A Theory of Financing Constraints
and Firm Dynamics
Gian Luca Clementi and Hugo Hopenhayn
forthcoming on the Quarterly Journal of Economics
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Introduction
Objective: Study the effect of borrowing constraints on
firms’ growth and survival.
Framework: Long-term contractual relationship
between an entrepreneur (borrower) and an investor
(lender) with asymmetric information.
Results:
Borrowing constraints as endogenous "products" of
the optimal contract; less stringent the higher the
entrepreneur’s share of total firm value (Equity);
Optimal contract generates firm dynamics that match
stylized facts qualitatively.
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Model I
Time: Discrete; infinite horizon.
Players:
A risk-neutral entrepreneur has a project "in mind" at
t = 0. She owns M but needs I0 > M to start. If
started, the project takes working capital kt as input
and returns output y˜t , ∀t ≥ 1.
A risk-neutral investor (with the same discount factor
as the entrepreneur) covers the start-up cost and
provides working capital. She cannot monitor the
project, must rely on entrepreneur’s reports.
Liquidation: The project can be scrapped at any time,
generating value S .
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Model II
Productivity shocks: θ ǫ Θ ≡ {H, L}; θ takes value H
with probability p. Accordingly,
(
R(kt ) with prob. p
y˜t =
0
with prob. 1 − p
Reporting strategy: θˆ = {θˆt (θt )}∞
t=1 , where
θt = (θ1 , . . . , θt ); history of reports thus ht = (θˆ1 , . . . , θˆt ).
Contract: At t = 0, the investor offers a contract to the
entrepreneur. The terms specify liquidation policy,
payments and capital advancements and are made
contingent on common information:
σ = {αt (ht−1 ), Qt (ht−1 ), kt (ht−1 ), τt (ht )}
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Timing
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Equity and Debt
σ feasible if, ∀t ≥ 1 and ∀ht−1 ǫ Θt−1
αt (ht−1 ) ǫ [0, 1];
Qt (ht−1 ) ≥ 0;
τt (ht−1 , H) ≤ R(kt (ht−1 )) and τt (ht−1 , L) ≤ 0.
ˆ implies expected discounted cash
∀ht−1 , the pair (σ, θ)
flows for:
ˆ ht−1 ) or Equity;
Entrepreneur: Vt (σ, θ,
ˆ ht−1 ) or Debt.
Investor: Bt (σ, θ,
σ incentive compatible if, ∀θˆ
ˆ σ, h0 )
V1 (θ, σ, h0 ) ≥ V1 (θ,
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Pareto Frontier
Set of equity values that can be generated by feasible
and incentive compatible contracts:
V ≡ {V |∃ σ s.t. feas., i.c. and V1 (θ, σ, h0 ) = V }
Can define a Frontier of values thus: for any V ǫV , the
optimal contract such that
B(V ) = sup{B|∃ σ s.t. V1 (θ, σ, h0 ) = V and B1 (θ, σ, h0 ) = B}
Each point on the frontier corresponds to a capital
structure (V, B(V )).
In general, the Modigliani-Miller Theorem does not
hold: W (V ) = V + B(V ) not invariant to capital structure.
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Symmetric Information
Benchmark: With symmetric information, the contract
simply maximizes the expected discounted surplus of
the project:
k ∗ = arg max pR(k) − k
k
π ∗ = pR(k ∗ ) − k ∗
∗
π
˜ = π ∗ [1 + δ + δ 2 + . . .] =
W
1−δ
˜ > I0 . Any division of W
˜
The project is undertaken if W
between investor and entrepreneur is feasible as long
as the latter gets a nonnegative value.
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Asymmetric Information I
Suppose liquidation has not occurred at the
beginning of the period. The problem is then:
ˆ (V ) =
W
max [pR(k) − k] + δ[pW (V H ) + (1 − p)W (V L )]
k,τ,V H ,V L
subject to
PK: V = [pR(k) − τ ] + δ[pV H + (1 − p)V L ]
IC: R(k) − τ + δV H ≥ R(k) + δV L
LL: τ ≤ R(k)
V H, V L ≥ 0
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Asymmetric Information II
Step back at the beginning of the period. Liquidation
decision to be made:
ˆ (Vc )
W (V ) = max αS + (1 − α)W
α,Q,Vc
subject to
PK: V = αQ + (1 − α)Vc
Q, Vc ≥ 0
Solution:
Set Q = 0 ⇒ α(V ) =
Vc (V )−V
Vc (V )
;
Set Vc (V ) = Vr such that 0 ≤ α(V ) ≤ 1, for V ≤ Vr ;
Three regions emerge (see graph below).
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Value Function
W (V ) linear for V ≤ Vr , where random liquidation occurs;
pR(k ∗ )
˜
˜
˜
.
strictly incr. for V < V and equal to W for V ≥ V ≡
(1−δ)
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Optimal Contract I
Capital Advancement:
k(V ) single valued and continuous;
for V < V˜ , k(V ) < k ∗ , i.e. borrowing constraint binds;
for V ≥ V˜ , k(V ) = k ∗ ;
Intuition when R(k) = τ : in this case IC binds, i.e.
R(k) = δ(VH − VL ). Higher capital advancement requires
higher spread in equity values: as standard with moral
hazard, the lender wants to make the borrower more
sensitive to the realization of output. Given the
concavity of W (V ), this spread is costly for the investor.
Borrowing constraints stem from this trade-off.
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Optimal Contract II
Repayment: For V < V˜ , given risk neutrality and the
common discount factor δ , it is optimal for both the
entrepreneur and the investor to set R(k) = τ until V
reaches V˜ and providing the efficient level of capital k ∗
becomes incentive compatible. (For V H < V˜ , R(k) = τ
is necessary for optimality).
Intuition: For given k and V L , there are infinite
combinations of τ and V H that are feasible and
incentive compatible. Instantaneous profits are not
affected by the choice. Continuation value, though, is
weakly increasing in V H .
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Optimal Contract III
Evolution of Equity when R(k) = τ and Vr ≤ V < V˜ :
VL <V <VH
(V L =
V −pR(k)
,
δ
VH =
V +(1−p)R(k)
);
δ
V < pV H + (1 − p)V L .
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Simulation
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Conclusions
Borrowing constraints emerge endogenously from
the optimal contract; tend to be less stringent the higher
the entrepreneur’s share of total firm value (V );
Optimal contract generates firm dynamics that match
stylized facts qualitatively; in a regression of investment
on some controls (Tobin’s average q ’s, cash flows, . . . ),
would expect positive and significative cash flow
coefficients.
Extensions:
"Hunger";
Multiple shocks;
General equilibrium.
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