Private Debt and Income Inequality: A Business Cycle Analysis

Transcription

1 Private Debt and Income Inequality: A Business Cycle Analysis Matteo Iacoviello Boston College May 23, 2005 [ PRELIMINARY. COMMENTS WELCOME ] Abstract I study a heterogeneous agents economy which is calibrated to mimic the business cycle behavior of the US income distribution. Agents face aggregate and idiosyncratic risk, issue collateralized and uncollateralized debt, work and accumulate assets over time. I investigate whether the model can replicate two empirical facts: the large increase in private debt over GDP in the last 30 years, and the diverging trends in consumption and wealth inequality. When the driving processes are calibrated according to data on idiosyncratic volatility, aggregate productivity and financial liberalization, I find that within-group income inequality can account for the trend and the cyclical behavior of private debt. In addition, I also show that the model can provide a good characterization of the recent trends in wealth and consumption inequality. Keywords: Credit constraints, Incomplete Markets, Income Inequality, Private Debt, Volatility Jel : E31, E32, E44, E52, R21 iacoviel@bc.edu. Address: Department of Economics, Boston College, Chestnut Hill, MA , USA.

2 This paper uses a dynamic general equilibrium model with heterogenous agents to examine the causes of the cyclical and trend behavior of private debt over GDP in the period Private debt (the sum of households and business debt) has since the 1980s jumped out of proportion with real activity, reaching 144 percent of GDP in This phenomenon has occurred alongside two separate and striking changes in economic volatility. On the one hand, aggregate volatility has fallen: in the US, the standard deviation of GDP growth has roughly halved between the period and the period On the other, cross-sectional risk has risen, as well as earnings inequality between and within income groups. Explanations for the rise in debt have referred to a combination of factors, including smaller business cycle fluctuations, the reduced costs of financial leveraging, changes in the regulatory environment for lenders, new technologies to control credit risk. Explanations for the decline in macroeconomic volatility have referred to good monetary policy, good practice (like better inventory management) and good luck (reduced volatility of the underlying economic shocks). Finally, explanations for the rise in volatility at the household and firm level have included shifts in the relative supply of and demand for skilled workers, changes in economic institutions, and technological change. To date, no study has tried to connect the various patterns in economic volatility with the behavior of private debt over time. There are several reasons, however, to believe that the forces driving aggregate and idiosyncratic developments in the economy play a major role in affecting the need of households and business to access the credit market. This is the perspective adopted here. At the most basic level, one would expect that the level and the growth rate of GDP should increase credit growth, insofar as a richer country better allocates its resources between those who have funds and those who need them; yet it is hard to find convincing theoretical arguments that justify why credit growth might indefinitely exceed GDP growth. At the cross sectional level, the arguments become subtler, and can be best conveyed by the following example. Consider an economy with three agents: two of them are identical except for the scale of their income: think of them as a young, untenured assistant professor (Youngie: agent 1); a young, tenured professor (Fullie: agent 2); a low-skilled agent (Lowe: agent 3) who has less education, behaves as a rule-of-thumb/impatient consumer, discounts heavily the future and is unable to save: such agent would always like to borrow at the going interest rate but is bound in the ability to do so by the amount of his pledgeable assets. Assume that, at some date 0, Lowe earns one dollar and has an outstanding debt of the same amount with 1 Throughout the paper, private (nonfinancial) debt is defined as the total outstanding debt of households and businesses. The discussion here does not consider public debt nor does it take into account the net foreign asset position of the United States, which is about -22% of GDP as of year-end

3 Fullie; that Youngie earns one dollar and owes three dollars to Fullie; and that Fullie earns three dollars (and lends four). The economy s total income is 5 dollars, and its debt to GDP ratio is 80 percent. If, at date 1, Lowe suffers a bad income shock at the expense of the others, income inequality between groups will increase: at the same time, Lowe will have less collateral to pledge for loans and will be forced to borrow less: debt over GDP will fall. Consider, on other hand, the scenario in which Youngie suffers a bad income shock vis-à-vis the two other agents. Income inequality within groups will increase. However, since Youngie does not face binding borrowing constraints, he will be able to smooth consumption borrowing from Fullie, and debt over GDP will increase. The above example is very stylized, but serves to illustrate the main question of the paper: how do the shocks hitting the economy s income distribution affect the behavior of its credit flows? I address this question by constructing a dynamic general equilibrium model of the interaction between income volatility, private sector financial balances, and the distribution of expenditure and wealth. The model ingredients are extremely simple: heterogeneity in discount rates and market incompleteness (in the form of borrowing constraints for some of the agents). The economy is hit by idiosyncratic income shocks, financial shocks and aggregate shocks. Three different agents are assumed to be representative of the US economy: they issue bonds, produce output using capital and labor and consume durable and non-durable goods. Two agents only differ in the scale of their productivity and in the idiosyncratic shocks that hit them, which give rise to a desire to borrow and lend. The third agent differs in preferences and its ability to access the credit market: being more impatient, such agent would always like to borrow at the equilibrium interest rate, but is constrained to do so by the amount of collateral that she can be pledge. Using annual observations on between and within-group income inequality, I estimate the stochastic processes for the idiosyncratic income shocks which are able to replicate the behavior of income inequality over time. Using data on loan-to-value ratios and productivity, I estimate processes for financial shocks and aggregate productivity shocks. I then consider the role of these shocks in explaining the patterns in the data, in particular the trend and the cyclical behavior of private debt and the distribution of consumption and wealth across the population. The key finding of the paper lies in the ability of a heterogeneous agents model to explain two salient features of the data: 1. On the one hand, the model can explain the timing and the magnitude of rise in private debt over GDP. Debt increases when within-group inequality increases and when borrowing constraints become looser. Of the total variation in debt, a large fraction is due to increased idiosyncratic volatility, and a smaller one to financial liberalization (time variation in the tightness of the borrowing constraints). 3

4 2. On the other, the model can reconcile the sharp increase in income inequality over the last 30 years with a smaller rise in consumption inequality and a very strong rise in wealth inequality. The model is solved using linear approximation methods and approximating the decision rules around the deterministic steady state. This approximate solution technique, albeit convenient, is subject to some caveats: 1. In the non-stochastic steady state of the model, the initial distribution of debt within agents with equal discount rates is indeterminate. Instead, agents with high discount rates hit their borrowing limits, and this pins down their total debt holdings. To match the observed distribution of debt at the beginning of the sample period, I assign values to the income share of impatient agents and calculate their equilibrium debt holdings. I then calibrate the credit-debt positions within patient agents in order to match the total debt to income in the data: however, a straightforward extension of arguments in Clarida (1990) would establish that, in a stochastic stationary equilibrium, the unconditional mean of debt within these agents should be zero. 2. In the non-stochastic steady state, I rule out precautionary saving motives. If uncertainty becomes large, impatient agents might want to keep a buffer of resources in order not to hit the borrowing limit in all periods. 3. In a neighborhood of the non-stochastic steady state, even if the model shocks are trend stationary, aggregate debt and the distribution of expenditure are not. For instance, while total income is trend stationary, the consumption gap between patient agents follows a random walk, and so does the ratio of aggregate debt to GDP. 2 While perhaps unappealing from a theoretical perspective (especially in light of the fact that this property is an outcome of the certainty-equivalence/linear approximation solution), this feature of the model is supported in the data. In fact, while the data seem to favor trend-stationarity for GDP and its components, they indicate that the private debt to GDP ratio is better described as a difference stationary variable In a neighborhood of the steady state, the assumption of certainty equivalence implies that patient agents behave like permanent income consumers. Impatient agents, instead, 2 Interestingly, this implication of the model mirrors analogous results which are standard in two-country models of the business cycle with incomplete markets. See Baxter (1995) for a discussion. 3 These claims are based on a set of standard unit-root tests conducted in sequence using the urauto.src procedure in Rats. Using annual data from 1952 to 2002, one concludes that the ratio of nominal debt to nominal GDP contains a unit root with no drift, whereas logged real GDP per worker is better described as trend-stationary. 4

5 being borrowing constrained, behave in a rule-of-thumb fashion, consuming a constant fraction of their income and rolling their debt holdings over forever. How would the results change if one were to calculate the exact equilibrium of the model without resorting to linear approximations? Computational complexity is a major hurdle here. In a separate appendix, I provide preliminary evidence based on simulations of non-linear, recursive, equilibria for two stripped-down versions of the above economy: an economy with two patient agents only, each bound by a natural debt limit; an economy with one patient and one impatient agent, bound by an ad-hoc collateral constraint. In the patient agents economy, the consumption differential between the two agents follows approximately a random walk, unless one of the two agents approaches (because of a series of bad income realizations) his natural debt limit. 4 When one of the agents start with a debt to income ratio close to 1, for realistic income processes the natural debt limit is almost never approached. 5 In the patient-impatient economy, if the impatient agent starts at the borrowing constraint, he might escape from the constraint after a sufficiently long series of positive income shocks. How often this happens depends on impatience, income volatility, and risk aversion. However, if the agent is impatient enough, he hits the borrowing limit with probability one. These findings, of course, warrant further investigation. Taken together, however, they justify why the certainty-equivalence solution of the model can offer a good approximation of the full, non-linear model. The structure of the paper is as follows. Section 1 briefly describes the patterns in the data. Section 2 presents the model. Section 3 describes the calibration and the simulation of the model. Section 4 presents the results. Section 5 concludes. 1. Patterns in private debt, volatility and inequality 1.1. Private debt The US financial markets and the links between financial transactions and economic activity have changed in striking ways over time. Figure 1 illustrates the behavior of household and business debt to gross domestic product from 1970 to The ratio of private debt to GDP was stable throughout the 1970s, and expanded at a fast pace from the early 1980s on. The ratio fell in the early 1990s, but began a gradual increase from 1994 on. At the end of 2002, the combined sum of household and business debt was 144% of GDP. 4 Loosely speaking, the natural debt limit is the present discounted value of the worst income realization. 5 In Zhang s (1997) two agents bond economy, agents rarely hit (the frequency is less than 1%) no default borrowing constraints which are much tighter than the constraints assumed here. 5

6 Series Std Std Std Relative volatility Real GDP growth % Real Debt growth % Nominal Debt growth % Inflation % Real M2 growth % Fed Funds rate % Real Fed Funds rate % Table 1.1: Standard Deviations of Annual Growth Rates of Selected Macroeconomic Time Series Note: GDP, Debt and M2 were normalized by the civilian non-institutional population aged 16 and over. Debt and M2 were transformed in real terms dividing by the GDP deflator. Exact names and data definitions are in the Appendix. The last column is the ratio of column 4 to column 3. Interestingly, the upward trend in private sector credit has occurred alongside an increase in its cyclical volatility. Stock and Watson (2002) document how most macroeconomic time series exhibit a decline in their cyclical volatility throughout the 1980s and 1990s, starting from around While the volatility decline characterizes many macroeconomic variables, it seems to escape volatility of the credit flows. Table 1.1 reports the annual sample standard deviation (in percentage terms) of some macroeconomic time series, as well as that of debt growth (in real and nominal terms). As illustrated by the last column - showing the volatility in the second subperiod relative to the first -, all series were less volatile in the second subsample, with the exception of debt growth Volatility and Inequality Several papers have documented upward trends in income and earnings inequality in the US in the last 30 years (see for instance Katz and Autor, 1999, and Moffitt and Gottschalk, 2002). Increased earnings dispersion has been apparent in every dimension of the data. High frequency changes in volatility appear however harder to measure, and while the trend is robust across studies and dataset, there appears to be less consensus on fluctuations at business cycle frequencies. It appears however that inequality was little changed in the 1960s, increased slowly in the 1970s and sharply in the early 1980s, and continued to rise, although at a modest pace, 6 The increase in aggregate debt volatility is however not common across subsectors. In particular, most of the increase in the volatility of debt growth is accounted for by the business (rather than the business) sector. If one were to consider household debt only, the standard deviation of nominal debt growth falls in the second subperiod to 2.4 from 2.7, while it increase from 1.9 to 4.2 for businesses. 6

7 throughout the 1990s. Once the wage structure is decomposed in between and within-components, both between and within-group changes in wages seem to have contributed to higher dispersion. As documented by Katz and Autor using CPS data, between-group inequality (as measured by the college wage premium) rose between 1963 and 1971, fell from 1971 to 1980, and rose sharply until around 1988, before leveling off. Residual (within) wage inequality is rather stable during 1960s, starts to increase in the 1970s, and accelerates in the 1980s until the mid-1990s. Recent work by Krueger and Perri (2003a and 2003b), using data from the Consumer Expenditure survey, documents analogous trends in the data for the period Krueger and Perri use after-tax earnings inequality measures for most of their analysis, thus excluding capital income from the income sources: because their study covers a fairly long period, I use their data on earnings inequality and their decomposition in within and between-group differences as an input in my calibration exercise A kitchen-sink regression Is there any connection between the patterns in inequality and the behavior of private debt? As a first pass, I estimate a reduced-form regression of real private debt growth ( D) on its own lag, real GDP growth ( Y ), changes in measures of between-group and within-group inequality ( σ W and σ B ), loan-to-value (LTV) ratios (m). 8 These are all likely candidates in an explanation of the cyclical behavior of debt. 9 All the variables are in percentage terms, and their exact definitions are in Appendix A. For the period , the resulting estimated equation, omitting the constant term, is given by (standard errors are in parentheses): D t =0.72 (0.09) D t (0.28) Y t (0.25) σw t 0.58 (0.43) σb t (0.24) m t. Real interest rates, house price inflation and consumer price inflation were not significant in this specification. The regression fits the data well (its adjusted R-squared is 0.84). The coefficient on within-group inequality is positive and strongly significant, while the coefficient on betweengroup inequality is negative but not significant at conventional levels. 10 GDP growth enters 7 See Appendix A for details. Several authors have decomposed earnings/wage inequality data in between and within components. For a discussion on how hard it is to measure earnings inequality, see Lemieux (2004). See also Autor, Katz and Kearney (2004), Card and DiNardo (2002) and Heathcote, Storesletten and Violante (2004) for recent evidence on income inequality. 8 To control for obvious endogeneity problems, I use two lags of each of the right-hand variables as instruments. I also use three-year centered moving averages of changes in inequality to remove noise from the inequality data. 9 At the cross-country level, one would expect that creditor protection through the legal system also plays a role. This is however less likely to matter in the time-series dimension. 10 Taken at face value, this estimates indicate that a one percent increase in the within-group standard deviation of log-income raises log debt permanently by 0.69/ (1 0.72) = percent. Once one observes that the 7

8 positively and significantly the regression, and that LTV ratios enter the regression with the expected positive sign, although they do not appear significant. 11 The next section sketches a general equilibrium model of debt, between and within-group inequality and business cycles that can help making sense of these results as well as being consistent with the observations of the introduction. 2. The model 2.1. The environment and the nature of income inequality Time is discrete. The economy consists of three classes of infinitely-lived agents who are distinguished by the scale of their production function, by their discount rates and by their access to the credit market. Agents are indexed by i =(1, 2, 3). Each agent works, produces and accumulates assets over time. Agents are subject to idiosyncratic and aggregate production risk. Thecreditmarketworksasfollows. Agents1 and 2 can trade one-period consumption loans between them and with agent 3. Agent3 cannot commit to repay her loans, and needs to post collateral to secure access to the credit market. Instead, unbacked claims are enforceable within patient agents, who are bounded only by their natural debt limits. For all agents, the amounts that they are allowed to borrow can be repaid with probability one, and there is no default. On the production side, agents differ in the scale of their total factor productivity which, absent shocks, can be thought as the source of permanent inequality in the economy. On the preference side, agents 1 and 2 are assumed to discount the future less heavily than agent 3; for this reason, I refer to them as patient. Agent 3 thus differs from the patient agents along two key dimensions: he discounts the future more heavily, and faces borrowing constraints. 12 As the model structure makes clear, productivity differentials across agents are completely exogenous. Yet heterogeneity in discount rates implies that impatient agents will have in steady state lower wealth to income, higher debt to income, and lower consumption to output ratios. within-group standard deviation of log-income has risen by 15 percent in the last 30 years, this illustrates why income inequality might quantitatively matter for credit flows. 11 As argued by Bernanke and Lown (1991), measuring debt growth in real terms can be misleading if prices are very volatile and debt contracts are not indexed. One way to get around this problem is to measure debt growth in nominal terms. When doing so, the coefficients are roughly similar, and, interestingly, also the coefficient on between group inequality is significant. 12 It would be easy to allow for two groups of impatient agents. However, given that impatient agents cannot share any idiosyncratic risk through borrowing and lending, their opportunities for intertemporal smoothing would still be limited to the financial trades with the patient agents. The only implication of the three-agents assumption is that any change in within-group inequality will be assigned by the model to agents who can buffer against income shocks. 8

9 To the extent that these characteristics are correlated with individual traits such as age and education, an econometrician aiming at explaining income inequality would attribute income differences between the impatient agent 3 and the rest of the economy to observables. For this reason, following widely used practice in the inequality literature, I refer to the differences between impatient and patient agents as between-group differences. The remaining income inequality in the economy (within the two patient agents) would be attributed to unobserved factors: therefore, I refer to agents 1 and 2 differences as within-group (or residual ) differences. For each agent, the production function used to produce output y combines beginning-ofperiod capital k and hours l through a standard constant returns to scale technology. 13 The resulting output can be either consumed, invested in business capital k (which depreciates at rate δ k ) or in durables / housing capital h (which depreciates at rate δ h and provides utility services). Finally, each agent is subject to idiosyncratic production risk. With three agents, there are at most two shocks which are not perfectly correlated which do not affect aggregates (keeping inputs fixed). The specific properties of these shocks, which would be completely diversified away in a complete market setting, are described in more detail below Patient Agent 1 At time t, patient agent 1 (indexed by a prime) chooses consumption c 0, durables h 0,capital k 0 and hours l 0 in order to solve: X max E 0 β 0! t Ãlog c 0t + j log h 0t τ 0 (l0 t) 1+η0 1+η 0. t=0 The flow wealth constraint and the production function are respectively: c 0 t + h 0 t (1 δ h ) h 0 t 1 + k 0 t (1 δ k ) kt b 0 t d t = yt 0 + Rt 1b 0 0 t 1 R t 1 d t 1 (1) Ã 1 µ yt 0 = A t k 0µ X t t 1 y 1 W t Z ζ lt! 0 (2) t where b 0 t are loans made at the riskless gross interest rate R t to impatient agent 3, whose behavior will be described below, and d is borrowing of agent 1 from agent In the production function above, y 1 is an individual specific fixed effect; A t denotes total factor productivity (TFP), which is assumed to follow an AR (1) process in logs. Shocks to TFP are perfectly correlated across all agents. To capture secular growth in output, I assume 13 The model agents can thus be thought of as independent producers each running a backyard technology. 14 In equilibrium, we will have d>0 and b>0, so that patient agent 1 will lend to impatient agent 3 and will borrow from patient agent 2. 9

10 that there is a deterministic component to productivity X t whichisassumedtoexpandata constant rate over time. That is: X t =(1+g) X t 1. Finally, W t and Z t are instead idiosyncratic shocks (which follow AR (1) processes in logs). The correlation structure of the idiosyncratic shocks is such that, if the agents were to keep their factor inputs unchanged, total output in the economy would not change. Shocks to W t affect the relative productivity of patient agent 1 vis-à-vis patient agent 2: given that patient agents are identical except that for the scale of their production function, shocks to W t can be interpreted as a source of within-group income volatility. Conversely, shocks to Z t affect all agents: when these shocks hit positively the patient agents, they also hit negatively the impatient agent: such shocks can be interpreted as a source of between-group income inequality. The first order conditions for this problem involve two standard Euler equations for consumptions, a durable good demand, capital demand and the labor supply condition as follows: µ 1 β 0 = E t (3) c 0 t 1 c 0 t 1 c 0 t 1 c 0 t c 0 Rt 0 t+1 µ β 0 = E t c 0 t+1 µ = j0 1 h 0 + β 0 δh E t t c 0 t+1 µ β µy 0 = t+1 +1 δ k c 0 t+1 1 c 0 (1 µ) y0 t t lt 0 = τ 0 lt 0 η 0. (7) In the solution procedure, I obtain the decision rules by assuming that these agents asset position is such that they are never close to their natural borrowing constraints. This procedure is safe if their maximum borrowing limit (the one which is consistent with positive steady state consumption) is large enough relative to their wealth, a condition which is assumed to hold throughout the paper. R t k 0 t (4) (5) (6) 2.3. Patient agent 2 Patient agents 2 are indexed by a double prime ( 00 ). They are identical to agent 1 (same discount factor, in particular), except for the scale of their production function (as proxied by the fixed effect y 2 ): in steady state, the ratio between fixed effects y 2 /y 1 is chosen to match observed within-group inequality in the data. 15 These agents can lend d t to patient agent 1 15 In the paper, I adopt the normalization that agent 1 is poorer than agent 2, and that the impatient agent 3 is poorer than the average between patients 1 and 2. Starting from the steady state, this implies that a 10

11 and can lend b 00 t to impatient agent 3. Their production function is µ yt 00 = A t k 00µ X t W θ 1 µ t t 1 y 2 Z γ lt 00. t The parameters θ and γ are constants which are chosen to guarantee that, keeping capital and hours fixed, the aggregate effects of given idiosyncratic shocks Z t and W t are zero (the restrictions on θ and γ are spelled out in Appendix B) Impatient agent 3 Impatient agents are assumed to discount the future more heavily than agents 1 and 2 and to face a liquidity constraint that limits the amount of borrowing to a time-varying fraction of their physical assets (durables and variable capital). With this simple assumption, I want to capture the idea that real assets are more likely to be used as a form of collateral. The problem the impatient agents solve is: Ã! X max E 0 β t log c t + j log h t τ (l t) 1+η 1+η t=0 subject to the following budget constraint, where b 0 t and b 00 t denote respectively borrowing from patient agent 1 and patient agent 2: c t +(h t (1 δ h ) h t 1 )+(k t (1 δ k ) k t 1 )+R 00 t 1b 00 t 1 + R 0 t 1b 0 t 1 = y t + b 0 t + b 00 t (8) where y t = A t k µ t 1 (y 0X t Z t l t ) 1 µ and the borrowing constraints are respectively: b 0 t α m ht h t /Rt 0 + m kt k t /Rt 0 (9) b 00 t (1 α) m ht h t /Rt 00 + m kt k t /Rt 00. (10) For each unit of h and k they own, impatient agents can borrow respectively at most m ht and m kt (to simplify matters, I will assume that m ht = m kt ): exogenous time variation in m proxies for any shock to the economy-wide supply of credit which is independent of income, as in Ludvigson (1999). In addition, α is the fraction of collateral which is pledged respectively to agents 1 and 2. Whileα could be potentially made a choice variable, it is obvious that any value of α would maximize the agent s borrowing capacity. To get around this problem, I will positive realization of W t widens within-group and total inequality, whereas a positive realization of Z t reduces between-group and total inequality. 11

12 calibrate α to be equal to the relative income of each patient agent. 16 I also assume that there is economy-wide variation in the amount that agents can borrow relative to their assets. The first order conditions can be written as: µ 1 βr 0 = E t t + λ 0 t (11) c t c t+1 µ 1 βr 00 t = E t + λ 00 t (12) c t c t+1 1 = j µ µ 1 δh αλ 0 + βe t + m t (1 α) λ00 t ht c t h t c t+1 Rt 0 + Rt 00 (13) µ 1 β = E t µ y µ t+1 αλ 0 +1 δ k + m t (1 α) λ00 t kt c t c t+1 k t Rt 0 + Rt 00 (14) 1 (1 µ) y t = τ (l t ) η. (15) c t l t The first-order conditions for agent 3 are isomorphic to those of agent 1, with the addition of λ 0 and λ 00, the Lagrange multipliers on the borrowing constraints. It is straightforward to show that, in a neighborhood of the non-stochastic steady state, agent 3 will be borrowing constrained so long as β<β 0 and the multiplier λ on the borrowing constraint will be strictly positive Equilibrium In equilibrium, all the markets clear and the interest rates work to equate demand and supply in the goods market and in the market for bonds. Operationally, I solve for the recursive equilibrium in a neighborhood of its deterministic steady state when the variances of all shocks are zero. In addition, I rule out Ponzi schemes by imposing the appropriate transversality conditions. The economy is hit by four shocks: an aggregate productivity shock, a financial shock, and two types of idiosyncratic shocks hitting differently the different agents in the economy. All shocks are assumed to vary according to an AR(1) process, that is: log A t = ρ A log A t 1 + σ A 1 ρ 2 A 1/2 eat log W t = ρ W log W t 1 + σ W 1 ρ 2 W 1/2 ewt log Z t = ρ Z log Z t 1 + σ Z 1 ρ 2 Z 1/2 ezt log m t = ρ m log m t 1 + σ m 1 ρ 2 m 1/2 emt 16 That is, I will pick a so that α = y 0 / (y 0 + y 00 ). See Iacoviello and Minetti (2005) for an example of how the borrowing decision of these agents could be endogenized, in the context of a two-country model of the business cycle. 17 See Iacoviello (2005) for a related application and for a discussion in the context of a monetary business cycle model with heterogeneous agents. 12

13 where the e s are normally distributed with zero mean and unit variance. As mentioned above, it is instructive to think of the two patient agents as otherwise identical agents. Shocks to the income of one patient agent relative to the other are therefore a source of within-group earnings inequality in the population. Instead, shocks to the income of constrained agents relative to the unconstrained agents are a source of between-group earnings inequality. I extract measures of the shocks from the data as follows: I take a total measure of income inequality over time (broken down by within-group and between-group) and recover from such measure the implied time series for the idiosyncratic shocks that matches the observed pattern in income inequality which emerges from the data. Appendix B explains the procedure in detail. Notice that the assumption of a Cobb-Douglas production function implies, through the constancy of the factor shares, a one-to-one mapping between changes in total income (including capital income) inequality and changes in total labor income inequality. 3. Calibration and simulation 3.1. Overview The model is solved using a certainty-equivalence approximation and standard linearization procedures. At best, this is an approximation: in particular, the non-stochastic state is such that the distribution of financial assets between the patient agents (as summarized by d) is indeterminate. While unappealing for some purposes, this property of the model gives me one degree of freedom, since it allows pinning down d in a way to match the private debt to GDP ratio which is observed in the data. To check whether the model can account for the main stylized facts in the data, I use the following procedure: 1. I calibrate the structural parameters of the model, so that the initial steady state matches key observations of the US economy in the year In detail, I set the parameters describing preferences and technology so that in the initial steady state the total wealth and private debt to GDP ratio, the ratios of the components of spending to output and the distribution of income replicate the data. 2. I estimate from the data sequences of technology shocks, financial shocks and between and within-group income shocks for the period in exam I feed the estimated shocks into the model decision rules (calculated under the assumption that the shocks that hit the economy are drawn by the same distribution from which the 18 In this, I depart from the traditional RBC approach that normally treats exogenous shocks as unobservable. 13

14 decision rules are calculated) starting from the year 1970, and check whether the time series generated from the model can replicate the cyclical and trend behavior of financial assets, income inequality and consumption inequality which is observed in the data Calibration The time period is set equal to one year. This reflects the lack of higher frequency measures of income inequality over time, which are needed to recover the processes for the idiosyncratic shocks. Table 3.1 summarizes the calibrated parameters. As explained above, these parameters are meant to capture the initial steady state distribution of income and financial assets, as well as the consumption, business fixed and housing investment to output ratios. Given that patient agents are unconstrained in the nonstochastic steady state, I set their discount factor to 0.985; together with the deterministic growth rate of the economy g, which is estimated to be 1.6 percent per annum, this pins down the steady state real interest rate at 3% per year. The Frisch labor supply elasticity is assumed to be 1/η =1/5 for all agents, a number which is in the range of microeconometric estimates. The labor disutility parameters (the τ s in the utility function) are chosen so that in steady state labor supply is unity for all agents, a convenient normalization. 20 The capital share µ and the durable/housing preferences parameter j are chosen to match the steady state stock of structures and flow of investment that are found in the data. It is assumed that housing depreciates more slowly than business capital. The discount factor for impatient agents is set at 0.9 (see Iacoviello, 2005 for a discussion). In the aggregate, these choices result in 29% fixed investment over output ratio, 4% durable goods investment over output ratio, 67% non-durable consumption output ratio. In addition, they imply that the total stock of durables/housing and business capital over GDP are respectively 134 and 245 percent, as shown in Table 3.2. These data are roughly consistent with US post-world war II averages (see for instance Greenwood, Hercowitz and Krusell, 2000, and Castaneda, Diaz-Gimenez and Rios-Rull, 2003). In order to match income dispersion in the data, I pick the individual-specific fixed effects y 0,y 1 and y 2 so as to match cross-sectional income volatility at the beginning of the sample. That is, the fixed effects are chosen so that in the initial steady state y =1,y 0 =0.76 and y 00 =3.07. Given these numbers, the initial between group standard deviation of log income 19 One caveat is in order here. The model linearized decision rules depend among other things on m, the steady state loan-to-value ratio, and d, the steady state distribution of within-agents debt. As m and d change over time, the linearized decision rules become less and less precise. To examine the bias due to time variation in m and d, I recomputed the model decision rules using the end-of-period steady state. The impulse responses calculated in the new steady state are broadly similar to those calculated using the initial values. 20 See for instance Browning, Hansen and Heckman (1999). 14

15 g = β = 0.9 β 0 = β 00 =0.985 µ = 0.33 j = j 0 = j 00 =0.1 δ k = 0.1 δ h = 0.01 η = η 0 = η 00 =5 V b0 = 0.2 V w0 = 0.57 m h = 0.6 m k = 0.6 Table 3.1: Calibrated Parameter Values is V b0 =0.2; the initial within-group standard deviation of log income is V w0 =0.58. Among other things, these numbers imply that credit constrained agents account for 20.8 percent of total income. These values also imply that moving from the initial steady state, an increase in y 00 (coming from positive realizations to W t )relativetoy 0 leads to an increase in within-group income inequality, whereas an increase in y (coming from positive realizations to Z t )relative to y 0 and y 00 leads to a decrease in between-group income inequality. 21 I then calibrate the loan-to-value ratios. In 1970, the ratio of private debt over GDP was 91%. Letting the beginning of period loan-to-value ratios m h and m k to be equal to 60%, a plausible number, implies a ratio of constrained debt to total output of 35%. The unconstrained debt cannot be pinned uniquely down in the deterministic steady state of the model: I choose an amount such that total debt matches its 1970 value: that is, unconstrained debt is chosen to be 91% 35% = 56%. A final decision has to be made as to whether patient agent 1 (poorer) or patient agent 2 (richer) is a borrower or a lender in the initial steady state. Since the data show an increase in debt parallel to an increase in within-group income inequality, it is self-evident that the model can explain a rise in debt starting from 1970 on only 21 Income dispersion here refers to the dispersion in the amount of goods that agents produce. In and out of the steady state, part of this income is redistributed from one agent to another through interest payments: I exclude this income sources from my income definition. 15

16 y i c i /y i h i /y i k i /y i d/y i b/y i (d + b) /y i Impatient agent Patient agent Patient agent Aggregate economy Table 3.2: Initial Income, Wealth and Financial Positions of the Agents. Note: For Each Agent, all variables (except income) are expressed relative to individual income. Recall that positive values of d or b indicate a negative financial position. if the poorer agent is a borrower at the beginning of the sample period. This is also consistent with evidence showing how rich agents have, in general, higher wealth to income ratios (see for instance Dynan, Skinner and Zeldes, 2004). Table 3.2 illustrates the distribution of income, financial assets and real assets of the three agents at the beginning of the sample period (year 1970) Recovering the stochastic processes for the shocks The technology shock I extract the technology shock from the BLS Manufacturing Multifactor Productivity Series. To decompose this series into deterministic trend and stochastic cycle, I fit alineartrendto the log of this series to extract g, the deterministic productivity component. The residual from this regression is then assumed to follow an AR (1) process and used to construct log (A t ) The financial shock It is hard to construct a single indicator of the degree of financial intermediation and of the ability of impatient agents to access the credit market. Likely candidates would include: 1. Observed measures of loan-to-value ratios: these are also readily identifiable from the data. 2. Measures of the willingness of banks to make loans. 3. Measures à-la-ludvigson (1999) of the ratio (Mix) between bank loans and total loans. 22 See King and Rebelo (1999) for details. The Manufacturing Multifactor Productivity (MFP) is the most widely reported measure of technical change and is perhaps the best readily available proxy for A t. 16

17 In reality, financial liberalization in the United States has been a combination of a variety of forces which no single indicator can easily capture. Therefore, any indicator is likely to proxy only imperfectly for the time-variation in the degree of tightness of the borrowing constraint. Because it comes closest to proxying for the model counterpart, I take the logged loan-toprice ratio for newly-built homes as a measure of financial shocks and assume that it varies stochastically over time according to an AR (1) process. This way, I can construct a measure of time-varying liquidity constraints, which gives me the process for log (m t ). As shown by Figure 2, loan-to-value ratios have increased (albeit at an uneven pace) over the last 30 years, rising by a total of 8% over the sample period. A sharp increase occurred in the early 1980s, when the Monetary Control Act of 1980 and the Garn-St. Germain Act of 1982 expanded households options in mortgage markets, thus relaxing collateral constraints The idiosyncratic shocks Appendix B describes in detail how one can use observed measures of time-varying within and between-group income inequality (when measured by the cross-sectional variance of log incomes) in order to recover the primitive idiosyncratic shocks that, keeping unchanged labor input, capital input, total factor productivity and deterministic productivity, would be consistent with given variations in within and between-group variance. The idea behind the calculation is the following: in presence of labor-augmenting technological progress and temporary aggregate productivity shocks, any change in the relative income scale of patient agent 1 with respect to patient agent 2 can be attributed to random changes in their idiosyncratic ability to transform inputs into outputs. Such changes are the within-group shocks. Likewise, any observable change in the relative income scale of the impatient agent relative to the patient agents must reflect changes in relative ability of each group to produce goods. Several studies (see for instance Lawrence, 1991, Samwick, 1998, and Becker and Mulligan, 1997) have documented how discount rates and ability to access the credit market decline significantly with age and income level. Given that low discount rates and borrowing constraints are the element of differentiation between impatient and patient agents, I attribute any change in the earnings gap patient-impatient agents to between-group shocks See Campbell and Hercowitz (2005) for a discussion of these issues as well as for a model of the interaction between financial liberalization, borrowing constraints, and business cycles. 24 For a similar attempt to recover idiosyncratic shocks from earnings inequality data, see Castaneda, Diaz- Gimenez and Rios-Rull (2003). 17

18 The estimated series Figure 2 plots the implied time series for the shock processes normalized to zero in the year 1970, which is taken to be the base year. The increase in W t over time reflects an increase in within-group income inequality, since income of patient agent 2 (who starts richer) rises over time at the expense of income of agent 1. The decrease in Z t over time reflect instead a decline over time of the income share of the impatient agents, which feeds back into a widening between-group income inequality. For log (m), log (W ), log (Z) and log (A) the estimated autocorrelations are respectively ρ m =0.71, ρ w =0.94, ρ z =0.97, ρ A =0.84, whereas the unconditional standard deviations are: σ m =2.0%, σ w =12.7%, σ z =11.9%, σ A =3.7%. It is interesting to notice how the idiosyncratic shocks which are consistent with given observations on changes in income inequality are estimated to be very persistent: interestingly, using adifferent methodology, Storesletten, Telmer and Yaron (2004) estimate an autocorrelation for idiosyncratic earnings risk using PSID data which ranges from 0.94 to Evidence on the behavior of the model 4.1. Impulse Responses Figure 3 presents the basic workings of the model, by showing the impulse responses of its key variables to a one percent innovation in each of the shocks constructed above. Some observations are in order: 1. Favorable financial shocks (first row of Figure 3) generate an increase in the amount of debt held by impatient agents. By transferring resources from agents with a low steadystate marginal productivity (propensity to consume) to agents with a high marginal productivity (propensity to consume), they also generate a small but positive effect on aggregate output (consumption). 2. Patient agents can effectively share the risk of within-group income shocks by trading one security only; the second row of Figure 3 shows how their consumption is virtually unaffected by the idiosyncratic shock. Not shown in the figure, the housing consumption closely mimics non-durable consumption for unconstrained agents, while it is smoother than non-durable consumption by a factor of about three for constrained agents: clearly, for these agents, who are impatient as well as borrowing constrained, preferences are tilted towards consumption goods. 25 Krueger and Perri (2003b) retrieve the persistence of the idiosyncratic income processes by estimating income processes for classes of individuals in the Consumer Expenditure Survey. Their measure of persistence (see Table 1) is the second largest eigenvalue of a transition matrix and cannot be directly compared to ours. 18

19 An interesting pattern emerges in the dynamics of borrowing in response to a withingroup income shock. Consider patient agents. In response to a positive (negative) realization of y 00 (y 0 ), the income of the richest (poorest) agent, inequality widens. Because the effects of the shock persist over time, the richer agent initially borrows from the poorer one to take advantage of the higher return on capital (d falls). As resources are shifted from agent 1 to 2, debt is reduced. Over time, however, the permanent income hypothesis kicks in. As the effects of the shock die out, agent 2 is willing to finance a permanently higher level of consumption by lending to agent 1. In the new steady state, debt d is permanently higher. Notice that very persistent idiosyncratic shocks coupled with endogenous capital accumulationarethereforekeyelementsingeneratingavolatilityofdebtgrowthwhichis of similar magnitude to what is observed in the data. Were endogenous capital accumulation absent, debt would not have the mean-reverting property described above, and would be less volatile than in the data, although the main results of the paper would be unaltered. 3. Between-group shocks generate slightly larger aggregate effects, because risk sharing is less effective between patient and impatient agents. In response to a positive shock to the productivity of the impatient agent, his borrowing demand rises, and his debt keeps rising over time as he builds up collateral. Because the patient agent is constrained, his consumption and housing demand follow output very closely. There is a small but positive effect on aggregate consumption. 4. Aggregate shocks generate an increase in borrowing, mostly by relaxing the collateral constraints of constrained agents. Because they are perfectly correlated across agents, the effects on borrowing within patient agents are very limited, and work mainly though the variations in the interest rate Model simulations You get what you put into it I start by presenting the findings which are too dependent on the properties of the estimated shocks to be taken as a valid measure of the success of the model in explaining the data. 1. The model captures the behavior of GDP over the last 30 years Recall that the original data series for GDP has been detrended with the linear deterministic trend extracted from the multifactor productivity process. As shown in the top 19

20 portion of Figure 4, the remaining cyclical component moves together with the model counterpart. This is not surprising, since output behavior is mostly driven by the exogenous technology shock, something which is well known in the real business cycle literature (see for instance King and Rebelo, 1999). It is also noteworthy that the model tracks reasonably well the behavior of the real interest rate, as shown in the bottom of Figure The model captures the evolution of income inequality over the last 30 years. Income inequality is mainly driven by the idiosyncratic shocks and by the heterogeneous responses in capital and hours worked from the agents. So long as labor supply is not too elastic and given the small share of fixed capital in production, it is not surprising that the model replicates the behavior of between and within income inequality extremely well. ThetwoleftpanelsofFigure5illustratetheincomedynamicsofthethreeagents as well as the cross-sectional distribution of total income over time The (successful) empirical properties I now report the properties of the model that I have not used as part of my calibration input. Therefore, any similarity of the model and the US data along the dimensions that follow can be considered as overidentifying restriction of my model and evidence of the model success in accounting for the properties of the data. 1. The model successfully captures the trend behavior of debt over GDP. In the data, the debt to GDP ratio rises from 91 percent in 1970 to 144 percent in In the model simulations, it rises from 91 percent to a very similar magnitude: As shown by the top-left panel of Figure 6, according to the model predictions, total debt should have been 148 percent of GDP in Below, I decompose the total variation in debt in its four candidate causes. 2. The model captures the cyclical behavior of debt. The bottom panel of Figure 6 compares year on year debt growth in the model and in the data. In the benchmark calibration, the correlation coefficient between the two series is positive (0.30) and different from zero at conventional significance levels. 27 The 26 In each year, the real interest rate variable was constructing by subtracting the annualized change in the GDP deflator from the Federal Funds rate. 27 Clearly, it would be possible to match the data even better by carefully choosing the model parameters. For instance, one can pick the structural parameters of the model in a way to maximize the correlation between actual debt growth and simulated debt growth: by doing so, the correlation between simulated and actual debt 20