Online Appendix to Accompany Household Search and the Aggregate Labor Market

Online Appendix to Accompany Household Search and the Aggregate Labor Market Jochen Mankart (Deutsche Bundesbank) Rigas Oikonomou (UC Louvain) September 6, 2016 This online appendix includes four sections. In Section 1, we show detailed results for some models presented in the main text. We also show additional output and robustness exercises based on different parameterizations of the model. Finally, we derive the program of the bachelor household. In Section 2, we describe in detail the CPS data and provide further evidence on the AWE. In Section 3, we present the numerical algorithm we used to solve the incomplete market models and discuss the properties of the solution. Finally, in Section 4 we present the derivation of the solution of the complete market model and describe in detail the numerical procedure we propose to solve this model. A separate document which is posted online contains the appendix for the unemployment insurance extension considered in Section 8 of the paper. 1

1 Further Results, Program of Bachelor Agents and Definitions 1.1 The Bachelor Household Model We define the household s program in the bachelor model. Consider first an employed agent. Let X denote the state variables. Let V E (X) be the lifetime utility of the agent. It solves the following Bellman equation: ( [ ]) V E (X) = max u(c, 1 h) + β φ R V R (X ) + (1 φ R ) (1 χ(λ))q e (X ) + χ(λ)q n (X ) dπ ɛ c, a ɛdπ λ λ ɛ,λ subject to a = (1 + r (Γ,λ) )a + w (Γ,λ) hɛ c, Γ = T (Γ, λ, λ ), and a 0 Now consider the program in state S {O, U}. We have ( [ ] V S (X) = max u(c, 1 κ(s)) + β (1 φ R ) p(s, λ)q e (X ) + (1 p(s, λ))q n (X ) c, a ɛ,λ ) +φ R V R (X ) dπ ɛ ɛdπ λ λ subject to a = (1 + r (Γ,λ) )a c, Γ = T (Γ, λ, λ ), and a 0 Finally for retired households we have s = s if S = O and s = s if S = U ( ) V R (X) = max u(c, 1) + β (1 φ A )V R (X ) + φ A Q n (X ) dπ ɛ c, a ɛdπ λ λ ɛ,λ subject to a = (1 + r (Γ,λ) )a c, Γ = T (Γ, λ, λ ), and a 0 Moreover, the option values are given by (1) Q k (X) = max {V S (X)} if (k = n) S {O,U} max {V S (X)} if (k = e) S {O,U,E} For the sake of brevity we omit the definition of the competitive equilibrium in the bachelor model. 1.2 Additional Moments and Calibrated Parameters We now discuss the calibrated parameters for the robustness exercises we performed in Section 7 of the paper. We also show additional moments from the simulations with aggregate shocks. Table 1 shows the calibrated parameters for the log-utility model. As in the case of non-separable preferences, under the bachelor model the equilibrium β is lowest since the incentive of households to save for precautionary purposes is strongest. The calibration of other parameters (frictions and idiosyncratic productivity) is the same as in the benchmark model and hence it is omitted from the 2

table. Table 2 reports the average labor market flows across the three states E, U and O under log utility. Recall that in the text we claimed that the flows do not change significantly (under incomplete markets) when we alter the specification of preferences; under complete markets they do not change at all. The results in the table confirm this claim. 1 [Tables 1 and 2 About Here] [Figure 1 About Here] In Figure 1 we show the wealth distributions under the benchmark model (solid lines) and the log utility model dashed lines. The top panels refer to the bachelors model; on the left we show the distributions conditional on the individual being employed, on the right we show the analogous objects when the individual is not employed. In the bottom panels we show the wealth distributions in the couples model and denote the joint status of the household members for each sub-figure. 2 Note that when we assume log utility the wealth distributions do not change significantly relative to the benchmark model. This proves the claim we made in the text. We now report our findings for the behavior of wages in the model with log utility. Under the bachelor model we find that the relative standard deviation of wages and GDP is 0.27 and the correlation coefficient ρ w,y equals 0.39. In the complete market model these quantities are 0.28 and -0.18 respectively. Finally, in the couples model we obtain 0.56 for the relative standard deviation and 0.78 for the correlation. These findings are essentially the same as in the benchmark model: the bachelors and complete market models predict a very procyclical entry into the LF which reduces the cyclicality of wages. Table 3 shows additional output from the model of Section 7.1 ( Reducing Non-Searchers ). For convenience, we also repeat the labor market statistics from the main text. Notice that the behavior of aggregate consumption and investment is similar to the benchmark. Wages are the only model quantity affected by the increase in the arrival rate of offers to out of the labor force individuals. Wages now become more procyclical in the models, this is a consequence of the fact that the correlation between the LF and aggregate output is weaker. However, since the correlation remains substantial under bachelors and complete markets, the cyclical correlation of wages with GDP remains higher than the couple households model. 1.3 The specification of the labor productivity process Recall that the specification of the idiosyncratic productivity process was (2) log(ɛ i t) = ρ ɛ log(ɛ i t 1) + v i ɛ,t with innovations v i ɛ,t N (0, σ ɛ ), i = 1, 2. We made the following two assumptions: i) Cov(v 1 ɛ,t, v 2 ɛ,t) = 0 (consistent with the estimation procedure employed by Chang and Kim (2006)) ii) The parameters 1 In the subsequent section which is devoted to the complete market allocation we discuss explicitly why the flows are identical under separable and non-separable preferences in the case of complete markets. 2 To save notation we denote non-employed agents using the symbol n (either unemployed or out of the LF). 3

ρ ɛ and σ ɛ were given values based on the male population estimates provided by Chang and Kim (2006). These assumptions have been made for a number of reasons: First, for parsimony; the estimates of Chang and Kim (2006) are the most recent ones in the literature which condense all of the labor market idiosyncratic uncertainty into a one shock process; this is convenient for us to run the large scale application of the couple model over the business cycle. Other recent estimates either separate idiosyncratic shocks into persistent and transitory components (Heathcote, Storesletten and Violante, 2010) or assume that idiosyncratic residuals follow a random walk process (Attanasio, Low and Sanchez-Marcos, 2008). 3 Second, since our model is one of ex ante identical household members, and we compare its performance with the bachelor household model and the model of complete markets, we use the estimates of the male population. We vary the household size from bachelors to complete markets, without placing any additional structure on the joint process in multi-member households. The same assumption is also employed by Heathcote et al. (2010) and Attanasio et al. (2008), even though these are papers which account explicitly for gender. For the same reason, that household members are identical, we have chosen to treat the case where Cov(v 1 ɛ,t, v 2 ɛ,t) = 0 as our benchmark. Heathcote et al. (2010) and Attanasio et al. (2008) use the empirical estimates of Hyslop (2001) to calibrate the within household correlation of the shocks to a positive number. As we will demonstrate in this subsection, our results are not impacted when we make a similar assumption. Finally, Chang and Kim (2006) correct for selection effects in their estimation through accounting for the participation margin. Note that though this may suggest that using the estimates from the female population (where selection issues are clearly more significant) along with the male estimates is appropriate, since we can claim that the female estimates of process (2) are unbiased. We can nonetheless find reasons to be sceptical about this: First, Chang and Kim (2006) have specified their selection equation outside their structural model. It may be (for example) that a structural model of intrahousehold decisions gives us a different (and non-linear) selection equation than the one they employed in their estimation. 4 Second, if the right selection correction varies with model specification (e.g. bachelors vs couples, models with frictions vs models without frictions) then we cannot make the claim that using the selection-corrected process is appropriate across all the models we consider. 5 For these reasons we opted to use the estimates of the male process for both household members, this is also in the spirit of Heathcote et al. (2010) and Attanasio et al. (2008) though the exact specification of the process we consider is different from these papers. 3 The process specified in (2) fits better into the infinite horizon setup we use. Both Attanasio et al. (2008) and Heathcote et al. (2010) consider models with an explicit life cycle structure. This makes the random walk process in Attanasio et al. (2008) manageable computationally. It also makes the mapping from the estimates of Heathcote et al. (2010) to their model more straightforward, since these authors assume (in both the estimation and the model) that at the beginning of the working life idiosyncratic productivity is drawn from a distribution with high variance. In our model with aggregate uncertainty and couples this requires to abandon the simplistic assumption that households retire and at constant rate become again economically active. 4 We are grateful to an anonymous referee for drawing our attention to this issue. 5 This requires a separate (structural) estimation of the female process to allow the estimates to vary across model specification. A recent attempt to estimate structurally a linearized model of the behavior of couples was done by Blundell, Pistaferri and Saporta-Eksten (2016). As the authors show the specification of utility, and the availability of insurance margins beyond joint labor supply, exerts an influence on the estimated variances and covariances of the processes of idiosyncratic productivity. Blundell et al. (2016) is a sizable step in the right direction; it is evident however, that further research is needed to unravel these forces in dynamic non-linear models such as ours. 4

In order to give insights on how the calibration of the parameters in (2) influences our results we now consider alternative assumptions to replace i) and ii). In particular, we experiment with a correlation of idiosyncratic shocks within the household equal to 0.15 as in Heathcote et al. (2010) and subsequently we replace the calibrated parameters ρ ɛ and σ ɛ with the estimates for the female population reported in Chang and Kim (2006). In a final experiment we consider the impact of increasing σ ɛ keeping the persistence parameter as in the baseline calibration. 1.3.1 Positive correlation of the shocks We chose a value of 0.15 for the correlation as a robustness exercise since we have temporary shocks in the idiosyncratic productivity process, and following the estimates of Hyslop (2001) for these types of shocks. As discussed, the value 0.15 is also employed by Heathcote et al. (2010). 6 The main findings are shown in Table 4. The cyclical properties of the aggregate labor market variables are very close to the benchmark analogues. We continue to obtain a participation which is not strongly correlated with aggregate output at business cycle frequencies. The correlation coefficient now is 0.33 vs. 0.34 in the data and 0.25 in the benchmark model. The ratio of standard deviations is very close to the value reported in the text and to the data moment. All other aggregates (including wages and consumption) behave similarly as in the text. 7 In Tables 5 and 6 we show the cyclical behavior of the distribution of households across S and the behavior of the flow rates in the model with correlated shocks. The business cycle properties are very close to the baseline model. [Tables 4, 5, 6 About Here ] To conclude, in the baseline model in the paper, we have assumed that the innovations in the productivity process are uncorrelated between household members. Assuming correlated shocks and calibrating the correlation coefficient to the values we find in the recent empirical literature (and other quantitative macro papers with couple households) does not alter our findings. 1.3.2 Applying the Estimates for Married Females In Table 7 we report the business cycle moments for employment, unemployment and participation when we assume ρ ɛ = 0.724 and σ ɛ = 0.341. As the table shows, calibrating the idiosyncratic productivity process to the female population moments does not impact the properties of the models. We continue to have a very procyclical labor force participation in the bachelor and complete market 6 Note that it is not straightforward to map the estimates of Hyslop (2001) to our specification of idiosyncratic productivity, or to the analogous specifications used in Heathcote et al. (2010) and Attanasio et al. (2008). The latter papers force the idiosyncratic processes to have identical persistence and standard deviation for men and women (as we do) and chose different values for these objects than the ones reported in Hyslop (2001). Partly this reflects the fact that Hyslop (2001) estimates the wage process in levels whereas most papers in the literature of quantitative macroeconomics specify idiosyncratic productivity assuming log-normal errors. As discussed the estimates in the empirical literature are scarce for the joint process of wages within the household. 7 When we increase the correlation of the idiosyncratic shocks within the household, there are two opposing forces. i) the higher correlation brings closer ɛ 1 and ɛ 2, therefore in response to an unemployment shock suffered by the primary earner, the secondary earner can make up for a larger fraction of the lost income when she works. ii) on the other hand, households become more like bachelors (under identical preferences); they accumulate more assets, and there are more families in S = (E, E) or S = (O, O) (for example). This increases the correlation of the labor force with GDP. The fact that our results are not sensitive to the alternative (reasonably calibrated) covariance structure of shocks within the household, reveals that i) and ii) offset each other in the model. 5

models. The labor force is acyclical under couples households. In fact, the correlation coefficient between the labor force and aggregate output now becomes slightly negative (-0.08). [Tables 7, 8 and 9 About Here ] This property is due to two features of our model with ex ante identical agents. First, the higher standard deviation of idiosyncratic shocks, makes business cycle fluctuations less important for labor supply decisions. The larger are the shocks to the household s income, the less cyclical fluctuations matter since agents will join the labor force when ɛ is high and withdraw when it is low, independent of the phase of the cycle. This reduces the cyclicality of participation. Second, the cyclicality also drops when we lower the persistence of the shocks. To understand why consider the case where ρ ɛ 0. In this case the family becomes indifferent between having agent 1 in the labor market or agent 2, no matter if today ɛ 1 > ɛ 2. In the next period when the new shocks are drawn, agents 1 and 2 will have on average the same productivity. In this case the insurance value of the joint labor supply is magnified and the AWE increases in the model. This also lowers the cyclicality of participation. 8 Notice that the second impact becomes important since we have not adjusted any of the model parameters to bring the insurance value of the joint labor supply to the same level as we have in the benchmark model. Since our model abstracts from gender and heterogeneity in preferences, the persistence of the shocks may exert a more significant influence on the behavior of households, if we keep (for instance) the frictions constant. To show that under the female calibration the model implies more insurance within the household than what the aggregate data tells us, in Table 8 we look at the distribution of households across S. Notice that now the fraction of households with one member in E and the other member in O becomes more procyclical than in the data. As we showed in Section 6 of the paper, the fact that this fraction is procyclical is explained by the incentive of households to bring their secondary earners in the labor force in recessions. The fact that the moment is now more procyclical reveals the stronger potential for insurance when the persistence parameter ρ ɛ is lowered. The model overshoots the data moment under the female calibration. These observations are in line with previous remarks that a structural model is needed to unravel the joint impact of the idiosyncratic shocks and frictions, and at the same time match (as key moments) the cyclical behavior of participation and employment we observe in the US data. The reader should note that if we wanted to keep the female calibration and also to reduce the insurance value of joint search (thus bringing the aggregate moments closer to the data) we could (for instance) reduce the value of p(s, λ s ) relative to the benchmark. In this case fewer households would be presented with the opportunity to bring their secondary earners in the labor market during recessions, and the procyclicality of the (E, O) (O, E) families would be lowered to match the US data moment. 9 8 This should be obvious since in the above example there is (essentially) no primary and secondary earner in the household. Differences in productivity last (on average) for one period. 9 In a previous version of the paper we performed such an exercise experimenting with higher values of ρ ɛ. To change the p(s, λ s ) parameter accordingly we targeted the fraction of households which benefit from the AWE in the model vs the data (see subsequent paragraph). We obtained correlation coefficients between the labor force and GDP in the order of the coefficient reported in the text. 6

1.3.3 Increasing the Variance of Earnings Shocks In the couples model, labor supply decisions are made jointly in the household. Individual supply is therefore influenced by the productivity and the labor market status of the spouse, it is now the joint process of productivity and employment shocks which determine the overall income of the household. It is interesting to investigate whether the distribution of household earnings is different in the couples model than in the bachelor households model. When we summarize the distribution into the earnings GINI coefficient we find the following: First, under bachelors, the earnings GINI is 0.52. In the case of couples, the coefficient for individual earnings is 0.52, but for total family income it is 0.40. Clearly, earnings inequality between households is lower since shocks to productivity and employment shocks are not perfectly correlated within the household. 10 [ Table 10 About Here ] We ask what would be the behavior of the couple households model if the inequality of income between households was similar to the bachelor model? It turns out that if we attempt to increase the GINI coefficient through increasing the variance of the productivity shocks (following De Nardi (2004)) it takes extreme values of the variance to hit the 0.52 target. We find that this happens when σ ɛ is 1.75 times the value we assigned under the benchmark calibration. 11 This prediction is explained by the fact that the model possesses an endogenous participation margin: Household members which have low productivity, do not work and their income always equals zero. It takes a considerable increase in the variance to hit the target. Rather than running the model with incredible calibrations, we want to give the reader a sense of the effects that we find if the standard deviation is increased by 10% relative to the benchmark. We report the business cycle moments in Table 10 where we also show that the new GINI index remains low (0.42) in spite of the substantial increase in the variance of the shocks. The LF participation now has a correlation coefficient of 0.03 with GDP at business cycle frequencies, lower than the benchmark. As we explained in the previous subsection, when the standard deviation of the shocks increases, business cycle fluctuations matter less for labor supply decisions. Individuals join the LF when their productivity is high and they withdraw when productivity is low. This impact complements the family insurance effect in the couples model making participation even less procyclical. 10 This should not be construed to mean that the bachelor model is preferable in terms of matching income inequality. Household income is even more unevenly distributed in the data than the bachelor model predicts. However, in the bachelor model we made the assumption that (many) household heads are out of the labor force, which is inconsistent with the data. Hence the bachelor model generates a more uneven income distribution but through the wrong mechanism. These observations simply tell us that the interactions between household members in terms of the optimal labor supply, have non trivial implications for the earnings processes that we may observe. 11 Note that to match the data GINI coefficient of 0.63 (Census Data) it does not suffice to increase the variance assuming a symmetric distribution (as we do). A substantially larger GINI can only derive from making the distribution skewed to reflect top coded earnings (e.g. Castaneda, Díaz-Giménez and Ríos-Rull, 2003). As discussed in the text, it seems unlikely that adjusting the process to incorporate individuals with 6 digit annual earnings is important for the participation margin. Moreover, compared to the rest of the literature of infinite horizon incomplete market models, our benchmark GINI is already quite high (see Table 1 in Castaneda et al. (2003) for a summary of different papers). 7

1.4 The Specification of the Frictions Recall that in the text we assumed that χ(λ t ) fluctuates in the interval [0.72χ(λ s ), 1.28χ(λ s )]. These parameters have been taken from the data and in particular they were based on the fluctuations of the EU transition probability in the CPS. In the literature of search and matching models of the labor market (e.g. Pissarides (1984, 2000)) it is not uncommon to assume that exogenous separations happen at a constant rate over the business cycle. We consider two alternative parameterizations of the model: we first assume that separations do not fluctuate, χ t = χ(λ s ) = 0.02 for all t, and second that the range of fluctuations is 50% then in the baseline model, [0.86χ(λ s ), 1.14χ(λ s )]s. [Table 11 About Here] In Table 11 we show the model output, summarized (for brevity) in the moments of three variables: employment, unemployment and participation. There are several noteworthy features: Consider first the top panel of the table which shows the case of constant separations. We see that across all models the volatility of aggregate employment has dropped, and labor force participation has become less procyclical. In the case of complete markets, the contemporaneous correlation between the labor force and output is 0.81, in the bachelors model it is 0.77. In the couple households model we record the largest drop on the cyclicality of participation: we now obtain -0.64. Recall that the movements in χ t over the business cycle make the labor force more procyclical, because non-searchers drop to O at faster pace during recessions. Because the arrival rate of job offers is lower in bad times, it takes considerably more time for these agents to flow back into employment. The fact that the cyclicality of participation in the bachelor and complete market models is not considerably impacted even if we assume constant separations, tells us that the movements in the arrival rates alone can account for a large part of the cyclicality in participation in these models. Now consider the behavior of the couples household model. Since with constant separations secondary earners join the LF in recessions but do not experience more frequent subsequent transitions to non-employment, the correlation of participation with GDP becomes very negative. This explains also why aggregate employment is now weakly correlated with economic activity (relative to the data). The moments reported in the bottom panel of Table 11 are for the case when the range is half as wide as in the baseline model. As expected, the results are in between the benchmark model and the constant separations model. We conclude this section with an important observation. Assuming constant separations is inappropriate for all the models we consider because in this case the EU rate becomes very procyclical and not volatile in all models. This is a very much at odds with the empirically observed EU rate. 12 1.5 Modeling Retirement Income We now extend the baseline version of the model to include positive income in retirement. We only consider the impact of this feature in the case of incomplete markets; under complete markets all 12 Constant exogenous job destruction shocks are typically assumed in search and matching models to reproduce a sufficiently negative cyclical correlation between equilibrium unemployment and vacancies (i.e. the Beveridge curve). In models with two labor market states (employment and unemployment) the EU rate coincides with the total separation rate from employment (EO + EU) which in the data is not very cyclical. 8

income in the economy is pooled hence retirement benefits do not matter at all for the optimal allocation. We add retirement income to the model assuming a net replacement ratio of 45% of the individual s current income in employment whɛ t. 13 To finance the benefits we assume that a (payroll) tax is levied on the employed population. The tax rate is set to ensure that the government runs a balanced budget each period. We obtain a value equal to 9.2% in the steady state. In the model with aggregate shocks, we adjust the tax rate to achieve a balance the government s budget in every period. 14 In Table 12 we show the business cycle properties of the model with social security. The parameters p(s, λ s ),p(s, λ s ) and χ(λ s ) are as in the baseline version of the model. The preference parameters are again recalibrated to hit the targets for employment, unemployment and interest rates. Panel A of the table shows the results of the couples model and panel B of the bachelors model. Notice that in the case of the couples model the labor force becomes mildly countercyclical. In the bachelor model it remains strongly procyclical. To understand these results, note that in both models social security crowds out private savings (even though we have recalibrated β). In the couples model, households hold a lower buffer stock of savings and when the unemployment shock hits, they will use family insurance more to compensate for the idiosyncratic risk. In the bachelors model lower savings will not reduce the labor force procyclicality because a larger mass of households is concentrated at the unemployment to out of the labor force margin, bestowing a strong intertemporal substitution effect which keeps participation correlated with the business cycle. Thus, including social security does not alter our qualitative results and it does not weaken the family insurance mechanism. To the contrary, the importance of family insurance is increased, making the labor force in the couples model even less procyclical. [Table 12 About Here] 1.6 AWE: Model vs. Data As discussed in the text, the model gives rise to AWEs when primary earners become unemployed (after the arrival of a separation shock) and secondary earners accept job offers they would otherwise reject- or flow to unemployment to engage in joint search. We had mentioned in the text that whereas in the US data families frontload the AWE (we observe a sizable response in the month the unemployment shock occurs) the model gives rise to a different pattern, whereby secondary earners increase their participation progressively as unemployment persists and household wealth is run down. We now compare the AWEs from the model to the estimates from the US data to give sense of how the different timing impacts the α τ for τ = 2, 1, 0, 1, 2 coefficients (see Section 2 in 13 This is a simplification made for computational purposes. Since ours is essentially an infinite horizon model it is difficult to keep track of the lifetime earnings of individuals (some individuals become retired after a few periods and others after many periods). Moreover, since the retirement period is short in the model and productivity is persistent and has zero correlation across household members, households will not suffer considerable changes in their total income over the retirement period. When we simply assume that retirement income is a constant fraction of average labor income in the economy our results do not change considerably. 14 To find the equilibrium tax rates, we again follow the Krusell-Smith approach. We specify a tax function that is linear in the aggregate states, initialize its coefficient to the steady-state tax rate and update the coefficients until the budget is in balance (up to an approximation error) in each period. 9

paper) and the percentage of households that benefit from the AWE. As we shall show, though in the model the α τ s fall short of the data, in the broader picture the dynamic response of secondary earner labor supply to an unemployment shock suffered by the primary earner is consistent with the US data. [Table 13 About Here] In Table 13 we show the coefficients. The data moments are in Column 1 and the model analogues in Column 2. Note first, that the model yields α 2 = α 1 = 0 since the job destruction shocks are i.i.d and therefore not predictable. Second, the model generates a contemporaneous value for the AWE equal to 1.6 percentage points, an effect after one month (α +1 ) equal to 1.5% and an effect after two months equal to 2.0%. These numbers are smaller than their data counterparts. In rows 6-10 of the table we extended the horizon to characterize the response of the AWE in the model for τ = 3, 4,..., which we do not observe in the data. Notice that the model continues to generate a significant AWE for several months after the unemployment shock occurs and indeed the AWE increases over time. Therefore, families progressively add secondary earners to the labor market, as unemployment continues and wealth is run down. Unfortunately these moments cannot be estimated with the US data, however, a simple inspection of the estimated coefficients α 0, α 1 and α 2 suggests that the AWE decreases over time after the unemployment shock occurs. There are several hypotheses which may explain why the AWE is much more likely to be frontloaded in the data than in the model. Recall that the model assumes that all household wealth is liquid, therefore households can run down smoothly their endowment to buffer shocks to unemployment. In reality a large part of wealth is likely to be illiquid; US households may therefore use joint labor supply more readily since assets offer to them fewer insurance opportunities against unemployment than in the heterogeneous agent model. 15 Second, some unemployment shocks may lead to persistently lower earnings (if human capital is run down or agents lose part of their search capital). This mechanism, which has been left outside the model, ought to induce a larger and more frontloaded AWE. 16 Finally, as we argued in the text, the model requires large search costs to match the average unemployment rate. If search (and participation) costs are smaller for some households than for others, then low cost households will join the labor force when the unemployment shock occurs, in subsequent periods, high cost households become less likely to join. The α τ coefficients that we observe in the data are influenced by selection effects which are not included in the model. [Table 13 About Here] 15 The argument is similar to Kaplan and Violante (2014). They show that illiquid wealth can be useful against large income shocks, but not temporary and small shocks such as unemployment. In the same vein US households may have consumption commitments such as rents, mortgages etc. The presence of such commitments will make the adjustment of spousal labor supply more urgent. Notice that implicit in this argument is that the distribution of wealth is endogenously determined in the model. Theoretically, we can increase the size of the α τ coefficients if we can lower the wealth level of marginal households. The general equilibrium discipline that we have put to our model does not allow us to do so. In partial equilibrium we can for example lower r and thus make precautionary savings more costly, this will frontload the AWE bringing the model closer to the data. 16 We are quite confident that this mechanism is plausible. Recall that in the text we argued that quits lead to sizable AWEs owing to the fact that both quits and layoffs lead to permanent earnings losses. Though this may seem not likely in the case of quits, in reality many worker s quit their jobs after suffering a worsening of the work conditions. 10

In Columns 3 and 4 of the table we report fractions of families which (across different horizons) benefit from the AWE (data moments are in Column 3). The model numbers are constructed by simulating a population of households which at the beginning of period -2 have primary earners employed and secondary earners O. At τ = 0 primary earners become unemployed. We then calculate the fractions of couples in which the secondary earner joins the labor force. 17 Notice that these fractions drop with τ for two reasons: First, because when the secondary earner joins the household is dropped from the sample. Second, many unemployment spells are resolved by a transition of the primary earner back into employment. In this case, the couple moves back to state S = (E, O) and hence is also dropped from the sample (see data appendix for a discussion of the sample selection we employed to estimate the AWE). Columns 3-4 reveal that the model matches the data moments. At τ = 0 for instance we have that 13% of households benefit from the presence of the secondary earner in the data, in the model this fraction is 9.1%. At τ = 1 we obtain 9.1% and 6.8% in data and model respectively. At τ = 2 the model produces 4.4% whereas the fraction in the data is 3.2%. If we add all the rows reported in the table we obtain the cumulative effect: The total fraction of households that benefit from joint labor supply is 47.4% in the data. In the model the analogous number is 38.4% if we stop at τ = 2. If we account for the fact that the AWE takes longer to materialize in the model and add the numbers for τ = 3, 4,... we will obtain 44.7% as the model fraction. The main takeaway from this subsection is that both in the model and the US data, a significant fraction of households mobilize the secondary earner to join the LF in response to an unemployment shock suffered by the primary earner. 1.7 Averaging As discussed in the text we used Monte Carlo simulations to construct objects Υ(ɛ) (to compute primary and secondary earners in the model). For each value of ɛ we pick a retirement date T. We then simulate the sequences of idiosyncratic productivity for many households from date 0 to T 1 and count the number of times that agent 1 has the highest average productivity. This gives us the probability that agent 1 is the primary earner conditional on the retirement date. To obtain the unconditional probability we perform the simulations many times (each time for a different T up until a sufficiently large value of T so that our results are robust) and average across retirement dates (using the formula described in the text). To construct the populations of primary and secondary earners we proceed as follows: For each realization of the ɛ vector we let y primary t (ɛ) = Υ(ɛ)yt 1 (ɛ) + (1 Υ(ɛ))yt 2 (ɛ), where y is a variable of interest (e.g. y = E-pop, U-rate, LF etc). Then, we construct y primary t averaging across productivity levels. y secondary t can be created in a similar fashion. Notice that using the above procedure is equivalent to simulating populations and looking ex post at the realized productivities of the individuals. As we later explain, when we solve the models we do not simulate panel data of households. To avoid sampling errors we use the approach of Young (2010) (we work with histograms). This is a further reason why averaging is convenient. 17 We set the entry rates at τ = 1 and τ = 2 equal to their unconditional means in the model. Even though unemployment shocks are unanticipated they are mitigated if both earners are in the labor force. 11

Table 1: Log-Separable Utility: Preference Parameters Preferences β η κ Couples 0.993 0.464 0.283 Bachelors 0.992 0.424 0.251 Complete Markets 0.996 0.511 0.250 Note: The table shows the specification of preferences in the three models when we assume logseparable utility. The preference parameters β, κ and η are such that the models produce an E pop ratio of 62% and an unemployment rate of 6.2% in steady state. All other calibrated parameters of the log-utility models (e.g. frictions and idiosyncratic productivity) have the same values as in the benchmark version of the model (see text). Table 2: Steady State Flows in the Models Log Utility A: Couples B: Bachelors C: Complete Markets To To To From E U O E U O E U O E 0.957 0.009 0.034 0.958 0.008 0.034 0.959 0.011 0.030 U 0.258 0.672 0.071 0.258 0.684 0.058 0.257 0.672 0.068 O 0.046 0.024 0.930 0.046 0.024 0.930 0.045 0.020 0.935 Note: The table shows average transition probabilities across labor market states from the 3 models with log utility: Panel A shows the baseline with couple households, panel B shows the model of bachelor households, and panel C shows the complete market model. E represents employment, U unemployment and O out of the labor force. 12

Table 3: Changing the frictions: Additional Results κ NS E-pop U-rate LF ρe pop,lf w C I σe pop,y ρe pop,y σu rate,y ρu rate,y σlf,y ρlf,y σw,y ρw,y σc,y ρc,y σi,y ρi,y p ( s, λ ) = 0.19 Couples 0.19 0.080 0.61 0.85 7.92-0.95 0.23-0.03 0.45 0.58 0.83 0.41 0.93 3.26 0.99 Bachelors 0.16 0.080 0.89 0.96 7.94-0.96 0.33 0.91 0.97 0.28 0.51 0.62 0.75 3.20 0.90 Complete Markets 0.16 0.071 1.09 0.97 8.65-0.96 0.48 0.90 0.97 0.28-0.03 0.39 0.93 3.31 0.98 p ( s, λ ) = 0.22 Couples 0.10 0.072 0.60 0.85 8.31-0.95 0.21-0.24 0.26 0.58 0.85 0.41 0.94 3.24 0.99 Bachelors 0.09 0.070 0.86 0.96 8.49-0.96 0.27 0.84 0.94 0.31 0.58 0.62 0.74 3.22 0.90 Complete Markets 0.08 0.064 1.00 0.96 9.30-0.96 0.36 0.82 0.93 0.28 0.11 0.38 0.94 3.30 0.98 p ( s, λ ) = 0.25 Couples 0.02 0.064 0.60 0.86 8.66-0.96 0.22-0.35 0.15 0.57 0.85 0.41 0.94 3.23 0.99 Bachelors 0.02 0.063 0.82 0.95 8.94-0.97 0.22 0.68 0.85 0.34 0.64 0.63 0.73 3.25 0.89 Complete Markets 0.02 0.056 0.97 0.96 9.52-0.96 0.31 0.74 0.89 0.29 0.23 0.39 0.94 3.28 0.98 Note: The table shows the results of the three models when the probability of receiving a job offer for out of the labor force individuals is increased. 13

Table 4: Correlated Productivity Shocks: Business Cycle Moments U-rate E-pop LF w C I σ x,y 7.44 0.68 0.27 0.53 0.41 3.27 ρ x,y -0.94 0.87 0.33 0.77 0.93 0.98 Note: The table shows business cycle moments in the couples models when productivity shocks are positively correlated. The preference parameters are β = 0.992, η = 0.451 and κ = 0.274 are calibrated to replicate the same steady state as the baseline model, i.e. E pop ratio of 62%, U rate ratio of 6.2% and a risk-free interest rate of 0.41%. Table 5: Correlated Productivity Shocks: Joint Labor Market Status Data Model σ S ρ x σ x,y σ Y ρ x x,y σ Y (E,E) 0.72 1.06 0.82 1.29 (E,O) (O,E) 0.49 1.12 0.46 1.06 (E,U) (U,E) -0.90 11.75-0.90 7.69 (U,U) -0.81 21.47-0.97 8.34 (U,O) (O,U) -0.88 12.66-0.97 6.36 (O,O) -0.03 0.80-0.80 0.34 Note: The table shows the contemporaneous correlation ρ x,y and the relative standard deviation σx σ Y between the fraction of households in state S (joint status) and aggregate output. The data corresponds to married couples in the US in the period 1994-2014. The model is the couples model with positively correlated productivity shocks. (E,O) (O,E) denotes households where one member (either agent 1 or agent 2, husband or wife in the data) is employed and the other member is out of the labor force. We define analogously (E,U) (U,E) and (U,O) (O,U). Finally, all series are logged and HP filtered with smoothing parameter equal to 1600. Table 6: Correlated Productivity Shocks: Flow Rates ρ x,y Data Model σ x σ Y ρ x,y σ x σ Y EU -0.83 6.41-0.71 3.66 EO 0.49 2.62 0.02 3.70 UE 0.87 7.11 0.91 7.72 UO 0.74 4.18-0.41 4.32 OE 0.62 3.30 0.66 2.59 OU -0.81 6.73-0.84 7.53 Note: The table shows the contemporaneous correlation ρ x,y and the relative standard deviations σx σ Y between labor market flows and de-trended GDP. Columns 2 and 3 report the US data moments. Columns 4 and 5 show the results of the couples model with positively correlated productivity shocks. 14

Table 7: Business Cycle Properties: Female calibration Couples Bachelors Complete ρ x,y U-rate -0.94-0.95-0.96 E-pop 0.79 0.96 0.96 LF -0.08 0.93 0.89 σ x,y U-rate 7.48 7.39 8.57 E-pop 0.61 0.88 1.07 LF 0.29 0.36 0.47 Note: The table shows the business cycle properties of the three models when the productivity process is calibrated using the estimates for ρ ɛ and σ ɛ from the married women sample, reported in Chang and Kim (2006). Table 8: Female Calibration: Joint Labor Market Status Data Model σ S ρ x σ x,y σ Y ρ x x,y σ Y (E,E) 0.72 1.06 0.72 1.15 (E,O) (O,E) 0.49 1.12 0.62 1.30 (E,U) (U,E) -0.90 11.75-0.90 8.13 (U,U) -0.81 21.47-0.97 8.62 (U,O) (O,U) -0.88 12.66-0.95 5.99 (O,O) -0.03 0.80-0.36 0.31 Note: The table shows the contemporaneous correlation ρ x,y and the relative standard deviation σx σ Y between the fraction of households in state S (joint status) and aggregate output. The data corresponds to married couples in the US in the period 1994-2014. The model is the couples model with the productivity process based on the estimates for married women in Chang and Kim (2006). (E,O) (O,E) denotes households where one member (either agent 1 or agent 2, husband or wife in the data) is employed and the other member is out of the labor force. We define analogously (E,U) (U,E) and (U,O) (O,U). Table 9: Female Calibration: Flow Rates ρ x,y Data Model σ x σ Y ρ x,y σ x σ Y EU -0.83 6.41-0.67 3.64 EO 0.49 2.62 0.18 4.40 UE 0.87 7.11 0.92 8.06 UO 0.74 4.18-0.44 4.28 OE 0.62 3.30 0.72 2.82 OU -0.81 6.73-0.86 7.72 Note: The table shows the contemporaneous correlation ρ x,y and the relative standard deviations σx σ Y between labor market flows and de-trended GDP. Columns 2 and 3 report the US data moments. Details on the data can be found in the online data appendix. Columns 4 and 5 show the results of the couples model with the productivity process based on the estimates for married women. 15

Table 10: Higher Uncertainty Model A: Benchmark B: Higher uncertainty (Gini=0.40) (Gini=0.42) E-pop U-rate LF E-pop U-rate LF σ x σ y 0.67 7.53 0.26 0.62 7.46 0.26 ρ x,y 0.86-0.94 0.25 0.83-0.95 0.03 Note: The table shows the results of the benchmark couples model in panel A. The Gini coefficient of total household earnings in this model is 0.40. Panel B shows the results of the model when we increase variance of the idiosyncratic productivity process by 10%. Table 11: Lower Variance of χ shocks: Business Cycle Properties Couples Bachelors Complete A: Constant Separations ρ x,y U-rate -0.93-0.94-0.96 E-pop 0.41 0.91 0.93 LF -0.64 0.77 0.81 σ x,y U-rate 6.91 7.01 7.66 E-pop 0.47 0.73 0.94 LF 0.40 0.25 0.41 B: Low Variance Separations ρ x,y U-rate -0.94-0.95-0.96 E-pop 0.74 0.95 0.95 LF -0.25 0.89 0.87 σ x,y U-rate 7.26 7.29 8.20 E-pop 0.56 0.84 1.03 LF 0.28 0.32 0.46 Note: The table shows the cyclical behavior of labor market aggregates in model simulations when we assume that separation shocks i) are constant over the cycle (top panel) and ii) have 50% of the variance in the benchmark calibration (bottom panel). Table 12: Retirement Income Model: Business Cycle Properties A: Couples B: Bachelors E-pop U-rate LF E-pop U-rate LF σ x σ y 0.58 7.47 0.33 0.94 7.76 0.31 ρ x,y 0.72-0.94-0.25 0.97-0.97 0.93 Notes: The table shows the behavior of labor market aggregates (employment, unemployment rate and LF participation) when we assume a positive income for retired individuals. Panel A shows the result for the couples model and Panel B for the bachelors model. 16

Table 13: Dynamic Added Worker Effect: Data and Model α τ % of Households τ Data Model Data Model -2 1.9% 0 11.4% 9.5% -1 3.2% 0 10.7% 8.6% 0 7.8% 1.6% 13.0% 9.1% +1 5.1% 1.5% 9.1% 6.8% +2 3.9% 2.0% 3.2% 4.4% +3-2.5% - 3.0% +4-2.9% - 2.0% +5-3.3% - 1.3% +6-3.6% - 0.9% +7-3.9% - 0.6% Note: The Table shows the estimates of the AWE in the data (Column 1) and the model (Column 2). Columns 3 and 4 show fractions of households that benefit from the AWE across horizons (see text for further details). 17

Figure 1: Wealth Distributions: Benchmark Model vs. Log Utility 2.5 x 10 3 Not Employed Bachelors 7 x 10 3 Employed Fractions of Households 2 1.5 1 0.5 0 0 200 400 600 Wealth Thousands US Dollars Fractions of Households Couples 6 5 4 3 2 1 0 0 200 400 600 Wealth Thousands US Dollars Baseline Log Utility Fractions of Households 4 x 10 3 3 2 1 2.5 x 10 3 8 x 10 4 (E,E) (E,n) (n,n) 2 6 Fractions of Households 1.5 S=(E,E) 1 0.5 Fractions of Households 4 2 0 0 200 400 600 0 0 200 400 600 Wealth Thousands US Dollars 0 0 200 400 600 Notes: The figure plots the steady state distributions of wealth in the benchmark model (solid line) and the log utility model (dashed line) for the bachelor and the couple households models. The top graphs (left and right) show the case of bachelors. The top left panel shows the wealth distribution for employed households and the top right the distribution for non-employed households. The bottom graphs represent the couples model. The joint status of household members in this case is denoted on each graph. Notice that to fit the graphs we denote the joint status using symbols E (employed agents) and n (agents without a job offer). 18

2 Data appendix 2.1 CPS: Description, Definitions and Filtering The Current Population Survey (CPS) is a monthly survey of about 60,000 households (56,000 prior to 1996 and 50,000 prior to 2001), conducted jointly by the Census Bureau and the Bureau of Labor Statistics. Survey questions cover employment, unemployment, earnings, hours of work, and a variety of demographic characteristics such as age, sex, race, marital status, and educational attainment. Although the CPS is not an explicit panel survey it does have a longitudinal component that allows us to construct the monthly labor market transitions in Section 2 of the paper. Specifically the design of the survey is such that the sample unit is interviewed for four consecutive months and then, after an eight-month rest period, interviewed again for the same four months one year later. Households in the sample are replaced on a rotating basis, with one-eighth of the households introduced to the sample each month. Given the structure of the survey we can match roughly three-quarters of the records across months. 18 Using these matched records, we calculate the gross worker flows (for the aggregate and by gender age group and marital status) that we report in Section 2 of the paper. Our sample covers the period 1994 (January) 2014 (October). The flows are estimates of a Markov transition matrix where the three states are employment, unemployment and out of the labor force. We also broaden the definition of the labor market status variable when we explicitly consider non searchers, this gives a four by four matrix of transitions. 19 state. We use the CPS classification rule to assign each member of a household to a labor market This rule is as follows: Employed agents are those who did (any) work for either pay or profit during the survey week. Unemployed are those who do not have a job, have actively looked for work in the month before the survey, and are currently available for work. Actively looking means that respondents have used one (or more) of the nine search methods considered by the CPS (6 methods prior to 1994) such as sending out resumes, responding to job adds, being enrolled with a public or private employment agency etc. Individuals who search Passively by attending a job training program or simply looking at adds are not considered as unemployed because these methods, according to the CPS, do not result in a sufficiently high arrival rate of job offers. The exception is workers on temporary layoff, i.e those workers who expect to be recalled by their previous employer. Those are counted as unemployed even if they do not search actively. Finally, out of labor force are all agents who are neither employed nor unemployed (based on the above definitions). Amongst these we can (in the post 1994 period) separate individuals by reason of inactivity (for example, schooling, family responsibilities, disability, retirement etc) when we are interested in a finer selection of participants in this group. We obviously can also distinguish between individuals who are out of the labor force as non searchers. Given this information we calculate the conditional probability that an agent who is in state i in the previous month (interview date) is in state j this month, where (i, j) {E, U, O}. We use the 18 Unfortunately, there is some sample attrition from individuals who abandon the survey (see for example Nagypál (2005) for a discussion of these issues). 19 The survey allows us to identify non searchers in the post 1994 period, in previous years we can only identify an individual s labor market status as {E, U, O}, but not the distinction between individuals who want jobs but do not search and those who do not want jobs. 19