Capital Structure and Investment Dynamics with Fire Sales

Capital Structure and Investment Dynamics with Fire Sales Douglas Gale Piero Gottardi NYU April 23, 2013 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 23, 2013 1 / 55

Introduction Corporate nance and the Modigliani Miller theorem Indeterminacy of capital structure Tradeo between taxes and bankruptcy cost Our objective: understand GE e ects of debt nance on investment and growth Competitive equilibrium when markets are incomplete Distortionary taxes and transfers Second best policies Related concern: capital adequacy regulation Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 2 / 55

Results The optimal capital structure is determinate, although individuals are indi erent between debt and equity Introducing a corporate income tax implies bankruptcy is (a) costly and (b) occurs with positive probability Equilibrium with a corporate income tax is constrained ine cient: welfare would be higher with I I a higher level of investment, other things being equal or a higher probability of default (in the limit, higher debt-equity ratio) In fact, we can approach the rst best by imposing near 100% debt nance The introduction of a safe technology to increase the liquidity of the asset market can make everyone worse o Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 3 / 55

Outline The planner s problem An economy with frictions, debt, and equity Equilibrium Steady states Transition dynamics Constrained ine ciency Ine cient hedging Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 4 / 55

The Planner s Problem Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 5 / 55

Production of capital Time is discrete and divided into a sequence of dates t = 0, 1,... There is a single perishable good which can be used for consumption or production Capital is produced using the good as the sole input An input of I 0 units of the good produces ϕ (I ) 0 units of capital goods at the same date, where ϕ is C 2 and satis es ϕ 0 (I ) > 0 and ϕ 00 (I ) < 0, for any I 0 Investment is irreversible (capital cannot be consumed) Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 6 / 55

Production of consumption goods The good is produced using capital as the sole input, subject to constant returns to scale Time to build: one unit of capital produced at date t is available to produce A > 0 units of goods at date t + 1 Depreciation: one unit of capital used for production at date t is transformed into θ units of capital available for production at date t + 1 Boundedness: There exists a constant 0 < ˆk < such that ϕ (Ak) < 1 θ k, for every k > ˆk There is an initial endowment of k 0 > 0 units of capital at date 0 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 7 / 55

Consumption There is a unit mass of identical, in nitely-lived consumers A consumption stream c = (c 0, c 1,...) 0 generates utility where 0 < δ < 1 U (c) = δ t u(c t ), t=0 The function u () has the usual properties: u (c) is C 2 and satis es u 0 (c) > 0 and u 00 (c) < 0 for any c 0. Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 8 / 55

The planner s problem Choose f(c t, k t, I t )g t=0 to maximize δ t u(c t ) t=0 subject to the constraints c t + I t = Ak t, t = 0, 1,..., k t+1 = θk t + ϕ (I t ), t = 0, 1,..., (c t, k t, I t ) 0, t = 0, 1,..., and k 0 = k 0 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 9 / 55

The planner s solution Suppose that f(ct, kt, It )g t=0 is a solution to the planner s problem and suppose that (ct, kt, It ) 0, t = 0, 1,... Then there exist non-negative multipliers f(λ t, µ t )g t=0 such that δ t u 0 (c t ) = λ t, t = 0, 1,..., λt+1a + µ θ t+1 = µ t, t = 0, 1,..., µ t ϕ0 (I t ) = λ t, t = 0, 1,... Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 10 / 55

Tranversality Feasibility and Boundedness imply that ϕ (I t ) ϕ (Ak t ) < 1 θ k t, 8k t > ˆk Hence, the law of motion k t+1 = θk t + ϕ (I t ) implies that fk t g is bounded and lim t! s=t k t, 8k t > ˆk δ s u (c s ) = 0 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 11 / 55

Steady states A steady-state solution satis es and (λ t, µ t, k t+1, c t, I t ) = δ t λ, δ t µ, k, c, I, 8t = 0, 1,... u 0 (c ) = λ, δλ A + δµ θ = µ, µ ϕ 0 (I ) = λ The second and third conditions can be rewritten as δa 1 δ θ = µ λ = 1 ϕ 0 (I ) Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 12 / 55

E cient steady states The feasibility conditions become c + I = Ak k = θk + ϕ (I ) Thus, the e cient steady state capital stock is where I is determined by k = ϕ (I ) 1 θ ϕ 0 (I ) δa 1 δ θ = 1 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 13 / 55

An Economy with Frictions, Debt, and Equity Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 14 / 55

An economy with frictions In a frictionless world, the planner s solution could be decentralized in the usual way; to make things more interesting, we impose a number of frictions Markets are incomplete: there are only spot markets for goods and assets Firms are nanced using (short-term) debt and equity Firms pay a distortionary corporation tax but interest income is exempt In the event of default, rms are forced into bankruptcy and their assets are liquidated Liquidation is subject to a nance constraint Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 15 / 55

Firms The capital goods sector consists of a continuum of producers with identical technologies ϕ (I ) Producers maximize pro ts and pay dividends to consumers at the end of each date Since production is instantaneous, nance is not required The consumption goods sector consists of a continuum of producers with identical technologies Ak Capital is subject to random depreciation, where θ t v F (θ) is i.i.d. across producers and R θ t df = θ (no aggregate uncertainty) Capital is long lived so goods producers nance capital purchase with (short-term) debt and equity Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 16 / 55

Bankruptcy, liquidation and settlement Each date t is divided into three sub-periods, labeled A, B, and C. A. In sub-period A, a rm either renegotiates the debt (rolls it over) or defaults and declares bankruptcy B. In sub-period B, bankrupt rms sell their capital at the market-clearing price C. In sub-period C, rms issue debt and equity to purchase newly produced capital goods; they choose the capital structure so as to maximize the market value of the rm Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 17 / 55

Sub-period A: The renegotiation game The renegotiation game at date t consists of two stages: 1 The entrepreneur makes a take it or leave it o er to the bond holders to rollover the debt, replacing the maturing debt with face value d t with new assets (a combination of equity and debt maturing at the following date with face value d t+1 ). 2 The creditors simultaneously accept or reject the rm s o er. The renegotiation succeeds if a majority of creditors accept and those who do not are paid d t There are multiple equilibria; we focus on equilibria in which renegotiation succeeds whenever possible Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 18 / 55

Sub-period A: The renegotiation game Without loss of generality, we consider the behavior of a representative rm with one unit of capital. If renegotiation succeeds, a creditor can obtain at least d t /q t units of capital The rm s manager can ensure the rms ends the period with A/q t + θ t units of capital Thus, the manager can make an acceptable o er if and only if d t A + q t θ t Renegotiation succeeds if and only if θ t z t, where the breakeven value z t is de ned by d t = A + q t z t Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 19 / 55

Sub-period B: Liquidation In sub-period B, the supply of capital from bankrupt rms is Z zt 0 θ t k t df The amount of cash available to purchase this capital is Z 1 A k t df = A (1 F (z t )) k t z t The market clears at a price q t, where Z zt q t θ t k t f (θ t ) dθ t A (1 F (z t )) k t, 0 and equality holds if q t < v t Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 20 / 55

Sub-period C: The capital market A rm in the capital goods sector chooses I t 0 to maximize v t ϕ (I t ) I t, where v t is the price of capital A rm in the consumption goods sector chooses z t to maximize the value of the rm The goods market clears if c t + I t = Ak t The capital market clears if household wealth w t is su cient to purchase all of the securities issued by rms v t θk t + ϕ (I t ) plus the consumption c t c t + v t θk t + ϕ (I t ) = w t Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 21 / 55

Taxes Tax base = value of the rm - renegotiated value of the debt: A dt v t + θ t v t q t q t The tax rate τ > 0 and the tax on equity at date t is then: vt τ max (A + q t θ t d t ), 0 q t Let z denote the breakeven point de ned by A + q t z t = d t ; then tax on equity is τ max fv t (θ t z t ), 0g Taxes are returned to consumers as a lump sum transfer T t in sub-period C Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 22 / 55

Asset pricing Let v e t and v b t denote the market price of this equity and debt at t The return on diversi ed equity and debt is known with certainty at date t because there is no aggregate uncertainty The one-period returns on debt and equity are equal: 1 + r t = R 1 0 min n R zt+1 0 vt+1 A + q t+1 θ t+1, v t+1d t+1 q t+1 τ v b t q t+1 o f (θ t+1 ) dθ t+1 (A + q t+1 θ t+1 d t+1 ) f (θ t+1 ) dθ t+1. v e t = Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 23 / 55

The optimal capital structure In a symmetric equilibrium, I 100% debt nance implies rms default with probability one and q t = 0 I 100% equity nance implies default with probability zero and q t = v t Thus, if τ > 0, equilibrium requires costly default q t < v t and equilibrium default occurs with probability strictly between zero and one: 0 < F (z t ) < 1. Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 24 / 55

Equilibrium Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 25 / 55

Pro t maximization in the capital goods sector In what follows, we consider only symmetric equilibria In the capital goods sector, the rm s decision is trivial: rms simply choose I t 0 to maximize pro ts v t ϕ (I t ) I t The optimal output is determined uniquely by the rst-order condition v t ϕ 0 (I t ) 1, with strict equality if I t > 0 The pro ts π t (v t ) = max fv t ϕ (I t ) sub-period C I t g are paid to consumers in Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 26 / 55

Value maximization in the consumption goods sector Suppose a rm has one unit of capital and the breakeven level z t+1 If θ t+1 < z t+1 the rm is bankrupt at t + 1 and the liquidated value is A + q t+1 θ t+1 If θ t+1 > z t+1 the rm is solvent and, w.l.o.g., the rm retains all its earnings, uses them to purchase liquidated capital, and the rm s value in sub-period C is A v t+1 + θ t+1 q t+1 The rm pays the corporation tax τv t+1 (θ t+1 C if it is solvent z t+1 ) in sub-period Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 27 / 55

Optimal capital structure The expected future value of the rm is Z zt+1 0 (A + q t+1 θ t+1 ) df + Z 1 vt+1 (A + q t+1 θ t+1 ) τv t+1 (θ t+1 z t+1 ) df q t+1 z t+1 The breakeven level z t+1 is chosen to solve δu v t = 0 Z (c t+1 ) zt+1 max z t+1 u 0 (A + q t+1 θ t+1 ) df + (c t ) 0 Z 1 vt+1 (A + q t+1 θ t+1 ) τv t+1 (θ t+1 z t+1 ) df. q t+1 z t+1 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 28 / 55

Optimal capital structure The derivative of the rm s value with respect to z t+1 is dv t = δu0 (c t+1 ) dz t+1 u 0 (A + q t+1 z t+1 ) f (z t+1 ) (c t ) v t+1 q t+1 (A + q t+1 z t+1 ) τv t+1 (θ t+1 z t+1 ) f (z t+1 ) + τv t+1 (1 F (z t+1 ))g The rst-order condition can be written as 1 (A + q t+1 z t+1 ) q t+1 v t+1 f (z t+1 ) 1 F (z t+1 ) = τ. The solution to this equation will be unique and satisfy 0 < z < 1 if q the hazard rate is increasing in z t+1 and 1 t+1 v t+1 Af (0) < τ. Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 29 / 55

The consumption decision Consumers maximize subject to where t=0 p t c t = v 0 k 0 + p t = δ t u (c t ) t=0 t=0 t 1, s=1 1 + r s p t (T t + π t ), the initial capital stock is k 0, π t are the pro ts of rms in the capital sector and T t are lump sum transfers Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 30 / 55

Market clearing In sub-period B, market clearing requires Z zt q t θ t k t f (θ t ) dθ t = A (1 F (z t )) k t, 0 and equality holds if q t < v t. In sub-period C, the goods market clears if c t + I t = Ak t By Walras Law, the securities market clears if the goods market clears Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 31 / 55

Equilibrium conditions 1 fct g t=0 maximizes the consumers discounted utility subject to the budget constraint 2 For every t, vt ϕ 0 (It ) 1 and equality holds if It > 0 3 For every t, z t maximizes the value of the consumption-producing rm 4 For every t, the asset market clears in sub-period B: q t Z z t 0 Z 1 θ t f (θ t )dθ t = A dθ t, zt 5 For every t, the goods market clears in sub-period C Ak t = c t + I t 6 For every t, fk t g satis es the law of motion k t+1 = θk t + ϕ (I t ) and k 0 = k 0 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 32 / 55

Steady States Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 33 / 55

Steady-state equilibrium conditions Optimal consumption requires the rst-order condition and the budget constraint 1 1 δ c = v k + 1 1 δ 1 1 + r = (c ) δu0 u 0 (c ) = δ, τk q Z 1 Pro t maximization requires the rst-order condition v ϕ 0 (I ) = 1 (θ z ) df + v ϕ (I ) I z Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 34 / 55

Steady-state equilibrium conditions Value maximization requires Z z v = δ 0 and the rst-order condition 1 (A + q θ) df + Z 1 v q (A + q θ) τv (θ z ) q t+1 v t+1 z (A + q t+1 z t+1 ) f (z t+1 ) 1 F (z t+1 ) = τ. df Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 35 / 55

Steady-state equilibrium conditions The asset-market clears The goods-market clears q Z z 0 Z 1 θdf = A df z Ak = c + I. The law of motion k = θk + ϕ (I ) is satis ed, where k = k 0 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 36 / 55

Existence and uniqueness Theorem There exists a unique steady-state equilibrium, obtained as a solution to the system of equations: 1 q = A (1 F (z )) R z, θdf 0 v δa = 1 δ θ + τ R 1 z (θ z ) df q t+1 f (z t+1 ) (A + q t+1 z t+1 ) v t+1 1 F (z t+1 ) = τ Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 37 / 55

Transition Dynamics Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 38 / 55

Non-steady-state paths Suppose that consumers are risk neutral, u (c) c; then the equilibrium conditions reduce to v t ϕ 0 (I t ) = 1 q t = A(1 R F (z t)) zt 0 θdf v t = q t + τ (q t ) 2 1 F (z t ) f (z t )(A + q t z t ) These equations yield solutions for I t, q t and v t as a function of z t We are left with a of two-equation system Z 1 v(z t ) = δ A + v(z t+1 ) θ τv(z t+1 ) (θ z t+1 )df (1) z t+1 k t+1 = θk t + ϕ(i (z t )) (2) where v t = v (z t ) is the solution for v t as a function of z t and equation (1) can be solved for z t+1 in terms of z t Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 39 / 55

Instability and uniqueness If we let w (z) denote the right hand side of (1), then we are interested in the behavior of the dynamical system v (z t ) = w (z t+1 ) We note that v (z t ) and w (z t+1 ) have the properties lim v (z t) = 0 and lim w (z t+1 ) = δa z!1 z!1 and lim v (z t) = lim w (z t+1 ) = z!0 z!0 We can show that, for any z, v 0 (z) < w 0 (z) < 0 and, as before, there is a unique solution 0 < z < 1 to the equation v (z ) = w (z ) Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 40 / 55

Instability and uniqueness These properties are illustrated below, where the blue line is v (z) and the red line is w (z) Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 41 / 55

Instability and uniqueness From the picture, it is clear that z t > z implies that for some T > t, v (z T ) < w (1) so the equilibrium condition (1) cannot be satis ed at T For z t < z, the equilibrium condition (1) can be satis ed for all t, however, z t! 0 which implies v (z t )! and q (z t )! Now v (z t )! implies I t! and k t! but this is clearly in violation of the boundedness of fk t g So any equilibrium must satisfy z t = z for all t Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 42 / 55

Constrained Ine ciency Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 43 / 55

The First Best From the planner s problem we know that If τ = 0 then z = 1 and k = ϕ (I ) 1 θ ϕ 0 (I δa ) 1 δ θ = 1 v = δ A + v θ = δa 1 δ θ Then the equilibrium condition v ϕ 0 (I ) = 1 implies that ϕ 0 (I δa ) 1 δ θ = 1 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 44 / 55

Temporary increase in investment Suppose di t = ε > 0 and di τ = 0 for τ 6= t The e ect on welfare of the policy change is given by! ϕ 0 (I )Aδ t=0 δ θ t The term in parenthesis is positive i A δ 1 δ θ > 1 ϕ 0 (I ) = v 1 u 0 (c ) ε + o (ε) But this follows from the equilibrium condition v = δ Z 1 A τq (θ z )f (θ)dθ 1 δ θ z if τ > 0 and z < 1 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 45 / 55

Increase in debt-equity ratio Consider a permanent increase starting from some date t + 1 The future value of q τ is constant and future values of v τ are given by a rst-order di erence equation that is divergent, so the only admissible solution is the new steady-state value The remaining variables are determined by k t+1 = θk + ϕ (I t+1 ), v t+1 ϕ 0 (I t+1 ) = 1. Then I τ is constant and greater than I i v τ > v and we can show that dv τ dz > 0 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 46 / 55

Increase in debt-equity ratio What happens as z! 1 (equilibrium is not de ned when z = 1)? From the market-clearing condition, z! 1 implies q! 0 As q! 0, we see that and hence I! I FB v! Also, as z! 1 we can show that δa 1 δ θ = v FB v b v e! so the debt-equity ratio is increasing in the limit. Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 47 / 55

Intuition Equilibrium is constrained ine cient because agents take prices as given and ignore the pecuniary externality of changes in v and q At the margin, z balances the two costs but the tax on investment distorts I An exogenous increase in z increases the v and decreases q but the tax on equity declines τv Z 1 z (θ z) df In fact, it is the lower tax that explains the increase in v; the fall in q does not a ect v An increase in v increases k and hence increases welfare Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 48 / 55

Ine cient Hedging Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 49 / 55

Introducing a safe technology What is the e ect of introducing a safe technology? Suppose there is an alternative technology for producing the consumption good: one unit of capital produces B units of the good and leaves θ units of capital The risky technology is dominated unless B < A It is never optimal for rms to use both technologies A fraction ` use the risky technology and 1 technology ` use the safe Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 50 / 55

Equilibrium with a choice of risk Risky rms choose z to maximize the value of the rm v Safe rms use 100% debt nancing Firms are indi erent between the two technologies if v = δ v q B + q θ () q = δb 1 δ θ If ` is the fraction of the capital stock devoted to the risky technology, the market-clearing condition at sub-period B is q `k Z z Z 1 θdf = A`k 0 z θdf + B (1 and the market-clearing condition at sub-period C is `) k c = A`k + B (1 `) k I Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 51 / 55

Welfare Let (k, c, z, q, v ) be the steady-state equilibrium with only the risky technology and let B satisfy q = δb 1 δ θ Then ` = 1 is an equilibrium with two technologies for B B and 0 < ` < 1 for B < B < A. A small increase in B at B will lead to the allocation of a small amount of capital to the safe technology Under the usual assumptions of risk neutrality and uniform distribution of θ, dv < 0 db B = B The change in c evaluated at B = B and ` = 1 is dc = 4A dv 2v dv db db db + (A d` B)4v db B = B B = B,`=1 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 52 / 55

Welfare Assume that Then v ϕ 0 (I ) = 1 implies that and Then dc db Then dc db B = B = 4A dv db = (4A 2v ) dv dv < 0 since db ϕ (I ) = 2I 1 2 I = (v ) 2 k = ϕ (I ) 1 θ = 4v 2v dv db + (A db + (A d ` < 0 and db v < 0 and δ 2 δ 2A d` B)4v db B = B,`=1 d` B)2v db B = B,`=1 Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 53 / 55

Numerical example Parameters: A = 2, δ = 0.9, τ = 0.35, θ v U [0, 1] Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 54 / 55

Conclusion Introducing a corporate income tax implies that bankruptcy is costly and occurs with positive probability The optimal capital structure is determinate, although individuals are indi erent between diversi ed debt and equity Equilibrium with a corporate income tax is constrained ine cient: welfare would be higher with I I a higher level of investment, other things being equal or a higher probability of default (in the limit, higher debt-equity ratio) In fact, we can approach the rst best by imposing near 100% debt nance If rms have a choice of a safe and a risky technology, the use of the safe technology makes them worse o Douglas Gale, Piero Gottardi (NYU) Capital Structure April 27, 2013 55 / 55