"A Theory of Financing Constraints and Firm Dynamics" - - PowerPoint PPT Presentation

"A Theory of Financing Constraints and Firm Dynamics"

G.L. Clementi and H.A. Hopenhayn (QJE, 2006) Cesar E. Tamayo Econ612- Economics - Rutgers April 30, 2012

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Program

I Summary I Physical environment I The contract design problem I Characterization of the optimal contract I Firm growth and survival I Contract maturity and debt limits

2/21

What they do in a nutshell

I The paper develops a theory of endogenous …nancing constraints.

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What they do in a nutshell

I The paper develops a theory of endogenous …nancing constraints. I Repeated moral hazard problem

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What they do in a nutshell

I The paper develops a theory of endogenous …nancing constraints. I Repeated moral hazard problem I The optimal contract (under asymmetric info) determines non-trivial

stochastic processes for …rm size, equity and debt.

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What they do in a nutshell

I The paper develops a theory of endogenous …nancing constraints. I Repeated moral hazard problem I The optimal contract (under asymmetric info) determines non-trivial

stochastic processes for …rm size, equity and debt.

I This in turn implies non-trivial …rm dynamics even under simple i.i.d.

shocks.

3/21

Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

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Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

I Borrower (E) and lender (L) are risk neutral, discount future at δ.

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Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

I Borrower (E) and lender (L) are risk neutral, discount future at δ. I Both agents can fully commit to a long term contract.

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Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

I Borrower (E) and lender (L) are risk neutral, discount future at δ. I Both agents can fully commit to a long term contract. I At each t 1, E can re-scale the project by investing additional kt.

4/21

Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

I Borrower (E) and lender (L) are risk neutral, discount future at δ. I Both agents can fully commit to a long term contract. I At each t 1, E can re-scale the project by investing additional kt. I Returns are stochastic and equal to R (kt) if state of nature is H

(with prob. p) and zero if state is L (prob. 1 p)

4/21

Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

I Borrower (E) and lender (L) are risk neutral, discount future at δ. I Both agents can fully commit to a long term contract. I At each t 1, E can re-scale the project by investing additional kt. I Returns are stochastic and equal to R (kt) if state of nature is H

(with prob. p) and zero if state is L (prob. 1 p)

I At the begining of each t, project can be liquidated (αt = 1)

yielding S 0 and resulting in payo¤s Q to E and S Q to L

4/21

Physical environment:

I At t = 0 entrepreneur (E) has a project that requires initial

investment I0 > M where M in his net worth.

I Borrower (E) and lender (L) are risk neutral, discount future at δ. I Both agents can fully commit to a long term contract. I At each t 1, E can re-scale the project by investing additional kt. I Returns are stochastic and equal to R (kt) if state of nature is H

(with prob. p) and zero if state is L (prob. 1 p)

I At the begining of each t, project can be liquidated (αt = 1)

yielding S 0 and resulting in payo¤s Q to E and S Q to L

I If project is not liquidated, E repays τ to L.

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Physical environment:

I Assumption: R () is continuous, increasing and strictly concave.

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Physical environment:

I Assumption: R () is continuous, increasing and strictly concave. I Assumption: R () is private information; the state of nature at t is

θt 2 Θ fH, Lg but E reports ˆ θt.

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Physical environment:

I Assumption: R () is continuous, increasing and strictly concave. I Assumption: R () is private information; the state of nature at t is

θt 2 Θ fH, Lg but E reports ˆ θt.

I Assumption: E consumes all proceeds R (kt) τ (i.e. no storage)

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Physical environment:

De…nition (reporting strategy)

A reporting strategy for E is ^ θ = ˆ θt

• θt∞

t=1 where θt = (θ1, θ2, ..., θt)

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Physical environment:

De…nition (reporting strategy)

A reporting strategy for E is ^ θ = ˆ θt

• θt∞

t=1 where θt = (θ1, θ2, ..., θt)

De…nition (contract)

A contract is a vector σ =

• αt
• ht1

, Qt

• ht1

, kt

• ht1

, τt (ht)

• where ht =

ˆ θ1, ..., ˆ θt

• 6/21

Physical environment:

De…nition (reporting strategy)

A reporting strategy for E is ^ θ = ˆ θt

• θt∞

t=1 where θt = (θ1, θ2, ..., θt)

De…nition (contract)

A contract is a vector σ =

• αt
• ht1

, Qt

• ht1

, kt

• ht1

, τt (ht)

• where ht =

ˆ θ1, ..., ˆ θt

• De…nition (feasible contract)

A contract σ is feasible if αt 2 [0, 1] , Qt 0, τt

• ht1, L

0, τt

• ht1, H

R (kt) .

6/21

Physical environment:

De…nition (reporting strategy)

A reporting strategy for E is ^ θ = ˆ θt

• θt∞

t=1 where θt = (θ1, θ2, ..., θt)

De…nition (contract)

A contract is a vector σ =

• αt
• ht1

, Qt

• ht1

, kt

• ht1

, τt (ht)

• where ht =

ˆ θ1, ..., ˆ θt

• De…nition (feasible contract)

A contract σ is feasible if αt 2 [0, 1] , Qt 0, τt

• ht1, L

0, τt

• ht1, H

R (kt) .

De…nition (equity and debt)

Expected discounted cash ‡ows for E is called equity, Vt(σ,^ θ,ht1) and for L is called debt, Bt(σ,^ θ,ht1)

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Physical environment:

De…nition (reporting strategy)

A reporting strategy for E is ^ θ = ˆ θt

• θt∞

t=1 where θt = (θ1, θ2, ..., θt)

De…nition (contract)

A contract is a vector σ =

• αt
• ht1

, Qt

• ht1

, kt

• ht1

, τt (ht)

• where ht =

ˆ θ1, ..., ˆ θt

• De…nition (feasible contract)

A contract σ is feasible if αt 2 [0, 1] , Qt 0, τt

• ht1, L

0, τt

• ht1, H

R (kt) .

De…nition (equity and debt)

Expected discounted cash ‡ows for E is called equity, Vt(σ,^ θ,ht1) and for L is called debt, Bt(σ,^ θ,ht1)

De…nition (incentive compatibility)

A contract σ is incentive compatible if 8 ^ θ, V1

• σ, θ,h0 V1(σ,^

θ,h0)

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The …rst-best (symmetric info)

I Since both are risk neutral and share δ, the optimal contract

maximizes total exp. discounted pro…ts of the match (E, L).

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The …rst-best (symmetric info)

I Since both are risk neutral and share δ, the optimal contract

maximizes total exp. discounted pro…ts of the match (E, L).

I In equilibrium L provides E with the unconst. e¤cient k in every t:

max

k

[pR (k) k] (1)

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The …rst-best (symmetric info)

I Since both are risk neutral and share δ, the optimal contract

maximizes total exp. discounted pro…ts of the match (E, L).

I In equilibrium L provides E with the unconst. e¤cient k in every t:

max

k

[pR (k) k] (1)

I So that R () strictly concave) 9 unique k 0 that solves (1).

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The …rst-best (symmetric info)

I Since both are risk neutral and share δ, the optimal contract

maximizes total exp. discounted pro…ts of the match (E, L).

I In equilibrium L provides E with the unconst. e¤cient k in every t:

max

k

[pR (k) k] (1)

I So that R () strictly concave) 9 unique k 0 that solves (1). I Assume k > 0 so that per-period total surplus is:

π = max

k

[pR (k) k] = pR (k) k

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The …rst-best (symmetric info)

I Since both are risk neutral and share δ, the optimal contract

maximizes total exp. discounted pro…ts of the match (E, L).

I In equilibrium L provides E with the unconst. e¤cient k in every t:

max

k

[pR (k) k] (1)

I So that R () strictly concave) 9 unique k 0 that solves (1). I Assume k > 0 so that per-period total surplus is:

π = max

k

[pR (k) k] = pR (k) k

I And PDV of total surplus is W = π

1δ > S by assumption.

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The …rst-best (symmetric info)

I Since both are risk neutral and share δ, the optimal contract

maximizes total exp. discounted pro…ts of the match (E, L).

I In equilibrium L provides E with the unconst. e¤cient k in every t:

max

k

[pR (k) k] (1)

I So that R () strictly concave) 9 unique k 0 that solves (1). I Assume k > 0 so that per-period total surplus is:

π = max

k

[pR (k) k] = pR (k) k

I And PDV of total surplus is W = π

1δ > S by assumption.

I Project is undertaken if W > I0 and once project is started, …rm

does not grow, shrink or exit.

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Private information: the Pareto frontier

I L designs a contract that gives her B (V ) and gives E a value V .

8/21

Private information: the Pareto frontier

I L designs a contract that gives her B (V ) and gives E a value V . I The Pareto frontier of the problem is given by Gr (B(V )) and each

(V , B(V )) implies a value for the match W (V ) = V + B (V ) .

8/21

Private information: the Pareto frontier

I L designs a contract that gives her B (V ) and gives E a value V . I The Pareto frontier of the problem is given by Gr (B(V )) and each

(V , B(V )) implies a value for the match W (V ) = V + B (V ) .

I Begin by …nding the equilibrium of the subgame that starts after L

decides not to liquidate the project.

8/21

Private information: the Pareto frontier

I L designs a contract that gives her B (V ) and gives E a value V . I The Pareto frontier of the problem is given by Gr (B(V )) and each

(V , B(V )) implies a value for the match W (V ) = V + B (V ) .

I Begin by …nding the equilibrium of the subgame that starts after L

decides not to liquidate the project.

I Upon continuation, the evolution of equity is given by:

V = p (R (k) τ) + δ h pV H + (1 p) V Li (2)

8/21

Private information: the Pareto frontier

I L designs a contract that gives her B (V ) and gives E a value V . I The Pareto frontier of the problem is given by Gr (B(V )) and each

(V , B(V )) implies a value for the match W (V ) = V + B (V ) .

I Begin by …nding the equilibrium of the subgame that starts after L

decides not to liquidate the project.

I Upon continuation, the evolution of equity is given by:

V = p (R (k) τ) + δ h pV H + (1 p) V Li (2)

I While the evolution of debt (not in the paper):

B (V ) = pτ k + δ h pB

• V H

+ (1 p) B

• V Li

8/21

Private information: the Pareto frontier

I L designs a contract that gives her B (V ) and gives E a value V . I The Pareto frontier of the problem is given by Gr (B(V )) and each

(V , B(V )) implies a value for the match W (V ) = V + B (V ) .

I Begin by …nding the equilibrium of the subgame that starts after L

decides not to liquidate the project.

I Upon continuation, the evolution of equity is given by:

V = p (R (k) τ) + δ h pV H + (1 p) V Li (2)

I While the evolution of debt (not in the paper):

B (V ) = pτ k + δ h pB

• V H

+ (1 p) B

• V Li

I The value of equity e¤ectively summarizes all the information

provided by the history itself (Spear and Srivastava, 1987; Green, 1987) so it’s the appropriate state variable in a recursive formulation

• f the repeated contracting problem.

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Recursive formulation upon continuation

I The optimal contract upon continuation maximizes the value for the

match ˆ W (Vc), subject to LL, IC and PK constraints.

9/21

Recursive formulation upon continuation

I The optimal contract upon continuation maximizes the value for the

match ˆ W (Vc), subject to LL, IC and PK constraints.

I In recursive form, the program to be solved upon continuation is:

ˆ W (V ) = max

k,τ,V H ,V L pR (k) k + δ

h pW

• V H

+ (1 p) W

• V Li

s.t. V = p (R (k) τ) + δ h pV H + (1 p) V Li (PK) τ

• δ
• V H V L

(ICC) τ

• R (k) ,

V H 0, V L 0 (LL)

9/21

Recursive formulation upon continuation

I The optimal contract upon continuation maximizes the value for the

match ˆ W (Vc), subject to LL, IC and PK constraints.

I In recursive form, the program to be solved upon continuation is:

ˆ W (V ) = max

k,τ,V H ,V L pR (k) k + δ

h pW

• V H

+ (1 p) W

• V Li

s.t. V = p (R (k) τ) + δ h pV H + (1 p) V Li (PK) τ

• δ
• V H V L

(ICC) τ

• R (k) ,

V H 0, V L 0 (LL)

I Authors show that V 7! ˆ

W (V ) is increasing and concave.

9/21

Recursive formulation upon continuation

I The optimal contract upon continuation maximizes the value for the

match ˆ W (Vc), subject to LL, IC and PK constraints.

I In recursive form, the program to be solved upon continuation is:

ˆ W (V ) = max

k,τ,V H ,V L pR (k) k + δ

h pW

• V H

+ (1 p) W

• V Li

s.t. V = p (R (k) τ) + δ h pV H + (1 p) V Li (PK) τ

• δ
• V H V L

(ICC) τ

• R (k) ,

V H 0, V L 0 (LL)

I Authors show that V 7! ˆ

W (V ) is increasing and concave.

I Solving this problem yields policy functions k (V ) , τ (V ), V H (V )

and V L (V ) .

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Recursive formulation before liquidation decision

I If project is liquidated, E receives Q while L receives S Q. If

project is not liquidated, they get Vc, B (Vc) .

10/21

Recursive formulation before liquidation decision

I If project is liquidated, E receives Q while L receives S Q. If

project is not liquidated, they get Vc, B (Vc) .

I Pure strategies may not be optimal for some values of V so

α 2 [0, 1] and L o¤ers a "lottery" to E.

10/21

Recursive formulation before liquidation decision

I If project is liquidated, E receives Q while L receives S Q. If

project is not liquidated, they get Vc, B (Vc) .

I Pure strategies may not be optimal for some values of V so

α 2 [0, 1] and L o¤ers a "lottery" to E.

I Thus, in recursive form, the program to be solved prior to liquidation:

W (V ) = max

α2[0,1],Q,Vc

• αS + (1 α) ˆ

W (Vc)

• s.t.

: αQ + (1 α) Vc = V (PK) : Vc 0, Q 0 (LL)

10/21

Recursive formulation before liquidation decision

I If project is liquidated, E receives Q while L receives S Q. If

project is not liquidated, they get Vc, B (Vc) .

I Pure strategies may not be optimal for some values of V so

α 2 [0, 1] and L o¤ers a "lottery" to E.

I Thus, in recursive form, the program to be solved prior to liquidation:

W (V ) = max

α2[0,1],Q,Vc

• αS + (1 α) ˆ

W (Vc)

• s.t.

: αQ + (1 α) Vc = V (PK) : Vc 0, Q 0 (LL)

I Notice that W () preserves the properties of ˆ

W () .

10/21

Regions for V (Propositions 1 & 2)

The domain of V can be partitioned in three regions:

I Region I: When 0 V Vr , liquidation is posible and randomizing

is optimal with α (V ) = (Vr V ) /Vr

11/21

Regions for V (Propositions 1 & 2)

The domain of V can be partitioned in three regions:

I Region I: When 0 V Vr , liquidation is posible and randomizing

is optimal with α (V ) = (Vr V ) /Vr

I Sketch of argument: α = 1 ) W (V ) = S while α = 0 ) W (V ) =

ˆ W (V ). Now W > S ) 9! Vr and α 2 (0, 1) s.t.V Vr implies that αS + (1 α) ˆ W (Vr ) > max

• S, ˆ

W (V )

• .

11/21

Regions for V (Propositions 1 & 2)

The domain of V can be partitioned in three regions:

I Region I: When 0 V Vr , liquidation is posible and randomizing

is optimal with α (V ) = (Vr V ) /Vr

I Sketch of argument: α = 1 ) W (V ) = S while α = 0 ) W (V ) =

ˆ W (V ). Now W > S ) 9! Vr and α 2 (0, 1) s.t.V Vr implies that αS + (1 α) ˆ W (Vr ) > max

• S, ˆ

W (V )

• .

I Intuition: As V ! Vr expected value ˆ

W (V ) rises above S and L liquidates with low probability (draw graph).

11/21

Regions for V (Propositions 1 & 2)

The domain of V can be partitioned in three regions:

I Region I: When 0 V Vr , liquidation is posible and randomizing

is optimal with α (V ) = (Vr V ) /Vr

I Sketch of argument: α = 1 ) W (V ) = S while α = 0 ) W (V ) =

ˆ W (V ). Now W > S ) 9! Vr and α 2 (0, 1) s.t.V Vr implies that αS + (1 α) ˆ W (Vr ) > max

• S, ˆ

W (V )

• .

I Intuition: As V ! Vr expected value ˆ

W (V ) rises above S and L liquidates with low probability (draw graph).

I Region III: When V V = pR (k) / (1 δ) the total surplus is

the same as under symmetric information (…rst-best), i.e., W (V ) = W .

11/21

Regions for V (Propositions 1 & 2)

The domain of V can be partitioned in three regions:

I Region I: When 0 V Vr , liquidation is posible and randomizing

is optimal with α (V ) = (Vr V ) /Vr

I Sketch of argument: α = 1 ) W (V ) = S while α = 0 ) W (V ) =

ˆ W (V ). Now W > S ) 9! Vr and α 2 (0, 1) s.t.V Vr implies that αS + (1 α) ˆ W (Vr ) > max

• S, ˆ

W (V )

• .

I Intuition: As V ! Vr expected value ˆ

W (V ) rises above S and L liquidates with low probability (draw graph).

I Region III: When V V = pR (k) / (1 δ) the total surplus is

the same as under symmetric information (…rst-best), i.e., W (V ) = W .

I Intuition: equivalent to E having a balance of k/(1 δ) in the

bank at interest rate (1 δ) /δ that is exactly enough to …nance the project at its optimum scale. Then L advances k and collects τ = 0 every period.

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The borrowing constraint region (Propositions 1 & 2 cont.)

I Region II: When Vr V < V :

12/21

The borrowing constraint region (Propositions 1 & 2 cont.)

I Region II: When Vr V < V :

(a) There is no liquidation in current period and V 7! W (V ) is strictly increasing and,

12/21

The borrowing constraint region (Propositions 1 & 2 cont.)

I Region II: When Vr V < V :

(a) There is no liquidation in current period and V 7! W (V ) is strictly increasing and, (b) The optimal capital advancement policy is single-valued and s.t. k (V ) < k (the …rms is debt-constrained)

12/21

The borrowing constraint region (Propositions 1 & 2 cont.)

I Region II: When Vr V < V :

(a) There is no liquidation in current period and V 7! W (V ) is strictly increasing and, (b) The optimal capital advancement policy is single-valued and s.t. k (V ) < k (the …rms is debt-constrained)

I Sketch of argument: suppose that the optimal repayment policy for

region II was τ = R (k) implying that the ICC binds (see below the proofs for both of these results). Then: R (k) = δ(V H V L) which implies that increasing k is only incentive compatible if V H V L also increases. But W (V ) concave implies that doing so is costly! (draw graph)

12/21

Optimal repayment policy (proposition 3)

I The optimal repayment function satis…es τ = R (k) for V < V and

τ = 0 for V V .

13/21

Optimal repayment policy (proposition 3)

I The optimal repayment function satis…es τ = R (k) for V < V and

τ = 0 for V V .

I Intuition:

13/21

Optimal repayment policy (proposition 3)

I The optimal repayment function satis…es τ = R (k) for V < V and

τ = 0 for V V .

I Intuition:

I We know that maxV 2V W (V ) = W and from props 1&2 we know

that W (V ) = W .

13/21

Optimal repayment policy (proposition 3)

I The optimal repayment function satis…es τ = R (k) for V < V and

τ = 0 for V V .

I Intuition:

I We know that maxV 2V W (V ) = W and from props 1&2 we know

that W (V ) = W .

I Now, at given t, L delivers promised utility Vt either by allowing

τ < R (k) or by promising higher future value.

13/21

Optimal repayment policy (proposition 3)

I The optimal repayment function satis…es τ = R (k) for V < V and

τ = 0 for V V .

I Intuition:

I We know that maxV 2V W (V ) = W and from props 1&2 we know

that W (V ) = W .

I Now, at given t, L delivers promised utility Vt either by allowing

τ < R (k) or by promising higher future value.

I Risk neutrality and common δ ) V ! V in the shortest time

possible is optimal.

13/21

Optimal repayment policy (proposition 3)

I The optimal repayment function satis…es τ = R (k) for V < V and

τ = 0 for V V .

I Intuition:

I We know that maxV 2V W (V ) = W and from props 1&2 we know

that W (V ) = W .

I Now, at given t, L delivers promised utility Vt either by allowing

τ < R (k) or by promising higher future value.

I Risk neutrality and common δ ) V ! V in the shortest time

possible is optimal.

I Limited liability then implies τ = R (k) until V = V 13/21

Evolution of equity when V < V

I From prop. 3 we know that τ = R (k) for Vr V < V is optimal.

14/21

Evolution of equity when V < V

I From prop. 3 we know that τ = R (k) for Vr V < V is optimal. I Next, notice that τ = R (k) implies that the ICC binds (lemma 2).

14/21

Evolution of equity when V < V

I From prop. 3 we know that τ = R (k) for Vr V < V is optimal. I Next, notice that τ = R (k) implies that the ICC binds (lemma 2).

I To see this, recall that from prop. 2, V < V ) k (V ) < k. Thus,

suppose that the optimal k is s.t. the ICC is slack: τ = R (k) < δ

• V H V L

. Then one could increase k thereby increasing the total surplus without violating the ICC, contradicting

• ptimality.

14/21

Evolution of equity when V < V

I From prop. 3 we know that τ = R (k) for Vr V < V is optimal. I Next, notice that τ = R (k) implies that the ICC binds (lemma 2).

I To see this, recall that from prop. 2, V < V ) k (V ) < k. Thus,

suppose that the optimal k is s.t. the ICC is slack: τ = R (k) < δ

• V H V L

. Then one could increase k thereby increasing the total surplus without violating the ICC, contradicting

• ptimality.

I Next, ICC binding ) R (k) = δ(V H V L). Recall that from prop.

2 Vr V ) α (V ) = 0. Summarizing: V = αQ + (1 α) Vc = Vc = δ h pV H + (1 p) V Li

14/21

Evolution of equity when V < V

I From prop. 3 we know that τ = R (k) for Vr V < V is optimal. I Next, notice that τ = R (k) implies that the ICC binds (lemma 2).

I To see this, recall that from prop. 2, V < V ) k (V ) < k. Thus,

suppose that the optimal k is s.t. the ICC is slack: τ = R (k) < δ

• V H V L

. Then one could increase k thereby increasing the total surplus without violating the ICC, contradicting

• ptimality.

I Next, ICC binding ) R (k) = δ(V H V L). Recall that from prop.

2 Vr V ) α (V ) = 0. Summarizing: V = αQ + (1 α) Vc = Vc = δ h pV H + (1 p) V Li

I And we obtain the policy functions:

V L (V ) = V pR (k) δ , V H (V ) = V + (1 p) R (k) δ

14/21

Evolution of equity when V < V

I The authors show that V L (V ) , V H (V ) are nondecreasing.

15/21

Evolution of equity when V < V

I The authors show that V L (V ) , V H (V ) are nondecreasing. I Moreover, starting from any equity value V0 2 [Vr , V ) after a …nite

sequence of good shocks V0 ! V .

15/21

Evolution of equity when V < V

I The authors show that V L (V ) , V H (V ) are nondecreasing. I Moreover, starting from any equity value V0 2 [Vr , V ) after a …nite

sequence of good shocks V0 ! V .

I Likewise, after a …nite sequence of bad shocks V0 ! Vr or below,

triggering randomized liquidation.

15/21

Evolution of equity when V < V

I The authors show that V L (V ) , V H (V ) are nondecreasing. I Moreover, starting from any equity value V0 2 [Vr , V ) after a …nite

sequence of good shocks V0 ! V .

I Likewise, after a …nite sequence of bad shocks V0 ! Vr or below,

triggering randomized liquidation.

I There is an asymmetry between change in equity following good and

bad shocks: if p, δ large ) V V L > V H V

15/21

Dynamics of equity

16/21

Dynamics of equity

I Finally, it is easy to see that fVtg is a submartingale with two

absorving sets: Vt < Vr and Vt > V 8 t

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Dynamics of equity

I Finally, it is easy to see that fVtg is a submartingale with two

absorving sets: Vt < Vr and Vt > V 8 t

I Simulations (R (k) = k2/5, p = 0.5, δ = 0.99, S = 1.5):

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Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

18/21

Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

I Now, it is di¢cult to characterize V 7! k (V ) in general (it is

nonmonotonic).

18/21

Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

I Now, it is di¢cult to characterize V 7! k (V ) in general (it is

nonmonotonic).

I However, simulations show that conditional on success, capital grows

at a positve rate, i.e. k(V H )/k (V ) > 1 while k(V L)/k (V ) < 1.

18/21

Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

I Now, it is di¢cult to characterize V 7! k (V ) in general (it is

nonmonotonic).

I However, simulations show that conditional on success, capital grows

at a positve rate, i.e. k(V H )/k (V ) > 1 while k(V L)/k (V ) < 1.

I This contributes to the "cash-‡ow coe¢cient" debate; if Tobin’s q is

a su¢cient statistic for investment, then cash ‡ows should not matter.

18/21

Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

I Now, it is di¢cult to characterize V 7! k (V ) in general (it is

nonmonotonic).

I However, simulations show that conditional on success, capital grows

at a positve rate, i.e. k(V H )/k (V ) > 1 while k(V L)/k (V ) < 1.

I This contributes to the "cash-‡ow coe¢cient" debate; if Tobin’s q is

a su¢cient statistic for investment, then cash ‡ows should not matter.

I But if the optimal contract of this model was the DGP:

18/21

Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

I Now, it is di¢cult to characterize V 7! k (V ) in general (it is

nonmonotonic).

I However, simulations show that conditional on success, capital grows

at a positve rate, i.e. k(V H )/k (V ) > 1 while k(V L)/k (V ) < 1.

I This contributes to the "cash-‡ow coe¢cient" debate; if Tobin’s q is

a su¢cient statistic for investment, then cash ‡ows should not matter.

I But if the optimal contract of this model was the DGP:

• 1. Cash ‡ows will matter for investment, k (V ) < k

18/21

Optimal k advancement policy

I We’ve seen that V0 2 [Vr , V ) ) k (V ) < k (the …rms is

debt-constrained).

I Now, it is di¢cult to characterize V 7! k (V ) in general (it is

nonmonotonic).

I However, simulations show that conditional on success, capital grows

at a positve rate, i.e. k(V H )/k (V ) > 1 while k(V L)/k (V ) < 1.

I This contributes to the "cash-‡ow coe¢cient" debate; if Tobin’s q is

a su¢cient statistic for investment, then cash ‡ows should not matter.

I But if the optimal contract of this model was the DGP:

• 1. Cash ‡ows will matter for investment, k (V ) < k
• 2. Cash ‡ows will matter more, the more constrained is the …rm.

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Firm growth and survival

I Firm size is captured by k

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Firm growth and survival

I Firm size is captured by k I Starting frm the same V0 2 [Vr , V ) simulate many di¤erent shock

paths.

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Firm growth and survival

I Firm size is captured by k I Starting frm the same V0 2 [Vr , V ) simulate many di¤erent shock

paths.

I As seen before, …rms eventually either exit or reach the

unconstrained optimum size.

19/21

Firm growth and survival

I Firm size is captured by k I Starting frm the same V0 2 [Vr , V ) simulate many di¤erent shock

paths.

I As seen before, …rms eventually either exit or reach the

unconstrained optimum size.

I Since k < k w/e Vt < V surviving …rms grow with age; size and

age are positively correlated in accordance with empirical evidence.

19/21

Firm growth and survival

I Firm size is captured by k I Starting frm the same V0 2 [Vr , V ) simulate many di¤erent shock

paths.

I As seen before, …rms eventually either exit or reach the

unconstrained optimum size.

I Since k < k w/e Vt < V surviving …rms grow with age; size and

age are positively correlated in accordance with empirical evidence.

I Mean and variance of equity growth decrease sistematically.

19/21

Firm growth and survival

I Firm size is captured by k I Starting frm the same V0 2 [Vr , V ) simulate many di¤erent shock

paths.

I As seen before, …rms eventually either exit or reach the

unconstrained optimum size.

I Since k < k w/e Vt < V surviving …rms grow with age; size and

age are positively correlated in accordance with empirical evidence.

I Mean and variance of equity growth decrease sistematically. I Conditional probability of survival increases with V . Given that V

increases over time (for surviving …rms), survival rates are positively correlated with age and size.

19/21

Firm growth and survival

I Firm size is captured by k I Starting frm the same V0 2 [Vr , V ) simulate many di¤erent shock

paths.

I As seen before, …rms eventually either exit or reach the

unconstrained optimum size.

I Since k < k w/e Vt < V surviving …rms grow with age; size and

age are positively correlated in accordance with empirical evidence.

I Mean and variance of equity growth decrease sistematically. I Conditional probability of survival increases with V . Given that V

increases over time (for surviving …rms), survival rates are positively correlated with age and size.

I The advantage of this model is that requires little structure on the

stochastic process driving …rm productivity; a simple i.i.d. process is enough to generate the rich dynamics described above.

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Firm growth and survival

20/21

Beyond …rm dynamics (not presented here)

I The authors show that the optimal (long term) contract can be

replicated by a sequence of one-period contracts i¤ it is renegotiation-proof.

21/21

Beyond …rm dynamics (not presented here)

I The authors show that the optimal (long term) contract can be

replicated by a sequence of one-period contracts i¤ it is renegotiation-proof.

I The contract is renegotiation-proof i¤ collateral S is greater than

the maximum sustainable debt.

21/21

Beyond …rm dynamics (not presented here)

I The authors show that the optimal (long term) contract can be

replicated by a sequence of one-period contracts i¤ it is renegotiation-proof.

I The contract is renegotiation-proof i¤ collateral S is greater than

the maximum sustainable debt.

I Risk aversion?

21/21

Beyond …rm dynamics (not presented here)

I The authors show that the optimal (long term) contract can be

replicated by a sequence of one-period contracts i¤ it is renegotiation-proof.

I The contract is renegotiation-proof i¤ collateral S is greater than

the maximum sustainable debt.

I Risk aversion? I No capital accumulation is WLOG?

21/21

Beyond …rm dynamics (not presented here)

I The authors show that the optimal (long term) contract can be

replicated by a sequence of one-period contracts i¤ it is renegotiation-proof.

I The contract is renegotiation-proof i¤ collateral S is greater than

the maximum sustainable debt.

I Risk aversion? I No capital accumulation is WLOG? I General equilibrium?

21/21