Modigliani-Miller Theorem and Complete Markets: A Comprehensive Analysis, Lecture notes of Economic Analysis

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Modigliani-Miller Theorem and Complete Markets

Shengxing Zhang LSE

October 10, 2016

Definition of a financial contract

I (^) a financial contract is a promise of some payment in the future

I (^) payment contingent on the state of the economy, c(s)

I (^) example from lecture 1

I (^) Alice

I (^) receives today £100, I (^) save £100 in a bank I (^) receives next year £

I (^) Bob

I (^) needs £100 to buy seeds and borrows from the bank I (^) with probability 0.5, earns £119 if the weather is good I (^) with probability 0.5, earns £101 if the weather is bad I (^) repays £101 to the bank

I (^) a deposit contract for £100 pays £101 in both states

Modigliani-Miller Theorem

I (^) A firm’s debt-equity ratio does not affect its market value.

I (^) market value of a firm is pE ⇥ E + pD ⇥ D

“Think of the firm as a gigantic tub of whole milk. The farmer can sell the whole milk as it is. Or

he can separate out the cream, and sell it at a considerably higher price than the whole milk would

bring. ... The Modigliani-Miller proposition says that if there were no costs of separation, (and, of course,

no government dairy support program), the cream plus the skim milk would bring the same price as the

whole milk.

Modigliani-Miller Theorem

The essence of the argument is that increasing the amount of debt

(cream) lowers the value of outstanding equity (skim milk) – selling

off safe cash flows to debt-holders leaves the firm with more lower valued equity, keeping the total value of the firm unchanged. Put

differently, any gain from using more of what might seem to be cheaper debt is offset by the higher cost of now riskier equity.

Hence, given a fixed amount of total capital, the allocation of capital between debt and equity is irrelevant because the weighted

average of the two costs of capital to the firm is the same for all possible combinations of the two.

Complete Markets

Preferences and endowments

I (^) There are two types of agents, i = 1 , 2.

I (^) Future payoff-relevant state, s 2 S.

I (^) Probability that state s takes place, ⇡(s). (⇡(s) 0, P s 2 S ⇡(s) =^ 1). I (^) Endowment of agent i in state s, y i^ (s).

I (^) Consumption of agent i in state s, ci^ (s).

Let ci^ = {ci^ (s), for all s 2 S}.

I (^) Payoff of an agent

U(ci^ ) =

X

s

us (ci^ (s))⇡(s),

where us (·) is increasing and concave, with

limc# 0 u^0 s (c) = + 1.

Planner’s solution and Pareto optimal allocation

I (^) The planner’s solution:

max (c^1 ,c^2 )

1 U(c^1 ) + 2 U(c^2 )

s.t.(c^1 , c^2 ) 2 C

where 1 , 2 0.

I (^) An allocation (c^1 , c^2 ) is Pareto optimal if and only if (c^1 , c^2 )

is the planner’s solution for some nonnegative 1 and 2.

Pareto optimal allocation

I

max c^1 ,c^2

X

s

⇡(s)us (c

1 (s)) + 2

X

s

⇡(s)us (c

2 (s))

subject to

c^1 (s) + c^2 (s)  y 1 (s) + y 2 (s), for all s 2 S,

c^1 (s), c^2 (s) 0 , for all s 2 S.

I (^) The Lagrangian for the problem is

L =

X

s

(

⇡(s)

⇥ 1 us (c^1 (s)) + 1 us (c^1 (s))

⇤

• ✓(s)

X

i

h y i^ (s) ci^ (s)

i

)

✓(s) is the Lagrangian multplier fore the resource constraint for

state s.

Welfare Theorems

The First Theorem states that a market will tend toward a

competitive equilibrium that is weakly pareto optimal when the market maintains the following three attributes:

I (^) complete markets - No transaction costs and because of this

each actor also has perfect information, and

I (^) price-taking behavior

I (^) No monopolists and easy entry and exit from a market.

I (^) Furthermore, the First Theorem states that the equilibrium will

be fully pareto optimal with the additional condition of local nonsatiation of preferences - No two market allocations give

any market actor equal satisfaction.

The Second Theorem states that, out of all possible Pareto optimal

outcomes, one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take

over.

Complete markets – decentralization of an efficient

allocation

I (^) At time 0, before the realization of states, there is a financial

contract for consumption at each state, s.

I (^) a security s pays 1 consumption good at state s.

I (^) There is a competitive market for each of these contracts, with

price q(s).

I (^) The budget constraint of agent i,

X

s

q(s)ci^ (s) 

X

s

q(s)y i^ (s)

Equilibrium definition

I (^) A competitive equilibrium is a feasible allocation, (c^1 , c^2 ), and

a price system, q = {q(s), for all s 2 S}, such that

I (^) given the price system, the allocation ci^ solves each type i agent’s optimal consumption choice; I (^) markets for all securities clear.

Solving for the equilibrium

Assume us (c) = u(c).

u^0 (c^1 (s)) u^0 (c^2 (s)) =^

μ 1 μ 2. Let^

c^1 (s) c^2 (s) =^ ⇢. I (^) Using aggregate resource constraints,

c^1 (s) + c^2 (s) = y 1 (s) + y 2 (s), we have

c

1 (s) =

y

1 (s) + y

2 (s)

c

2 (s) =

y

1 (s) + y

2 (s)

I (^) Let μ 1 = 1, from the first order condition of agents.

q(s) = ⇡(s)

u^0 s (c^1 (s))

μ 1

I (^) Solve for ⇢ by substituting eq. (1), (2) and (3) into the budget

constraint. (^) X

s

q(s)c

1 (s) =

X

s

q(s)y

1 (s)

Example 2: periodic endowment process

I (^) Infinite horizon.

I (^) ust (c) = t^ c

1 1 , I

y 1 (st ) =

1 , t is odd, 0 , t is even.

y 2 (st ) =

0 , t is odd, 1 , t is even.

I

c

1 (st ) =

c^2 (st ) =

q(st ) =

t u

0

Discussion

Modigliani-Miller theorem revisited I (^) What is the value of a firm under complete markets?

V =

X

s

q(s)y (s)

V : the value of the firm, y (s): the cash flow of the firm in

state s) I (^) Does the capital structure of a firm affect the value? No.

The value of the outstanding debt: VD ; The value of the

equity: VE

VD =

X

s

q(s) min{y (s), D}

VE =

X

s

q(s) max{y (s) D, 0 }

V = VD + VE , for all D.

The capital structure of the firm, measured by leverage, defined as VD /VE , does not affect the value of the firm.