Intergenerational Persistence and Misallocation. Section 4

4 Quantitative Model In this section we develop a quantitative model that extends the previous model in a number of dimensions. First, the model presented here is dynamic, with individuals facing a stochastic ageing process. Second, we have O dierent occupations. Third, we introduce nonpecuniary benets (preferences) for occupations, which are allowed to be correlated across generations. Fourth, we introduce occupation specic human capital. We retain the assumption that social contacts are occupationspecic, and that fathers help their ospring to nd jobs in their occupations, without internalizing that in their own occupational choice. Both social contacts and occupationspecic human capital evolve over time. Finally, we also introduce shocks to nonpecuniary benets, in order to generate occupational mobility over the lifecycle. 4.1 Model Environment Time is discrete of submarkets t = 0, 1, 2, ... O, and goes on forever. The economy is divided into a discrete number which represent the dierent occupations. A measure 2 of workers populate the economy. All agents (both workers and rms) are riskneutral and discount the future at rate λ. There are two phases of life: young and old. Every period young (old) individuals age (die) with probability ζ. All individuals who die are replaced by young unemployed workers. We assume that these shocks are perfectly correlated within an household (fatherson pair). This is equivalent to assuming that individuals stop being connected to their parents when they have children, so that at each point in time only two generations are connected Workers are indexed by parative advantage erences 21 This φit τ i, i 21 . and dier along several dimensions: preferences for occupations occupationspecic human capital are represented by a vector of size O, hio,t where the and social capital oth assumption is made for simplicity and does not alter our results. 37 nio,t φit , com- (networks). Pref- element is the level of nonpecuniary benets associated to occupation component, so that the period i,T φio,t = φi,P o + φo,t , advantage τ i, for each o. t o. We assume that preferences have both a permanent and a transitory nonpecuniary benets are equal to the sum of the two components: The permanent preferences component φi,T t , as well as the comparative are drawn at birth (i.e. at the entry in the labor market) and do not change over time, sticking to the worker for the rest of his life. Instead, the temporary component of preferences as well as occupationspecic human and social capital, evolve over time according to law of motions to be i specied in the following. Each worker is either employed or unemployed (et to some occupation oi . ∈ {0, 1}), Unemployed workers receive an unemployment benet equal to and is attached b per period. Let us denote parental variables with an F superscript, so that the occupation of individual father will be denoted oi,F , parental networks ni,F o i's and so on. Employed parents help their unemployed ospring by passing vacancies of which they become aware on to them. We assume that unemployed fathers do not help their ospring, as they are actively seeking a job for themselves. Upon matching, the surplus generated is split according to a linear sharing rule, according to which the wage is set to a χ share of the worker's productivity. 4.2 Search and Reallocation across Occupations We assume that search is costless and directed across occupations. Workers start their worklife as unemployed, and have to decide in which occupation to look for vacancies. The lack of rm heterogeneity usually implies that there is no scope for onthejob search. Nonetheless, in our model employed parents do engage in onthejob search, in order to help their unemployed ospring to nd a job. Whenever they nd a vacancy in their occupation, they will pass it on to their ospring. As a consequence, unemployed ospring nd vacancies in their father's occupation with higher probability than anywhere else, ceteris paribus. Each occupation consists of a single labor market, where the number of matches between unemployed 38 workers and vacancies is governed by the following constant returns to scale technology: Mo,t = (Uo,t )η (Vo,t )1−η where Mo,t exerted and (37) denotes the total number of matches produced, Vo,t is the measure of vacancies posted at time matching function with respect to Uo,t is given by t Uo,t is the total eciency units of search in occupation o. The elasticity of the η. Search eort is exerted both by unemployed workers and employed fathers whose ospring are currently unemployed. When searching for a job, workers exploit their social networks. We assume that information can ow within a network at zero cost and that there is no competition among workers belonging to the same network, as everyone can get the information at the same time. Thus, social networks can help workers nd a job, and having a larger network represents an advantage for unemployed workers. This is reected in the model by an increase in the eciency units of search that a worker can exert. In particular, it is assumed that a worker with network nio,t can exert (1 + nio,t ) eciency units of search. Z Uo,t = 1+ | nio,t
1{e = 0; o = o}di i {z i } Search Eort of Unemployed workers where ξ Z +   i i i,F ξ 1 + ni,F = o; ei,F = 1}di o,t 1{e = 0; o = o | {z } (38) Search Eort of Employed Fathers captures possibly dierent levels of eectiveness of the father's network, compared to the personal network. The occupationspecic individual job nding rate pio,t is the sum of the probability of meeting a vacancy himself (either through his own eort or through his social network), and the probability of his father meeting a vacancy for him, if employed in the same occupation: pio,t = i  i,F Mo,t h i,F (1 + nio,t ) + ξ(1 + ni,F ) 1 e = 1; o = o o,t Uo,t 39 (39) At each time t, workers face one single jobnding rate, in the occupation to which they are attached. We assume freeentry of rms, and that posting a vacancy costs meet with a worker with probability qo,t = κ per period. Firms in occupation Mo,t . Matches get exogenously destroyed at rate Vo,t δ o per period. When an employed worker gets separated from his match, he remains attached to his occupation. Workers (both employed and unemployed) can freely reallocate across occupations. It is assumed that when an employed worker wants to reallocate, he is separated from the current match (the match is destroyed) and moves into the unemployment pool of his new occupation. Also, his occupationspecic human capital and social contacts stocks fully depreciate. 4.3 Intergenerational Transmission and Laws of Motion P We assume that upon entry in the labor market the duplet {φ , τ } is imperfectly inherited from the father: φP ∼ F φP |φF ,P τ ∼ G τ |τ F
(40)
(41) The initial level of occupationspecic human and social capital is assumed to be zero in all occupations. Both human and social capital, as well as preferences for occupations, evolve over time according to the following laws of motion: ho,t+1 = Fh (ho,t |et ) (42) no,t+1 = Fn (no,t |et ) (43) Last, the temporary preferences vector is drawn each period from a distribution φT t ∼ Fφ 40 Fφ : (44) As already mentioned, upon changing occupation we assume that ho and no start again from zero. We do this for computational reasons, even though in principle it would be interesting to keep track of all occupationalspecic variables and make them decay over time if the worker is not attached to that occupation anymore. 4.4 Timing The timing of our model is as follows: 1. Old (young) workers die (age) with probability ζ. The old ones who die are replaced by young unemployed workers. 2. Preference shocks are realized; reallocation of unemployed and employed workers takes place. 3. Wages and unemployment benets are paid, occupationspecic utility ow is realized. 4. Exogenous separations take place. Unemployed workers nd jobs or not. 5. The state variables of the workers are updated, according to the laws of motion. 4.5 Workers' Problem At the beginning of his life, the problem of a worker consists in choosing in which occupation to search. Besides this initial choice, at the beginning of each period workers have the option of reallocating into another occupation. In what follows, we suppress the although in the model all variables (except for φP and i t subscript for readability, τ ) do change over time. We denote next period's superscript and the state variables with a prime. All functional equations are conditional on the worker's state variables. Denote the state of a worker by Γ = {φ, τ, ho , no , o, e}, where o is set for simplicity equal to 0 in the case of a worker who has to choose an occupation for the rst time. A young worker's choices are inuenced by his father as he can get help in nding jobs from him, so that his own state includes also 41 his father's state ΓF. We have to track all of father's state variables because ospring take into account that: i) even if unemployed (and therefore, not directly relevant for jobnding rate) today, their fathers will eventually help them nd a job in the future; ii) fathers also change occupations over their lifecycle. In the following, we make explicit the dependance of workers' value functions on their employment status and occupation. Hence, conditional on employment status and occupational aliation, we denote the state variable of young workers by Ξ, Ω = {φ, τ, ho , no } ∪ Γ F . All Bellman equations are conditional on the aggregate state variables, even though we omit this dependance for readability. We rst write down the value functions for old workers (denoted by a subscript workers are characterized by F ), with the understanding that old Γ F = ∅. Fathers' problem We denote by WR the value of reallocation across occupations: WFR (Ω) where φTj,F = n o U T max Wj,F (Ω) + φj,F , represents the temporary preference shock for occupation draw a vector of size Oby1 (45) j∈{1,...,O} j. Note that unemployed workers of preference shocks each period. The deterministic component of the value of unemployment in occupation U o, Wo,F , includes the value of unemployment benets for the current period and the expected discounted value of the future U Wo,F (Ω) 22 h   R i  E 0 0 = b + λ̃ po (Ω) E Wo,F (Ω ) + (1 − po (Ω)) E Wo,F (Ω ) . An unemployed worker is matched with a vacancy in his occupation with probability remains unemployed with the remainder : (46) po (Ω), and (1−po (Ω)) probability, in which case he can decide to reallocate 22 In this case, the value function for unemployment has to be interpreted at the stage immediately after the reallocation decision. That is, the worker has to spend the whole period unemployed in occupation o. 42 next period. The future is discounted at the rate λ̃ = λ(1 − ζ), in order to account for the risk of dying. Employed workers face the reallocation decision at the beginning of the period. After that, if they decide to stay on the job, they receive the ow utility, earn the wage and keep with the same job next period, unless their match gets exogenously destroyed (this happens with probability δ ). In this case, they will then choose their new occupation next period. n h o  E   R 0 i R E 0 (Ω) = max φPo,F + φT,E + w(Ω) + λ̃ (1 − δ) E W (Ω ) + δ E W (Ω ) , W (Ω) . Wo,F o,F F F o,F Note that employed workers draw a vector of size draws a jobspecic shock φT,E o,F (O + 1)by1 (47) of preference shocks (that is, he on top of the occupationspecic ones). 23 4.6 Wage Determination Upon matching, wages are set by a linear sharing rule, that is the worker is paid a xed share of his productivity. 24 Let y(τ, h) be the productivity level of a worker of type (τ, h). Then, the equilibrium wage is: w(τ, h) = χ y(τ, h). Importantly, we assume that all payorelevant information are common knowledge within the match. We also assume that wages are perfectly exible and that they are renegotiated every period, upon changes in the worker's level of human capital. 23 This is done to preserve the symmetry between the occupational reallocation choices of unemployed and employed workers. 24 Assuming other forms of wage determinations, such as Generalized Nash Bargaining, would be a rather extreme assumption in our setup, as in that case we would have to assume that all networks, preferences and the productivity levels in all other occupations, are common knowledge within the match. 43 4.7 Reallocation Across Occupations We denote as j∗ the preferred occupation for search, that is the occupation that maximizes the value of reallocation: jF∗ (Ω) Next, let us dene n o U T ∈ argmaxj∈{1,...,O} Wj,F (Ω) + φj,F . (48) ŴoE (Ω) the value of staying employed in occupation o (that is, the value of being employed and choosing not to reallocate): E (Ω) Ŵo,F Notice that j ∗ (Ω) = φPo,F + φT,E o,F h   R 0 i  E 0 + w(Ω) + λ̃ (1 − δ) E Wo,F (Ω ) + δ E WF (Ω ) . (49) might be the same or not as the current occupation of a worker. In case it is not, an unemployed worker will always want to reallocate, while for an employed worker the choice will depend on the dierence between the value functions We dene when k Ro,F (Ω) (for k ∈ {E, U }) E (Ω) Ŵo,F and WFR (Ω). the policy function with respect to the reallocation decision: k Ro,F (Ω) = 1, then a worker of type Ω with employment status k in occupation o optimally decides to reallocate. Optimality implies: U Ro,F (Ω) = 1{jF∗ (Ω) 6= o}. E E Ro,F (Ω) = 1{WFR (Ω) > Ŵo,F (Ω)}. Sons' problem Sons face a very similar problem to the one of fathers. The only dierence is that they do internalize their father's decisions. As a result, young workers can decide to reallocate as a consequence of a change in their own state variables (preferences) or because their father's state variables have changed, in which case they might want to follow their father in order to benet from a higher probability of nding a job. 44 The expression for the value of reallocation stays exactly the same: WSR (Ω) = max j∈{1,...,O} o n U (Ω) + φTj,S , Wj,S The value of unemployment and employment are augmented by the fact that with (50) ζ probability, next period the worker will become a father: h 
 E   E U (Ω 0 ) Wo,S (Ω) = b + λ po (Ω) ζ E Wo,F (Ω 0 ) + (1 − ζ)E Wo,S (51)  R   R 
i +(1 − po (Ω)) ζ E Wo,F (Ω 0 ) + (1 − ζ)E Wo,S (Ω 0 ) . n h  E   E 
E Wo,S (Ω) = max φPo,S + φT,E + w(Ω) + λ (1 − δ) ζ E Wo,F (Ω 0 ) + (1 − ζ)E Wo,S (Ω 0 ) o,S (52) o    
i R +δ ζ E WFR (Ω 0 ) + (1 − ζ)E WSR (Ω 0 ) , WS (Ω) . Reallocation decisions E U (Ω) are isomorphic to the ones of the fathers, and are dened (Ω) and Ro,S Ro,S according to the above specied value functions. 4.8 Firms' Problem Each rm is represented by one job that is either lled or vacant. The value function for a job lled with a worker of type Ω is denoted by Jo,k (Ω), where k ∈ {F, S} denotes the age of the worker. Provided that the worker does not choose to leave the rm, this value function includes the current prot (given by production net of the wage payment) and the continuation value of holding the worker. The value 45 of holding an old worker reads as: h i  E (Ω) = 0} y(τ, ho ) − w(Ω) + λ̃ (1 − δ)E [Jo,F (Ω 0 )] + δVo0 + Jo,F (Ω) = 1{Ro,F (53) E 1{Ro,F (Ω) = 1} Vo . The production of the match is given by the function period, with δ y(τ, ho ), for which we assume y1 ,y2 > 0. Next probability the match is exogenously destroyed (in which case, like in the case of the endogenous separation, the rm is left with the value of a vacancy Vo0 ). With the remainder (1 − δ) probability, the match continues, and the state variables of the worker are updated. Instead, the value of holding a young worker reads as: Jo,S (Ω) = 1 E {Ro,S (Ω) δVo0 i h  = 0} y(τ, ho ) − w(Ω) + λ (1 − δ) (ζ E [Jo,F (Ω 0 )] + (1 − ζ)E [Jo,S (Ω 0 )]) + (54) E + 1{Ro,S (Ω) = 1} Vo . This equation has the same interpretation as the one for the old worker, except that it allows for the possibility that a worker turns old (when he is hit by the The value of a vacancy Vo ζ shock) and the match continues. is given by the expected prots minus the cost of the posting Vo = −κ + qo E [Jo (Ω 0 )] + (1 − qo )Vo0 , κ. (55) where the expectation is taken over the distribution of unemployed workers attached to occupation both across all possible types Ω and possible ages {F, S}. 46 o, 4.9 Worker Flows The evolution of the stock of unemployed and employed workers is the result of optimal reallocation decisions, age shocks and labor market shocks (creation of new matches and exogenous destruction of existing ones). Ω̃ Let us dene with employment status k k gΩ (Ω̃) = P (Ω 0 = Ω|Ω̃) becomes of type multidimensional distribution of Ω. Ω the probability measure that a worker of type next period. This probability is dened over the In particular, it involves changes in: the temporary preference vectors (his own or his father's), occupationspecic human capital and networks stocks (his own or his father's), father's occupation or employment status. Let us dene u0o,F (Ω) next period's measure of unemployed fathers of type Ω attached to occupation o: u0o,F (Ω) = Z h i U U E E ûo,F (Ω̃) (1 − Ro,F (Ω̃)) (1 − po,F (Ω̃)) gΩ (Ω̃) + êo,F (Ω̃) (1 − Ro,F (Ω̃)) δ gΩ (Ω̃) dΩ̃ (56) Ω i XZ h U E U ûõ,F (Ω̃) Rõ,F (Ω̃) + êõ,F (Ω̃) Rõ,F (Ω̃) 1{jF∗ (Ω̃) = o} (1 − po,F (Ω̃)) gΩ (Ω̃) dΩ̃, + õ6=o where Ω ûo,F = uo,F (1 − ζ) + uo,S ζ , and êo,F = eo,F (1 − ζ) + eo,S ζ . These are the measures of workers after the age shock, that is this stock includes fathers who do not die, as well as sons who become fathers. Equation 56 is composed of four dierent terms: the rst two refer respectively to unemployed workers attached to occupation in occupation o o who decided not to reallocate and did not nd a job, and employed workers who did not reallocate and lost their job; the last two terms are (unemployed and em- ployed) workers who decided to reallocate into occupation 47 o but did not nd a job in the previous period. Turning to employed fathers, e0o,F (Ω) Z h = e0o (Ω) is dened as follows: i U U E E (Ω̃) + ûo,F (Ω̃) (1 − Ro,F (Ω̃)) po,F (Ω̃) gΩ (Ω̃) dΩ̃ (Ω̃)) (1 − δ) gΩ êo,F (Ω̃) (1 − Ro,F (57) Ω + XZ h õ6=o i U E U ûõ,F (Ω̃) Rõ,F (Ω̃) + êõ,F (Ω̃) Rõ,F (Ω̃) 1{jF∗ (Ω̃) = o} po,F (Ω̃) gΩ (Ω̃) dΩ̃. Ω The stock of employed is given by workers who were already employed last period in the same occupation and neither did not lose the job nor found protable to reallocate, plus the mass of unemployed workers who did not want to reallocate and found a vacancy, plus all workers who had just reallocated into occupation o and found a job. For what concerns the distribution of sons, the one of the employed is exactly symmetric to the fathers': e0o,S (Ω) Z h i E E U U = (1 − ζ)eo,S (Ω̃) (1 − Ro,S (Ω̃)) (1 − δ) gΩ (Ω̃) + (1 − ζ)uo,S (Ω̃) (1 − Ro,S (Ω̃)) po,S (Ω̃) gΩ (Ω̃) dΩ̃ Ω (58) + XZ h i U E U (1 − ζ)uõ,S (Ω̃) Rõ,S (Ω̃) + (1 − ζ)eõ,S (Ω̃) Rõ,S (Ω̃) 1{jS∗ (Ω̃) = o} po,S (Ω̃) gΩ (Ω̃) dΩ̃. Ω õ6=o Finally, the distribution of unemployed sons reads as follows: u0o,S (Ω) Z h i U U E E = (1 − ζ)uo,S (Ω̃) (1 − Ro,S (Ω̃)) (1 − po,S (Ω̃)) gΩ (Ω̃) + (1 − ζ)eo,S (Ω̃) (1 − Ro,S (Ω̃)) δ gΩ (Ω̃) dΩ̃ Ω (59) + XZ h õ6=o i U E U (1 − ζ)uõ,S (Ω̃) Rõ,S (Ω̃) + (1 − ζ)eõ,S (Ω̃) Rõ,S (Ω̃) 1{jS∗ (Ω̃) = o} (1 − po,S (Ω̃)) gΩ (Ω̃) dΩ̃ Ω 1{Ω ∈ Ω N B } , 1{Ω ∈ Ω N B } Ω +ζ R where the only dierence comes from the last term, which represent the ow of newborns, randomly 48 directed to the subset ΩN B of the entire state space. 4.10 Equilibrium Denition We focus on a symmetric Steady State equilibrium (SS) in which all value functions and reallocation decisions of workers are constant over time. In particular, value functions and optimal decisions only depend on the individual state Denition: Vo , A SS equilibrium is a set of value functions reallocation decisions wages wo (Ω), Ω. U U E E Wo,F (Ω), Wo,S (Ω), Wo,F (Ω), Wo,S (Ω), U U E E Ro,F (Ω), Ro,S (Ω), Ro,F (Ω), Ro,S (Ω), jF∗ (Ω), jS∗ (Ω), labor market tightness θo , laws of motion for the individual state variables, and laws of motion of unemployed and employed workers over all occupations, such that: • Value functions for workers follow from equations 46, 47, 51, 52 and reallocation decisions are optimal for them. • The value of a vacancy is equal to zero in all occupations: Vo = 0 (that is, there is freeentry in all occupations). • Labor market tightness is consistent with freeentry and pinned down by equation 55. • Wages are pinned down according to equation 35. • Individual state variables evolve according to equations 42, 43 and 44. • Distributions of workers evolve according to equations 56, 57, 58 and 59. • Measures of employed and unemployed workers of each type • Flows of employed and unemployed workers of each type 49 Ω Ω are constant over time. are constant over time.