Web Appendix for: On-the-Job Search and. Precautionary Savings

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1 Web Appendix for: On-the-Job Search and Precautionary Savings Jeremy Lise University College London and IFS July 212 B Web Appendix (Not for Publication) B.1 Additional IID Income Uncertainty This section explores the implications of a more general version of the model presented in the body of the paper in hich uncertainty in non-labor income is included, specifically the addition of a Bronian motion component to the budget constraint. In general the variance of this component may be a function of the current state, so in hat follos σ may be read as σ(a, ). Indeed, σ(a, ) = is a necessary condition to guarantee positive consumption hen unemployed. The addition of an additional Bronian motion income process can be expressed compactly in the asset accumulation equation as: da = [ra + c]dt + σdz. The corresponding value functions can be ritten as: ρu(a) = max {u(c) e(s) + U a (a)[ra + b c] + σ2 c a a, s 2 U aa(a) ˆ } + λs max{w (a, x) U(a), }df (x). (32) ρw (a, ) = max {u(c) e(s) + W a (a, )[ra + c] + σ2 c a a, s 2 W aa(a, ) ˆ } + λs max{w (a, x) W (a, ), }df (x) + δ[u(a) W (a, )]. (33) 1

2 Note that the first order conditions are unchanged ith the addition of the diffusion term, but the value function ill be loer due to the terms 1 2 σ2 U aa (a) and 1 2 σ2 W aa (a, ). The differential equation characterizing consumption, analogous to equation (11), has an additional precautionary term that is sensitive to the variance of the iid random shock (σ 2 ): ċ c = 1 ( ( ˆ r ρ λs F () γ(c) u ) ( (ĉ) u (c) )) u df (x) + δ (c) u (c) 1 σ2 2 ( u (c) u (c) c a c c ) aa, c here γ(c) is the coefficient of relative risk aversion. For unambiguous additional precaution e require that the third derivative of the utility function u (c) be positive (the consumption function is concave so c aa is negative). The term σ 2 2 ( u (c) u (c) c a c c ) aa c is exactly the Kimball (199) definition of the degree of prudence. B.2 Optimal Search Effort Derivation of Equation (1): Given the reservation age strategy, and the fact that employed orkers accept any age higher than their current age, equation (7) can be reritten as ˆ e (s) = λ max{w (a, x) W (a, ), }df (x) ˆ = λ [W (a, x) W (a, )] df (x) [ˆ ] = λ W (a, x)df (x) F ()W (a, ) = λ = λ = λ ˆ ˆ ˆ W (a, x)f (x)dx W a + W a [ra + x c] F (x)dx ρ + δ + λsf (x) u (c) + [u (c)c ][ra + x c] F (x)dx (34) ρ + δ + λsf (x) here the second line follos from W (a, ) being increasing in, the third line follos directly, the fourth line uses the fact that integration by parts implies that ˆ the fifth line substitutes for ˆ W (a, x)f (x)dx = W (a, x)f (x) W (a, x)( f(x))dx = ˆ W (a, x)df (x) F ()W (a, ), W = W a + W a [ra + c], ρ + δ + λsf () and the final line uses the first order condition for consumption, to substitute in W a (a, ) = u (c) and W a = u (c)c (a, ). 2

3 B.3 Optimal Consumption Groth Derivation of Equation (11): Applying the envelope theorem to the value of employment, e can rite: ρw a (a, ) = W aa (a, )[ra + c] + rw a (a, ) or equivalently, + λs ˆ W a (a, x)df (x) λsf ()W a (a, ) + δ[u a (a) W a (a, )] ˆ W aa (a, )[ra + c] = [ρ r + δ + λsf ()]W a (a, ) δu a (a) λs W a (a, x)df (x) (35) Recall that the state variables evolve according to da = [ra + c]dt (36) d = max{, }dq λs + [ ]dq δ, (37) here F (). Proceed by totally differentiating W a, making use of equations (36) and (37): dw a = W aa [ra + c]dt + [W a (a, max{, }) W a (a, )]dq λs + [W a (a, ) W a (a, )]dq δ. (38) No, multiply equation (35) by dt, add [W a (a, max{, }) W a (a, )]dq λs + [W a (a, ) W a (a, )]dq δ, and substitute using equation (38): ˆ dw a = [ r + ρ + δ + λsf ()]W a (a, )dt λs W a (a, x)df (x)dt δw a (a, )dt + [W a (a, max{, }) W a (a, )]dq λs + [W a (a, ) W a (a, )]dq δ. Concentrating on the time hen neither an acceptable job offer is received, nor is the job destroyed (dq λs = dq δ = ) e can obtain some insight into savings behavior at different age levels. Making use of the first order condition W a (a, ) = u (c), and dw a (a, ) = u (c)dc, e can rite: u (c(a, ))dc = [r ρ δ λsf ()]u (c(a, ))dt + λs ˆ u (c(a, x))df (x)dt + δu (c(a, ))dt, dq λs = dq δ =. Rearranging e obtain an expression for consumption groth (alternatively asset groth) for a given age as: ċ c = 1 ( ( ˆ u ) ( (ĉ) u (c) )) r ρ λs F () γ u df (x) + δ (c) u (c) 1, (39) here ċ = dc/dt, γ(c) = u (c)c/u (c) is the coefficient of relative risk aversion, and I make use of the shorthand ĉ = c(a, x), here x is a dummy of integration, and c = c(a, b), ith corresponding notation for s. We can similarly derive the differential equation governing optimal consumption hen un- 3

4 employed as ċ c = 1 ( ( ˆ u )) (ĉ) r ρ λs 1 γ b u df (x), (4) (c) indeed equation (4) is nested by equation (39) since = b. Optimal consumption and search are characterized by equations (34), (36), (37), (39), and the present-value budget constraint lim t e rt a(t) (a.s.). 1 B.4 Equation (5) as the Limit of the Discrete Time Bellman Equation Consider a discrete time model here the length of a time period is t. The continuous time Bellman equation (5) can be derived as the limit here t goes to zero. { W (a,, t) = max (u(c(a,, t)) e(s(a,, t))) t c a a, s 1 ( ˆ + λs t max{w (a + a, x, t + t), W (a + a,, t + t)}df (x) 1 + ρ t )} + δ tu(a + a, t + t) + [1 λs t δ t]w (a + a,, t + t) + o( t), here o( t) are terms that go to zero faster than t (for example, the probability of a job offer and a job destruction in the same period). Multiply by (1 + ρ t) and rearrange, ρ tw (a,, t) = ˆ + λs t max c a a, s { (1 + ρ t) (u(c(a,, t)) e(s(a,, t))) t max{w (a + a, x, t + t) W (a + a,, t + t), }df (x) + δ t[u(a + a, t + t) W (a + a,, t + t)] } + [W (a + a,, t + t) W (a,, t)] + o( t). Dividing by t and taking the limit as t goes to zero, ρw (a,, t) = { max u(c(a,, t)) e(s(a,, t)) c a a, s ˆ + λs max{w (a, x, t) W (a,, t), }df (x) } + δ[u(a, t) W (a,, t)] + W a (a,, t)[ra + c], here the last line follos from lim t [W (a+ a,, t+ t) W (a,, t)]/ t = dw (a,, t)/dt = W a (a,, t)da/dt, and lim t o( t)/ t =. 1 For other derivations of optimal consumption in the face of Poisson uncertainty see Merton (1971) for a model in hich earnings are incremented upard by a fixed amount at random intervals, Wälde (1999) for a model here the capital gain from an R&D investment is realized at random intervals, and Toche (25) for a model of precautionary savings here the risk to labor income takes the form of exogenous retirement. 4

5 B.5 Solving the Model Given a set of structural parameters, the model described in Section 2 is solved numerically. Given an estimate of the age offer distribution, I solve the orkers dynamic programming problem (equations (4) (7)). The solution to the orkers problem provides policy functions for consumption and search effort. The dynamic programming problem laid out in Section 2 is expressed in continuous time, and is continuous in the state variables. Since it is not possible to solve an infinite dimensional problem numerically, I use a set of interpolation techniques that are ell suited to addressing this problem. The value functions are approximated using Chebyshev collocation. This method is discussed in detail by Judd (1998, chs. 6,11). The basic idea behind the method is to approximate the unknon value function by the closest function in polynomial space. The approximations to the value functions in equations (4) and (5) are then ritten as zero functions and solved for exactly at a discrete set of grid points. Working in continuous time in this context has several advantages. First, it enables me to derive an analytical form for the consumption dynamics in equation (11). Second, the first order conditions (equations (6) and (7)) are expressed solely in terms of instantaneous marginal utility of consumption and the instantaneous marginal value of assets. This alleviates the need for a numerical search algorithm over optimal consumption and search effort that ould be required in a discrete time frameork. The policy functions can be computed directly from the first order conditions. The grid points are chosen as the zeros of the Chebyshev polynomials; this choice of polynomial representation and choice of grid points ensures the numerical problem is ell conditioned. B.5.1 Value Function Approximation The value function is approximated using Chebyshev collocation (See Judd (1998, chs. 6, 11)): ˆ W (a, ) = Φ(a, )β, W a (a, ) = Φa (a, )β, ˆ W (a, x)df (x) = Φ(a, x)βdf (x) ˆΦ(a, )β, here Φ(a, ) is a k a k k a k matrix comprising the tensor product of basis functions for a and : Φ(a, ) = Φ(a) Φ(). Similarly, Φ a (a, ) = Φ a (a) Φ(). β is a k a k 1 vector of coefficients on the basis functions. The basis is given by the Chebyshev polynomials. The policy functions are given by the first order conditions (equations (8) and (9)): c(a, ) = φ (Φa (a, )β), (41) )) s(a, ) = ϕ (λ (ˆΦ(a, ) F ()Φ(a, ), (42) here φ and ϕ are the inverse functions for the marginal utility of consumption u (c) and the marginal cost of search effort e (s). Reriting the value function from equation (33) using the polynomial approximations e have: ρφ(a, )β = u(c) e(s) + [ra + c]φ a (a, )β + λsˆφ(a, )β λsf ()Φ(a, )β + δ[φ(a, )β Φ(a, )β]. (43) The collocation method proceeds by solving equation (43) at the k a k nodes by choosing the 5

6 k a k coefficients β. This can be accomplished by iteratively solving a system of linear equations as follos: 1. Use equations (41) and (42) to find the policy functions for consumption and search effort, given a guess for the polynomial coefficients β j, 2. Update the guess to β j+1 by solving the linear system of equations specified in equation (43): ( ) (ρ+λsf ()+δ)φ(a, ) (ra+ c)φ a (a, ) λsˆφ(a, ) δφ(a, ) β j+1 = u(c) e(s) 3. Stop if β j+1 β j ε tolerance, otherise increment j and repeat step one using β j+1. One additional advantage of the collocation approach is that the residual error can be calculated for points off the grid used in the solution. Specifically, I find the k a k coefficients β that solve equation (43) at the k a k nodes. Thus at these nodes the residual function is less than numerical tolerance ε = 1 1 : R (a, ) = u(c) e(s) + Φ a (a, )β[ra + c] + λsˆφ(a, )β λsf ()Φ(a, )β + δ[φ(a, )β Φ(a, )β] ρφ(a, )β ε. I then create a grid ten times finer over (a, ) and evaluate the residual function at these points. The maximal (scaled) residual error is max R (a, ) /Φ (a, ) β. With a grid comprising 11 nodes each for ages and assets, the maximum scaled residual error is on the order of 1 5, indicating that any approximation errors are at most.1 percent of the value function. B.5.2 Change of Variables When solving the model, the folloing change of variables allos a much more accurate solution, even near the borroing constraint. Define the change of variables in assets: A = log(a a + 1) here the linear adjustment, a + 1, ensures that A is bounded belo by zero. No define the ne value function W such that: W(A, ) = W (a, ), hich, along ith the change of variables in assets implies: here W A (A, )e A = W a (a, ), da da = 1 a a + 1 = e A. We no ork ith the transformed value function: { ρw(a, ) = max c,s [ u(c) e(s) + W A (A, )e A r ˆ + λs [ ] ] e A + a 1 + c } max{w(a, x) W(A, ), }df (x) + δ[w(a, ) W(A, )]. 6

7 There are several advantages of orking ith this transformation. First, W(A, ) has much less curvature than the original function W (a, ), alloing for a loer degree polynomial approximation than the original value function. Second, e end up placing many more nodes near the borroing constraint hich is the part of the function that displays the most curvature. B.6 Some Further Implications and Robustness B.6.1 Implications for Consumption Inequality The model parameters have been estimated using the NLSY data, hich as argued above has excellent information on ages, job changes, and asset stocks. Since the theory is developed in terms of consumption, one ould like to kno hat the model s implications for consumption look like relative to the data. The moments that the model fit best are the groth moments, or the micro moments. Unfortunately e do not have access to panel data on consumption that ould permit the construction of consumption groth at the individual level, effectively alloing the differencing out of individual fixed effects. The Consumer Expenditure Survey (CEX) is, hoever, useful in constructing an age profile for the variance of consumption, so e are effectively considering the change in the variance of consumption as the cohort ages, rather than our preferred variance of consumption groth. I plot the age profiles from 25 to 6 for the variance of consumption constructed from the CEX data and the model in Figure 1. The simulated variance is normalized to have the same value as the CEX data at age 65. The stationarity implied in the model produces a strong concave relationship beteen the variance of consumption and age. Looking at the age profile for the lo education group e see that the total increase from age 25 to 6 is the same in the model as it is in the data, although the pattern is quite different. Most of the increase happens at early ages in the model, during the first five years in the labor market, hile most of the increase occurs during the older ages in the data. Looking at the high education group, the model and the data line up quite ell after the initial five years in the labor market. It is somehat difficult to compare these results directly to the existing literature, here results are not reported separately by education groups, and the age range is generally narroer (see, for example, Heathcote, Storesletten, and Violante (29)). B.6.2 Ho important is on-the-job search? In the model orkers face to sources of risk. First, there is uncertainty about ages in the job finding process and ho quickly the orker ill climb the age ladder. The second is the risk of job loss, and the need to start again at the bottom of the ladder. Looking again at the equation characterizing consumption groth (11), e have a convenient ay to decompose the variance of consumption groth into the part due to on-the-job search and the part due to the risk of job loss. Define the terms S and D as representing on-the-job search and potential job destruction respectively: S = λs ( F () γ D = δ ( u ) (c) γ u (c) 1. ˆ u ) (ĉ) u df (x) (c) Then the variance of consumption groth can be ritten as: var (ċ/c) = var (S) + var (D) 2cov (S, D). 7

8 (a) Lo Education (b) High Education Figure 1: Implications for Variance of Log Consumption Note: The dashed line is the age profile of variance of log consumption computed on the CEX data Consumption is the residual from a regression of log average household consumption on indicators for sex, race, rural, region of residence, and dummies for birth cohort and year, here year effects are constrained to be orthogonal to a linear time trend. The dotted lines represent to standard errors. The solid line is the variance of log consumption computed from the model simulated data. I have normalized the variance to be equal to the variance computed in the data at age 6. Table 6: Decomposition of the variance of consumption groth Share of total variance due to o-t-j search risk of job loss covariance Lo Education High Education Note: The variance of consumption groth is calculated by pooling over ages 25 to 6 for the cohort. This can be interpreted as representing an overlapping generations economy ith a stationary population structure. The first term involves the job search component λs, and the second involves the risk of job loss δ. Table 6 gives the share of the total variance of consumption groth due to job search and due to the risk of job loss, ith the residual component due to the covariance term. The risk of moving beteen employment and unemployment does account for some of the variance, 11 and 24 percent, for lo and high education groups respectively. Hoever, it is on-the-job search that accounts for the majority of the variance, 8 and 64 percent respectively for the lo and high education groups. The fact that orkers are climbing the age ladder, and not simply moving beteen employment and unemployment, adds substantially to the variance in consumption groth across orkers, and similarly, to the variance in assets across orkers. B.6.3 Robustness to Alternative Specifications In Tables (7) I present the conditional transition probabilities for different definitions of employment (35 or 4 hours per eek) and for different definitions for assets (net orth or net financial orth). The regression coefficients are quite robust to these alternative definitions. In 8

9 Table 7: Effect of Assets and Wages on Labor Market Transitions (LPM Results). Lo Education Unemployment to Employment Job-to-Job I II III IV I II III IV Total Assets (.32) (.27) (.2) (.2) Financial Assets (.77) (.59) (.4) (.4) Wage (.2) (.2) (.2) (.2) Intercept (.11) (.96) (.111) (.96) (.1) (.11) (.1) (.1) Person-eeks 68,368 85,531 67,491 84, , ,39 5, ,749 Persons High Education Unemployment to Employment Job-to-Job I II III IV I II III IV Total Assets (.46) (.29) (.1) (.1) Financial Assets (.58) (.39) (.2) (.2) Wage (.2) (.2) (.2) (.2) Intercept (.178) (.142) (.179) (.143) (.1) (.1) (.1) (.1) Person-eeks 22,55 33,37 21,856 33, ,29 37,2 317,926 36,719 Persons Notes: In columns I and III employed is defined as 35 hours per eek, in columns II and IV employed is defined as 4 hours per eek or more. All estimates are from random effects regressions, conditioning on marital status, number of children, region, and rural or urban location. 9

10 Table 8: Effect of Assets and Wages on Labor Market Transitions (Probit Results). Lo Education Unemployment to Employment Job-to-Job I II III IV I II III IV Total Assets (.348) (.342) (.247) (.25) Financial Assets (.76) (.73) (.539) (.544) Wage (.162) (.172) (.161) (.171) Intercept (.612) (.598) (.66) (.592) (.982) (.14) (.974) (.132) Person-eeks 68,368 85,531 67,491 84, , ,39 5, ,749 Persons High Education Unemployment to Employment Job-to-Job I II III IV I II III IV Total Assets (.395) (.352) (.22) (.26) Financial Assets (.527) (.476) (.269) (.274) Wage (.191) (.198) (.19) (.198) Intercept (.899) (.787) (.91) (.794) (.1124) (.116) (.1118) (.1154) Person-eeks 22,55 33,37 21,856 33, ,29 37,2 317,926 36,719 Persons Notes: In columns I and III employed is defined as 35 hours per eek, in columns II and IV employed is defined as 4 hours per eek or more. All estimates are from random effects Probits, conditioning on marital status, number of children, region, and rural or urban location. 1

11 Table 9: Parameter Estimates III IV III IV r: risk free rate.3 µ: search costs scale 1. High school College δ: job destruction rate [.223,.2295] [.215,.223] [.97,.12] [.89,.93] λ: job contact rate [1.51,1.531] [.334,.4] [.7,1.24] [.139,.151] η: elasticity of search costs.r.t. effort - [1.54,1.96] - [1.33,1.37] γ: relative risk aversion [.845,.883] [.814,.97] [.57,.6] [1.67,1.79] ρ: time preference [.34,.37] - [.6,.67] Note: 95 percent confidence intervals in square brackets (2.5 and 97.5 quantiles of the quasi-posterior distribution). Specification III uses auxiliary models (15) and (18) to (26). Specification IV uses all 12 auxiliary models (15) to (26). Table (8) I present Probit results for the same alternative variable definitions. In estimation I choose to use the linear probability model for speed of computation. In Table (9) and Figures (2) to (5) I present estimates for to alternative specifications of the model. The first alternative assumes search effort is exogenously fixed at s = 1. The second alternative estimates ρ along ith the other parameters. The model ith exogenous search effort does substantially orse at fitting the moments associated ith age groth. Adding ρ as an additional parameter to be estimated allos for a somehat better fit to the age profile of employment, but separate identification of ρ and γ is dubious given the assumption of stationarity (there is no variation in the interest rate). My preferred specification, used in the body of the paper, is to condition the estimates and the interpretation on a fixed value of ρ that is common across education groups. 11

12 (a) t (b) σ (c) cov ( t, t 1 ) (d) a t (e) var ( a t ) (f) cov ( a t, a t 1 ) (g) cov ( a t, t ) (h) E t (i) a t (j) var (a t ) (k) var ( t ) (l) cov (a t, t ) Figure 2: Data and Model Moments: Exogenous Search Intensity, Lo Education Note: The thin dashed line and the thin dotted lines are the data, plus and minus to standard errors. The thick solid lines are from specification III. 12

13 (a) t (b) σ 2 (c) cov ( t, t 1 ) (d) a t (e) var ( a t ) (f) cov ( a t, a t 1 ) (g) cov ( a t, t ) 4 (h) E t.2.2 (i) a t (j) var (a t ) (k) var ( t ) (l) cov (a t, t ) Figure 3: Data and Model Moments: Exogenous Search Intensity, High Education Note: The thin dashed line and the thin dotted lines are the data, plus and minus to standard errors. The thick solid lines are from specification III. 13

14 (a) t (b) σ 2 (c) cov ( t, t 1 ) (d) a t (e) var ( a t ) (f) cov ( a t, a t 1 ) (g) cov ( a t, t ) (h) E t (i) a t (j) var (a t ) (k) var ( t ) (l) cov (a t, t ) Figure 4: Data and Model Moments: Estimated ρ, Lo Education Note: The thin dashed line and the thin dotted lines are the data, plus and minus to standard errors. The thick solid lines are from specification IV. 14

15 (a) t (b) σ 2 (c) cov ( t, t 1 ) (d) a t (e) var ( a t ) (f) cov ( a t, a t 1 ) (g) cov ( a t, t ) 4 (h) E t.2.2 (i) a t (j) var (a t ) (k) var ( t ) (l) cov (a t, t ) Figure 5: Data and Model Moments: Estimated ρ, High Education Note: The thin dashed line and the thin dotted lines are the data, plus and minus to standard errors. The thick solid lines are specification IV. 15

16 References Heathcote, J., K. Storesletten, and G. L. Violante (29): Consumption and Labor Supply ith Partial Insurance: An Analytical Frameork,. Judd, K. L. (1998): Numerical Methods in Economics. MIT Press. Kimball, M. S. (199): Precautionary Savings in the Small and in the Large, Econometrica, 58, Merton, R. C. (1971): Optimum consumption and Portfolio Rules in a Continuous- Time Model, Journal of Economic Theory, 3, Toche, P. (25): A Tractable Model of Precautionary Saving in Continuous Time, Economics Letters, 87, Wälde, K. (1999): Optimal Saving under Poisson Uncertainty, Journal of Economic Theory, 87,