1. Borrowing Constraints on Firms The Financial Accelerator

Transcription

1 Part 7 1. Borrowing Constraints on Firms The Financial Accelerator The model presented is a modifed version of Jermann-Quadrini (27). Earlier papers: Kiyotaki and Moore (1997), Bernanke, Gertler and Gilchrist (1999). Here we focus on the role of borrowing constraints for firms decisions and its macroeconomic implications. Without borrowing constraints, the firm equates the marginal productivity of borrowing (for investment of some sort) to the interest rate as in the basic BC. With a borrowing constraint, the availability of funds also plays a role.this link is referred to as the"financial accelerator". We use the monopolistic competition model, widely used in macroeconomics. A closer look at the firms Production and market structure There are two goods, intermediate products and final goods. Monopolistic competitive firms produce intermediate inputs x i using labor and constant capital. Production takes one period. Theroleofimperfectcompetitionistogeneratepositiveprofits and dividends. Perfectly competitive firms produce a final good goods using intermediate goods The financial market There are two assets: 1. Shares in intermediate goods producing firms. The representative individual holds s t shares, and their market price is q t. Shares in final good firms have zero profits; hence their market value is zero. 2. isk free debt b t+1 which pay r t next period. Firms are subject to a subsidy on interest payments (a tax deductibility) and they pay the rate t =1+r t (1 τ). 1 A key modelling feature is a borrowing constraint: Firms cannot borrow more than a fraction of their market value. 1 See Hennesy and Whited, Journal of Finance (25), about the tax benefits of debt for the firms.

2 The final good firm The final good Y is produced with the technology Z 1 1/η Y t = x η it di, where x it,i [, 1] are intermediate products purchased at prices v it and η<1. This assumption implies diminishing marginal productivities. This firm s optimization problem is max The optimality conditions are 1 η Z 1 Z 1 1/η x η it di Z 1 v it x it di. 1 η η x η it di ηx η 1 it = v it, i, or Y 1 η t x η 1 it = v it, i. Hence, v i = Y 1 η x η 1 i is the demand function for intermediate good x i The intermediate good firm Firm i produces intermediate good x i with the technology x it+1 = f(z t+1,l it )=z t+1 l 1 θ it, <θ<1, where z is an exogenous productivity shock, following an A(1) with coefficient ξ. The current gross revenue of the firm is where S t =[z t,y t ]. ρ t = v it x it = Y 1 η t x η 1 it x it, = Y 1 η t x η it, = Y 1 η t zt l 1 θ it 1 η ρ (lit 1 ; S t ), 2

3 Because production takes time, the firm needs working capital, i.e., liquid funds to pay wages. The borrowing constraint is: b t+1 φe t m t+1 V t+1, where V t+1 is the value of the firm next period, <φ<1 is the collateralizable fraction of the firm s value and m t+1 is a stochastic discount factor. Thecurrentvalueofthefirm is V t = E t X j= m t+j d t+j, (1.1) (m t =1)andd is the net payment to shareholders i.e., d can be negative if the firm issues equity. There is an institutionally based friction in equity financing (issue and repurchasing of shares, and paying dividends). This is captured as follows: The total expenditure on dividend payments net of funds raised by issuing equity is ϕ(d t )=d t + κ(d t ˆd) 2, where ˆd is the long-run dividend level. The intermediate good firm s problem is subject to V (b, ρ; S) =Max d,l,b [d + Em V (b,ρ ; S )], ρ = ρ (l; S ) = ρ (l 1 ; S)+b wl b 1 ϕ(d), φem V (b,ρ ; S ) b. Technical comment: The recursive formulation is convenient because (1) dividends do not enter linearly in the budget constraint, (2) the value function is used in the borrowing constraint. Denoting with λ and μ the Lagrange multipliers on the two constraints, the first-order conditions are: d : 1 λϕ d (d) =, l : λw + Em V ρ (b,ρ ; S )ρ l (l; S )(1+μφ) =, b : λ +(1+μφ) Em V b (b,ρ ; S ) μ =. 3

4 Envelope conditions: V b (b, ρ; S) = λ 1 V ρ (b, ρ; S) = λ d : 1 λϕ d (d) =, l : λw + Em λ ρ l (l; S )(1+μφ) =, b : (1 + μφ) Em λ μ + λ =. Interpretation of the condition for the debt: Increasing the debt involves three costs: 1.repaymentnextperiod 2. tightening the borrowing constraint by reducing V for next period 3. tightening the borrowing constraint by increasing b this period The benefit is more resources this period. From the condition for the debt we get 1+μφ = λ μ Em λ (1.2) Substituting (1.2) into the labor condition: w =(1 μ/λ) Em λ ρ l (l; S ) 1 Em λ. To see the main feature of this labor demand condition, assume that ρ l (l; S ) is currently known. Then, w = (1 μ/λ) ρ l (l; S ), w + μ ρ l (l; S ) = ρ l (l; S ) λ. The marginal cost of labor is not only the wage, paid currently, but also the value of tightening the borrowing constraint, expressed as μ/λ times the amount that should be borrowed because of the production lag: the present value of the marginal revenue. Without a production lag, this amount would be received currently. 4

5 Main point: The borrowing constraint generates something similar to a tax on labor demand or production. For a relaxation of the borrowing constraint to increase labor demand, the ratio μ/λ should go down. Finally, note the following: 1. In general equilibrium, if x it = x t, i, then Z 1 1/η Z 1 1/η Y t = x η it di = x t di = x t. 2. Using the equation above, the marginal revenue from hiring labor is ρ l (l t 1 ; S t ) = Y 1 η t zt lt 1 1 θ η 1.2. The Households l t =(1 θ) ηy 1 η t = (1 θ) η z t lt 1 1 θ 1 η zt lt 1 1 θ η 1 zt lt 1, θ = (1 θ) ηz t lt 1. θ zt lt 1 1 θ η 1 zt lt 1, θ To stress the role of labor demand and the asset demand, we use here a utility function that yields a simple labor supply which depends on the wage only. Households maximize E ³ U(c t,l t ) = ln subject to the budget constraint: X t= β t U(c t,l t ), c t l 1+ψ t, ψ >. w t l t + b t (1 + r t 1 )+s t (d t + q t )=b t+1 + s t+1 q t + c t + T t. s t : equity shares q t : price of the stock before dividend, i.e., q t = V t d t. r t : risk-free rate T t : lump-sum taxes to finance the interest subsidy to the firms First-order conditions: 5

6 = wu c (c, l)+u l (c, l), = U c (c, l) β(1 + r)eu c (c,l ), = U c (c, l)q βe(d + q )U c (c,l ). The last equation can be used to determine the share price. It can be written as: U c (c t+1,l t+1 ) q t = βe t (d t+1 + q t+1 ). U c (c t ) Then, substituting q t+1 = βe t+1 U c(c t+2,l t+1 ) U c (c t+1,l t+1 ) (d t+2 + q t+2 ), and so on, and assuming that lim j β j E t U 1 (c t+j+1,l t+j+1 ) U 1 (c t+j,l t+j ) q t+j =, q t = βe t U c (c t+1,l t+1 ) U c (c t,l t ) q t = E t d t+1 + β 2 U c (c t+2,l t+2 ) E t d t U c (c t,l t ) X j=1 β j U c (c t+j,l t+j ) d t+j. (1.3) U c (c t,l t ) If we define m t+j = β j U c (c t+j,l t+j ), U c (c t,l t ) i.e., the firms use the stockholders subjective discount rate as theirs, then the link between the value of the firm above in (1.1) (V t = E t q t = V t d t. X j= m t+j d t+j ) and (1.3) is Note also that for j =1, the debt Euler equationu c (c, l) =β(1 + r)eu c (c,l ), can be written as 1.3. Equilibrium equations Labor market: Em = βeu c(c,l ) U c (c, l) = 1 1+r. 6

7 µ w =(1 μ/λ) E w = U l(c, l) U c (c, l) m λ ρ l (l; S ) =(1+ψ) lψ 1 Em λ Intermediate and final goods markets (all intermediate goods firms produce in Z 1 1/η Z 1 1/η equilibrium the same amount): Y = x η i di = x di = x = f(z,l 1 ) f(z,l 1 )=c + ϕ(d) d. Financial market: λ =(1+μφ) 1 1+r Eλ + μ, U c (c, l) =β(1 + r)eu c (c,l ), =1+r(1 τ), ρ (l 1 ; S) =Y 1 η zl 1 1 θ η = f(z,l 1 ) ϕ(d) =f(z,l 1 ) wl + b b 1, b φ 1 1+r EV (b,ρ(l, S ); S )), Lagrange multiplier: λ = 1 ϕ d (d). The equation =U 1 (c, 1 l)q βe(d + q )U 1 (c, 1 l ) is eliminated, along with the variable q. This is a set of 9 equations. The current unknowns are: c, l, d, w, r,, b,μ,λ. This, given z 1,z,l 1,b 1, and the expectations of future values Steady State (1 θ) ηl θ w =(1 μ/λ), w =(1+ψ) l ψ, c = l 1 θ, 7

8 λ =(1+μφ) βλ + μ, µ 1 =1+ β 1 (1 τ), d = zl 1 θ wl ( 1)b, b = φd X j=1 β j b = φ β 1 β d, 8 equations in λ,, μ, d, b, w, l, c. λ = Discussion of the Steady State τ : =1+ µ 1 β 1 (1 τ) 1 β β γ 1 1 = (1+μφ) γ + μ, μ = 1 γ φγ +1. If τ =, γ =1, μ =. The borrowing constraint does not bind. Then, labor demand is standard, except for the production lag: (1 θ) ηl θ w =. If τ>, μ >. The borrowing constraint binds. Then, labor demand includes the financial factor: w =(1 μ) (1 θ) ηl θ, 8

9 1.4. Partial equilibrium intuition Assume a starting situation where the borrowing constraint binds, i.e., μ>. Consider then a deterministic temporary productivity shock, given constant interest rates r, and wage w. The shock increases the firm s current resources. Hence, the multiplier on the resource constraint goes down: 1 λ = 1+ κ(d 2 ˆd). The condition for optimal debt can be written as: μ λ = 1+r ³ 1 φ + 1+r λ 1 1 λ ³. 1 λ 1 The financial accelerator: A temporary increase in resources lowers λ more than λ 1, and then μ /λ declines. Hence, labor demand "curve" goes up: w =(1 μ/λ) (1 θ) ηl θ. Importance of the friction in dividend payment/equity issue: If κ =,λis constant no financial accelerator effect on labor demand. Shocks are freely absorbed by paying more or less dividends no effect on the borrowing constraint. In general equilibrium, however, even when κ =, there are 2 effects on labor demand. Consider for example a highly temporary shock. (1) Interest rates go down, present value of marginal revenue goes up. (2) Because =1+r(1 τ), responds less than r. Hence, the decline in r reduces μ. The opposite for temporary, but more persistent shocks. 1+r μ = 1 φ + 1+r. What happens if τ =and hence the borrowing constraint doesn t bind? We will not address this case, but then there is an asymmetry: Positive shocks will not have a financial accelerator effect, but negative shocks will increase μ above zero. 9

10 1.5. Simulation: Effects of a fully temporary shock Impulse response of hours worked (let s focus on the shape): 9 x Discussion: Higher d μ/λ l. Is the latter increase fully temporary? The gradual decline in consumption (not shown) hints low interest rate (not sure because here also labor enters the marginal utility). If so, the expansion can persist. The rest of the analysis is left for the homework. 1