Precautionary saving or denied dissaving

Economic Modelling 28 (2011) 1559–1572

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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Precautionary saving or denied dissaving☆ James Feigenbaum Department of Economics and Finance, Utah State University, United States

a r t i c l e

i n f o

Article history: Accepted 6 February 2011 JEL classification: E21 Keywords: Borrowing constraints Anticipatory saving Dissaving Precautionary saving Quadratic utility Idiosyncratic risk Uncertainty Perturbation theory

a b s t r a c t Precautionary saving in response to uninsurable income risk can in principle explain the stylized fact that aggregate saving increases with the variance of income, but it is controversial how much of the observed variation in incomes is, in fact, unpredictable. Borrowing constraints offer an alternative explanation that does not require consumers to be uncertain about their future income. This paper employs a three-cohort, overlapping generations model with quadratic utility and no capital to show that, if agents are patient enough, heterogeneity alone can account for more than half the decrease in the equilibrium interest rate caused by a borrowing constraint. The possibility of facing a binding borrowing constraint in the future induces saving, and this saving increases with the cross-sectional variation in income. Another channel that pushes down the interest rate is the direct effect caused by currently constrained agents not being allowed to dissave. For patient agents, the two channels have roughly the same impact on the interest rate. © 2011 Elsevier B.V. All rights reserved.

Why do people save? According to the Lifecycle/Permanent-Income Hypothesis, people save in order to smooth their consumption over their lifespan, but there has been much skepticism about whether consumption smoothing alone can generate enough saving to account for the observed accumulation of capital. Another reason to save that has received wide attention is the precautionary motive. Precautionary saving can account for many empirical properties of consumption that cannot be explained by a perfect-foresight lifecycle model, such as the correlation between saving and the cross-sectional variance of income across groups of consumers Carroll and Samwick (1997).1 However, the quantitative significance of precautionary saving depends on how much risk consumers face. Some researchers argue this has been overestimated.2 Two 25-year-old individuals with the

☆ I would like to thank Jim Bullard, Frank Caliendo, Dave DeJong, Douglas Gale, Mark Huggett, Matt Mitchell, Gene Savin, Pedro Silos, and Dan Thornton for offering suggestions and advice, and I would especially like to thank B. Ravikumar, Chuck Whiteman, and Steve Williamson for their comments. Financial support was received from the Koch Foundation and the Federal Reserve Bank of St. Louis. All mistakes are my own. E-mail address: [email protected]. URL: http://huntsman.usu.edu/~jfeigenbaum/. 1 Other such properties include a large marginal propensity to consume Carroll (2000) and the hump in lifecycle profiles of consumption Attanasio et al. (1999), Carroll (1997), Carroll and Summers (1991), Feigenbaum (2009b), Hubbard et al. (1994) Gourinchas and Parker (2002). 2 Cunha et al. (2005) estimate nearly half the cross-sectional variation in income across American households is predictable. Feigenbaum and Li (2010) found that, if more of the data available about individuals in the in the Panel Study of Income Dynamics is taken into account, the variances of forecasting errors regarding future income will be reduced on average by a third compared to previous estimates. 0264-9993/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2011.02.006

same level of education and similar family backgrounds might look identical to an econometrician, but they may have different ambitions and thus have different expectations about their lifetime income streams. If agents have a 50% chance of receiving either $10,000 or $20,000, turning off uncertainty does not mean everyone gets $15,000.3 Rather, it means that half the agents know beforehand they will receive $10,000 and the other half know they will receive $20,000. Since the degree of heterogeneity does not change, this is not a valid measure of uncertainty. For many questions of concern to economists, such as understanding how consumption inequality evolves over the lifecycle, the distinction between uncertainty and heterogeneity may be innocuous since both factors will contribute.4 However, precautionary saving depends specifically on uncertainty, not heterogeneity. If the actual risk faced by individuals is significantly less than what we infer from the heterogeneity of the population, will this substantially change the quantitative predictions of precautionary saving models?5 Not necessarily because most models with uninsurable income risk also incorporate borrowing constraints, and borrowing constraints can generate many of the same properties independent of

3

This is what happens if we preserve uncertainty but allow agents to pool their risk. Guvenen (2007), Heathcote et al. (forthcoming) and Storesletten et al. (2004) consider the impact on consumption inequality of an individual-specific component to income that is realized (though not necessarily observed) before birth. 5 Eeckhoudt et al. (2005) and Feigenbaum (2008) have also shown that, even if income is initially uncertain, if information about the shock is revealed prior to the earning of the income then precautionary effects will be diminished. 4

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uncertainty. The purpose of this paper is to demonstrate in a stylized quantitative model how much of the correlation between saving and the cross-sectional variance of income depends on uncertainty and how much can be explained by heterogeneity alone.6 As in Huggett (1993), this is a model without capital, where bonds are in zero net supply, so it is actually a decrease in the equilibrium interest rate that will emerge as a consequence of an increased demand for saving. The traditional explanation for precautionary saving, established by Leland (1968) and Sandmo (1970), requires the period utility function to have a strictly positive third derivative. However, Aiyagari (1994) and Huggett and Ospina (2001) found that uncertainty increased aggregate saving even when utility was quadratic. Carroll and Kimball (2001) accounted for this by showing that borrowing constraints open another channel for saving that does not depend on the third derivative of utility. Agents who are presently unconstrained but who may be constrained in the future can only smooth consumption up until they hit the constraint. Consequently, these agents behave as though they will only live until they become borrowing constrained. Since they spread their resources over a shorter time horizon, their marginal propensity to save diminishes. If we graph saving as a function of income, the saving function becomes nonlinear as the lower end bends upward. Here we emphasize the fact that this bending of the saving function does not require uncertainty. Indeed, the saving function bends up more for those agents who know for certain they will be constrained than it does for agents who will only be constrained with some less than unit probability. In the aggregate, the bending of the saving function in the case of perfect certainty will be ameliorated by the presence of other agents who know for certain they will not be constrained and maintain linear saving functions. Nevertheless, some of the saving caused by the borrowing constraint is independent of how much information the agents in the model have. In addition, there is a second, more direct channel by which a borrowing constraint increases saving that has nothing to do with precautionary saving. Agents who presently are constrained cannot borrow, so any dissaving they might prefer to partake in is disallowed. While most of the aforementioned literature has focused on how agents save more in response to future events, it turns out that for small discount factors denied dissaving has a bigger effect on the interest rate than precautionary saving, with or without uncertainty. For discount factors close enough to one, the two channels contribute about equally to the decline in the interest rate. The paper is organized as follows. In Section 1, I describe the model. In Section 2, the resulting bond demand functions are specified, and the Carroll-Kimball mechanism is described both with and without uncertainty. In Section 3, I obtain the interest rate in general equilibrium and show how much of the interest-rate response to income heterogeneity persists in the absence of uncertainty. In Section 4, I discuss how these results would generalize to a model with capital. In Section 5, I summarize the results.

1. The model Although one only needs a two-period model to exhibit precautionary saving through the Leland–Sandmo mechanism, three periods are needed to get precautionary saving induced by a borrowing constraint via the Carroll–Kimball mechanism. Two periods are needed so agents have a consumption-saving decision that can be constrained,

6 Nirei (2006) considers the effects of a borrowing constraint and uncertainty on excess sensitivity of consumption. Cordoba (2008) considers the differing effects of a borrowing constraint vs uninsurable risk on the wealth distribution.

and then an additional period before this decision is required so agents can anticipate whether they will be constrained.7 We consider an overlapping generations (OLG) model where the population is constant and the economy is stationary with no aggregate uncertainty. Agents live for three periods, and at any time three cohorts will exist simultaneously: the young (age 0), the middle-aged (age 1), and the old (age 2). Nothing depends on absolute time, but quantities pertaining to a specific agent may depend on his age, which will be indexed with a subscript. Agents have the utility function "

2

# t

U = E ∑ β uðct Þ ;

ð1Þ

t =0

where ct is consumption at age t, β ∈ (0, 1] is the discount factor, and the period utility function u(⋅) is quadratic with bliss point M: 1 2 uðcÞ = − ðM−cÞ : 2

ð2Þ

At age t, the agent receives the possibly stochastic income yt. For both young and old agents, income is y0 = y2 = μ b M. For the middleaged, income is y1 = μ(1 ± δ) with equal probability, where 0 ≤ δ ≪ 1 is small enough so that agents always consume at or below the bliss point in equilibrium.8 Thus δ2μ2 is the cross-sectional variance of income for the middle-aged, and δ is the coefficient of variation. Risk-free bonds are the only available intertemporal asset, and they are in zero net supply with a gross return R N 0. The demand for bonds that pay off at age t is denoted bt. Note that b1 is the bond demand of a young agent while b2 is the bond demand of a middleaged agent. The optimization of Eq. (1) can then be expressed as a system of recursive Bellman equations. For t = 0, 1,     vt ðbt ; yt Þ = max uðct Þ + Et vt+1 bt+1 ; yt+1 bt+1 ;ct

ð3Þ

subject to bt+1 + ct = yt + Rbt b0 = 0

ð4Þ

while v2 ðb2 ; y2 Þ = uðy2 + Rb2 Þ: The heart of the paper is then to examine what happens when, in addition, we impose the borrowing constraint b2 ≥ −Q ;

ð5Þ

where Q ≥ 0, so middle-aged agents are limited in how much they can borrow. Note that there are potentially two frictions in this model: the borrowing constraint and uninsurable risk. Thus we introduce four models to consider what happens when we turn each friction on or 7 An alternative to this OLG economy would be a finite-horizon model where a single cohort lives for three periods and then the economy terminates for everyone. This would simplify the appearance of the market-clearing equations at each period, but solving for the equilibrium would not be any simpler since the interest rate will no longer be stationary. I prefer the OLG approach taken here because the stationary equilibrium allows a more straightforward comparison to infinite-horizon and lifecycle models. 8 This is necessary to maintain the budget constraint as an equality without abandoning free disposal. Under the equality constraint, we have a simple linearquadratic optimization problem with certainty-equivalent policy functions. Without the equality constraint, certainty equivalence generally will not hold even in the absence of borrowing constraints. (See Carroll and Kimball (2001) for more discussion of this point.)

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

Note that the smaller number of state variables and types makes the UM simpler to solve than the CM. This difference compounds exponentially as we increase the number of periods. Consider a more sophisticated model where income follows a first-order Markov process for T periods with S possible income states in each of these periods. If agents know their whole history than there will be ST types of agents for each age group and TST agents in total. If agents only know their past and current incomes, there will be only S types of agents for each age group and ST in total. Thus it would be computationally infeasible to solve the CM for large S or T. In addition to its transparency, this is another reason for doing the ensuing calculation with a three-period model.

FM

uninsurable risk

Frictionless Model

LSM

borrowing constraint CM Certainty Model

Leland-Sandmo Model

UM Uncertainty Model

Fig. 1. A graphical representation of the four models, showing which friction is active in each.

off. In the simplest, frictionless model (FM), each consumer knows his entire income stream with certainty, and there is no borrowing constraint for middle-aged agents. Next we have a Leland–Sandmo model (LSM) in which the consumer faces no borrowing constraints but does not learn his income in middle age until he gets there and he cannot insure against this risk. This is the model that Leland (1968) and Sandmo (1970) studied. In the certainty model (CM), the consumer knows his income stream with certainty as in the FM, but now the consumer faces a borrowing constraint in middle age. Finally, in the uncertainty model (UM), the consumer faces both uninsurable risk and the borrowing constraint. Fig. 1 is a graphical depiction of the four models, showing which friction is active in each.9 In the following, a subscript or superscript c is used to signify quantities in models that do not involve uncertainty. A subscript or superscript u refers to quantities in models that do involve uncertainty. A hat is used to signify quantities that arise in the absence of borrowing constraints whereas no hat is used when there is a borrowing constraint. With respect to their exogenous setups, the CM and the UM differ only in the information available to agents. In the CM (and FM), the realization of y1 occurs at age 0, and agents know their entire future income stream at birth. Consequently, within the CM (and FM), we must for all ages distinguish between two types of agents, those who know they will receive the high income shock and those who know they will receive the low income shock. An age-t agent who will receive middleperiod income y1 = μ(1± δ) will be denoted by the subscript t ±. An equilibrium in this OLG economy consists of consumptions {c0 +, c0 −, c1 +, c1 −, c2 +, c2 −} and bond demands {b1 +, b1 −, b2 +, b2 −} that solve the optimization problem (Eq. (3)) subject to Eq. (5) and satisfy the market-clearing equation b1+ + b1− + b2+ + b2− = 0:

ð6Þ

In contrast, in the UM (and LSM) the agent does not learn y1 until age 1. Young agents are all identical, so income histories can only be distinguished after agents reach middle age. An equilibrium in the UM (and LSM) consists of consumptions {c0, c1 +, c1 −, c2 +, c2 −}and bond demands {b1, b2 +, b2 −} that solve Eq. (3) subject to Eq. (5) and satisfy the market-clearing equation 2b1 + b2+ + b2− = 0:

9

1561

ð7Þ

As we will see in Section 2, with quadratic preferences and no borrowing constraint, there is no macroeconomic distinction between the FM and LSM, so we will focus exclusively on how differences between the FM and UM decompose into a difference between the FM and CM vs a difference between the CM and UM. More generally, there will be an equally valid decomposition into the difference between the FM and LSM and the difference between the LSM and UM. This alternate decomposition is pursued by Xu (1995) and Feigenbaum (2009).

2. Bond demands Before we can compute equilibrium interest rates, we need to determine the demand for bonds in each model. We can then see how the Carroll–Kimball mechanism works. 2.1. Frictionless model In the frictionless model, each household will set the marginal rate of substitution of consumption at ages t and t + 1 equal to the relative price of (t + 1)-consumption in terms of t-consumption. Thus β

M− cˆ1c M− cˆc2 1 =β = : R M− cˆc0 M− cˆc1

ð8Þ

Note that if βR = 1, we get a perfectly smooth consumption stream c c c with cˆ0 = cˆ1 = cˆ2 as predicted by the Lifecycle/Permanent-Income Hypothesis. The bond demand functions that satisfy Eqs. (8) and (4) are c bˆ 1 ðy0 ; y1 Þ =

  μ 1 2 y1 + R −m1 ð 1 + ϕ Þ ½ y −M −ϕ 0 R 1 + ϕ + ϕ2

ð9Þ

and c bˆ 2 ðx1 Þ =

  1 μ−M ½x1 −M−ϕ ; 1+ϕ R

ð10Þ

where xt = yt + Rbt is cash on hand (as defined by Deaton (1991)), y2 is fixed at μ, and mt = ∑ 2s = t Rt − sM is the present value of the stream of bliss points. The quantity ϕ = βR

2

ð11Þ

is the inverse of the marginal propensity to save for an agent with quadratic preferences and an infinite life span. All marginal propensities to save and consume in the various models of this paper will be functions of ϕ alone. Note that the young bond demand (Eq. (9)) is a linear function of then cash on hand x0 = y0 and the middle-aged bond demand (Eq. (10)) is also a linear function of cash on hand x1. An important macroeconomic variable that will be of interest to us in the next section is the aggregate bond demand of young agents (“young aggregate bond demand”). Viewed as a function of initial income, which is the same for all agents, this is defined as B1 ðy0 Þ ≡

1 ½b ðy ; μ ð1−δÞÞ + b1 ð y0 ; μ ð1 + δÞÞ: 2 1 0

ð12Þ

For the FM, this is c Bˆ 1 ðy0 Þ =

  μ 1 2 μ + R −m1 : ð1 + ϕÞ½ y0 −M −ϕ 2 R 1+ϕ+ϕ

ð13Þ

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J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

Note that if we set R = β− 1 and y0 and y1 to their possible values in the model, we find ∓βδμ 1 + β + β2

c bˆ 1 =

ð14Þ

middle-aged agents, Eq. (10) is followed unless it violates the borrowing constraint:

c

b2 ðx1 Þ =

8 > < > :

 1 ϕ x1 −M− ðμ−MÞ 1+ϕ R

x1 ≥ x1

−Q

x1 b x1

and c bˆ 2 + =

ð15Þ 

x1 = M +

These bond demands aggregate to the net supply of zero, and, indeed, Bc1(μ) = 0. Consequently, R = β− 1 gives an equilibrium. 2.2. Leland–Sandmo model When we introduce uncertainty to the frictionless model to obtain the Leland-Sandmo model, the Euler equations (Eq. (8)) become    u  u u  M− cˆt = βREt M− cˆt+1 = βR M−Et cˆt+1 :

ð16Þ

Given that consumption is a linear function of incomes, Eq. (16) u implies that cˆt depends only on the expected value of future incomes, a property known as certainty equivalence. We obtain the bond demands simply by replacing future incomes in Eqs. (9) and (10) by their expectation:

1 ð1 + ϕÞ½ y0 −M −ϕ 1 + ϕ + ϕ2

 2 μ +

μ R −m1

R

ð17Þ

ð18Þ

ð23Þ with a threshold value of initial income 

y0 ð y1 Þ = m0 −

u bˆ 2 =

1 1 + β−1

μ R

−

 1 + ϕ + ϕ2 M−μ +Q : R R

ð24Þ

"

# 1 1 ϕ2 1 + R ð 1 + ϕ Þ ð μ−M Þ− μ + δμ−m 1 R 2 1 + ϕ + ϕ2 R

c

B1 ðμ Þ =

ð25Þ

 1 1 ϕ + μ−M− ðμ ð1−δÞ + Q−M Þ : 21 + ϕ R

ð19Þ

4.4. Uncertainty model

ð20Þ

In the uncertainty model, all uncertainty will be resolved by middle age, so the bond demand of middle-aged agents is still given by Eq. (21):

and   βδμ −1 μ−M = ½ μ ð1  δÞ−M−β : 1+β β−1

y1 + R

In equilibrium, y0* (μ(1 + δ)) b μ b y0* (μ(1 − δ)), so the aggregate bond demand of young agents will be

Since there is only one type of young agent, young aggregate bond demand in the LSM is the same as the individual bond demand. Notice that Eqs. (17) and (13) are identical, as are Eqs. (10) and (18). If R = β− 1, u u c bˆ 1 = Bˆ 1 ðμ Þ = 0 = Bˆ 1 ðμ Þ

ð22Þ

8 ( ) i > 1 ϕ2 h μ >  > y ð 1+ ϕ Þ ð y −M Þ− + −m y0 ≥ y0 ðy1 Þ > 0 1 <1+ ϕ+ϕ2 R 1 R c b1 ðy0 ; y1 Þ=  > > 1 ϕ >  > y0 by0 ðy1 Þ y0 −M− ðy1 +Q −MÞ : 1+ ϕ R



  1 μ−M c = ½x1 −M−ϕ = bˆ 2 ðx1 Þ: 1+ϕ R



ϕ ðμ−MÞ−ð1 + ϕÞQ : R

Likewise, the bond demand function for young agents, derived in Appendix A.1, follows Eq. (9) unless it would lead to a violation of the borrowing constraint:

and u bˆ 2 ðx1 Þ

ð21Þ

where the threshold cash-on-hand can be shown to equal

2

β δμ : 1 + β + β2

u bˆ 1 ðy0 Þ =

;

−1

also gives an equilibrium in the LSM with the same Thus R = β aggregate bond demands for the young and the old. I refer to this model as the Leland–Sandmo model since this is the type of model that Leland (1968) and Sandmo (1970) studied when they first developed the theory of precautionary saving. However, their precautionary saving mechanism feeds off a positive third derivative of the utility function. With quadratic preferences, this mechanism plays no role, and there is no precautionary saving in this model. From a macroeconomic perspective, behavior in the FM and the LSM is identical, so in the sequel I will typically refer only to one frictionless model that actually encompasses both of these models. Where the distinction between the two models is irrelevant, I will refer to endogenous quantities from either model by just a hat, disregarding the c or u sub/superscripts.

u b2 ðx1 Þ

8 > < =

> :

 1 ϕ x1 −M− ðμ−MÞ 1+ϕ R

x1 ≥ x1

−Q

x1 b x1



:

As in the LSM, young aggregate bond demand is simply the bond demand of a young agent since there is only one type of young agent. In Appendix A.2, we show that this bond demand at y0 = μ is u

u

b1 ðμ Þ = B1 ðμ Þ =

f

1 2ð1 + ϕÞðy0 −MÞ 2 + 3ϕ + 2ϕ2

g

ϕ − ½ϕðμ−M Þð1 + RÞ=R + ð1 + ϕÞðμ + Q −M Þ−δμ  : R

ð26Þ

4.5. The Carroll–Kimball mechanism 2.3. The certainty model In the certainty model, we need to modify the bond demands Eqs. (9) and (10) to account for the borrowing constraint. In both cases, this modification makes the bond demands piecewise linear. For

How do the borrowing constraint and uninsurable risk affect the aggregate demand for bonds of young agents? To understand this, it is instructive to consider first what happens in a partial equilibrium where R = β− 1. The effects of the two frictions can be discerned from

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

Fig. 2, which plots young aggregate bond demand as a function of y0 for the FM, CM, and UM.10 In the FM, the aggregate bond demand is linear, both with and without certainty, since quadratic preferences confer certainty equivalence. Since we have assumed R = β− 1, the Lifecycle/Permanent-Income Hypothesis holds exactly so consumption will be perfectly smoothed, at least in expectation, over the lifecycle. Under this hypothesis, a FM household that will live for T more periods will have marginal propensity to save (MPS) m = 1−

1 : ∑Ts= 0 βs

1563

B1 (young aggregate bond demand)

precautionary saving

UM

anticipatory saving

ð27Þ

CM

For T = 2 as in this model, the FM MPS will be mhigh =

β + β2 : 1 + β + β2

FM ð28Þ

In the CM and UM, if y0 is large enough that the household need not worry about being borrowing constrained regardless of y1, young aggregate bond demand will be the same as in the FM. Thus all three graphs coincide for high incomes with MPS mhigh. Getting borrowing constrained at age 1 is analogous to dying after age 1 since decisions at ages 0 and 1 are unaffected by y2. If a CM or UM household is poor enough that it knows it will be borrowing constrained regardless of its middle-period income, its MPS will be mlow =

β ; 1+β

ð29Þ

which is smaller than mhigh since the household is saving to smooth its consumption over a smaller number of periods. For the UM, previously studied by Carroll and Kimball (2001), agents are only differentiated in middle age when they learn their income shock. Kinks occur in the bond demand for young agents at income thresholds where the probability of being borrowing constrained in middle age changes. For intermediate incomes, such as y0 = μ, a UM household will be constrained with 50% probability. Consequently, its MPS will be an average, specifically a harmonic mean, of mhigh and mlow: UM

mint =

β + β2 ; 1 + 32 β + β2

ð30Þ

as can be seen by differentiating the young bond demand in Appendix 2. The innovation of this paper is to also consider the aggregate effects of a borrowing constraint in a model without uncertainty, for the Carroll–Kimball saving mechanism is still present in the CM. Each individual has a bond demand with a single kink where he becomes borrowing constrained. Above the kink he has MPS mhigh while below the kink he has MPS mlow. For intermediate incomes, where half the population has one MPS and half has the other, the CM MPS is the arithmetic mean of mhigh and mlow: CM mint

! 2 1 β β+β : = + 2 1+β 1 + β + β2

Fig. 2. Comparison of young aggregate bond demand from the certainty model (CM) with the bond demand from the uncertainty model (UM) and the corresponding unconstrained (FM) aggregate bond demand for a partial equilibrium with βR = 1. The scale of differences is exaggerated for clarity.

The relative location of the upper kinks can be understood as follows. In the CM, agents know for certain if they will get the low shock. Holding y0 constant, they save more in anticipation of this shock than they would in the UM, where they are uncertain if they will get the shock. Consequently, agents in the CM who get the low shock can avoid the borrowing constraint at lower y0 than their counterparts in the UM. Precautionary saving is represented by the difference between the UM demand and the CM demand. We refer to the complementary difference between the CM and FM curves as “anticipatory saving”.11 Both differences are proportional to the coefficient of variation δ. In general equilibrium, the quantity of bonds that can be purchased will be fixed by the amount of borrowing. Thus increases in the demand for bonds will lead to higher bond prices and, therefore, lower interest rates, rather than a greater quantity of bonds in the market. In the next section, we will see how the equilibrium interest rate depends on δ in each of the three models. 3. The response of the interest rate to a borrowing constraint There is no exact analytic solution for the equilibrium interest rate even in this relatively simple model, but we can use perturbation theory to work out an approximate analytic expression by computing the first-order Taylor expansion of the interest rate with respect to the coefficient of variation δ for middle period income y1. Here, I consider only the equilibrium to which the R = β− 1 equilibrium moves continuously as I increase δ from zero.12 3.1. Certainty model

ð31Þ

UM Since mCM int and mint are both obtained by averaging mhigh and mlow, differing only in the method of averaging, they cannot differ by more UM than 1 part in 48, although mCM int is always slightly greater than mint . This, together with the fact that the upper kink is higher in the UM than in the CM, accounts for why the UM aggregate bond demand is always weakly greater than the CM aggregate bond demand.

10

y0 (initial income)

In general equilibrium, we will fix y0 = μ, but for now we leave it a free variable.

First let us ascertain what circumstances the borrowing constraint will bind under since this will determine where the CM diverges from

11 Although he did not use the terminology of anticipatory saving, Flodén (2008) has also considered this decomposition but in an infinite-horizon setting. 12 Wealth effects can lead to multiple equilibria, which this perturbation approach cannot reveal since it explores only the local behavior of one solution to the marketclearing equation, in this case R = β− 1. However, since my primary focus is the interaction between the borrowing constraint and uncertainty, I disregard the possibility of these wealth effects here.

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J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

the FM. Agents who receive the low income shock will have the bond holding c bˆ 2−

in the absence of the borrowing constraint. Therefore, in the CM the borrowing constraint will not bind for low-income agents iff 2

β δμ ≤ Q; 1 + β + β2

ð32Þ

in which case there is no difference between the CM and FM in equilibrium. On the other hand, if β2 δμ N Q; 1 + β + β2

ð33Þ

the borrowing constraint will bind for low-income agents, and the bond demands in the market-clearing Eq. (6) will no longer be described by Eqs. (14)–(15). Instead, we have to solve the marketclearing equation c

c

b1+ ðRÞ + b1− ðRÞ + b2+ ðRÞ = Q;

ð34Þ

for which R = β− 1 is not a solution. We proceed by assuming the solution R(δ) to Eq. (34) is differentiable in a neighborhood of δ = 0, in which case R(δ) should be an approximately linear function of δ in this neighborhood. We can then write the gross interest rate as   −1 2 R = β ð1 + δr1 Þ + O δ :

ð35Þ

The derivative R′(0) is proportional to r 1, a constant to be determined.13 r1 can be called the first-order correction to the interest rate (since δβ− 1r1 “corrects” the estimate R ≈ β− 1 for the presence of the borrowing constraint). Once we have calculated r1, we can approximate the gross interest rate by R1 = β

−1

ð1 + δr1 Þ:

In the following, it is helpful to define (

) 2 β μ −q; 0 ; τc = max 1 + β + β2

ð36Þ

where q = Q = δ:

ð37Þ

We can interpret τc as a measure of how strongly the borrowing constraint binds at R = β− 1. The general solution for the equilibrium interest rate, derived in Appendix B.1, is described by the following proposition along with the resulting equilibrium bond demands. Proposition 1. In the CM, the equilibrium interest rate is   3   ð1 + 2βÞ 1 + β + β2 δτ 2 c 5 1− +O δ ; 2 3 M−μ 4β + 7β + 4β

2 Rc = β

13

  ð1 + 2βÞ2 2 δτc + O δ ; 2 4 + 7β + 4β

ð39Þ

2   2β + 2β−1 c c 2 δτc + O δ ; b1− = bˆ 1− + 2 4 + 7β + 4β

ð40Þ

  ð1 + 2βÞð2 + βÞ c c 2 b2+ = bˆ 2 + − δτc + O δ ; 2 4 + 7β + 4β

ð41Þ

c c b2− = −Q = bˆ 2− + δτc :

ð42Þ

c c b1+ = bˆ 1+ −

β2 δμ =− 1 + β + β2

c

and the equilibrium bond demands are

−1 4

r1 should not be confused with the net interest rate R − 1.

ð38Þ

Eq. (38) shows that the change in the interest rate is proportional both to the factor τc that measures the tightness of the borrowing constraint and to the coefficient of variation of middle-period income. 14 Thus if the distribution of income becomes more heterogeneous, the interest rate will fall more, and the equilibrium bond demands will diverge more from their frictionless equilibrium values. For explicative purposes, let us focus on the special case where β = 1 and Q = 0. In that case, Proposition 1 implies Rc = 1−

  1 δμ 2 +O δ ; 5 ðM−μ Þ

ð43Þ

which will be less than 1 since the bliss point M N μ. The equilibrium bond demands respond to the borrowing friction as follows:   δμ c c 2 b1+ = bˆ 1+ − +O δ 3

ð44Þ

  δμ c c 2 +O δ b1− = bˆ 1− + 15

ð45Þ

  δμ c c 2 b2+ = bˆ 2+ − +O δ 15

ð46Þ

  δμ c c 2 b2− = 0 = bˆ 2− + +O δ : 3

ð47Þ

Note that low-income agents of both young and middle age increase their savings from the frictionless equilibrium while high-income agents decrease their savings. However, the low-income middle-aged agents, i.e. the presently constrained agents, increase their saving (by not borrowing) five times as much as these same agents when they are young. Young, low-income agents are solely responsible for saving in anticipation of the borrowing constraint in the certainty model. That is to say, these are the agents who engage in “precautionary saving,” although there is nothing actually precautionary about this saving since these agents know for certain they will be borrowing constrained. In fact, Eq. (40) shows that the sign of the change in saving of the low-income young agents is ambiguous. For sufficiently large discount factors, it will be positive, but if consumers are impatient enough the saving of these presumed precautionary savers will actually decrease. The relative magnitude of the change in saving by young, lowincome agents gives a misleading picture of the size of Carroll– Kimball saving, which actually does contribute to the decrease in the interest rate. To see this, let us consider a hypothetical certainty model (CMh) in which young agents do not know that they cannot borrow in middle age so they cannot save in anticipation of being constrained. Quantities related to the hypothetical model are designated by an

14

The factor (M − μ)− 1 is the absolute risk aversion evaluated at c = μ.

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

1565

additional h subscript or superscript. In this case, the bond demand of low-income young agents will still be given by Eq. (9). Then the market-clearing condition for the equilibrium interest rate Rch is       c c ch ch c ch Q = b1+ R + bˆ 1− R + b2+ R : If we define rch 1 as in (35) for the CMh, we find for the case β = 1 and Q = 0 that r1ch 5 = : r1c 9 If we turn off Carroll–Kimball saving, the direct effect of the borrowing constraint will still decrease the interest rate by slightly more than half the total effect of heterogeneity. Though the direct channel is marginally favored, the two channels contribute roughly the same amount to the decrease in the interest rate. To see this graphically, let us consider the supply of bonds issued by low-income, middle-aged (LMA) agents: c

c

BS ðRÞ = −b2− ðRÞ; which is zero unless the interest rate is so high that no one wants to borrow. Since all bonds are equivalent, the demand for LMA bonds is just the sum of the demands for saving of all other types of agents: c

c

c

Fig. 3. Supply (upward-sloping) and demand (downward-sloping) curves for bonds issued by low-income, middle-aged (LMA) agents in the certainty model (CM), the frictionless model (FM), and the hypothetical certainty model (CMh) with β = 1, M = 2, μ = 1, and δ = 0.1. Note that the price of a bond is R− 1.

c

BD ðRÞ = b1+ ðRÞ + b1− ðRÞ + b2+ ðRÞ: We can analogously define these curves for the frictionless model

This is a decreasing function equal to 1 when β = 0, and always greater than one half.

c c Bˆ s ðRÞ = − bˆ 2− ðRÞ c c c c Bˆ D ðRÞ = bˆ 1+ ðRÞ + bˆ 1− ðRÞ + bˆ 2+ ðRÞ

and the CMh c Bch S ðRÞ = −b2− ðRÞ c c ˆc Bch D ðRÞ = b1+ ðRÞ + b1− ðRÞ + b2+ ðRÞ:

These are shown for the case where M = 2μ and δ = 0.1 in Fig. 3. Note that the price of a LMA bond is R− 1, and we plot the curves as a function of this so the supply curves have their usual positive slope and the demand curves have their usual negative slope. The supply and demand curves for the frictionless model intersect 1 = β = 1 but with a positive value for the quantity, meaning that at R− f middle-aged agents who get the low-income shock do borrow in the frictionless model. If we shut down their borrowing, the supply curve is truncated into a perfectly inelastic vertical curve at sufficiently high bond prices. If we ignore Carroll–Kimball saving for the moment, then the demand curve remains the same as in the frictionless model except for a kink at very high bond prices where even the highincome, middle-aged agents would like to borrow. The CMh (=CM) supply curve intersects the CMh demand curve and the coincident FM 1 −1 = 1.011 N β, so denied dissaving demand curve at R− ch = (1 − δ/9) alone causes bond prices to increase by 1.1%. If we include the Carroll– Kimball saving, the demand curve tilts up, intersecting the CM supply 1 = (1 − δ/5)− 1 = 1.020. curve at the even higher R− c The next result, proven in Appendix B.2, shows in general how much of the response of the interest rate to heterogeneity can be attributed to the direct effect of the borrowing constraint. Proposition 2. To first order in the coefficient of variation of middleperiod income, the fraction of the interest rate decrease that would occur if young agents behave as in the frictionless model is r1ch 4 + 7β + 4β2 : c = r1 ð1 + 2βÞð4 + 5βÞ

Since these results are based on an approximation, it is important to ask how accurate this approximation is. In fact, first-order perturbation theory gives an excellent approximation to the interest rate for values of δ ≤ 0.25. For example, let us calibrate the model with β = 0.4, which is appropriate if we imagine a period being equal to 20 years and households have an annual discount factor close to 0.96. If we set the other parameters to M/μ = 2.0, q = 0.0, and δ = 0.25, we get Rc1 = 2.43952 as computed via Eq. (38) while the solution to Eq. (34) is Rc = 2.43725, which differ by less than one part in a thousand.15 (These correspond to net annual rates of 4.55991% and 4.55505% respectively.) For a larger coefficient of variation of 0.5, we still get less than 1% error with Rc1 = 2.37903 and Rc = 2.36935.16 (These are 4.42873% and 4.0846% respectively.) For both values of δ, the first-order approximation is a clear improvement over the zerothorder approximation Rc0 = Rf = 2.5.

3.2. Uncertainty model Now let us see how things change when young agents do not know their middle-period income. From Eq. (20), we know that if R = β− 1, middle-aged agents who receive the low-income shock would optimally borrow, having the negative bond demand

u

b2− = −

β δμ: 1+β

15 This is an extreme example since people will be satiated with consumption of twice their mean income. If we increase M relative to μ so the bliss point is further out of reach, the perturbation approximation will do even better. 16 In a typical model where income multiplicatively decomposes into permanent and temporary shocks, Gourinchas and Parker (2002) estimate the variance of the log of the permanent shock to be 0.0212 and the variance of the log of the temporary shock to be 0.0440. With these parameters, the coefficient of variation of the present value of income over twenty years, assuming an interest rate of 4%, will be 0.37.

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J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

So in the UM, the borrowing constraint binds if Qb

β δμ: 1+β

ð48Þ

agents who do not get the low income shock. For these agents, Carroll–Kimball saving leads to an accumulation of precautionary wealth (Huggett and Vidon (2002)) that leads to further saving in middle age. Thus when we shut down Carroll–Kimball saving, we have to consider this effect too, so the market-clearing equation for the UMh equilibrium rate Ruh becomes

ð49Þ

    u u uh uh + bˆ 2+ R : Q = 2 bˆ 1 R

It is then natural to introduce

 β τu = max μ−q; 0 ; 1+β

which measures the tightness of the borrowing constraint in the UM. As was the case for the CM, if bu2 − = − Q, the market-clearing equation u

u

2b1 + b2+ = Q is a high-degree polynomial, so we approximate the solution in a neighborhood of δ = 0 as a linear function R1(δ) with first-order correction ru1 as in Eq. (35). The equilibrium interest rate for the general model is solved for in Appendix B.3. The properties of the equilibrium are summarized in the following proposition.

−1

   2β + 1 τ u 2 +O δ : 1− 4β M−μ

ð50Þ

u

    δτu δτ u 2 2 + O δ = bˆ 1 − u + O δ 4 4

u

u bˆ 2+

b2+ =

  δτ 2 − u +O δ 2

u u b2− = −Q = bˆ 2− + δτu :

ð51Þ

ð52Þ

ð53Þ

  3 δμ 2 +O δ 8 M−μ

u u b2+ = bˆ 2+

  δμ 2 − +O δ 4

u b2−

u bˆ 2+

=0=

  δμ 2 + +O δ 2

u

u

u

u

BD ðRÞ = 2b1 ðRÞ + b2

+

ðRÞ:

ˆ uS ðRÞ = − bˆ u2− ðRÞ B

u

Note that the demand curve for the UMh is Bˆ D ðRÞ while its supply curve is BuS (R). These curves are shown in Fig. 4 for the same parameters as in Fig. 3. The effects are somewhat more pronounced but otherwise are similar to what we found in the CM. Denied dissaving increases the 1 equilibrium bond price from the frictionless value of β = 1 to R− uh = (1 − δ/6)− 1 = 1.017. Accounting for the increase in demand caused by Carroll–Kimball saving increases the equilibrium bond price even 1 −1 = 1.039. further to R− u = (1 − 3δ/8) The following general result regarding the importance of Carroll– Kimball saving is proven in Appendix B.4. Proposition 4. To first order in the coefficient of variation of middleperiod income, the fraction of the interest rate decrease that would occur if young agents behave as in the frictionless model is

for the equilibrium interest rate, and   δμ u u 2 b1 = bˆ 1 − +O δ 8

u

BS ðRÞ = −b2− ðRÞ;

ˆ uD ðRÞ = 2 bˆ u1 ðRÞ + bˆ u2+ ðRÞ: B

Qualitatively, the results for the UM are similar to the CM. The decrease in the interest rate is again proportional to the tightness factor τu and the coefficient of variation of middle-period income. Focusing again on the case β = 1 and Q = 0, we have Ru = 1−

Thus denied dissaving can account for slightly less than half of the decrease in the interest rate due to the borrowing constraint. Again, this can be seen graphically if we define the supply of LMA bonds as

Likewise, we can define these functions for the Leland–Sandmo model:

The equilibrium bond demands are b1 = −

r1uh 18 4 = = r1u 63 9

which is again zero unless the interest rate is high enough that no one wants to borrow. Meanwhile, the demand for LMA bonds is

Proposition 3. In the UM, the equilibrium interest rate is R=β

If we define ruh according to Eq. (35) for the UMh, we find

ð54Þ ð55Þ ð56Þ

for the equilibrium bond demands. Notice that young agents, all of whom are the potential Carroll–Kimball savers in this model, always decrease their saving from the FM equilibrium, even when they are most patient. Indeed, Eq. (51) shows this is true in general. This might lead one to question whether the young agents engage in Carroll– Kimball saving. Again we can assess this by considering a hypothetical uncertainty model (UMh) in which young agents do not know the borrowing constraint will be imposed in middle age, so they follow the FM bond demand function (Eq. (9)). Unlike in the CM, we also have to consider the impact of this bond demand on the bond demand of middle-aged

uh

2

r1 1+β+β : =4 r1u ð1 + 2βÞð4 + 5βÞ This is a decreasing function equal to 1 when β = 0 with a minimum of 4/9 at β = 1. For β close to 1, we see that Carroll–Kimball saving actually has slightly more impact on the interest rate than denied dissaving. However, if consumers are less patient, denied dissaving dominates. u For β ≈ (0.96)20 = 0.4, ruh 1 /r1 = 0.578, so at a more standard calibration Carroll–Kimball saving accounts for less than half of the decrease in the interest rate. Regarding the accuracy of the approximation, the perturbative calculation for the UM is an even better approximation for δ ≤ 0.25 than (43) was for the CM. For example, if β = 0.4, M/μ = 2.0, q = 0.0, and δ = 0.25, Ru1 = 2.29911 while Ru = 2.29929, a difference of less than one part in ten thousand. (These correspond to annual rates of 4.25046% and 4.25071% respectively.) Even for δ = 0.9, the first-order

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

Fig. 4. Supply (upward-sloping) and demand (downward-sloping) curves for bonds issued by low-income, middle-aged (LMA) agents in the uncertainty model (UM), the Leland–Sandmo model (LSM), and the hypothetical uncertainty model (UMh) with β = 1, M = 2, μ = 1, and δ = 0.1. Note that the price of a bond is R− 1.

correction is accurate to one part in one thousand with Ru1 = 1.77679 and Ru = 1.77873. (These are respectively 2.91573% and 2.92136% in annual terms.) 3.3. Anticipatory versus uncertainty effects Now let us compare the total effect of the borrowing constraint in the certainty and uncertainty models. To see this graphically, let us define B(R) to be the aggregate demand for saving of all agents in a model—i.e. BðRÞ = b1+ ðRÞ + b1− ðRÞ + b2+ ðRÞ + b2− ðRÞ or BðRÞ = 2b1 ðRÞ + b2 + ðRÞ + b2− ðRÞ; depending on whether young agents know their middle-period income or not. These are shown for the case where β = 1, M = 2, μ = 1, δ = 0.1, and Q = 1 in Fig. 5. Since bonds are in zero net supply, the corresponding aggregate supply curve is just the y-axis. The frictionless model intersects this 1 = β = 1. If consumers know their entire income stream, axis at R− f the aggregate bond demand increases to the CM curve, which 1 = 1.02. Uncertainty then increases the intersects the y-axis at R− c bond demand further to the UM curve, which intersects the y-axis at 1 R− u = 1.039. Thus uncertainty contributes slightly less than one half to the effect of the borrowing constraint. More generally we get the following result.

1567

Fig. 5. Aggregate demand curve for bonds in the FM, CM, and UM for β = 1, μ = 1, M = 2, Q = 0, and δ = 0.1. The y-axis is the corresponding aggregate supply curve. Note that R− 1 is the price of a bond.

which is an increasing function of β with a maximum greater than one half. For the case Q = 0, the ratio rc1/ru1 is plotted as a function of β in Fig. 6. For agents who are sufficiently patient, heterogeneity alone can account for half of the response of the interest rate to the borrowing constraint. For a more common value of β = (.96)20 = 0.442, heterogeneity and the borrowing constraint can account for 0.324 or about a third of the reduction in the interest rate that the borrowing constraint and uncertainty together produce. Thus uncertainty is quantitatively important but not essential to account for a negative relationship between the equilibrium interest rate and the coefficient of variation of income. 4. Introducing capital Thus far we have focused on a stylized model without capital and production because this allowed us to obtain analytic results regarding the contributions of Carroll–Kimball saving, both anticipatory and precautionary, and denied dissaving to the interest rate. However, to study the effects of these three saving mechanisms on

Proposition 5. The ratio of the first-order correction to the interest rate of the CM to the UM is   2 c 4 1+β+β τ r1 c = ∈ ½0; 1: r1u 4 + 7β + 4β2 τ u For the special case when Q = 0, this reduces to r1c 4βð1 + βÞ = ; r1u 4 + 7β + 4β 2

Fig. 6. Ratio of rc1 to ru1 as a function of β. This is the fraction of the interest-rate reduction caused by the borrowing constraint which can be accounted for by anticipatory effects alone.

1568

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

aggregate saving, it is necessary to introduce capital so it is actually possible to increase saving in the aggregate. Even in this stylized three-period environment the introduction of capital would preclude us from getting analytic results. So a quantitative study of the effects of capital might as well be done whole hog with a more plausible utility function and a finer time structure, as in Huggett (1996) or Feigenbaum (2009). Nevertheless, with regards to the robustness of the correlation between the interest rate (or aggregate saving) and the coefficient of variation of income, we can see graphically that our main results should still qualitatively hold when capital is introduced. Since capital and bonds will be perfect substitutes in this world with no aggregate uncertainty, the market-clearing condition will be modified so the capital stock equals the aggregate demand for saving. With a Cobb– Douglas production function Y = AK

α

ð57Þ

and a depreciation rate of ρ N 0 for capital, the gross return on capital will be R = α AK

α−1

+ 1−ρ:

ð58Þ

Inverting this equation, we obtain the demand for capital by firms as a function of R: KD ðRÞ =



1 R−1 + ρ α−1 ; αA

In equilibrium, this must equal the supply of capital, which will equal the demand for aggregate saving since the supply of bonds is zero: KS ðRÞ = b1+ ðRÞ + b1− ðRÞ + b2+ ðRÞ + b2− ðRÞ

ð59Þ

for the frictionless and certainty models, and KS ðRÞ = 2b1 ðRÞ + b2+ ðRÞ + b2− ðRÞ

ð60Þ

for the LSM and uncertainty models. Our results without capital largely go through after introducing capital because the supply of capital–the analog of the demand for bonds in the previous sections–is completely unchanged. The demand for capital–the analog of the supply of bonds in the previous sections, which was simply the y-axis–has become nonlinear. An example of the supply and demand for capital is plotted in Fig. 7 for each model. As long as the equilibrium bond price does not get so large that the nonlinearity in the supply curve becomes especially pronounced, we still get a roughly 50–50 breakdown between the change in the bond price from the frictionless model to the certainty model and then from the certainty model to the uncertainty model.17 This is only one example, but we see that the essential message of the model without capital can still carry through to the model with capital. If consumers are sufficiently patient, and there is a borrowing constraint that binds for some agents, then the interest rate will depend on the coefficient of variation of income even if consumers can predict their future income.18 5. Concluding remarks This paper investigates the extent to which the relationship between the equilibrium interest rate and the cross-sectional variance ˆ = 1:0129, Rc = 1.0033, and Ru = 0.9923. Thus for these parameters, In the graph, R 47% of the response to the interest rate does not depend on uncertainty. 18 Feigenbaum (2009) shows that denied dissaving and anticipatory saving account for most of the “aggregate precautionary saving” in a fully calibrated model with CRRA utility. In that model the effects of introducing a borrowing constraint are essentially independent of risk aversion. 17

Fig. 7. Supply and demand curve for capital in the FM, CM, and UM for β = 1, μ = 1, M = 2, Q = 0, δ = 0.1, A = 0.01, α = 1/3, and ρ = 0.1. Note that R− 1 is the price of a bond.

of income depends on uncertainty in a model with heterogeneous agents and a borrowing constraint. If agents are sufficiently patient, we find that if agents are fully informed about their future income streams then the response of the interest rate to the borrowing constraint will be cut by only about half from what would occur if all variation in income was unpredictable. We also investigate how much of the response of the interest rate to the borrowing constraint can be attributed to the direct effect of constrained agents not being allowed to dissave and how much can be attributed to the Carroll–Kimball mechanism that unconstrained agents who know they may be borrowing constrained in the future save more. Again, for sufficiently patient agents, this decomposition is roughly fifty-fifty. Appendix A. Derivation of first-period bond demands A.1. Certainty model First, we compute the value function at age 1. Suppose that x1 ≥ x1*. Then we are in the situation that would arise in the unconstrained model. On the other hand if x1 b x1*, then b2 = − Q and c1 = x1 + Q. The value function is then 1 β 2 2 v1 = − ½M−ðx1 + Q Þ − ½M−ðμ−RQ Þ : 2 2 Combining these two cases, we obtain

v1 ðx1 Þ =

8 > > <

−

 1 ϕ h μ i2 m1 − x1 + 21 + ϕ R

> > : − 1 ½M−ðx + Q Þ2 − β ½M−ðμ−RQ Þ2 1 2 2

x1 ≥ x1

:

ð61Þ



x1 b x1

Let us now consider the Bellman equation at age 0. There are two cases to consider, differentiated by whether y1 + Rb1 is greater or less than x*1. Case I. y1 + Rb1 ≥ x*1. Inserting the unconstrained branch of the value function (61) into the Bellman Eq. (3) at age 0, the Lagrangian becomes  1 β ϕ h μ i2 2 : m1 − y1 + Rb1 + L0 = − ½M−ðy0 −b1 Þ − 2 21 + ϕ R

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

The first-order condition in b1 is M−ðy0 −b1 Þ = βR

 ϕ h μ i m1 − y1 + Rb1 + : 1+ϕ R

Solving for the bond demand, I b1 ðy0 Þ

1 = 1 + ϕ + ϕ2

) 2 i ϕ h μ y + −m1 ; ð1 + ϕÞðy0 −M Þ− R 1 R

) y  ϕ2 μ  ð1 + ϕÞ 1 + y0 −M − −m1 R R R   c  1+ϕ h μ =R y0 + 1 −m0 − −m1 : R R 1 + ϕ + ϕ2 =

R 1 + ϕ + ϕ2

(

However, this solution is only valid if x1 ≥ x1*. Since xI1 is an c increasing function of y0, there will exist a y 0 such that this branch of c c the solution will hold iff y0 ≥ y 0 . This y 0 will satisfy R

1+ϕ 1 + ϕ + ϕ2



  μ c  −m0 − −m1 = x1 : y0 + R R hc1

c

Solving for y 0 , we obtain c

y 0 = m0 −

Since b1(y0) must be strictly increasing and continuous, we must c !c have y 0 = y 0 , and one can verify that  hc 1 + ϕ + ϕ2 M−μ c c !c y 0 = y 0 = y0 ðy1 Þ = m0 − 1 − +Q : R R R

(

which is equivalent to Eq. (9), and the cash-on-hand next period will be x1I ðy0 Þ = y1 + Rb1

1569

i hc1 1 + ϕ + ϕ2 1 h  μ + x1 + −m1 : R 1+ϕ R R

Case II. y1 + Rb1 b x1*. Now, using the borrowing constrained branch of the value function (Eq. (61)), the Lagrangian is 1 β 2 2 L0 = − ½M−ðy0 −b1 Þ − ½M−ðy1 + Rb1 + Q Þ ; 2 2 ignoring a constant term that does not affect the optimization. The first-order condition with respect to b1 is

A.2. Uncertainty model Since nothing is different between the CM and UM after the uncertainty is resolved at t = 1, Eq. (61) correctly describes the value function of an agent in the UM at age 1. However when the veil of uncertainty remains for a young agent, the Bellman equation then becomes

 β v0 = max uðy0 −b1 Þ + ½v1 ðμ ð1−δÞ + Rb1 Þ + v1 ðμ ð1 + δÞ + Rb1 Þ ; 2 b1 ð62Þ where b1 is unconstrained. Since the domain of v1 divides into two subdomains, there are three possible cases to consider, depending on the value of b1: 1. x1* ≤ μ(1 − δ) + Rb1, 2. μ(1 − δ) + Rb1 b x1* ≤ μ(1 + δ) + Rb1, and 3. μ(1 + δ) + Rb1 b x1*.

In equilibrium, only the second case will arise, where the agent knows he has equal probability of being constrained or unconstrained. In this case, both branches of v1 show up in the Bellman equation (Eq. (62)), and the first-order condition becomes

M−ðy0 −b1 Þ =

 βR ϕ h μ i m1 − μ ð1+δÞ +Rb1 + 2 1 +ϕ R

f

g

+M−ðμ ð1−δÞ+Rb1 +Q Þ :

M−ðy0 −b1 Þ = βR½M−ðy1 + Rb1 + Q Þ; which has the solution II

b1 ðy0 Þ =

 1 ϕ y0 −M− ðy1 + Q −MÞ : 1+ϕ R

Cash on hand in the next period will be

Solving for b1, we obtain

b1 ðy0 Þ =



R ϕ y0 −M− ðy1 + Q −MÞ x1II ðy0 Þ = y1 + 1+ϕ R  R y ϕ y0 + 1 −M− ðQ −M Þ : = R 1+ϕ R Since x1II is an increasing function of y0, there will exist a yc0 such that !c !c this constrained branch will hold iff y0 b y 0 . The upper bound y 0 will satisfy

f

 1 ϕ  u 2ð1 + ϕÞðy0 −M Þ− ϕ h1 −m1 R 2 + 3ϕ + 2ϕ2 + ð1 + ϕÞðμ + Q−MÞ−δμ

½

g:

ð63Þ

Appendix B. Perturbative interest rate calculations Suppose we assume for both models that   2 ð1 + δr1 Þ + O δ

 R y ϕ  !c y 0 + 1 −M− ðQ −M Þ = x1 : R 1+ϕ R

R=β

!c Solving for y 0 , we get

for δ small relative to 1, and we disregard all terms of order δ2 or higher. For example, the ϕ factor is

 1 + ϕ  1+R y +Q !c y0 = x1 + Q −M + M− 1 : R R R

ϕ = βR = β

−1

2

−1

  2 ½1 + 2δr1  + O δ :

ð64Þ

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J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

B.1. Certainty model The threshold incomes where the bond demand function has kinks are y c0 yc0

 μ ð1  δÞ μ 1 + ϕ + ϕ2 M−μ − 2− +Q : R R R R    M−μ ϕ −1 2 1+ β +β δq; 1+R− ð1 + ϕÞ −β = μ ð1∓βδÞ + R R

= m0 −

ð65Þ

The first term is the bond demand without the borrowing constraint. The second term is the change in demand due to a change in the interest rate. Note that the borrowing constraint does not show up in this expression. Meanwhile a low-income young agent will be constrained in middle age, so 

1 ϕ ϕ ðμ−MÞ 1− + δμ−δq : 1+ϕ R R

c

b1− = Since

  ϕ 2 = −δr1 + O δ ; R

where R and ϕ can be set to β− 1 in terms that are already first order in δ. Note that

1−

  −1 1+R= 1+β 1+

we have

  δr1 2 +O δ ; 1+β

ð66Þ

    2δr1 −1 2 1+ +O δ : 1+ϕ= 1+β 1+β

ð67Þ

    ϕ −1 2 1 + R− ð1 + ϕÞ = − 1 + 2β δr1 + O δ : R

ð68Þ

Therefore Eq. (65) simplifies to

+ δβ

−1

ð69Þ

"   2 ∓ 1+β+β

#   β2 2 μ−q +O δ : 2 1+β+β

this model. Since yc0 and yc0 are functions of the interest rate we are solving for, we will have to check that this assumption holds later. A high-income young agent will be unconstrained in both periods, so 1 = 1 + ϕ + ϕ2

(

"

2

ϕ ðμ−MÞ 1 + ϕ− R

)

# 2 1 ϕ 1+ : ð70Þ −δμ R R

Since

we can work out that the factor proportional to μ − M inside the curly brackets is 2

ϕ R



    1 −1 2 δr1 + O δ 1+ =− 2+β R

ð71Þ

Everything in the curly brackets of Eq. (70) is then first order in δ, so we can ignore the δ dependence of the outside factor. Thus, c b1 +

  βδμ βð1 + 2βÞ c 2 =− + ðM−μ Þδr1 + O δ : 2 2 1+β+β 1+β+β

bc2+ =

 β δμ 1+2β c c + ð M−μ Þδr ðM−μ Þδr1 + δμ− 1 1+ β 1 +β + β2 1+ β +β2   β2 βðβ + 2Þ c 2 δμ + ðM−μ Þδr1 + O δ : 2 2 1+β+β 1+β+β ð75Þ

As was the case for Eq. (72), the first term is the bond demand without the constraint and the second term is the effect of a change in the interest rate. Finally, a middle-aged low-income agent will have bond demand bc2− = −δq =−

β2 δμ + δτ c : 1 + β + β2

ð76Þ

Here, the first term is the bond demand without the constraint, and the second term is the direct effect of the borrowing constraint. Combining Eqs. (72), (74), (75), and (76), we get the marketclearing equation

  ϕ 2 = 1 + 2δr1 + O δ ; 2 R 2

1 + ϕ−



1 ϕ ðμ−MÞ 1− + Rb1 + + δμ : 1+ϕ R

  2 +O δ =

In order for Eq. (34) to be the applicable market-clearing equation, we must assume yc0 ≤ μ ≤ yc0 since initial income y0 = μ for all agents in

c b1 +

c

b2 + =

Using (73) and (72), this simplifies to

c

= μ−ðM−μ Þð2 + βÞδr1

ð74Þ

Again, the first term is the bond demand without the constraint. The term proportional to rc1 represents the effect of the change in interest rates. The term proportional to τc corresponds to anticipatory saving. In middle age, the high-income agent will have bond demand

we can work out that

yc0

   β βδ  c 2 δμ + ðM−μ Þr1 + τc + O δ : 2 1+β 1+β+β

c

    ϕ β−1 ð1 + 2δr1 Þ 2 2 = −1 + O δ = 1 + δr1 + O δ ; R β ð1 + δr1 Þ

y

Using (36), we can write this b1− =

Since we also have

c 0

   βδ  c 2 ðM−μ Þr1 + μ−q + O δ : 1+β

c

b1− =

and

ð73Þ

ð72Þ

4β + 7β2 + 4β3 β c   ðM−μ Þr1 + τ c + τ = 0; 1+β c ð1 + βÞ 1 + β + β2

ð77Þ

where the first term is the effect of the change in interest rates, the second term is the direct effect of the borrowing constraint, and the third term is the effect of anticipatory saving. Solving Eq. (77), the first-order correction to the interest rate is   2 ð 1 + 2β Þ 1 + β + β τc c : r1 = − M−μ 4β + 7β2 + 4β3

ð78Þ

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

B.3. Uncertainty model

According to Eq. (69), c

y0 = μ−ðM−μ Þð2 + βÞδr1 + δβ

−1

    2 2 1 + β + β τc + O δ :

The threshold value of y0 below which agents are always borrowing constrained in middle age is

If the borrowing constraint binds, τc N 0. Since we also have M N μ, and r1 b 0, we must have that μ ≤ yc0 , so one of our assumptions holds true. Substituting (36) and (78) into (69), we also have

y c0 = μ + δ

 " ð2 + βÞð1 + 2βÞ 1 + β + β2 2

4β + 7β + 4β   −1 2 1 + β + β δq: −δβμ−β

β2 μ−q 1 + β + β2

3

2

ð2 + βÞð1 + 2βÞ 1+β+β −1 = −2 b 0; 4 + 7β + 4β2 4 + 7β + 4β2 c

c

u

y0 = μ +

c

so y 0 ≤ μ, and our assumption that y 0 ≤ μ ≤ y0 holds true. Thus, even with the interest rate correction, an agent who receives the low income shock will be liquidity-constrained, and the agent who receives the high income shock will not be. The borrowing constraint is too tight to be evaded by the low-income agents.

    u u −1 2 y 0 = μ−ð2 + βÞðM−μ Þδr1 − β + 1 + β δq−ð1 + βÞδμ + O δ : ð79Þ Similarly, the threshold value of y0 above which agents are never borrowing constrained in middle age is     u u −1 2 y0 = μ−ð2 + βÞðM−μ Þδr1 + β + 1 + β δτu + O δ :



 1 1 2 μ−M 2 δμ ð 1 + ϕ Þ ½ μ−M −ϕ 1 + + ϕ R R R 1 + ϕ + ϕ2

   1 2 2 3 2 2 2 δμ ð μ−M Þ + ϕ 1 + βR −β R −β R = R 1 + β−1 + β−2       −1 ch −1 ch ch 1+β 1 + 2δr1 −β 1 + 3δr1 − 1 + 2δr1 ðμ−M Þ = −1 1 + β + β−2 β β + 2β2 β ch δμ = ðM−μ Þδr1 + δμ: + 2 1+β+β 1 + β + β2 1 + β + β2

Thus using the other bond demands (72) and (75), the marketclearing equation becomes β + 2β2 βð1 + 2βÞ ch ch ðM−μ Þδr1 + ðM−μ Þδr1 1 + β + β2 1 + β + β2 2

+

β βðβ + 2Þ ch δμ + ðM−μ Þδr1 1 + β + β2 1 + β + β2

4β + 5β2 ch −δτc = ðM−μ Þδr1 1 + β + β2 ch

r1 = −

1 + β + β 2 τc : 4β + 5β2 M−μ

Thus, using Eq. (78), r1ch 4 + 7β + 4β2 : c = r1 ð1 + 2βÞð4 + 5βÞ

ð80Þ

Proceeding as in the previous section, we assume yu0 ≤ μ ≤ yu0 , where initial income y0 = μ, so the 2 − types will be borrowing constrained and the 2 + types will not be, an assumption that must be checked after we determine Ru. From Eq. (26), we have the young bond demand bu1 =

ch b1− =

δμ : R

Using Eq. (68), this simplifies to

B.2. Certainty model without Carroll–Kimball saving Now let us consider an experiment where we shut down the Carroll–Kimball saving mechanism. Suppose we imagine that young agents do not know they will be borrowing constrained in middle age and so they always follow the frictionless bond demand (Eq. (9)). The bond demands of high-income agents will be unchanged since they already followed the frictionless bond demands.

   ðM−μ Þ ϕ 2 δq ð1 + RÞ− ð1 + ϕÞ − 1 + ϕ + ϕ R R R

−ð1 + ϕÞ

#

The coefficients of q are all negative. The coefficient of δμ is proportional to

dq =

1571

1 2 + 3ϕ + 2ϕ2 −

f

" 2ð1 + ϕÞ−

g

# ϕ2 ϕ ð 1 + R Þ− ð 1 + ϕ Þ ðμ−M Þ R R2

ϕð1 + ϕÞ ϕ δq + δμ : R R

Using Eqs. (71) and (73), we work out that the factor proportional to μ − M inside the curly brackets is 2

ϕ ϕ 2ð1 + ϕÞ− 2 ð1 + RÞ− ð1 + ϕÞ R R     = − 3 + 2β−1 δr1 + O δ2 ; so u

b1 =

    β u 2 ð2 + 3βÞðM−μ Þδr1 + ð1 + βÞδτu + O δ : 2 2 + 3β + 2β ð81Þ

Note that young bond demand is zero in the absence of the borrowing constraint. Here, the first term is the effect on the bond demand of a change in the interest rate while the second term is saving (both precautionary and anticipatory) induced by the borrowing constraint. The high-income middle-aged bond demand is the same as in the CM except bond holdings held over from youth will differ: u

b2+ =



  1 ϕ u 2 ðμ−MÞ 1− + Rb1 + δμ + O δ 1+ϕ R

Using Eqs. (75) and (83), u

b2+ =

  βδμ 2βðβ + 2Þ βδτu u 2 ðM−μ Þδr1 + +O δ : + 1+β 2 + 3β + 2β2 2 + 3β + 2β2 ð82Þ

The first term is the bond demand without the borrowing constraint, and the second term is the effect of a change in interest rates. The third term is precautionary wealth accumulation, additional saving

1572

J. Feigenbaum / Economic Modelling 28 (2011) 1559–1572

that occurs in middle age as a result of anticipatory and precautionary saving that proved unnecessary. Finally, for the low-income middle-aged, u

b2− = −

β δμ + δτ u ; 1+β

ð83Þ

where the first term is the bond demand without the borrowing constraint and the second term is the direct effect of the borrowing constraint. Inserting Eqs. (81)–(83) into the market-clearing Eq. (7), we obtain the equation: 8βðβ + 1Þ 2β2 + 3β u ð M−μ Þδr + δτ + δτ u = 0: 1 u 2 + 3β + 2β2 2β2 + 3β + 2

Thus the market-clearing condition is 2

4β + 5β β uh ðM−μ Þδr1 + δμ = δq 1+β 1 + β + β2 2

4β + 5β uh ðM−μ Þδr1 = −δτ u 1 + β + β2 r1uh = −

2

1 + β + β τu : 4β + 5β2 M−μ

Thus, r1uh 1 + β + β2 4β 1 + β + β2 : =4 u = 2 r1 ð1 + 2βÞð4 + 5βÞ 4β + 5β 2β + 1

ð84Þ References

The first term is the effect of the change on interest rates, the second term is the direct effect of the interest rate, and the third term is the effect of anticipatory and precautionary saving, including precautionary wealth accumulation. Solving Eq. (85), we obtain the first-order correction to the interest rate: u

r1 = −

2β + 1 τu : 4β M−μ

ð85Þ u

Since ru1 b 0, Eq. (80) implies that μ ≤ y0 . Inserting Eq. (85) into Eq. (79), we get 2

u

y 0 = μ−

2

2β + 3β + 2 6β + 9β + 6 δμ− δq ≤ μ; 4ð1 + βÞ 4β

so our assumption that μ is in the partially constrained region was valid. B.4. Uncertainty model without Carroll–Kimball saving Now let us see what happens if we shut down the Carroll-Kimball mechanism by having young agents follow the frictionless bond demand (Eq. (13)). Then we will have

buh 1 = =

1 1 + ϕ + ϕ2

( ð1 + ϕÞ½ μ−M −

) 2 ϕ 1 1+ ðμ−M Þ R R

β + 2β2 uh ðM−μ Þδr1 : 1 + β + β2

Meanwhile  1 ϕ uh μ + δμ + Rb1 −M− ðμ−M Þ 1+ϕ R " # 2 1 −1 β + 2β ð 1−βR Þ ð μ−M Þ + β ð M−μ Þδr + δμ = 1 1+β−1 1 + β + β2 " # β 2 + 3β + β2 uh = ðM−μ Þδr1 + δμ 1 + β 1 + β + β2

buh 2+ =

=

2β + β2 β uh ðM−μ Þδr1 + δμ: 1+β 1 + β + β2

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