The Hard Core of the Modigliani Miller Programme,

Presentation for teachingpurposes of  part of Cools, K., hamminga, B. and Kuipers, T. (1994) "Truth Approximation by Concretization in Capital Structure Theory" in: hamminga B. and N.B. de Marchi (eds)   Idealization VI: Idealization in Economics. Poznan Studies in the Philosophy of Science and the Humanities, Vol 38. Amsterdam, Atlanta: Rodopi 205-28.

1 Interesting theorems

This section displays the logical skeleton of "A state-Preference Model of Optimal Financial Leverage" by Kraus and Litzenberger (1973).

As usual in theoretical economics, the strategy of model construction is focused on an interesting theorem (hamminga (1983)). Here, what is to be derived is a theorem on how the value of (debt plus equity of) a firm depends on the proportion of debt in that value.

Modigliani and Miller (1958)(1963) used a plausible model to arrive at implausible, and therefore intriguing theorems: the proportion of debt has no influence at all on the value of the firm if corporate tax is nonexistent, and should be 100% if corporate tax exists. The model of Kraus and Litzenberger based on a state preferance approach, could explain the existence of an optimal solution between 0% and 100%

2 State preference

The "state preference" approach consists of assuming a set J of all conceptually possible end- of- period "states of the world" j. In every single possible state of the world j there is a fixed and given return X_fj for every firm f, a given tax rate T_j and given bankruptcy cost C_fj (C_fj = 0 for firms f that survive in state j).

It is a one period analysis. The return (earnings before interest and taxes) X_fj of firm f in state j is identical to the end- of- period market value of firm f. If you would know what state j would be realized you would have no problems of uncertainty. Hence, the problem of uncertainty solely consists of investors not knowing what state will be realized, and having different probability beliefs concerning j � J.

A "market expectation" about the states j � J is negotiated among investors by assuming tradeable "primitive securities" P_j, which can be thought of as "lottery tickets" yielding 1$ if state j occurs and nothing if j` � j occurs. The equilibrium price P_j of P_j could be derived from 1) probability beliefs of investors concerning state j 2) their time preference for holding money, 3) their risk aversion and 4) their utility functions (Arrow (1969), Debreu (1959)). Kraus and Litzenberger do not perform this derivation, but take P_j, j � J as given, exogenous variables. Any other security bought by an investor can now be identified with a definite number of lottery tickets P_j for every state j (since investors are assumed to know the return of every security in every state).

x = < J,F,Y,R,P,D,X,T,C,P,Y,Z,B,S,V,>

The symbols J,F,Y,R, denote elementary sets. Their meanings are

J the set of states of the world j

F the set of firms f in the market system

Y the set of primitive securites

R the set of real numbers

The symbols P,D,X,T,C,P,Y,Z,B,S,V denote functions:

P: J � Y yields the primitive security P_j that has a return of 1 $ if state j occurs, and 0 $ in any state j` � j.

D: F � R⁺_D yields the debt of firm f, a promise to pay a fixed amount D_f, irrespective of the state that occurs, a nonnegative real number

X: F x J � R_X yields the return of f in state j, the end of period value of f, a real number X_fj, negative in case of bankruptcy.

T: J � R_T^[0,1] yields the tax rate over X in state j, a real number T_j in the closed interval [0,1].

C: F x J � R⁺_C is the bankruptcy cost C_fj of f in state j, zero if the firm survives in state j.

With the help of the primary functions P,D,X,T,C, the secundary functions P,Y,Z,B,S,V, are defined.¹²⁾

P: Y � R⁺_p yields the price P_j of primitive security P_j

Y: F x J x R⁺_D x R_X x R⁺_C � R⁺_Y is the return to holders of debt D_f of every firm f in state j, where  

Z: F x J x R⁺_D x R_X x R_T[0,1] � R⁺_Z is the return to the holders of the equities of f in state j, where

B: F x J x R⁺_Y x R⁺_P x R⁺_D x R_X x R[0,1]_T x R⁺_C � R⁺_B yields the present market value of the debt D_f of firm f

S: F x J x R⁺_Z x R⁺_P x R⁺_D x R_X x R[0,1]_T x R⁺_C � R⁺_S yields the present market value of the equities of firm f

V: F x J x R⁺_S x R⁺_B x R⁺_P x R⁺_D x R_X x R[0,1]_T x R⁺_C � R⁺_V yields the present total value of the firm, where V_f: � S_f + B_f

Since B and S have only their domains and ranges specified, nothing is yet said about how they could be determined from the variables introduced earlier. This is all we do to specify the set DETCp. A thing x is a potential structure of the Kraus and Litzenberger theory, or a potential model iff it satisfies all requirements mentioned thusfar, and hence B_f, S_f and V_f can be, in an x � DETCp, any nonnegative real number.

The set of models of the theory differs from the set of potential models by a specification of how the present market value of debt and equity is determined. This yields a subset DETC � DETCp of structures:

1) x is a DETCp

2) P is such that primitive securities in Y (P_j, j � J) have market equilibrium prices P_j (j � J). (An explicit definition of this as a defined function would require extension of DETCp with a set of investors, and functions mapping their probability beliefs on state j � J, risk aversions, time preference and utility functions)

For convenient notation of the requirements on the functions B and S, Kraus and Litzenberger (re)number the states j₁, j₂, .....,j_n in J for every firm in climbing order of the known return X_fj in each state. So we have, for every f:
X_f1 � X_f2 � ... � X_fj � ... � X_fn

This ends the list of additional requirements that a structure x�DETCp must meet in order to be an element of DETC � DETCp. The list is the result of a strategy meant to find the class of structures for which the following theorem T-DETC holds:

T-DETC: in all structures x�DETC, for all firms f, if T_j is identical in all states j (dropping subscript j),

This possibility arises as a result of the introduction of bankruptcy cost in the model. If bankruptcy costs are removed from the structures in DETC and DETCp, a poorer set of structures DET(p), results, where x�DET(p) is called (potential) debt equity market system with corporate tax. This brings us back to the original paradoxical Modigliani- Miller theorem on the optimal amount of corporate debt in case of the existence of corporate tax; setting C_fj � 0 in T-DETC immediately yields T-DET:

T-DET: in all structures x�DET, for all firms f, if T_j is identical in all states j:

If, finally, tax T_j is removed from the structures, we obtain DE(p), where x � DE(p) is called a (potential) debt equity market system. This yields the initial Modigliani-Miller Theorem T-DE.

T-DE: in all structures x�DE, for all firms f, V(D_f) = V(0)

However, Modigliani and Miller did not use the state preference framework. For what this example is meant to illustrate below, we do not need to go into the logical intricacies of reducing Modigliani and Miller's of introducing market forces to the method used by Kraus and Litzenberger.