰_ξ¯∈ intΠ_{j ¯y|¯}

ξ

o . By continuity of Πj, intΠj, ¯y|ξ¯ = Π_{j, ¯y|¯}

ξ,and by continuity of the scalar product, H_p⁺ =n

∀ y_j|⁰_ξ¯∈ Π_{j, yj}|_ξ¯ : p ¯y_j|_ξ¯≤ p y_j|⁰_ξ¯, ∀ y_j|⁰_ξ¯∈ Π_{j, ¯yj}|_ξ¯

o ⇔ p ∈ ∂Π( ¯y_j|_ξ¯)_j H_p⁻ =

n

∀ yj|ξ¯∈ Yj|ξ¯: p yj|ξ¯≤ p ¯yj|ξ¯, o

⇔ p ∈ N_Yj|_ξ¯( ¯yj|ξ¯)j

hence, there exists p such that ∂Π( ¯y_j|_ξ¯)_j∩ N_{j, Yj|¯}

ξ( ¯y_j|_ξ¯)_j 6= {0} is satisfied.

(ii) Suppose that ¯y_j|_ξ¯∈ Y_j|_ξ¯,and there exists p ∈ ∂Π( ¯y_j|_ξ¯)_j ∩ N_Yj|_ξ¯( ¯y_j|_ξ¯)_j, p 6= 0.

If ¯y_j|_ξ¯ is not a solution of the profit maximization problem (RFE), then there exists

¯

y_j|_ξ¯0 ∈ intΠ_{j, ¯yj}|_ξ¯∩ Y_j|_ξ¯.Since p ∈ ∂Π( ¯y_j|_ξ¯)_j, we have

p ¯y_j|_ξ¯0 > p ¯y|_z¯

But since p ∈ NYj|ξ¯( ¯y_j|_ξ¯) and ¯y_j|_ξ¯0 ∈ Y_j|_ξ¯, it follows that p ¯y_j|_ξ¯0 ≤ p ¯y_j|_ξ¯ which contradicts that ¯y_j|_ξ¯0 is preferred to ¯y_j|_ξ¯.

Proof 5.5 (Theorem 5.2) Part (i). Let us first show that the reduced form equilib-rium allocations ((¯x, ¯ξ), (¯y))satisfy the first order conditions (Lemma (5.1)) ∂u( ¯x_i|ξ¯)_i∩ NB_ξi( ¯xi|ξ¯) 6= {0}and ¯xi|ξ¯∈ Bξi,and (Lemma(5.2)) ∂Π( ¯y|ξ¯)j ∩ Yj|ξ¯( ¯yj|ξ¯) 6= {0}and

¯

y_j|_ξ¯∈ Y_j|_ξ¯,so that the conditions (i) and (ii) in the (RF E) are satisfied. The first order conditions for the consumer’s problem in the (EF E) are

p · x_i|_zˆ

i=1z¯_i(j)the problem of the producer reduces to

arg max (^y|¯ξ¯(s))_j

nXS

s=1p(s) y¯ _j|_ξ¯(s) : y_j|_ξ¯(s) ∈ Y_j|_ξ¯(s), s ∈ So

since feasible ˆz_j ⇒ Φ_j|_zˆ(s) =⇒ Y_j|_zˆ(s) for all s ∈ S, for which the first order con-ditions (lemma (5.2)) hold. The result follows ,since market clearing condition ¯ξ_i

zˆj = order conditions are satisfied. This implies that for any feasible ˆz_j 6Pm

i=1z¯_i(j)

arg max for which the first order conditions are satisfied (Lemma (5.2)), hence ¯y_j|_ˆz is a solution of (ii) in (EF E) for feasible ˆz_j. Pick any feasible ˆz_j and define

The second part of this chapter, sections (5.3) and (5.4), compares the model in-troduced in definition (5.1) with the classical GEI model of production. For this purpose, this section introduces assumptions, notation, and the asset structure of the classical GEI exchange-, and the classical GEI production model. Again, some nomenclature is unavoidable in order to economize on plain text when considering variations of the basic models, and when making comparisons be-tween them.

The main conclusion of this part of chapter 5 suggests that the way the model of the firm is introduced into general equilibrium with incomplete markets has non-trivial equilibrium consequences. It shows that the organization of pro-duction in the decentralized objective function model is more efficient relative