Appendices to: Revisiting the Welfare Effects of Eliminating Business Cycles

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1 Appendices to: Revisiting the Welfare Effects of Eliminating Business Cycles Per Krusell Toshihiko Mukoyama Ayşegül Şahin Anthony A. Smith, Jr. December 2008 A The integration principle applied to the baseline model The 4 4 Markov transition matrix for (ɛ, z), in its calibrated form, reads , with ordering (e, g), (e, b), (u, g), (u, b). This implies probabilities of finding a job, conditional on last period s employment status on the aggregate shocks in the current in the last period. These job-finding probabilities can be ranked from least to most lucky as follows: the probability of becoming employed next period (ɛ = 1) is, 1. conditional on ɛ = 0, z = g, z = b, / = ī 1 ; 2. conditional on ɛ = 0, z = b, z = b, / = ī 2 ; 3. conditional on ɛ = 0, z = g, z = g, / = ī 3 ; 4. conditional on ɛ = 0, z = b, z = g, / = ī 4 ; 5. conditional on ɛ = 1, z = g, z = b, / = ī 5 ; 6. conditional on ɛ = 1, z = b, z = b, / = ī 6 ; 7. conditional on ɛ = 1, z = g, z = g, / = ī 7 ; 8. conditional on ɛ = 1, z = b, z = g, / = ī 8. Princeton University, IIES, CEPR, NBER University of Virginia CIREQ Federal Reserve Bank of New York Yale University 1

2 Thus, our idiosyncratic shock i t can end up in 9 relevant subintervals, defined by the cutoff values ī 1 ī 9, in each period. Let us take an example: if i [ī 6, ī 7 ), the agent would have been employed currently only if the aggregate idiosyncratic shocks leading up to the last period made the agent employed in that period the current aggregate state would have been bad (independently of what the state was last period). For a realized sequence of idiosyncratic shocks {i s } t s=1 one can then compute an average employment outcome in period t by brute-force averaging across all {z s } t s=1 sequences (appropriately weighted by probabilities): given each such sequence of aggregate shocks, together with an employment status in period 0, the employment outcomes in all periods up to including t are known: they follow applying the cutoff values above in every time period. The resulting employment process will have long memory in terms of the idiosyncratic shocks: one generally needs to know all prior values of i s, s < t, in order to know what the average employment outcome is at t. However, it is possible to represent the new employment process recursively. To this end, let P gt denote the probability (or fraction of the time) that, among all possible outcomes of the aggregate process, (i) the individual would have been employed in time t, given his initial (time-0) employment status an initial (time-0) value for the aggregate state AND (ii) the aggregate state at time t would have been good. Similarly define P bt as the probability that the agent would have been employed in t jointly with a bad aggregate state in that period. Letting π t denote the probability of a good aggregate state in period t given z 0, these definitions imply that π t P gt is the probability that the agent would have been unemployed in t jointly with a good aggregate state in t similarly that 1 π t P bt is the probability that the agent would have been unemployed in t jointly with a bad aggregate state in t. The key insight now is that the variables P t (P gt, P bt ) summarize all there is to know from history in order to know the expected (average) value for employment in period t + 1 given a value for i t+1. I.e., P t summarizes all the relevant knowledge about {i 1, i 2,..., i t }. This representation is possible because the joint underlying process for employment the aggregate state is first-order Markov. The recursive structure needs to update P t into P t+1 given a value for i t+1, it needs to assign a value for the average employment outcome in period t + 1 conditional on the state variable P t summarizing the individual s idiosyncratic history i t+1, the new idiosyncratic shock. The latter is easy: the average value of employment across the aggregate shock outcomes will be ɛ w/o t+1 = P g,t+1 + P b,t+1, because g b are disjoint outcomes. To underst how to update P t given i t+1 (in order to obtain P t+1 ), note that in any given period in the economy with cycles the consumer is in one of four possible states: he is either employed or unemployed the aggregate state is either good or bad. Denote these states by (1, g), (1, b), (0, g), (0, b). (As noted above, the probabilities of these four states can be deduced from knowledge of P gt, P bt, the probability that the aggregate state is good in period t.) Suppose that the consumer is in state (1, g) in period t that the aggregate state in period t + 1 is also good. If the consumer s luck in period t + 1, i t+1, is sufficiently good, he will also be in state (1, g) in period t + 1. In this case, the consumer s luck is sufficiently good if i t+1 < π 1 1gg P (ɛ t+1 = 1 ɛ t = 1, z t = g, z t+1 = g). The conditional probability π 1 1gg, therefore, is a cutoff 2

3 that determines whether the consumer s luck is good or bad, given that he is employed in t that the aggregate state is good in both t t + 1. In total, there are eight such cutoffs, one for each of the eight possible permutations of (ɛ t, z t, z t+1 ), these eight cutoffs define the 9 regions described above. In period t, the consumer could also be in state (1, b), (0, g), or (0, b). In each case, if the aggregate state in period t + 1 is good the consumer s luck in t + 1 is below the relevant cutoff, he will be in state (1, g) in period t + 1. P g,t+1 is then a weighted average of four indicator functions (each of which indicates whether i t+1 is above or below the appropriate cutoff). The weights corresponding to the period-t probabilities of the four states (1, g), (1, b), (0, g), (0, b), multiplied in each case by the conditional probability of transiting to a good aggregate state in t + 1 given the aggregate state in t. Similarly, P b,t+1 is also the weighted average of four indicator functions, appropriately weighted. As described above, in the baseline model the variable i t+1 can fall into any one of the 9 regions defined by the cutoffs ī 1 ī 9 above. The exact updating formulas for the baseline model are: 1. i t+1 [0, ī 1 ): P g,t+1 = π t+1 P b,t+1 = 1 π t+1 ; 2. i t+1 [ī 1, ī 2 ): P g,t+1 = π t+1 P b,t+1 = P gt π b g + (1 π t )π b b 3. i t+1 [ī 2, ī 3 ): P g,t+1 = π t P b,t+1 = P gt π b g + P bt π b b ; 4. i t+1 [ī 3, ī 4 ): P g,t+1 = P gt π g g + (1 π t )π g b P b,t+1 = P gt π b g + P bt π b b ; 5. i t+1 [ī 4, ī 5 ): P g,t+1 = P gt π g g + P bt π g b P b,t+1 = P gt π b g + P bt π b b ; 6. i t+1 [ī 5, ī 6 ): P g,t+1 = P gt π g g + P bt π g b P b,t+1 = P bt π b b ; 7. i t+1 [ī 6, ī 7 ): P g,t+1 = P gt π g g + P bt π g b P b,t+1 = 0; 8. i t+1 [ī 7, ī 8 ): P g,t+1 = P bt π g b P b,t+1 = 0; 9. i t+1 [ī 8, 1]: P g,t+1 = P b,t+1 = 0. One needs to spell through these carefully to see that they are correct. We have, for verification we have also (i) simulated this process for various draws of the idiosyncratic process {i t } T t=1 (where T is large), based on the resulting {P t } T t=1 sequence, computed the associated employment outcomes (ii) made sure that the resulting values are replicated for the same {i t } T t=1 draws by a brute-force averaging across aggregate shock processes. They do. Appendix B displays the general updating formula (expressed as a weighted average of indicator functions), for the baseline case (expressed as a weighted average of indicator functions), Appendix D displays it for the model with short- long-term unemployment. 3

4 B Computational algorithm for the benchmark model B.1 General algorithm This section outlines the computational algorithm applying the integration principle to our model computing the transition path. 1 Note that the business cycle is eliminated in the beginning of period 1, after all the period-1 shocks are realized. Thus z 1 k 1 are given. At the individual level, the distribution of k 1, ɛ 1, β 1 are given from one point in the simulation that corresponds to z 1 k 1. ɛ 1 provides the initial conditions for each individuals: if z 1 = g ɛ 1 = 1, P g1 = 1 P b1 = 0; if z 1 = b ɛ 1 = 1, P g1 = 0 P b1 = 1; if ɛ 1 = 0, P g1 = 0 P b1 = 0. (Recall that P gt is the joint probability of z t = g ɛ t = 1 that P bt is the joint probability of z t = b ɛ t = 1.) The general computational strategy is to first postulate the time path of aggregate capital, solve for the agents decisions given this path, then verify that that the time path for aggregate capital implied by agents aggregated decisions matches the postulated time path. We postulate the time path for 600 periods. Then we divide the 600 periods into the first 125 periods the final 475 periods. We solve the consumer s problem backwards first solve for the final 475 periods then for the first 125 periods. After the optimization problem is solved, we simulate the economy with many consumers (we use 90,000 consumers) generate the path for aggregate capital by summing up individual savings. Finally, we check whether this simulated path of capital stock is the same as the initially postulated time path. The following explains these steps more in detail. 1. First, postulate the path of the aggregate capital stock for 600 periods. We use the average of the law of motions of the capital stock in the fluctuating economy to generate the initial guess. 2. We solve the consumer s problem backwards. In the final 475 periods, the exogenous variables, such as z, u, π z are set to their limit values (z = 1, u = (u g + u b )/2, π z = 0.5). Thus, we treat this economy as a stationary on except for the movement in k ( k settles much more slowly than do the exogenous variables). We summarize the movement of the capital stock by the law of motion k = H( k). In practice, we use a (log-)linear function for H( ), with the initial value of the law of motion obtained by applying ordinary least squares to the data on aggregate capital for the final 475 periods. The Bellman equation is: subject to V (k, P g, P b, β; k) = max c,k {U(c) + βe[v (k, P g, P b, β ; k ) P g, P b, β]} c + k = r( k, 1 ū)k + w( k, 1 ū)(p g + P b ) + g(1 P g P b ) + (1 δ)k, 1 Also see Mukoyama Şahin (2005, Appendix C D) for a detailed exposition of the implementation of the integration principle. Note that Krusell Smith (2002) Mukoyama Şahin (2005, 2006) use Markov approximations to the P processes while here we use the P s directly in the computation. 4

5 k = H( k). As is explained in Appendix A, P g P b are functions of a rom variable i U[0, 1]. P g(i ) = Pr[z = g, ɛ = 1 i ] = z=g,b [Pr[z = g, ɛ = 1 i, z, ɛ = 1]P z + Pr[z = g, ɛ = 1 i, z, ɛ = 0](1/2 P z )] = z=g,b [Pr[ɛ = 1 i, z = g, z, ɛ = 1]π g z P z + Pr[ɛ = 1 i, z = g, z, ɛ = 0]π g z (1/2 P z )] = z=g,b [I(i π 11 zg )π g z P z + I(i π 01 zg )π g z (1/2 P z )] P b (i ) = Pr[z = b, ɛ = 1 i ] = z=g,b [Pr[z = b, ɛ = 1 i, z, ɛ = 1]P z + Pr[z = b, ɛ = 1 i, z, ɛ = 0](1/2 P z )] = z=g,b [Pr[ɛ = 1 i, z = b, z, ɛ = 1]π b z P z + Pr[ɛ = 1 i, z = b, z, ɛ = 0]π b z (1/2 P z )] = z=g,b [I(i π 11 zb )π b z P z + I(i π 01 zb )π b z (1/2 P z )], where I( ) is the indicator function equaling 1 if the statement is true 0 if it is false. Note that this is just a compact way of writing the transition rules in Appendix A. The prices are: r( k, 1 ū) = α k α 1 (1 ū) 1 α w( k, 1 ū) = (1 α) k α (1 ū) α. The expectation operator in the Bellman equation is taken over i β values. In practice, we divide the [0, 1] into = 9 subintervals when we take an expectation with respect to the i s, as is explained in Appendix A. The dynamic-programming problem is solved in a similar way to that used in Krusell Smith (1998). We use 6 grids in each P direction (or, more precisely, on the conditional probability ˆP, as is detailed below) apply linear interpolation to the value function when evaluating the values under P (which are usually not on the grid). 3. For t = 1,..., 125, we solve backwards for the path. Now the exogenous parameters move over time we treat each period differently (for example, the value function has the index t). First, we provide the terminal value function: V 126 (k, P g, P b, β) = V (k, P g, P b, β; k 126 ), we then calculate the probability that the aggregate state is z at time t, π z t, for t = 1,..., 125: π g 1 = 1 if z 1 = g π g 1 = 0 if z 1 = b. It is always the case that π b t = 1 π g t, π g t+1 = πg t π g g + (1 π g t )π g b. z t can be calculated using z t = π g t g + (1 πg t )b. u t can also be calculated as z t = π g t u g + (1 π g t )u b. Thus the prices are r t = αz t kα 1 t (1 u t ) 1 α w t = (1 α)z t kα t (1 u t ) α. 5

6 Given these, we can solve the consumer s problem, working backwards. The Bellman equation is: V t (k, P g, P b, β) = max c,k {U(c) + βe t [V t+1 (k, P g, P b, β ) P g, P b, β]} subject to c + k = r t k + w t (P g + P b ) + g(1 P g P b ) + (1 δ)k, Again, P g P b are functions of a rom variable i U[0, 1]. P g(i ) = Pr[z = g, ɛ = 1 i ] = z=g,b [Pr[z = g, ɛ = 1 i, z, ɛ = 1]P z + Pr[z = g, ɛ = 1 i, z, ɛ = 0](π z t P z )] = z=g,b [Pr[ɛ = 1 i, z = g, z, ɛ = 1]π g z P z + Pr[ɛ = 1 i, z = g, z, ɛ = 0]π g z (π z t P z )] = z=g,b [I(i π 11 zg )π g z P z + I(i π 01 zg )π g z (π z t P z )] P b (i ) = Pr[z = b, ɛ = 1 i ] = z=g,b [Pr[z = b, ɛ = 1 i, z, ɛ = 1]P z + Pr[z = b, ɛ = 1 i, z, ɛ = 0](π z t P z )] = z=g,b [Pr[ɛ = 1 i, z = b, z, ɛ = 1]π b z P z + Pr[ɛ = 1 i, z = b, z, ɛ = 0]π b z (π z t P z )] = z=g,b [I(i π 11 zb )π b z P z + I(i π 01 zb )π b z (π z t P z )]. 4. Having solved the consumer s decision problem, we can simulate the economy. We assign an initial distribution for the individual state variables then simulate consumers decisions (we use 90,000 consumers). Adding up the implied saving choices, we obtain the time series for k t. Using this path, we check whether it reproduces the initially postulated path. If not, we update k 2,..., k 126 the law of motion H( k) by ordinary least squares. We repeat until convergence. B.2 Issues in actual implementation In practice, we work on the conditional probabilities for P s rather than with the joint probabilities. We define ˆP z as the probability of being employed conditional on the aggregate state z. Here, in the first step (since the probability of each aggregate state is 1/2), ˆP g = P g 1/2 = 2P g (1) ˆP b = P b 1/2 = 2P b. (2) The new problem becomes (with the new value function ˆV ) subject to ˆV (k, ˆP g, ˆP b, β; k) = max c,k {U(c) + βe[ ˆV (k, ˆP g, ˆP b, β ; k ) ˆP g, ˆP b, β]} ( 1 c + k = r( k, 1 ū)k + w( k, 1 ū) 2 ˆP g + 1 ) 2 ˆP b + g (1 1 2 ˆP g 1 2 ˆP b ) + (1 δ)k, 6

7 k = H( k). From (1) (2), ˆP g (i ), ˆP g (i ) can be calculated by ˆP g(i ) = 2P g(i ) = 2 z=g,b [I(i π 11 zg )π g z P z + I(i π 01 zg )π g z (1/2 P z )] = z=g,b [I(i π 11 zg )π g z ˆPz + I(i π 01 zg )π g z (1 ˆP z )] ˆP b (i ) = 2P b (i ) = 2 z=g,b [I(i π 11 zb )π b z P z + I(i π 01 zb )π b z (1/2 P z )] = z=g,b [I(i π 11 zb )π b z ˆPz + I(i π 01 zb )π b z (1 ˆP z )]. The advantage of using conditional probabilities is that we can ensure that the labor-income terms in the budget constraint, w( k, 1 ū)( ˆP g /2 + ˆP b /2) g(1 ˆP g /2 ˆP b /2), are positive as long as ˆP g, ˆP b [0, 1], we can utilize the entire [0, 1] domain for ˆP g ˆP b. (Here, it is not a big advantage since we can instead just restrict P g, P b [0, 0.5]. However, the advantage is much larger in the second step, since the corresponding domain becomes time-variant.) Similarly, in the second step of the optimization, define ˆP g = P g π g t (3) ˆP b = P b πt b. (4) Note that we have to be careful about the initial point we cannot divide when πt z = 0. To avoid this, we can start from π g t = 1 ε πb t = ε for a very small ε, for example. The problem becomes ˆV t (k, ˆP g, ˆP b, β) = max c,k {U(c) + βe t [ ˆV t+1 (k, ˆP g, ˆP b, β ) ˆP g, ˆP b, β]} subject to c + k = r t k + w t (π g t ˆP g + π b t ˆP b ) + g(1 π g t ˆP g π b t ˆP b ) + (1 δ)k. Again, P g P b are functions of a rom variable i U[0, 1]: ˆP g(i ) = 1 π g P g(i ) t+1 = [ z=g,b I(i π 11 zg ) π g z = z=g,b π g t+1 ] P z + I(i π 01 zg ) π g z π g (πt z P z ) t+1 [ I(i π 11 zg ) πz t π g π g z ˆPz + I(i π 01 zg ) πz t t+1 π g π g z (1 ˆP z ) t+1 ] ˆP b (i ) = 1 P πt+1 b b (i ) [ ] P z + I(i π 01 zb ) π b z (π z πt+1 b t P z ) = z=g,b I(i π 11 zb ) π b z πt+1 b = [ ] z=g,b I(i π 11 zb ) πz t π πt+1 b b z ˆPz + I(i π 01 zb ) πz t π πt+1 b b z (1 ˆP z ). 7

8 C Calculating the welfare gain As in Lucas (1987), our measure of the welfare gain from eliminating the business cycle, λ, satisfies (we now change the notation of the discount factor so that we can be explicit about its stochastic nature) [ ( t ) ] [ ( t ) ] E 0 β(j) log((1 + λ)c t ) = E 0 β(j) log( c t ), t=0 t=0 t=0 t=0 where β(j) is the discount factor from time j 1 to j (known at time j) β(0) = 1. β(1) is known at time 0 it is an initial condition. c t is consumption in the economy with business cycles c t is consumption in the economy with business cycles. λ can be calculated as follows. 2 where ( ) V Ṽ λ = exp 1, d [ ( t ) ] V = E 0 β(j) log(c t ) t=0 t=0 [ ( t ) ] Ṽ = E 0 β(j) log( c t ) t=0 t=0 can easily be calculated from the value functions. d is defined as [ ( t )] d E 0 β(j). t=0 t=0 Let d i be the value of d when β(1) = β i (i {h, m, l}). Let the vector D be defined as D d h d m dl. D satisfies the following equation: D = I + BΩD, where Ω I 1 1 1, ω hh ω hm ω hl ω mh ω mm ω ml ω lh ω lm ω ll, 2 The welfare measure in the main text is in percentage points, i.e. λ

9 Therefore, where I is the 3 3 identity matrix. B β h β m β l D = (I BΩ) 1 I,. D Algorithm for the model with short- long-run unemployment D.1 Notation Let ɛ {l, f, s, e} denote long-term unemployment after the second unemployment period, longterm unemployment in the first unemployment period ( fired ), short-term unemployment, employment, respectively. z = g is the good state z = b is the bad state (from the second consecutive period). The transition matrix for z is now The individual employment states evolve according to the following matrices. For (z, z ) = (g, g), π ll gg π lf gg π ls gg π le gg π fl gg π ff gg π fs gg π fe gg π sl gg π sf gg π ss gg π se gg = π el gg π ef gg π es gg π ee gg For (g, b), For (b, g), For (b, b), π ll gb π lf gb π ls gb π le gb π fl gb π ff gb π fs gb π fe gb π sl gb π sf gb π ss gb π se gb π el gb π ef gb π es gb π ee gb π ll bg π lf bg π ls bg π le bg π fl bg π ff bg π fs bg π fe bg π sl bg π sf bg π ss bg π se bg π el bg π ef bg π es bg π ee bg π ll bb π lf bb π ls bb π le bb π fl bb π ff bb π fs bb π fe bb π sl bb π sf bb π ss bb π se bb π el bb π ef bb π es bb π ee bb = = = Computation of equilibrium for the economy with aggregate shocks is more involved than in the homogeneous unemployment case, since now we have 3 aggregate states 4 individual states, but does not significantly depart in complexity or difficulty from Krusell Smith (1998). 9

10 D.2 Transition dynamics The general computational strategy is the same as for the case of homogeneous unemployment in Appendix B, though it is more involved because of the many state variables. 1. First we postulate the path for the aggregate capital stock. 2. As in the previous section, we start from t 126. The Bellman equation is: subject to V (k, P, β; k) = max c,k {U(c) + βe[v (k, P, β ; k ) P, β]} c + k = r( k, 1 ū)k + w( k, 1 ū) Pz e + ψ s (Pz s + Pz f ) + ψ l Pz l + (1 δ)k, z=g,b z=g,b z=g,b k = H( k). Here, P is the vector of Pz ɛ s (the joint probability that the aggregate state is z the individual state is ɛ, if there were aggregate fluctuations). We do not need to keep track of all the P s: it is sufficient to have Pg e, Pb e, P g s, Pb s, P f b as state variables.3 This is because Pz l = π z Pz e Pz s Pz f Pg f = 0. Pz ɛ evolves according to P ɛ z (i ) = Pr[z, ɛ i ] = z=g,b ɛ=e,s,f,l Pr[z, ɛ i, z, ɛ]pz ɛ = z=g,b ɛ=e,s,f,l Pr[ɛ i, z, z, ɛ]π z zpz ɛ. (5) Calculating Pr[ɛ i, z, z, ɛ] is harder in this case. Before, we set Pr[ɛ = 1 i, z, z, ɛ] = 1 if i π ɛɛ zz zero otherwise; Pr[ɛ = 0 i, z, z, ɛ] = 1 if i > π ɛɛ zz zero otherwise. Now we have four idiosyncratic states instead of two, so we adopt the following cutoff rule: { Pr[ɛ = e i, z 1 if i, z, ɛ] = [0, π ɛe zz ] 0 otherwise Pr[ɛ = s i, z, z, ɛ] = { 1 if i (π ɛe zz, π ɛe zz + π ɛs zz ] 0 otherwise { Pr[ɛ = f i, z 1 if i, z, ɛ] = (π ɛe zz + π ɛs zz, π ɛe zz + π ɛs zz + π ɛf zz ] 0 otherwise { Pr[ɛ = l i, z 1 if i, z, ɛ] = (π ɛe zz + π ɛs zz + π ɛf zz, 1]. 0 otherwise We keep the structure that a low i is lucky a high i is unlucky. In the computation, we divide the interval of possible is in [0, 1] into subintervals by these cutoff thresholds when we take expectations in the Bellman equation. 3 Note that here we do not need to make distinctions between the z b state the z d state. 10

11 Again, in the actual computation, we work with conditional probabilities (but not conditional only on the aggregate states); we define these conditional probabilities, ˆP ɛ z, as follows. ˆP g e P g e π g, (6) t ˆP b e P b e πt b, (7) ˆP g s Pg s π g t (1 ˆP g e ), (8) ˆP s b P s b πt b(1 ˆP (9) b e ), ˆP f b P f b πt b(1 ˆP b e)(1 ˆP (10) b s ). Then, the problem becomes subject to ˆV (k, ˆP, β; k) = max c,k {U(c) + βe[v (k, ˆP, β ; k ) ˆP, β]} c + k = r( k, 1 ū)k + I w + I u1 + I u2 + (1 δ)k k = H( k). Here, I w w( k, 1 ū) Pz e = w( k, 1 ū) πt z ˆP z e, I u1 ψ s (Pz s + Pz f ) = ψ s z=g,b z=g,b z=g,b z=g,b πt z (1 ˆP z e ) ˆP z s + πt b (1 ˆP b e )(1 ˆP b s ) ˆP f b, I u2 ψ l z=g,b P l z = ψ l z=g,b (πz t P e z P s z P f z ) = ψ l [ 1 π g t ( ˆP e g + (1 ˆP e g ) ˆP s g ) π b t ( ˆP e b + (1 ˆP e b ) ˆP s b + (1 ˆP e b )(1 ˆP s b ) ˆP f b ) ]. The evolution of ˆP ɛ z can be obtained from the above relationships. In particular, we can convert ˆP s into P s using the equations (6)-(10), calculate the transition of P s by (5), transform the P s back to ˆP s using (6)-(10) once again. Here, since all the exogenous variables are already settled, we can use π z t = 1/2. Again, the advantage of working on the conditional distributions is that we can ensure that I w, I u1, I u2 are all positive for ˆP g e, ˆP b e, ˆP g s, ˆP b s, ˆP f b [0, 1]. We put 5 grid points in each ˆP direction linearly interpolate the value functions when evaluating the value at ˆP. 3. The other steps are similar to those in Appendix B. For the first 125 periods, π z t changes over time: it evolves according to π z t+1 = z=g,b π z zπ z t. 11

12 E More tables E.1 One type of unemployment Utility gain in percentage consumption < > 99 All ɛ = ɛ = Average utility gains by wealth group ( k = 11.2, z = z b ) Utility gain in percentage consumption < > 99 All ɛ = ɛ = Average utility gains by wealth group ( k = 12.3, z = z g ) Utility gain in percentage consumption < > 99 All ɛ = ɛ = Average utility gains by wealth group ( k = 12.3, z = z b ) Wealth percentile Type of agent constr ɛ = 1, β = low ɛ = 1, β = middle ɛ = 1, β = high ɛ = 0, β = low ɛ = 0, β = middle ɛ = 0, β = high Utility gains for different types of agents ( k = 11.2, z = z b ) 12

13 Wealth percentile Type of agent constr ɛ = 1, β = low ɛ = 1, β = middle ɛ = 1, β = high ɛ = 0, β = low ɛ = 0, β = middle ɛ = 0, β = high Utility gains for different types of agents ( k = 12.3, z = z g ) Wealth percentile Type of agent constr ɛ = 1, β = low ɛ = 1, β = middle ɛ = 1, β = high ɛ = 0, β = low ɛ = 0, β = middle ɛ = 0, β = high Utility gains for different types of agents ( k = 12.3, z = z b ) 13

14 E.2 Short- long-term unemployment Utility gain in percentage consumption < > 99 All ɛ = e ɛ = s ɛ = f ɛ = l Average utility gains by wealth group ( k = 11.3, z = z d ) Utility gain in percentage consumption < > 99 All ɛ = e ɛ = s ɛ = l Average utility gains by wealth group 4 ( k = 11.3, z = z b ) Utility gain in percentage consumption < > 99 All ɛ = e ɛ = s ɛ = l Average utility gains by wealth group ( k = 12.1, z = z g ) Utility gain in percentage consumption < > 99 All ɛ = e ɛ = s ɛ = f ɛ = l Average utility gains by wealth group ( k = 12.1, z = z d ) 4 In the simulated data, there are no agents with ɛ = s asset holdings below the first percentile. 14

15 Utility gain in percentage consumption < > 99 All ɛ = e ɛ = s ɛ = l Average utility gains by wealth group ( k = 12.1, z = z b ) Wealth percentile Type of agent constr ɛ = e, β = low ɛ = e, β = middle ɛ = e, β = high ɛ = s, β = low ɛ = s, β = middle ɛ = s, β = high ɛ = f, β = low ɛ = f, β = middle ɛ = f, β = high ɛ = l, β = low ɛ = l, β = middle ɛ = l, β = high Utility gains for different types of agents ( k = 11.3, z = z d ) Wealth percentile Type of agent constr ɛ = e, β = low ɛ = e, β = middle ɛ = e, β = high ɛ = s, β = low ɛ = s, β = middle ɛ = s, β = high ɛ = l, β = low ɛ = l, β = middle ɛ = l, β = high Utility gains for different types of agents ( k = 11.3, z = z b ) 15

16 Wealth percentile Type of agent constr ɛ = e, β = low ɛ = e, β = middle ɛ = e, β = high ɛ = s, β = low ɛ = s, β = middle ɛ = s, β = high ɛ = l, β = low ɛ = l, β = middle ɛ = l, β = high Utility gains for different types of agents ( k = 12.1, z = z g ) Wealth percentile Type of agent constr ɛ = e, β = low ɛ = e, β = middle ɛ = e, β = high ɛ = s, β = low ɛ = s, β = middle ɛ = s, β = high ɛ = f, β = low ɛ = f, β = middle ɛ = f, β = high ɛ = l, β = low ɛ = l, β = middle ɛ = l, β = high Utility gains for different types of agents ( k = 12.1, z = z d ) 16

17 Wealth percentile Type of agent constr ɛ = e, β = low ɛ = e, β = middle ɛ = e, β = high ɛ = s, β = low ɛ = s, β = middle ɛ = s, β = high ɛ = l, β = low ɛ = l, β = middle ɛ = l, β = high Utility gains for different types of agents ( k = 12.1, z = z b ) References Krusell, P. & A.A. Smith, Jr. (1998), Income Wealth Heterogeneity in the Macroeconomy. Journal of Political Economy 106, Krusell, P. & A.A. Smith, Jr. (2002), Revisiting the Welfare Effects of Eliminating Business Cycles. Manuscript. University of Rochester Carnegie-Mellon University. Lucas, R.E., Jr. (1987), Models of Business Cycles. Basil Blackwell, New York. Mukoyama, T. & A. Şahin (2005), Costs of Business Cycles for Unskilled Workers, CIREQ working paper Mukoyama, T. & A. Şahin (2006), Costs of Business Cycles for Unskilled Workers, Journal of Monetary Economics 53,