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Asymmetry and the Amplitude of Business Cycle Fluctuations: A Quantitative Investigation of the Role of Financial Frictions Manoj Atolia∗ Florida State University
John Gibson† Georgia State University
Milton Marquis‡ Florida State University December 31, 2013 Abstract We examine the quantitative significance of financial frictions that reduce firms’ access to credit in explaining asymmetric business cycles characterized by disproportionately more severe and persistent downturns. Using rate spread data to calibrate the severity of these frictions, we successfully match both the magnitude and “relative” ordering of asymmetry in output and hours worked found in the U.S. data. We also demonstrate that financial frictions operating only during severe downturns can greatly amplify volatility. While the welfare cost associated with this amplification is modest when measured over the business cycle, a significant increase in output loss is detected during a downturn.
Keywords: Business Cycle Asymmetry, Moral Hazard, Agency Costs, Liquidity Shocks, Occasionally Binding Constraints.
JEL: C63, D82, E32, E44, E51.
∗
Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 1-850644-7088. Email:
[email protected]. † Department of Economics, Georgia State University, Atlanta, GA 30303, U.S.A. Telephone: 1-404-4130202. Email:
[email protected] ‡ Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 1-850645-1526. Email:
[email protected].
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Introduction
Empirical evidence suggests that U.S. business cycles are asymmetric, and that this asymmetry can be subdivided into two broad categories, steepness and deepness. Steepness captures the fact that sharp contractions are often followed by long protracted recoveries, while deepness captures the fact that business cycle troughs are often deeper than peaks are tall. Early works by Neftci (1984), Hamilton (1989), Sichel (1993), and Acemoglu and Scott (1997) clearly identify the presence of these forms of asymmetry in many macroeconomic aggregates, such as real output, hours worked, unemployment and investment. Also, some of these works, along with others, focus on key stylized facts regarding U.S. business cycle asymmetry, such as employment and hours worked being more asymmetric than output (see Sichel (1993), Hansen and Prescott (2005), and McKay and Reis (2008)). While the empirical evidence related to asymmetric business cycles has been observed for many years, the possible mechanisms that can generate quantitatively significant and realistic levels of asymmetry within a calibrated dynamic stochastic general equilibrium (DSGE) model are still being explored. Many mechanisms have been proposed for generating asymmetric business cycles. For example, Acemoglu and Scott (1997) use intertemporal increasing returns arising from endogenous variations in the profitability of firms’ investment choices to generate persistent, asymmetric, responses. Other works, such as Caballero and Hammour (1996) and McKay and Reis (2008), introduce similar mechanisms, but focus on the adoption of new technology and the optimal timing of creative destruction. Hansen and Prescott (2005) use occasionally binding capacity constraints to generate realistic levels of deepness in output and hours worked, while Van Nieuwerburgh and Veldkamp (2006) incorporate asymmetric learning over the business cycle to capture the degree of steepness observed in the data. This paper addresses the quantitative significance of financial frictions in amplifying shocks and generating realistic levels of business cycle asymmetry. Specifically, financial frictions enter our model through entrepreneur-run projects that face both firm-level moral hazard (see Holmstrom and Tirole (1997), Tirole (2006)) and idiosyncratic liquidity shocks (see Holmstrom and Tirole (1998)). These financial frictions give rise to incentive constraints that only bind when the economy is in a sufficiently adverse state. When these constraints bind, firms lose access to both current investment funding and future liquidity provisions, thereby exacerbating the severity and persistence of the downturn. We calibrate the extent of the agency problem to match features of the U.S. economy and observe that our model is capable of amplifying adverse productivity shocks in a way that generates realistic levels of 2
deepness asymmetry in output and hours worked.1 Our paper addresses the point raised by Kocherlakota (2000) which demonstrates that endogenous borrowing constraints may significantly amplify large adverse income shocks, but leave both positive and sufficiently small adverse income shocks unaffected. While Kocherlakota (2000) demonstrates that financial frictions arising from endogenous borrowing constraints have the potential for significantly amplifying and propagating large adverse income shocks, he reports that the degree of this amplification depends crucially on the parameters of the model. Given that a rigorous calibration was outside the scope of his paper, Kocherlakota (2000) leaves questions regarding the quantitative significance of financial frictions in generating business cycle asymmetry for future research. Therefore, the results of our calibrated model, which suggest that financial frictions significantly amplify business cycle fluctuations in a downwardly biased way giving rise to realistic levels of business cycle asymmetry, serve as a response to the open question posed by Kocherlakota (2000). Other papers have addressed the issue of credit constraints and amplification. Mendoza (2010) demonstrates that an occasionally binding leverage constraint is capable of drastically amplifying small shocks in a thoughtfully calibrated DSGE model, giving rise to sudden stops. Li and Dressler (2010) include an occasionally binding international borrowing constraint in a small open economy model and demonstrate that the degree of steepness asymmetry generated by the model depends on the initial debt level of the country. Our model differs from the previously mentioned works in three important ways. First, we focus on generating realistic levels of deepness asymmetry within our calibrated model. Mendoza (2010), while successful in generating significant amplification of adverse shocks, focuses on matching the properties of sudden stops, not long-run asymmetric behavior. Li and Dressler (2010) focuses on steepness rather than deepness, and find that they must use unrealistically large levels of international debt to generate statistically significant asymmetry. Second, we quantify the importance of financial frictions in amplifying business cycle volatility. And lastly, we provide two measures of the loss associated with the additional volatility driven by the financial frictions. The model presented in our paper follows the structure presented in Atolia, Einarsson, and Marquis (2010) but overcomes a salient shortcoming of their model: the absence of capital. By omitting capital, Atolia et al fail to match many basic empirical facts associated 1
There is an extensive literature that examines the role of financial frictions in amplifying business cycles, however these works have focused on “always binding” constraints and as such, they give rise to mostly symmetric amplifications. For a review of this extensive literature see Diamond (1984), Williamson (1986, 1987), Bernanke and Gertler (1989), Kiyotaki and Moore (1997), and Bernanke, Gertler, and Gilchrist (1998).
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with business cycles. For example, the lack of capital coupled with a small holding of the real liquid asset relative to consumption, results in limited savings opportunities for agents in their model. Therefore, they overestimate the volatility of consumption, indicating an absence of consumption smoothing. Similarly, their model makes the counter-factual prediction that the supply of labor and liquidity financing for projects are countercyclical. By including capital, we are able to fix these shortcomings and provide a better fit to the standard business cycle facts (see Cooley 1995) With our model structured to match the data more closely than Atolia et al (2010), we are able to provide a more credible quantitative assessment of the importance of financial fictions in both generating asymmetric business cycles and amplifying the volatility present within the model. In order to quantify the effects of the financial frictions on the performance of the model economy, we link the model’s friction to fluctuations in the spread between the rate paid on 3-month non-financial commercial paper and 3-month Treasury bills. Since financial frictions are the source of asymmetry in the model, we use the asymmetry evident in the data to calibrate the severity of these frictions.2 Both panels of Figure 1 present this spread along with a symmetric band computed from the spread’s minimum value, mean value, and minimum value reflected about the mean. Notice the asymmetric fluctuations in the rate spread with large spikes outside of this symmetric band during economic downturns. We use this asymmetric increase in the wedge between investors’ and firms’ valuations of funds to provide a target for the severity of the agency problem present in our model. In particular, the size of the agency rent present in our model is set so that the computed time path for the firms’ shadow price of funds, λR , spikes outside of its symmetric band with a frequency equal to that observed in the rate spread data. Our model’s output series shows a modest degree of asymmetry with a skewness of -0.55, which is similar to the value of -0.41 that we observe in the data. We also find that hours worked displays more asymmetry over the business cycle than aggregate output in our model. Specifically, the average percent deviation above trend relative to below trend for these two variables are 0.91 and 0.80 respectively. These values are very close to what we observe in the data, 0.97 and 0.79, and they also match the fact that hours worked is more asymmetric than output (see Sichel (1993), Hanson and Prescott (2005), and McKay and Reis (2008)). Ultimately, these results suggest that our benchmark model with financial frictions does well in capturing both the magnitude and relative ordering of the asymmetry in output and hours 2
Hansen and Prescott (2005) and Li and Dressler (2011) proceed in a similar way when calibrating the frictions in their models. The main differences between our calibration strategy and those outlined in the above mentioned papers will be discussed in our calibration section.
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worked that is evident in the U.S. data. Our financial frictions are also found to significantly amplify business cycle volatility. We quantify this effect by comparing the standard deviation of output under our benchmark model to that found when the effects of moral hazard are removed. When financial frictions are turned off, the standard deviation of output falls from 1.60 percent to 1.41 percent. Thus, our model suggests that financial frictions increase the volatility of output by about 13.5 percent. The effect for labor is much larger at approximately 40.6 percent. Therefore, the model suggests that financial frictions operating only during recessions add significantly to the volatility experienced over the business cycle. We also provide two quantitative measures of the cost of this excess volatility generated by the financial frictions. The first measure is found by computing the additional consumption required each period in order to make the agents indifferent between the two scenarios. This value is found to be 0.046%, which indicates a modest impact on the agent’s utility when averaged over the entire business cycle. The second measure is the percentage amplification in accumulated output loss due to the presence of moral hazard. This measure is calculated during a downturn, and is found to be approximately 16%, with the majority of this loss accounted for during the first five quarters. This shows that while financial frictions may only be triggered occasionally, they still impose a significant cost on the economy in the short-run. The remainder of the paper is organized as follows. Section 2 presents the structure of the model. Section 3 presents the solution to the model. Section 4 addresses the calibration of the model. Section 5 discusses the model’s results, and section 6 concludes. Also, a brief discussion of potential extensions is included at the end of the paper.
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The Model
The economy is populated by a continuum of households of measure one, where the members of each household pool and share risk perfectly. Specifically, a household consists of an investor, a continuum of entrepreneurs, and a continuum of workers each of measure one. At the beginning of each period, every entrepreneur is endowed with a plan for a socially beneficial project that requires outside funding to rent capital and hire labor from other households. The workers of a household all supply labor to the entrepreneurs of other households in exchange for the market clearing wage rate, while the investor manages the household’s portfolio. This portfolio consists of the household’s equity holdings in outside projects, its current capital position, and its current holdings of a real liquid asset which is
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held in expectation of future cost overruns (See Tirole (1998)).
2.1
Structure of Projects
Each entrepreneur starts a new project indexed by i ∈ [0, 1] every period. These projects take two periods to complete. During the first-period, time t, capital and labor must be acquired for use in the project. In order to finance his resource costs an entrepreneur sells shares, sit , in his project at price pit . Therefore, the first period financing constraint faced by entrepreneur i at time t is given by: i ≤ pit sit wt ni1,t + rt kt+1
(1)
i where ni1,t and kt+1 denotes the labor and capital inputs of the project, while wt and rt denotes their respective factor prices.3 Also, sit denotes the share of the project sold to outside investors, and pit denotes the share price. At the start of the second period, time t+1, the aggregate productivity of the economy, θt+1 , is realized. This value of productivity, along with the previously installed quantities of capital and labor determine the potential output of project i at time t + 1.
i i yt+1 = θt+1 (kt+1 )α (ni1,t )1−α
(2)
Each project also experiences an idiosyncratic cost overrun, ρit+1 , at the start of the second period that requires the entrepreneur to employ an additional ni2,t+1 hours of labor immediately or forgo the output of the project. The wage bill for these additional labor hours must be paid using the real liquid asset. This financing requirement is what facilitates the interpretation of this cost overrun as a liquidity shock (See Tirole (1998)). Investors are aware of the second-period liquidity shock when they make their initial investment decision. As such, they plan for it by allocating some of their household’s current resources to building up a balance of the real liquid asset that can be used to meet the liquidity need next period. After observing both the aggregate and idiosyncratic shocks, the investors form a rule to determine how they will finance the liquidity need. Since all projects are ex-ante identical, investors will invest equally in all projects in the first period and expect identical returns conditional on financing the liquidity need in the second period. Therefore, this rule takes the form of an endogenously determined liquidity financing threshold, ρ?t+1 . 3
Note that the capital rental is paid in advance in the model. This timing change will affect the form of the Euler equations (See below).
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All projects that have a liquidity need less than ρ?t+1 will have their need met, while the others will be terminated. The success of a project that has its liquidity need met is still uncertain. Entrepreneurs possess a hidden action, their choice of effort, which effects their project’s probability of success. If an entrepreneur is diligent, his project will succeed with high probability pH . If he chooses to shirk his responsibilities and engage in a privately beneficial activity, then his project’s probability of success will fall to pL . Both the pH and the cost of shirking, ∆p = pH − pL , are assumed to be sufficiently high so that all projects with diligent entrepreneurs have a positive expected net present value while all projects with non-diligent (shirking) entrepreneurs have a negative expected net present value. Therefore, it is never advantageous for the investor to allow the entrepreneur to shirk (See Tirole (2005)).
2.2
Household Sector - Worker and Investor Choices
The representative household derives utility from consumption and leisure where all goods are viewed as perfect substitutes. All of the household’s agents engage in separate income generating activities throughout the time period, and they reassemble at the end of the period to pool their resources and consume together. Workers generate wage income by supplying labor, nt , to the entrepreneurs of other households. Entrepreneurs generate income for the household through the returns they earn when their projects are successfully completed. Let Πlt denote the income accruing to the household from its entrepreneurs at time t4 . Lastly, the investor optimally allocates the household’s resources between current consumption, investment in other households’ projects started at time t, deepening their investments in others’ projects started at time t − 1, accumulating the real liquid asset for use next period and capital investment which appear, in that order, as items on the left-hand side of the household’s budget constraint:5 Z Ct + 0
1
pjt sjt dj
Z +
1
mkt skt−1 I(ρkt ≤ ρ?k t )dk + Mt+1 + kt+1 − (1 − δ)kt ≤
(3)
0
Z wt nt + Mt + rt kt+1 +
1
Z
l
Π dl + 0
1
ˆ tk skt−1 I(ρkt ≤ ρ?k pH R t )dk
0
4
The entrepreneur solves a separate maximization problem that is consistent with household’s incentives. Therefore, the income accruing to entrepreneur i at time t is taken as given. 5 Throughout the household’s problem, the superscripts j and k are used to denote projects started by other households in the current and previous period respectively, while superscripts i and l denote the current and previous period projects that are managed within the household.
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In equation (3) mkt =
wt ρkt skt−1
denotes the liquidity need per outstanding share, Mt denotes ˆ tk denotes the current return on the household’s current balance of the real liquid asset, R previous investment projects if successful, and I(ρkt ≤ ρ?k t ) denotes an indicator function that equals 1 when ρkt ≤ ρ?k t and 0 otherwise. The household also faces a time constraint that states that all time is spent either working or taking leisure: nt + lt ≤ 1 (4) and a cash-in-advance style liquidity financing constraint: Z
1
mkt skt−1 I(ρkt ≤ ρ?k t )dk ≤ Mt
(5)
0
This last constraint states that the amount of liquidity payments made at time t cannot exceed the current balance of the liquid asset on hand. The household’s state vector consists of its beginning of period real liquid asset balance, Mt , capital holdings, kt , shares held in external projects started last period, skt−1 , shares issued last period to other households, slt−1 , the current level of productivity, θt , and the aggregate level of productive labor employed last period, n1,t−1 . Using this state vector, the household’s problem can be written as the following dynamic program:
V (M, k, sl−1 , sk−1 ; θ, n1,−1 ) =
max 0 0
c,l,n,M ,k ,sj ,ρ
0 0 i j 0 U (C, l) + βE V (M , k , s , s ; θ , n 1) ?
(6)
where the maximization is subject to the equations (3)-(5).
2.3
Entrepreneur’s Problem
Moral hazard is present within the structure of the risky projects in that an entrepreneur can affect the probability with which his project succeeds through his choice of effort. If an entrepreneur chooses to be diligent, his project will succeed with high probability pH . Instead, if he chooses to shirk his responsibilities and engage in a privately beneficial activity, his project’s probability of success will fall to pL 6 . Investors are aware of this agency problem and they incorporate a mechanism into the equity contracts that will guarantee effort by the entrepreneur. Specifically, incentive compatibility (IC) constraints will be present to ensure 6
We will primarily be concerned with the cost of shirking, ∆p = pH − pL .
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that entrepreneurs are always diligent. Since all of a project’s inputs are purchased in advance, any revenue generated by a project is profit that must be divided between its shareholders. Outside investors are entitled to sit of this profit, leaving 1 − sit for the entrepreneurs. Conditional on being successful, the revenue, or profit for project i at time t + 1 is given by: i i i i )α (ni1,t )1−α θt+1 (kt+1 = qt+1 yt+1 Rˆi t+1 = qt+1
(7)
i where qt+1 denotes the price of project i’s output at time t + 1. Given the assumption that it is always optimal to induce the entrepreneur to be diligent, and that there exists a liquidity i ˆ t+1 F (ρ?t+1 ), financing threshold, the expected revenue from project i at time t + 1 is pH R where pH denotes the the project’s probability of success, and F (ρ?t+1 ) denotes the likelihood the project’s second-period liquidity need will be met. The entrepreneur issues shares, hires labor, and rents capital at time t in order to maximize his share, (1 − sit ), of his project’s expected present-discounted-value of the future revenue. This maximization problem is constrained by the entrepreneur’s first-period funding constraint (See equation (1)) and his incentive compatibility (IC) constraints.
max
i sit ,kt+1 ,ni1,t
βEt
Uct+1 i ? (1 − st )pH Rˆi t+1 F (ρt+1 ) Uct
(8)
s.t. pH (1 − sit )Rˆi t+1 ≥ pL (1 − sit )Rˆi t+1 + Jsit
(9)
Equation (9), the IC constraint, establishes that the entrepreneur will always prefer diligence over shirking by stating that the entrepreneur’s share of the revenue when diligent must be at least as large as the sum of his share of the revenue when shirking and his private benefit from shirking. This equation can be written more compactly as: (1 − sit )Rˆi t+1 ≥ Asit
(10)
J denotes the entrepreneur’s agency rent, and the right-hand side of equawhere A = ∆p tion (10) represents the minimum payment to the entrepreneur that would preserve the entrepreneur’s incentive to not shirk (See Holmstrom and Tirole (1997, 1998) and Tirole (2005)).7 7
The fact that the minimum, incentive compatible payment to an entrepreneur is a function of sit is
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3
Solving the Model
We will solve the Household’s and Entrepreneur’s problems separately in order to isolate the affect of moral hazard on the model economy.
3.1
Solution to the Household’s Problem
The trade-off between working and taking leisure yields the following familiar Euler equation: wt Uct = Ult (11) The household’s investment decision is more complicated. They must allocate their resources between current consumption and the four other competing uses: investing in capital, buying equity shares in external projects, providing liquidity in the current period, and building up a new balance of the real liquid asset that can be used to meet next period’s liquidity needs. The optimality conditions for these choices yield the following equations: rt = 1 − βEt ( Uct = βEt
Uct+1 (1 − δ) U ct !) ˆ t+1 pH R
Uct+1
m?t+1
ˆt U c pH R Uct + λLt = t ? mt ( " #" #) ? ˆj t+1 m ˆ t+1 F (ρ? ) ¯ pH R R t+1 Uct = βEt Uct+1 − ?t+1 ˆ pt mt+1 Rt+1
(12) (13) (14) (15)
R ρ? w ρ? where m?t+1 = t+1st t+1 , and m ¯ ?t+1 = 0 t+1 mt+1 Ff(ρ(ρ?t ) ) denotes the average liquidity need t+1 conditional on being financed, and λLt is the Lagrange multiplier on the liquidity financing constraint.
3.2
Solution to Entrepreneur’s Problem
Before we can solve the entrepreneur’s problem, we must first consider a few details regarding the IC constraint and the distribution of the idiosyncratic shock. Inspection of consistent with the standard assumption that agency costs are decreasing in one’s own stake in the project.
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equation (9) indicates that the IC constraint is satisfied as long as: θt+1 ≥ θ˜ =
Asit i )α (ni1,t )1−α (1 − sit )(kt+1
(16)
˜ then it will also be satisfied for any θ > θ. ˜ Thus, if the IC constraint is satisfied for some θ, Now, given that it is never optimal for the investor to allow the entrepreneur to shirk, the IC constraint must be satisfied for all possible future productivity levels, including the lowest possible productivity level that could be realized tomorrow given today’s conditions, denoted θL,t+1 . Therefore, the relevant IC constraint is given by: i (1 − sit )θL,t+1 (kt+1 )α (ni1,t )1−α ≥ Asit
(17)
The entrepreneur is aware that his current actions will effect his likelihood of receiving i , and ni1,t impact F (ρ?t+1 ) are liquidity financing next period. Thus, how his choices of sit , kt+1 taken into account when performing the maximization. An expression for ρ?t can be derived using equation (14) 1
1+
λL t Uct
pH sit−1 θt (kti )α (ni1,t−1 )1−α wt
(18)
Updating this expression and substituting it into equation (8) yields the following objective function: U α i 1−α i i p s θ (k ) (n1,t ) ct+1 i i α i 1−α H t t+1 t+1 E β (1 − s )p θ (k ) (n ) F max (19) t t H t+1 t+1 1,t λL i Uct sit ,kt+1 ,ni1,t 1 + t+1 w Uct+1
t+1
where the maximization is subject to equations (1) and (17). Note that in order to solve the entrepreneur’s problem, we must specify a functional form of F (·), the distribution of the second-period idiosyncratic liquidity shock. We assume that Fe (·) belongs to the family of truncated power-law distributions. In particular, F (ρ) = ρρ¯ , where ρ¯ is the upper limit of the support of the truncated distribution, zero being the lower limit. Parameter e ∈ (0, 1] controls the shape of the distribution with smaller values resulting in higher probabilities of smaller shocks. There are two possible solutions to the entrepreneur’s problem, one where the IC constraint binds and one where the IC constraint is naturally satisfied (non-binding). In the i binding IC constraint case, the entrepreneur chooses sit , kt+1 , and ni1,t to maximize (19)
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subject to (1) and (17). Solving this problem yields: i θL,t+1 (kt+1 )α (ni1,t )1−α i )α (ni1,t )1−α A + θL,t+1 (kt+1 (1 + e)pit I i λt = [(1 + 2e) − 2st (1 + e)] i i )α (ni1,t )1−α st θL,t+1 (kt+1 i θL,t+1 (kt+1 )α (ni1,t )1−α sit (1 + e)2 − e(1 + e) R I λt = + λt sit pit sit
sit =
i = αwt ni1,t (1 − α)rt kt+1
(20) (21) (22) (23)
In the non-binding case, the entrepreneur solves the same problem as before, but ignores the IC constraint. Solving this problem yields: sit =
1 + 2e = s¯ 2(1 + e)
(24)
λIt = 0 λR t =
(25)
s¯(1 + e)2 − e(1 + e) s¯
(26)
Equation (23) also holds in the non-binding case. Notice that if we substitute s¯ into equations (21) and (22) they would become equations (25) and (26). Thus the two solutions only differ in terms of the definition of sit . This shows that financial frictions arising from moral hazard in first-period lending operate through the amount of equity that can be credibly committed to outside investors without jeopardizing incentives. This in turn determines the amount of financing and the size of projects in terms of the quantity of factors employed. In order to verify which solution is correct, we solve the model assuming the IC constraint is non-binding and compare the value for ni1,t found using s¯ to the threshold value for ni1,t , denoted by n?1,t , that is derived from the IC constraint. n?1,t
=
A¯ s i (1 − s¯)θL,t+1 (kt+1 )α
1 1−α
(27)
If the value of ni1,t found using s¯ is greater than or equal to n?1,t , then the IC constraint is satisfied and the non-binding solution is correct. However, if the value for ni1,t is less than n?1,t , then the binding solution must be used.
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3.3
Equilibrium Conditions
All goods produced in equilibrium are from projects that have liquidity needs less than or equal to ρ?t . Thus for all projects/goods for which ρit ≤ ρ?t , we have the following equilibrium conditions: sit = st
(28)
pit = pt
(29)
qi = q = Q = 1 ˆ = y = θk α n1−α Rˆi = R
(30)
1
(31)
The labor market clearing condition is given by: n1,t + n ¯ 2,t (ρ?t )F (ρ?t ) = nt
(32)
R ρ?t
t) ρ Ff (ρ dρ denotes the average additional labor requirement, conditional (ρ?t ) e on receiving liquidity financing. Under the assumption that F (ρ) = ρρ¯ , the labor market clearing condition reduces to:
where n ¯ 2,t (ρ?t ) =
0
n1,t +
e 1+e
ρ?1+e t = nt ρ¯e
(33)
The household’s time constraint is given by: nt + lt = 1
(34)
The goods market clearing condition is given by: Ct + Mt+1 + kt+1 = Yt + (1 − δ)kt + Mt
(35)
where Yt = pH yt F (ρ?t ) denotes the aggregate output of the economy at time t. The final equilibrium condition is for the real liquid asset. It states that the liquidity provided in the current period cannot exceed the current balance of the real liquid asset on
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hand:8 Z
ρ?t
st−1 mt f (ρ)dρ ≤ Mt
(36)
0
Applying the distributional assumption for ρ and evaluating the integral, this equilibrium condition reduces to: ?1+e ρt e wt ≤ Mt (37) 1+e ρ¯e The model can be summarized by the following equations, (1), (11)-(15), (21)-(23), (30), (31), (33)-(35), and (37). These 15 equations contain the 16 distinct endogenous variables, ˆ n1 , ρ? , q, w, l, n, c, M , r, k, λL , λR , and λI . In order to close the system, an s, p, y = R, equation must be added. If the IC constraint is slack, equation (24) is added, but if the IC constraint binds, equation (20) is used to close the system.
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Calibration
In this section we provide an overview of the data targets used to bring the model in line with features of the aggregate U.S. economy.
4.1
Functional Forms and Distributional Assumptions
We choose the following standard functional forms for utility and the aggregate productivity shock process: U (Ct , lt ) = log Ct + η log Lt
(38)
log(θt+1 ) = Ψθ log(θt ) + t
(39)
where is drawn from a symmetrically truncated (+/- 2.5 std. dev.) normal distribution with mean 0 and variance σ2 . The truncation at the lower end is required to compute the worst possible productivity shock, θL,t+1 = θtΨ exp(L ), that is used in the relevant IC constraint. The truncation at the upper end is imposed to maintain the symmetry of the shock process as we are specifically interested in asymmetry generated endogenously by the financial frictions. 8
This equilibrium condition is written as an inequality constraint because it may be optimal for households to withhold liquidity in severely depressed times, thereby not exhausting their current stock of M every period. However, we track the lagrange multipliers on this constraint and they remain positive over our entire simulation.
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In order to calibrate the two parameters of the distribution of the idiosyncratic liquidity shock, namely, ρ¯ and e, we choose two targets. Following Atolia et al (2010) we set F (ρ?ss ) = 0.85 so that 85 percent of the projects receive their second period liquidity funding in steady state. However, our second target departs from Atolia et al. They assume that F (·) is uniform, which corresponds to e = 1. As a result, about half of the hours worked in a given period are due to the idiosyncratic cost overruns, which is very high. In order to avoid this = 32 so that only one-third of hours worked are due to the issue, we set e to target nn1,ss ss second-period need.
4.2
Preference and Production Parameters
The model is calibrated to a quarterly frequency with the discount rate, β, set to 0.99, implying a risk free annual interest rate of approximately 4%. The rate of capital depreciation, δ, is set to 0.02, indicating a 8% annual depreciation. The capital’s share of income at steady state is set to 1/3, which yields a steady state capital stock that is approximately five times as large as output. Lastly, the persistence in the productivity shock process, Ψθ , is set to 0.925. These values, while grounded in observable data, are also supported by the existing business cycle literature (see Kydland and Prescott (1982)). The production function parameter, α, is backed out using the capital income share restriction. So, while our value of α = 0.429 seems high, it is consistent with both a capital’s share of income equal to 31 and a labor’s share of income equal 23 . The reason for this is that, in our model, labor used in production is not the only source of labor’s factor payment. Wages paid to workers due to liquidity needs must also be considered in the calculation of labor’s share of income. Entrepreneurial income is also present in the model. The closest real-world analogue to this is proprietor’s income which is typically divided between capital and labor in a proportion similar to that found in the aggregate economy. A full description of the model’s parameters can be found in Table 1.
4.3
Calibrating the Severity of the Agency Problem
In order to provide a measure of the quantitative impact of moral hazard on the economy, we calibrate the severity of the agency problem in the model using data on the spread between the rate paid on three-month non-financial commercial paper and three-month U.S. Treasury bills. Both panels of Figure 1 present this spread along with a symmetric band computed from the spread’s minimum value, mean value, and minimum reflected about the mean. Notice
15
the asymmetric fluctuations in this rate spread, with large spikes outside of its symmetric band during economic downturns. Widening in this rate spread is traditionally viewed as representing an increase in the level of business or project risk. While this logic explains a basic or modest spread, it fails to address the presence of the large asymmetries or the extreme values of the spread that are realized during economic downturns. We interpret these extreme values as being indicative of the presence of moral hazard which exposes investors to a greater share of the increase in risk that accompanies downturns. This interpretation is consistent with an increase in risk that results in a reduction in the supply of loanable funds that far exceeds the reduction in demand. In the model, this asymmetric divergence in the valuation of funds by the ‘inside’ entrepreneurs and the ‘outside’ investors is captured by the variation in the entrepreneurs’ shadow price of funds, λR . J , so that the Given this interpretation, we set the entrepreneur’s agency rent, A = ∆p shadow price of funds mimics the asymmetric divergence found in the rate spread data J so that λR spikes outside its symmetric band mentioned above. In particular, we set A = ∆p about 12 percent of the time, matching the frequency found in the data. Figure 2 shows the time path of λR for the first 300 periods along with the data on the rate spread. Notice how similar the spikes in the shadow price are to those found in the rate spread data. Having J to match the asymmetry of the rate spread data we simply set pH = 0.9 so calibrated ∆p that the entrepreneur’s project succeeds with a high probability when the entrepreneur is diligent.9 As mentioned earlier, financial frictions are the source of asymmetry in our model. Therefore, the calibration strategy outlined above, which specifies the rate and severity with which our IC constraint binds, also determines the degree of asymmetry present in our model. This is a common feature among models with occasionally binding constraints. For example, Hansen and Prescott (2005) calibrate the frequency with which their model’s capacity constraint binds in order to target the level of deepness asymmetry found in U.S. output data. They proceed by demonstrating that, under this calibration, their model is capable of generating a time series for hours worked that is more asymmetric than output. This implies that our calibration strategy is more general, since we do not target the asymmetry of output directly. Instead, we target features of financial data and then judge the performance of our model (see below) based on it’s ability to replicate the level of asymmetry observed in both output and hours worked. The model has been re-solved under three alternate calibrations: F (ρ? ) = 0.7, nn1 = 0.8, and pH = 0.6. The results of these alternative calibrations are consistent with that presented below (tables available upon request). 9
16
The model presented in Li and Dressler (2011) generates asymmetry through the presence of an occasionally binding international borrowing constraint. The authors calibrate their baseline model to match Canadian data, but ultimately find that their model generates statistically insignificant levels of asymmetry under this calibration. They conclude that the level of asymmetry generated by their model rises with the country’s initial debt level, suggesting that countries with high levels of international debt may experience more volatile and asymmetric business cycles. This result is not surprising given the nature of borrowing constraints. High levels of debt push agents closer to their borrowing constraint, making it more likely to bind and thereby generating more asymmetry.
5
Results
We are now ready to investigate the effect of the financial frictions on the performance of our benchmark model. As our focus is on assessing the role of financial frictions, we will, when necessary, compare the results of our benchmark model with the “No-Frictions” model where J , to a very moral hazard has been shut down by setting the entrepreneur’s agency rent, ∆p low value (close to zero). In both cases, the Deterministic Extended Path (DEP) method is used to compute an approximate solution. This solution is used to estimate initial values for the parameters of the expectation functions for the Parameterized Expectation Algorithm (PEA) which is used to improve the accuracy of the approximation.10 In particular, our PEA approximation is based on simulations with a half a mi2621llion time periods.
5.1
Basic Characteristics of Business Cycle Fluctuations
Table 2 presents a summary of the model’s second moments. Inspection of the first two columns indicates that our benchmark model provides a reasonable match to the data in terms of the volatility of consumption and investment relative to output. The model also matches many of the key cyclical results found in the U.S. data. For example, aggregate labor is found to be strongly pro-cyclical, with a correlation value of 0.77. This finding is in line with U.S. data which suggests that a dominating substitution effect exists between consumption and leisure. The liquidity funding threshold, ρ? , is also found to be pro-cyclical, supporting the intuitive result that investors’ willingness to deepen their investments should 10
See Heer and Mausner (2004) for a textbook treatment of these techniques. Den Haan and Marcet (1990) present a detailed description of PEA. Also see Christiano and Fisher (1994) for applications related to models with occasionally binding constraints.
17
increase as conditions in the aggregate economy improve. Our model’s ability to generate consumption smoothing behavior and reproduce the procyclical supply of labor and liquidity funding found in the data is driven by the presence of capital. In Atolia et al (2010), capital was not included, and their model failed to capture these results. The differences in the cyclicality of the supply of labor and liquidity funding between the two models can best be explained as follows. In a model without capital, an adverse productivity shock induces a strong rightward shift in labor supply. Simultaneously, this adverse shock reduces labor demand, but this reduction in demand is smaller than the increase in supply. The net result of the adverse shock in a model without capital is a reduction in wages and an increase in labor. In contrast, the inclusion of capital mitigates the volatility of consumption, and hence that of labor, resulting in a reduction in labor demand that exceeds the increase in labor supply that accompanies the adverse shock. The net result of the adverse shock in a model with capital is a reduction in both wages and labor from their pre-shock levels.
5.2
Financial Frictions and the Severity of Downturns
The success of the model in matching the stylized business cycle facts lends more credibility to the assessment of the role of financial frictions to which we turn to in this section. It is intuitively clear that financial frictions arising from moral hazard will exacerbate the intensity and duration of downturns while leaving upturns unaffected. This one-sided response will, in most cases, induce business cycle asymmetries. While we have shown, in the previous section, that our model reproduces key business cycle statistics, we now turn–in steps–to establishing that financial frictions indeed lead to quantitatively significant business cycle asymmetry. We begin by presenting the impulse response functions of the key variables. Specifically, the economy is subjected to a shock of L in the first three periods, and allowed to recover at rate Ψθ thereafter. This three step shock process was chosen because starting from the steady state the first two shocks are needed to bring the IC constraint just past the point of binding and the third shock is used to cause the IC constraint to bind more severely. Figure 3 presents plots of these impulse response functions for several key variables of the benchmark model. To highlight the role played by the financial frictions, the figures also show (in dashed lines) the response of the no-frictions model where the effect of moral hazard J , to a very low value. As has been shut down by setting the entrepreneur’s agency rent, ∆p expected, the benchmark model’s impulse responses fall farther from their steady-state level, 18
and recover more gradually than their no-friction counterparts, indicating an exacerbation in both the intensity and duration of downturns. While the plots of these impulse response functions highlight the fact that downturns are exacerbated by the financial frictions, they are silent about the effect of the frictions during upturns. Figure 4 presents plots of the variables’ simulated time paths for the first 200 periods for both the benchmark and no-frictions models. During times of neutral or high productivity, the variable time paths of the two models lie on top of each other, but during periods of sufficiently low aggregate productivity, the time paths of the benchmark model with financial frictions fall below their no-frictions counterparts. Together, the plots of the models’ impulse response functions and time paths clearly indicate that financial frictions arising from moral hazard exacerbate the intensity and duration of downturns, while leaving upturns unaffected.
5.3
Asymmetry of Business Cycle Fluctuations
While the results of the previous subsection demonstrate that financial frictions exacerbate downturns relative to the no-frictions counterpart, they do not establish that the resulting asymmetry shows up in our benchmark model. It is conceivable, albeit unlikely, that the no-frictions model is, in fact, asymmetric, with disproportionately large upturns relative to downturns. In this case, financial frictions that exacerbate downturns will actually work to remove or mitigate asymmetry, rather than induce it. To conclusively make the case that financial frictions generate asymmetric fluctuations, we subject both the benchmark and no-frictions models to a pair of equal but opposite shocks. The difference in response of each model to this pair of shocks provides an assessment of the level of asymmetry present in the model. To be precise, both specifications are subjected to a short downturn and a short expansion comprised of shocks of magnitude L in the first three periods. The profiles for aggregate output, capital, and aggregate labor found from the downturn and expansion are converted to percent deviations from steady state. The downturn profiles are scaled by -1 so they can be plotted on top of the expansions (see figure 5). The first column of plots in Figure 5 confirms that little to no asymmetry is present in the no-frictions model, while the second column confirms its presence in the benchmark model with financial frictions. More specifically, the benchmark model’s paths clearly show that downturns are more severe and persistent than expansions. Taken together, these results demonstrate that the presence of moral hazard, and the resulting financial frictions, lead to asymmetric business cycles both by exacerbating 19
downturns and slowing down the recovery from these downturns. Having established that financial frictions are the source of asymmetry in the model, we now turn to quantifying the degree of this asymmetry. Table 3 presents both the deepness and skewness statistics of the model’s variables. Two main results stand out from Table 3. First, the asymmetry generated by the model is quantitatively significant. Second, it is similar to that found in the data. For example, the skewness of output is found to be -0.55, similar to -0.41 found in the data. Also, the deepness of output and hours worked are found to be 0.91 and 0.80 respectively which are similar to the values of 0.97 and 0.79 found in the data. In particular, the model reproduces the relative ordering of the asymmetry of labor and output. Therefore, our calibrated benchmark model can simultaneously capture both the standard business cycle facts mentioned earlier and reproduce the relative ordering of the asymmetry of output and hours worked that is found in the data.
5.4
Financial Frictions and Amplification of Business Cycles
The fact that financial frictions exacerbate downturns implies that they must also increase the volatility of business cycle fluctuations. The no-frictions results presented in Table 2 clearly indicate that the removal of financial frictions results in a significant reduction in volatility. For example, the volatility of output falls from 1.60 percent for the benchmark model to 1.41 percent for the no-frictions model, implying that the financial frictions increased the volatility of output by about 13.5 percent. The effect for labor is much larger at about 41 percent. The additional volatility in the benchmark model comes from the fact that when aggregate productivity is sufficiently low, the IC constraint binds, forcing investors to incentivize the entrepreneurs by giving them a higher stake in the project. To do so, investors must reduce their stake in the project, which in turn reduces the funds available to entrepreneurs at the start of a project. As resources fall, project size, and more importantly, the potential output of the project is reduced. Compounding this issue is the fact that the investors’ expected returns from a project influence their liquidity financing decisions in the next period. Thus, a reduction in shares today will not only lead to smaller firms on average, but these firms will be less likely to secure their liquidity financing in the future. Given the evidence that the financial frictions have a measurable impact on both the asymmetry and volatility of business cycles, we now investigate their respective costs. We propose two strategies for measuring the costs of these frictions. The first strategy is a standard welfare loss calculation that is often found in business cycle literature and follows Lucas 20
(1987). Given that we are interested in quantifying the impact of financial frictions on the welfare of the agents in the economy, we use the solutions to the benchmark and no-frictions models to compute the present discounted value of future utility for both cases. Using these values we compute the average level of consumption an agent must be given in each period in order to be indifferent between the benchmark model and the no-frictions model. This P PT B t−1 NF is done by finding the ξ such that Tt=1 β t−1 U ((1 + ξ)cB U (cNF t , lt ) = t , lt ), where t=1 β T = 500, 000. Given that the utility function is linear in logs this value of ξ is given by: ! PT t−1 NF NF t−1 B B β U (c , l ) − β U (c , l ) t t t t t=1 t=1 −1 PT t−1 β t=1
PT ξ = exp
(40)
ξ = 0.00046, indicating that an agent would require an additional 0.046% units of consumption each period in order to be indifferent between the two scenarios. This value is close to that reported in Lucas (1987), indicating that business cycle asymmetries induced by financial frictions have a modest impact on the agent’s welfare. The previous result indicates that the average welfare loss that accompanies asymmetric cycles is of a similar magnitude to the average welfare loss that accompanies symmetric cycles. However, it is clear that financial frictions amplify the effects of severe economic downturns and the cost associated with these infrequent events is likely to be large. While there is no standard way to measure this cost during isolated time periods in the literature, we present two crude but simple measures as an attempt to do this. To measure this cost in terms of consumption for a severe downturn, we simply compute ξ for the impulse response paths found in Figure 3. We find that the value of ξ increases from 0.046% to 0.054% when we focus on the loss of utility arising from financial frictions during a severe downturn. We can take this analysis a step further by assessing the welfare cost over time during the downturn. Figure 6 presents these results. The top panel plots an instantaneous measure of this welfare cost over time. That is, it plots ξt which solves B NF NF U ((1 + ξt )cB t , lt ) = U (ct , lt )
(41)
over the impulse response paths. Its hump-shaped response is indicative of both a lagged welfare loss associated with financial frictions and the fact that short-run costs are more significant for this calculation. It reaches a maximum value of 0.11%, almost two and a half times the value found over the business cycle. The second panel of Figure 6 presents the profile for ξ(T ) in equation (40) computed for different time horizons, T = 1, 2, · · · , 100.
21
As shown, this profile is hump-shaped as well, reaffirming that short-run costs are quite substantial. The second measure for computing the costs of financial frictions is based on the difference between the response of output during a downturn under the two models. This approach, while unconventional in the literature, represents a more popular notion of recessions. Again, to isolate a downturn for this calculation we make use of the impulse response functions computed earlier. Using these impulse response functions, we create plots of the output cost of the severe downturn in both period-by-period (instantaneous) and accumulated terms. The first two panels of Figure 7 present these profiles and clearly show that output falls considerably farther when financial frictions are present. For example, the total accumulated loss of output in the benchmark model was found to be 88.7% (of steady state), while this value for the no-frictions model was found to be 76.7%. Figure 7 also presents the profile of the amplification in output losses over the downturn. We find that over the entire impulse response period, there is an extra accumulated output loss of 13% which implies an amplification of aggregate output loss of approximately 16%, with a majority of this loss occurring in the first five quarters.
6
Conclusion
This paper addresses the two open questions posed by Kocherlakota (2000): to what extent do financial frictions amplify business cycle fluctuations, and are these amplifications sufficient to generate realistic levels of asymmetry? We introduce financial frictions into our model through entrepreneur-run firms that face both moral hazard and idiosyncratic cost overruns. The basic structure of our model follows Atolia et al (2010), but overcomes many salient shortcomings of the previous work. Most notably, we provide a strategy for connecting the level of moral hazard in the model to characteristics found in the U.S. data. With the severity of the agency problem calibrated to a realistic level, we are able to address the question regarding the quantitative impact of financial frictions in a model that is able to replicate stylized business cycle facts. Using our calibrated model, we examine the role played by financial frictions in generating asymmetries and exacerbating the fluctuations found in the business cycle. For our benchmark calibrated model not only do we find quantitatively significant levels of asymmetry in key variables, but this asymmetry replicates stylized facts found the data. Specifically, we are able to closely match the asymmetries present in output and hours worked. Also, while
22
the financial frictions are only operational occasionally, they are found to significantly increase the volatility of the model’s variables. For example, the presence of financial frictions was shown to increase the volatility of output and hours worked by about 13.5% and 41% respectively. Taken together, these results indicate that the presence of financial frictions leads to asymmetric business cycles by exacerbating model volatility in a way that leads to deeper and more persistent downturns while leaving upturns unaffected. We also present two measures of the cost of financial frictions in the model economy. In terms of consumption, we find this cost to be 0.046%. In terms of output, it amounts to a long-run amplification in output loss of almost 16%, with a majority of this loss occurring in the first five quarters. The basic framework of this model can be extended in many directions. One interesting extension would be the inclusion of labor market search which would allow for the examination of the effect of fluctuations in credit access on behavior of labor market variables. Another natural extension is the inclusion of long-lived firms and firm-level heterogeneity.
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Table 1: Parameters and Steady State Values Preference Parameters β = 0.99 η = 0.8293 Production Parameters α = 0.4286 1 − α = 0.5714 δ = 0.02 Ψθ = 0.925 σθ = 0.0028 L = −2.5 ∗ σθ pH = 0.9 pL = 0.4 ρ¯ = 1.1220 Calibrated Steady State n1 =2 n = 0.3600 n1 = 0.2400 n ? e3 ρ ρ? = 0.6353 e = 0.2857 = 0.8500 ρ¯ c Y
= 0.8647 w = 0.7060 s = 0.6111
k Y
= 6.7673 r = 0.0298 Y = 0.6301
24
M Y
= 0.1344 p = 0.4852 y = 0.8237
Table 2: Summary of Second Moments std(log(X))*100 Dataa Benchmark Y c inv n k M w r ρ? p λR s a
1.59 1.27 5.05 0.84 1.05 -
1.60 1.26 5.31 0.45 1.92 1.74 1.41 1.11 0.31 1.58 1.02 0.23
corr(X,Y)
No-Frictions 1.41 1.10 4.65 0.32 1.62 1.38 1.18 0.96 0.27 1.38 0.00 0.00
Dataa Benchmark 1.00 0.89 0.88 0.78 0.28 -
1.000 0.93 0.83 0.77 0.80 0.99 0.97 0.10 0.66 0.98 -0.68 0.68
No-Frictions 1.000 0.93 0.83 0.69 0.78 0.99 0.97 0.05 0.55 0.99 0.00 0.00
Data on Y , c, and inv comes from the National Income and Product Accounts tables and covers the period of 1946-2011. Specifically, Y is computed as GDP less consumer durables, c is computed as non-durables plus services, and inv is computed as Gross Private Domestic Investment. Data on k comes from the BEA, covers the same time period, and is measured as Private Fixed Assets.
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Table 3: Asymmetry Statistics Skewness(X)a Datab Benchmark Y c inv n k M w r ρ? p λR s
-0.41 -0.23 -0.47 -0.72 -0.29 -
-0.55 -0.54 -0.64 -1.30 -0.68 -0.99 -0.72 0.13 -1.04 -0.55 4.29 -4.32
Deepness(X)
No-Frictions -0.00 0.04 -0.10 -0.02 0.00 0.00 0.00 0.02 -0.00 0.00 NA NA
Datab Benchmark 0.97 0.98 0.84 0.79 0.95 -
0.91 0.91 0.91 0.80 0.89 0.84 0.88 1.04 0.83 0.91 6.09 0.16
No-Frictions 1.01 1.01 1.01 1.00 1.02 1.01 1.01 1.01 1.00 1.01 NA NA
Deviation Above Trend Deepness(X) = % % Deviation Below Trend b Data on Y , c, and inv comes from the National Income and Product Accounts tables and covers the period of 1946-2011. Specifically, Y is computed as GDP less consumer durables, c is computed as non-durables plus services, and inv is computed as Gross Private Domestic Investment. Data on k comes from the BEA, covers the same time period, and is measured as Private Fixed Assets.
a
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Kiyotaki, Nobuhiro and John Moore. 1997. “Credit Cycles,” Journal of Political Economy 105(2):211-241. Kocherlakota, Narayana R. 2000. “Creating Business Cycles Through Credit Constraints,” Federal Reserve Bank of Minneapolis Quarterly Review Summer 2-10. Kydland, Finn E., and Edward C. Prescott. 1982. “Time to Build and Aggregate Fluctuations,” Econometrica 50:1345-1370. Li, Shuyun M. and Scott Dressler. 2011. “Business Cycle Asymmetry via Occasionally Binding International Borrowing Constraints,” Journal of Macroeconomics 33:33-41. Lucas, Robert E., Jr. Models of Business Cycles. New York: Blackwell, 1987. Neftci, Salih N. 1984. “Are Economic Time Series Asymmetric Over the Business Cycle?,” Journal of Political Economy 92(2):307-328. McKay, Alisdair and Ricardo Reis. 2008. “The Brevity and Violence of Contractions and Expansions,” Journal of Monetary Economics 55:738-751. Mendoza, Enrique G. 2010. “Sudden Stops, Financial Crises, and Leverage,” American Economic Review 100:1941-1966. Sichel, Daniel E. 1993. “Business Cycle Asymmetry: A Deeper Look,” Economic Inquiry 31:224-236. Tirole, Jean. 2006. The Theory of Corporate Finance, Princeton University Press: Princeton and Oxford. Van Nieuwerburgh, Stijn and Laura Veldkamp. 2006. “Learning Asymmetries in Real Business Cycles,” American Economic Review 53(4)753-772. Williamson, Stephen D. 1986. “Costly Monitoring, Fiancial Intermediation, and Equilibrium Credit Rationing,” Journal of Monetary Economics 18:159-179. Williamson, Stephen D. 1987. “Financial Intermediation, Business Failures, and Real Business Cycles,” Journal of Political Economy 95:1196-1216.
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Figure 1: Lending over the Business Cycle
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Figure 2: Shadow Price of Funds and Rate Spread Data
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Figure 3: Impulse Response Functions
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Figure 4: Simulated Time-Paths
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Figure 5: Asymmetry Plots
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Figure 6: Utility Loss Profiles
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Figure 7: Output Loss Profiles
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