Job Matching Within and Across Firms: Supplementary Appendix Elena Pastorino∗

A

Omitted Proofs

Proof of Proposition 1: Using w(p, θ) = max{f k|θf k =θ} y(p, θf k , yf Hk , yf Lk ) from equation (9) in the paper, the worker’s problem can be written as X   ˜ d(θ) (1 − δ) max{f k|θf k =θ} y(p, θf k , yf Hk , yf Lk ) + δEV (p0 |p, θ) . (1) V (p) = max{d(θ)} ˜ θ∈Θ

Since EV (p0 |p, θ) in (1) does not depend on either f or k, I can further express V (p) as X  ˜ d(θ) max{f k|θf k =θ} [(1 − δ)y(p, θf k , yf Hk , yf Lk ) + δEV (p0 |p, θf k )] V (p) = max{d(θ)} ˜ θ∈Θ

or, equivalently, as n X X ˜ max{d(θ)} d(θ) max{df k (θ)} ˜ θ∈Θ

o df k (θ) [(1 − δ)y(p, θf k , yf Hk , yf Lk ) + δEV (p |p, θf k )] , 0

f,k

where df k (θ) = 1 denotes the choice of job k of firm f with informativeness θf k = θ, with the convention that df k (θ) = 0 if job k of firm f has informativeness θ0 6= θ. Denoting by {df k } the unconditional choice of job k of firm f induced by V (p), it follows that I can also rewrite V (p) as X V (p) = max{df k } df k [(1 − δ)y(p, θf k , yf Hk , yf Lk ) + δEV (p0 |p, θf k )] , f,k which clearly equals W (p). Proof of Proposition 2: By Proposition 1, equilibrium job assignment solves the planning problem. Since firms are identical, the subscript f can be dropped, and given that all jobs are equally informative, W (pt ) reduces to W (pt ) = (1 − δ) maxk∈K¯ y(pt , k) + δEW (pt+1 |pt ), which implies that the assignment rule solves maxk∈K¯ y(pt , k). As argued, the equilibrium wage is given by w(pt ) = maxk∈K¯ y(pt , k). To prove that the optimal policy is increasing with p, note that by ∗

University of Minnesota and Federal Reserve Bank of Minneapolis, U.S.: [email protected]. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

1

monotonicity of expected output in p and by the property of increasing differences, if a higherindexed job k 0 is preferred to a lower-indexed job k at some prior p, then k 0 must also be preferred to k at any higher prior p0 ≥ p. The rest of the claim is an immediate implication of this result and the fact that αf k > βf k implies that the prior increases after a success and decreases after a failure. Proof of Implication 1: For (i) suppose that a worker in job k at prior pt ∈ [ρk , ρk+1 ) experiences a success, so the new prior is PH (pt ) ≥ pt . If PH (pt ) < ρk+1 , then the worker stays in job k, and since expected output at each job increases with the prior, the wage increases from y(pt , k) to y(PH (pt ), k). If PH (pt ) ≥ ρk+1 , the worker is promoted to job k + 1 and the new wage is y(PH (pt ), k + 1). That this wage is greater than y(pt , k) follows by the monotonicity of expected output in pt , which implies y(PH (pt ), k) ≥ y(pt , k), and by the increasing differences property of expected output, which implies y(pt , k + 1) > y(pt , k) if pt > ρk . The proof of (ii) is symmetric. Result (iii) immediately follows from the monotonicity of expected output in pt and the increasing differences property of expected output. For (iv), I formalize the statement that conditional on current wages, wage increases are positively serially dependent as Pr (wt+2 − wt+1 > 0|wt+1 − wt > 0, wt ) ≥ Pr (wt+2 − wt+1 > 0|wt ) .

(2)

Since wages increase from any period t to t + 1 only if a success occurs, an event denoted by zt = H, and wages increase with the prior, (2) can be rewritten as Pr (zt+1 = H|zt = H, pt ) ≥ Pr (zt+1 = H|pt ) .

(3)

Given that the left side of (3) equals Pr(zt+1 = H|pt+1 = PH (pt )), the result follows since the probability of success increases with the prior. Next, note that (v) means that, given kt+2 = k 00 > kt+1 = k 0 > kt = k and that a worker is assigned to job k and receives the wage wt in t, the following inequality holds Pr(kt+2 = k 00 > kt+1 |kt+1 = k 0 > kt = k, wt ) ≥ Pr (kt+2 = k 00 > kt+1 |kt+1 = k 0 , wt ) .

(4)

Here, I restrict attention to wages wt such that the probabilities in (4) are well defined, that is, wages at job kt associated with priors such that the probability of promotion to job kt+1 > kt is positive. Let pt denote a prior consistent with wt and job kt in this sense. The conditioning statement on the left side of (4), that a promotion occurred at t + 1 starting from some prior pt in t, implies that pt+1 = PH (pt ). The event that kt+2 = k 00 > kt+1 = k 0 requires that a success also occurred in t + 1 at job kt+1 so that pt+2 = PH (pt+1 ) with PH (pt+1 ) ≥ ρkt+2 > pt+1 ≥ ρkt+1 . Then, given the conditioning on wt , either both sides of (4) are zero or two successes from pt lead to pt+2 = PH (PH (pt )) ≥ ρkt+2 , in which case the result clearly holds, since the statement can be rewritten Pr(zt+1 = H|pt+1 = PH (pt )) = Pr (zt+1 = H|zt = H, pt ) ≥ Pr (zt+1 = H|pt ), since PH (pt ) ≥ pt and the probability of success increases with the prior. As for (vi), note that for a promotion to occur from k to k + 1 in period t + 1, pt must be such that PH (pt ) ≥ ρk+1 > pt . As β approaches α, the difference PH (pt ) − pt becomes arbitrarily small. Hence, promotions from k to k + 1 occur only at pt just below ρk+1 leading to PH (pt ) just above ρk+1 . Since wages equal expected output, by the definition of ρk+1 it follows that as β approaches α, wage changes become arbitrarily small upon promotion. At level k, instead, wages range from w(ρk , k) = y(ρk , k) to (just below) w(ρk+1 , k + 1) = y(ρk+1 , k + 1). Since it is possible to adjust yHk and yLk to keep this range fixed while β is made closer to α, the result follows. Lastly, (vii) 2

is an immediate implication of (i) through (v). Proof of Implication 3: Suppose that G(p1 ) is a Beta distribution with parameters λ1 and λ2 , positively skewed (λ2 > λ1 ), with support [0, 1], and associated density g(p1 ; λ1 , λ2 ). For simplicity, assume that the population of workers (of measure 1) consists of newborn ‘young’ workers and older workers, who have experienced one success or failure when young. In this case, the cross-sectional distribution of priors among workers is a mixture of Beta distributions, with weight 1/2 on the density g(p1 ; λ1 , λ2 ) and weight 1/2 on the density r(p1 )g(PH (p1 ); λ1 + 1, λ2 ) + [1 − r(p1 )]g(PL (p1 ); λ1 , λ2 + 1), since, after observing t − 1 output signals, the density of the posterior is P Pt−1 λ + t−1 x −1 p1 1 τ =1 τ (1 − p1 )λ2 +(t−1)− τ =1 xτ −1 g(pt ; λ1 , λ2 ) = , P Pt−1 B(λ1 + t−1 τ =1 xτ , λ2 + (t − 1) − τ =1 xτ ) where B(·) is the Beta function, with mean (µt ) and variance (σt2 ) given by
  P P P λ1 + t−1 λ2 + (t − 1) − t−1 λ1 + t−1 τ =1 xτ τ =1 xτ 2 τ =1 xτ and σt = , µt = λ1 + λ2 + (t − 1) [λ1 + λ2 + (t − 1)]2 [λ1 + λ2 + (t − 1) + 1] and mode given by P λ1 + t−1 τ =1 xτ − 1 mt = . λ1 + λ2 + (t − 1) − 2 Note that if a random variable is subjected to a linear or affine transformation, so are its mean, median, and mode. I will use this property in what follows. I will also use the notion of nonparametric skew of a distribution, ηt = (µt − mt )/σt , for a measure of skewness simpler to compute. Following Kim and White (2003), the distribution of priors of old workers is the mixture g(p2 ; λ1 , λ2 ) = r(p1 )g(PH (p1 ); λ1 + 1, λ2 ) + [1 − r(p1 )]g(PL (p1 ); λ1 , λ2 + 1). Let µj , mj , and σj2 , j = H, L, be, respectively, the mean, mode, and variance of the distribution G(PH (p1 ); λ1 + 1, λ2 ), when j = H, and of the distribution G(PL (p1 ); λ1 , λ2 + 1), when j = L. Then, the mean, µm , and mode, mm , of the distribution of priors of old workers are given by µm = r(p1 )µH + [1 − r(p1 )]µL =

r(p1 ) + λ1 r(p1 ) (λ1 + 1) + [1 − r(p1 )]λ1 = λ1 + λ2 + 1 λ1 + λ2 + 1

and mm = r(p1 )mH + [1 − r(p1 )]mL =

r(p1 )λ1 + [1 − r(p1 )] (λ1 − 1) r(p1 ) + λ1 − 1 = , λ1 + λ2 − 1 λ1 + λ2 − 1

2 2 2 2 whereas the variance σm is given by σm = r(p1 )(σH + δH ) + [1 − r(p1 )](σL2 + δL2 ), where 2 σH =

λ1 (λ2 + 1) (λ1 + 1) λ2 and σL2 = , 2 (λ1 + λ2 + 1) (λ1 + λ2 + 2) (λ1 + λ2 + 1)2 (λ1 + λ2 + 2) δH = µ H − µ m =

r(p1 ) + λ1 1 − r(p1 ) λ1 + 1 − = , λ1 + λ2 + 1 λ1 + λ2 + 1 λ1 + λ2 + 1

3

and δL = µ L − µ m = which leads to

λ1 r(p1 ) + λ1 r(p1 ) − =− , λ1 + λ2 + 1 λ1 + λ2 + 1 λ1 + λ2 + 1

2 2 σH + δH =

(λ1 + 1) λ2 [1 − r(p1 )]2 + (λ1 + λ2 + 1)2 (λ1 + λ2 + 2) (λ1 + λ2 + 1)2

σL2 + δL2 =

λ1 (λ2 + 1) r2 (p1 ) + . (λ1 + λ2 + 1)2 (λ1 + λ2 + 2) (λ1 + λ2 + 1)2

and

Note that the mixture of g(p1 ; λ1 , λ2 ) and g(p2 ; λ1 , λ2 ) with weight 1/2 (distribution H(p) in the general setup) has mean   1 λ1 r(p1 ) + λ1 1 + µtot = (µ1 + µm ) = 2 2 λ1 + λ2 λ1 + λ2 + 1 and mode mtot

  1 1 r(p1 ) + λ1 − 1 λ1 − 1 = (m1 + mm ) = + , 2 2 λ1 + λ2 − 2 λ1 + λ2 − 1

which implies that µtot − mtot

  λ1 − 1 r(p1 ) + λ1 − 1 1 λ1 r(p1 ) + λ1 − − = + . 2 λ1 + λ2 λ1 + λ2 + 1 λ1 + λ2 − 2 λ1 + λ2 − 1

With δ1 = µ1 − µtot

    1 r(p1 ) + λ1 r(p1 ) + λ1 λ1 λ1 1 λ1 − + − = = λ1 + λ2 2 λ1 + λ2 λ1 + λ2 + 1 2 λ1 + λ2 λ1 + λ2 + 1

and δm = µm − µtot

    r(p1 ) + λ1 1 λ1 r(p1 ) + λ1 λ1 1 r(p1 ) + λ1 = − + − = , λ1 + λ2 + 1 2 λ1 + λ2 λ1 + λ2 + 1 2 λ1 + λ2 + 1 λ1 + λ2

2 2 2 + δm ) /2 is given by it follows that σtot = (σ12 + δ12 + σm

2 σtot

 2 λ1 λ2 1 λ1 r(p1 ) + λ1 = + − 2(λ1 + λ2 )2 (λ1 + λ2 + 1) 8 λ1 + λ2 λ1 + λ2 + 1

 2 (λ2 + 1) [λ1 + 2r(p1 )] − r2 (p1 ) (λ1 + λ2 + 2) 1 r(p1 ) + λ1 λ1 + − + 8 λ1 + λ2 + 1 λ1 + λ2 2 (λ1 + λ2 + 1)2 (λ1 + λ2 + 2) and increases with r(p1 ) if, and only if, 2λ2 (λ2 + 1) + λ1 (λ2 − λ1 ) > r(p1 ), (λ1 + λ2 + 2) (λ1 + λ2 ) which, for λ1 close enough to zero, reduces to (2 + 2λ2 )/ (λ2 + 2) > r(p1 ), a condition that is

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satisfied when λ2 > 1. Hence, ηtot can be expressed as the ratio of λ1 r(p1 ) + λ1 λ1 − 1 r(p1 ) + λ1 − 1 + − − λ1 + λ2 λ1 + λ2 + 1 λ1 + λ2 − 2 λ1 + λ2 − 1 and

√ 2



λ1 λ2 (λ2 + 1) [λ1 + 2r(p1 )] − r2 (p1 ) (λ1 + λ2 + 2) + (λ1 + λ2 )2 (λ1 + λ2 + 1) (λ1 + λ2 + 1)2 (λ1 + λ2 + 2)  2 ) 21 1 r(p1 ) + λ1 λ1 + − . 2 λ1 + λ2 + 1 λ1 + λ2

Note that, for λ1 close enough to zero, the numerator of ηtot is positive if, and only if, r(p1 ) 1 1 − r(p1 ) + + > 0, λ2 + 1 λ2 − 2 λ2 − 1 for which a sufficient condition is λ2 > 2. Also, the numerator of ηtot decreases with r(p1 ). Recall that for λ1 small enough, the denominator of ηtot increases with r(p1 ). Thus, when λ1 is close to zero and λ2 > 2, the greater r(p1 ), the greater the mean, the smaller the difference between the mean and the mode, the greater the variance, and, hence, the smaller the positive skew of the aggregate distribution of priors. Recall that wages at any job k of a firm f are an affine transformation of the prior in that w(pt ) = cf k + df k pt . It is immediate that if the mean of pt increases, the difference between the mean and the mode of pt decreases, and the variance of pt increases, then the same changes occur for the corresponding moments of the distribution of wages at each job. Proof of Implication 6: Recall that the planner is given two technologies to produce output: job 1, the mailroom job, and job 2, the executive job. The planner has the ability to decide how informative to make each job by choosing α1 and α2 subject to the information capacity constraint (α1 −α)2 +(α2 −α)2 ≤ κ. Note that the further αk is from 1/2, the more information is yielded by job k. Indeed, by Blackwell’s criterion of informativeness, for any two α1 ≥ 1/2 and α2 ≥ 1/2, job 1 is more informative than job 2 if, and only if, α1 > α2 . Recall that a worker’s expected output at job k is y(p1 , k) = ck + dk p1 , given the prior p1 that a worker is of skill type a, and that the static cutoff prior at which expected output at the two jobs is equal is ρ2 = (c1 − c2 )/(d2 − d1 ). Assume that the economy lasts for two periods. To establish the implication, it is sufficient to show that the optimal choice of informativeness by the planner satisfies (α1 − α)/(α2 − α) = g/(1 − g). The planner knows agents’ beliefs about each worker’s skill type. Note that once α1 and α2 are set, the value of welfare in the economy is given by the planning problem V1 (p1 ; α1 , α2 ) = max [(1 − δ)y(p1 , k) + δ(r(p1 , αk ) max {y(PH (p1 , αk ), 1), y(PH (p1 , αk ), 2)} k∈{1,2}

+[1 − r(p1 , αk )] max {y(PL (p1 , αk ), 1), y(PL (p1 , αk ), 2)})]. Let there be a fraction g of workers with prior pL and a fraction 1 − g of workers with prior pH that their skill type is a, where pL < ρ2 < pH . Clearly, if a high enough fraction of workers have prior pL , then it is optimal to make job 1 as informative as possible. Likewise, if a high enough fraction of workers have prior pH , then it is optimal to make job 2 as informative as possible. That is, when g ≥ g for some g, then α1 = α + κ1/2 and α2 = α, whereas when g ≤ g for some g, 5

then α1 = α and α2 = α + κ1/2 . Consider the case in which g ∈ (g, g). If PH (pL , α) > ρ2 and PL (pH , α) < ρ2 , then the planner’s problem simplifies to V1 (p1 ; α1 , α2 ) = max [(1−δ)y(p1 , k)+δ{r(p1 , αk )y(PH (p1 , αk ), 2)+[1−r(p1 , αk )]y(PL (p1 , αk ), 1)}. k∈{1,2}

With pL low enough and pH high enough, the expected value of output of a worker with prior pL is V1 (pL ; α1 , α2 ) = (1 − δ)y(pL , 1) + δ{r(pL , α1 )y(PH (pL , α1 ), 2) + [1−r(pL , α1 )]y(PL (pL , α1 ), 1)}, (5) whereas the expected value of output of a worker with prior pH is V1 (pH ; α1 , α2 ) = (1−δ)y(pH , 2)+δ{r(pH , α2 )y(PH (pH , α2 ), 2)+[1−r(pH , α2 )]y(PL (pH , α2 ), 1)}. (6) Thus, V1 (p1 ; α1 , α2 ) = gV1 (pL ; α1 , α2 )+(1−g)V1 (pH ; α1 , α2 ), with V1 (pL ; α1 , α2 ) and V1 (pH ; α1 , α2 ) as in (5) and (6), respectively. Given V1 (p1 ; α1 , α2 ), the planner chooses α1 and α2 to solve max [gV1 (pL ; α1 , α2 ) + (1 − g)V1 (pH ; α1 , α2 )]

{α1 ,α2 }

s.t. (α1 − α)2 + (α2 − α)2 ≤ κ, with associated multiplier µ. Using the fact that y(p, k) = ck + dk p, y(PH (p, αk ), 2) = c2 + d2 PH (p, αk ) = c2 +

d2 αk p , (2αk − 1)p + 1 − αk

y(PL (p, αk ), 1) = c1 + d1 PL (p, αk ) = c1 +

d1 (1 − αk )p , αk − (2αk − 1)p

r(p, αk ) = (2αk − 1)p + 1 − αk , and 1 − r(p, αk ) = αk − (2αk − 1)p, I can rewrite gV1 (pL ; α1 , α2 ) + (1 − g)V1 (pH ; α1 , α2 ) = (1 − δ)gy(pL , 1) + δgr(pL , α1 )y(PH (pL , α1 ), 2) +δg[1−r(pL , α1 )]y(PL (pL , α1 ), 1) + (1 − δ)(1 − g)y(pH , 2) +δ(1 − g)r(pH , α2 )y(PH (pH , α2 ), 2) + δ(1 − g)[1−r(pH , α2 )]y(PL (pH , α2 ), 1) or, equivalently, I can express gV1 (pL ; α1 , α2 ) + (1 − g)V1 (pH ; α1 , α2 )   d2 α 1 pL = (1 − δ)g(c1 + d1 pL ) + δg[(2α1 − 1)pL + 1 − α1 ] c2 + (2α1 − 1)pL + 1 − α1   d1 (1 − α1 )pL +δg[α1 − (2α1 − 1)pL ] c1 + α1 − (2α1 − 1)pL   d2 α2 pH +(1 − δ)(1 − g)(c2 + d2 pH ) + δ(1 − g)[(2α2 − 1)pH + 1 − α2 ] c2 + (2α2 − 1)pH + 1 − α2   d1 (1 − α2 )pH +δ(1 − g)[α2 − (2α2 − 1)pH ] c1 + . α2 − (2α2 − 1)pH 6

Hence, after collecting and simplifying terms, gV1 (pL ; α1 , α2 ) + (1 − g)V1 (pH ; α1 , α2 ) = (1 − δ)[g(c1 + d1 pL ) + (1 − g)(c2 + d2 pH )] +δg[(c1 + d1 )pL + c2 (1 − pL )] + δ(1 − g)[(c1 + d1 )pH + c2 (1 − pH )] +δg[(c1 − c2 )(1 − 2pL ) + (d2 − d1 )pL ]α1 + δ(1 − g)[(c1 − c2 )(1 − 2pH ) + (d2 − d1 )pH ]α2 . The first-order conditions for α1 and α2 are δg[(c1 − c2 )(1 − 2pL ) + (d2 − d1 )pL ] = 2µ(α1 − α) δ(1 − g)[(c1 − c2 )(1 − 2pH ) + (d2 − d1 )pH ] = 2µ(α2 − α). By taking the ratio of the left and right sides of these two conditions, using the fact that d2 − d1 = 2(c1 − c2 ) so that ρ2 = 1, and simplifying terms, it follows g α1 − α g[(c1 − c2 )(1 − 2pL ) + (c1 − c2 )2pL ] = = (1 − g)[(c1 − c2 )(1 − 2pH ) + (c1 − c2 )2pH ] 1−g α2 − α with c1 > c2 , as desired. In general, when ρ2 is not equal to one, the condition (α1 −α)/(α2 −α) = g/(1 − g) depends on the size of pL and pH . Finally, with ρ2 = (c1 − c2 )/(d2 − d1 ), note that (c1 − c2 )(1 − 2p) + (d2 − d1 )p > 0 if, and only if, ρ2 + (1 − 2ρ2 )p > 0. When ρ2 ≤ 1/2, ρ2 + (1 − 2ρ2 )p > 0 since ρ2 > 0. When, instead, ρ2 > 1/2, ρ2 + (1 − 2ρ2 )p > 0 since 1 − ρ2 > 0. Thus, (c1 − c2 )(1 − 2p) + (d2 − d1 )p > 0 for any p. Proposition A1. In the entry job economy, if job 1 is sufficiently more informative than the other jobs and ρk < ρk+1 for k = 1, . . . , K − 1, then there exists φk∗ satisfying ρ2 < φk∗ < 1 such that the optimal job assignment rule is dynamic and is given by the interval characterization [0, φk∗ ), [φk∗ , ρk∗ +1 ), [ρk∗ +1 , ρk∗ +2 ), . . ., [φK , 1]. In particular, there exist dynamic cutoffs φk∗ and φK such that at priors in [0, φk∗ ) the entry job is assigned, where k ∗ is defined as the smallest job k such that ρk+1 is strictly greater than φk∗ whereas at priors in [φK , 1] job K is assigned. Proof : Note that the requirement ρk < ρk+1 simply amounts to a no-redundancy condition in that it is necessary and sufficient to ensure that all jobs are selected at some prior under a statically optimal policy. Since jobs k = 2, . . . , K have the same informativeness, the planner’s value W (pt ) can be interpreted as the value  of the choice between job 1 or the statically best job ¯ (pt ) , where among jobs 2, . . . , K at t. So, W (pt ) = max W1 (pt ), W W1 (pt ) = (1 − δ)y(pt , 1) + δEW (pt+1 |pt , θ1 )

(7)

¯ (pt ) = (1 − δ) max y(p, k) + δEW (pt+1 |pt , θ). W

(8)

k=2,...,K

Suppose first that job 1 is perfectly informative so that α1 = 1 and β1 = 0. Then, EW (pt+1 |pt , θ1 ) = pt W (1) + (1 − pt )W (0) = pt y(1, K) + (1 − pt )y(0, 1), where I have used the fact that r(pt , θ1 ) = pt , that PH (pt , θ1 ) = 1 and PL (pt , θ1 ) = 0 at pt ∈ (0, 1), that priors of 0 and 1 are absorbing states, and, finally, that under complementarity, job 1 is the

7

best assignment at pt = 0 and job K is the best assignment at pt = 1. In particular, W1 (pt ) = (1 − δ)y(pt , 1) + δ [pt y(1, K) + (1 − pt )y(0, 1)] .

(9)

¯ (0) = y(0, 2) and W1 (1) = (1 − δ)y(1, 1) + Note that by complementarity, W1 (0) = y(0, 1) > W ¯ δy(1, K) < W (1) = y(1, K). Moreover, as is apparent from (9), W1 (pt ) is an affine function of pt , ¯ (pt ) is an increasing convex function whereas by complementarity and the convexity of W (pt ), W ¯ (pt ) once from below to above at φk∗ . of pt . Therefore, W1 (pt ) crosses W Next, I claim that φk∗ > ρ2 . First I show that φk∗ ≥ ρ2 . To see why, suppose by way ¯ (φk∗ ) by the definition of φk∗ . By singleof contradiction that φk∗ < ρ2 with W1 (φk∗ ) = W ¯ crossing of W1 (pt ) and W (pt ), this implies that there exists p ∈ (φk∗ , ρ2 ) such that W1 (p) < ¯ (p). This is impossible, since by complementarity and the definition of the static cutoff ρ2 , W y(p, 1) > y(p, 2) at any p < ρ2 , and by convexity, EW (pt+1 |pt , θ1 ) ≥ EW (pt+1 |pt , θ) since job 1 ¯ (p). Contradiction. is more informative than the other jobs. Hence, at p ∈ (φk∗ , ρ2 ), W1 (p) > W Second, φk∗ cannot equal ρ2 . The reason is that since ρ2 is interior and job 1 is strictly more informative than the other jobs, job 1 has a strict information gain over the other jobs in that EW (pt+1 |ρ2 , θ1 ) − EW (pt+1 |ρ2 , θ) > 0. Since job 1 is tied statically with job 2 at ρ2 , it follows ¯ (ρ2 ). that W1 (ρ2 ) > W ¯ , at any p such that W ¯ (p) > W1 (p), the assigned job is Now, note that by construction of W given by k ∗ (p) = arg maxk=2,...,K y(p, k), which is the statically optimal job. Finally, the same results hold when the entry-level job is sufficiently informative by continuity, for α1 sufficiently close to 1 and β1 sufficiently close to zero. Proof of Proposition 5: Observe that w(pt , θ1 ) = y(pt , 1) and w(pt , θ) = maxk=2,...,K y(pt , k) by the equilibrium assignment rule and the associated wage function. By (7), (8), and the definition ¯ (φk∗ ). Hence, using the expressions for w(pt , θ1 ), w(pt , θ), W1 (pt ), of the cutoff φk∗ , W1 (φk∗ ) = W ¯ (pt ) with pt = φk∗ yields the displayed expression in the statement of the proposition. and W Next, note that since φk∗ > ρ2 it follows that at any pt in (ρ2 , φk∗ ], y(pt , 1) is strictly smaller than y(pt , 2) and over the interval (ρ2 , φk∗ ], the difference y(pt , 2) − y(pt , 1) is positive and largest at φk∗ . Hence, wages discontinuously increase as pt crosses the cutoff φk∗ . Proof of Implication 7: Note first that the expected wage change, E(wt+1 − wt |wt , kt+1 = k + 1 > kt = k), with wt = cf k + df k pt and wt+1 = cf k+1 + df k+1 pt+1 , can be expressed as E(cf k+1 +df k+1 pt+1 −cf k −df k pt |pt , kt+1 = k +1 > kt = k) = cf k+1 −cf k +df k+1 PH (pt , θf k )−df k pt . Since priors change relatively less for an individual with large labor market experience, PH (pt ) cannot be much larger than pt , so E(wt+1 − wt |wt , kt+1 = k + 1 > kt = k) ≈ cf k+1 − cf k + (df k+1 − df k ) pt . Consider first the case in which jobs are equally informative. For a promotion to occur for such an individual, pt must be close to the threshold ρk+1 at which job k and k + 1 have the same expected output. But if so, then E(wt+1 − wt |wt , kt+1 = k + 1 > kt = k) must be small, possibly negligible, since E(wt+1 − wt |wt , kt+1 = k + 1 > kt = k) ≈ cf k+1 − cf k + (df k+1 − df k )ρk+1 when PH (pt , θf k ) ≈ pt ≈ ρk+1 , and the difference cf k+1 − cf k + (df k+1 − df k ) pt in expected output between jobs k + 1 and k is zero when pt = ρk+1 by definition of ρk+1 . In particular, the closer is pt to ρk+1 , the smaller is the wage change at promotion. Consider, in contrast, the case in which jobs are differentially informative with ρ∗k+1 > ρk+1 . In this case, as Figure 4 in the paper shows, even if PH (pt , θf k ) ≈ pt , when promoted individuals 8

have starting prior pt with ρk+1 < pt < ρ∗k+1 , their expected output at job k + 1 is higher than at job k. Therefore, E(wt+1 − wt |wt , kt+1 = k + 1 > kt = k) ≈ cf k+1 − cf k + (df k+1 − df k ) pt > 0. The more informative is job k compared to job k + 1, the larger is the difference ρ∗k+1 − ρk+1 and, then, the larger is the wage increase at promotion, even if pt and PH (pt , θf k ) are very close to each other. Since workers reach priors at which higher-level jobs are optimal only over time, the second part of the result immediately follows. If df k+1 − df k increases with k fast enough—given that changes in the prior tend to be smaller and smaller as pt becomes large—wage jumps at promotion can also be larger at higher levels.

B

Promotion and Demotion Rungs

This interval-based rule for job assignment can be further partitioned into rungs of promotion and demotion, depicted in Figure A.1. Specifically, suppose that PH (ρk+1 ) < ρk+2 and PL (ρk ) ≥ ρk−1 , so that all promotions and demotions are by one level. For promotion rungs, given the cutoff ρk+1 , qk2 ) = ρk+1 , and so on, where PH2 (p) define the priors q¯k1 , q¯k2 , . . . so that PH (¯ qk1 ) = ρk+1 , PH2 (¯ ¯ k be the smallest integer such denotes PH (PH (p)) and similar notation is used for PHn (p). Let n that PHn¯ k (ρk ) ≥ ρk+1 . Thus, job k has n ¯ k promotion rungs given by [ρk , q¯k,¯nk −1 ), . . ., [¯ qk2 , q¯k1 ), [¯ qk1 , ρk+1 ), with the interpretation that a worker in promotion rung n ∈ {1, . . . , n ¯ k } of job k, namely, with a prior in [¯ qkn , q¯k,n−1 ), will be promoted to job k + 1 exactly after n successes. Demotion rungs can be constructed analogously. Given the cutoff prior ρk for demotion to job k − 1, define the priors q k1 , q k2 ,. . . so that PL (q k1 ) = ρk , PL2 (q k2 ) = ρk , and so on, where n nk is the smallest integer such that PL k (ρk+1 ) < ρk ; the equalities PLn (q kn ) = ρk are intended to hold for priors just below the cutoff ρk . Then, job k has nk demotion rungs given by [ρk , q k1 ), [q k1 , q k2 ), . . ., [q kn , ρk+1 ), with the interpretation that a worker in demotion rung n ∈ {1, . . . , nk } k of job k, namely, with a prior in [q k,n−1 , q kn ), will be demoted to job k − 1 after n failures. This construction can be easily modified to produce multi-step promotions or demotions. I focus on one-step transitions simply for consistency with the data.

C

Example: Heterogeneous Priors and Promotion

Here I show that with multiple initial priors, the model with general ability can naturally produce hazards of promotion nonmonotone with tenure. For simplicity only, assume that the output distribution is symmetric across types of workers in that type a workers succeed with probability α and type b workers succeed with probability β = 1 − α at any job. This symmetry ensures that one failure and one success cancel each other in that PH (PL (p)) = PL (PH (p)) = p. Next, in a slight abuse of notation, let the initial distribution of priors be discrete and correspond to just two groups of workers, denoted by A and B, with mass g(pL ) on prior pL and mass g(pH ) on prior pH , respectively, where pH > pL . Suppose the initial priors are such that both groups start at job 1 in period 1 but that group B workers need one success to be promoted, whereas group A workers need two successes, that is, PH (pL ) < ρ2 < PH (pH )

(10)

and PH (PH (pL )) > ρ2 , where ρ2 is the cutoff prior for promotion to job 2. This example is illustrated by Figure A.2. I will focus on promotion from job 1 to job 2. 9

After one year, only the group B workers who were successful get promoted. The successful group A workers have updated prior PH (pL ), which by (10) falls short of the promotion cutoff. Hence, after one year the fraction r(pH )g(pH ) (11) of all workers get promoted, where r(p) = αp + β(1 − p) denotes the fraction of successes for a group with prior p. After two years, only the group A workers who received two successes get promoted, so the fraction of promoted workers is r(PH (pL ))r(pL )g(pL ). (12) The group B workers who failed in the first year and then succeeded in the second have priors PH (PL (pH )), which by symmetry equals the initial prior pH below the cutoff for promotion. Likewise, the group B workers who succeeded in the first year and then failed in the second have priors PL (PH (pH )), which also equals pH . Adding together these two sets of group B workers gives that after two years there is a measure of {r(pH ) [1 − r(PH (pH ))] + [1 − r(pH )] r(PL (pH ))} g(pH )

(13)

workers at prior pH . After three years, the group B workers of measure (13) with priors pH who receive a success have priors PH (pH ). By (10) these workers are promoted, and using (13) these workers are fraction r(pH ) {r(pH ) [1 − r(PH (pH ))] + [1 − r(pH )] r(PL (pH ))} g(pH )

(14)

of the workers. Note that no worker from group A can be promoted in period 3: any group A worker with three successes has already been promoted in period 2, and any group A worker who experienced at least one failure has a prior no greater than PH (PH (PL (pL )) = PH (pL ), which by (10) falls short of the cutoff level for promotion, ρ2 . Clearly, if g(pL ) is sufficiently larger than g(pH ) = 1 − g(pL ), then the promotion rate initially increases and then decreases, in that (11) is smaller than (12), which is larger than (14).

D

Details of the Two-Period Example in Section 6.5

Recall the two-period economy in Section 6.5 with three jobs. I will maintain that job 1 is the most informative, job 2 the next-most informative, and job 3 the least informative. I assume that these jobs are ordered as in Figure 4 in the paper. Job Assignment. In the second period, since there is no benefit to information, the allocation rule is static: a worker is assigned to jobs 1, 2, and 3, respectively, as the prior ranges from [0, ρ2 ) to [ρ2 , ρ3 ) to [ρ3 , 1]. In the first period, the allocation rule is dynamic: a worker is assigned to jobs 1, 2, and 3, respectively, as the prior ranges from [0, φ2 ) to [φ2 , φ3 ) to [φ3 , 1]. Under three simple conditions, this dynamic path of job assignments is optimal and entails promotions by one step but no demotions. The first two, PL (φ2 , θ2 ) ≥ ρ2 and PL (φ3 , θ3 ) ≥ ρ3 ,

(15)

rule out demotions. The first inequality implies that even if a worker fails at the lowest prior at which job 2 is assigned in the first period, φ2 , the resulting posterior PL (φ2 , θ2 ) is still above the 10

static cutoff for job 2, ρ2 . Hence, all workers assigned to job 2 in the first period who fail will still be assigned to job 2. The second inequality analogously rules out demotions from job 3. The third condition, PH (φ2 , θ1 ) < ρ3 , (16) rules out promotions by more than one level: even if a worker succeeds at the highest prior at which job 1 is assigned in the first period, a prior just below φ2 , the resulting posterior PH (φ2 , θ1 ) is still below the static cutoff for job 3, ρ3 . Under these restrictions, the dynamic cutoff φ2 satisfies W1 (φ2 ) = W2 (φ2 ), where W1 (φ2 ) = (1 − δ)y(φ2 , 1) + δ{r(φ2 , θ1 )y(PH (φ2 , θ1 ), 2) + [1 − r(φ2 , θ1 )]y(PL (φ2 , θ1 ), 1)}

(17)

and W2 (φ2 ) = y(φ2 , 2). To understand W2 (φ2 ), note first that since job 1 is more informative than job 2, PH (φ2 , θ2 ) ≤ PH (φ2 , θ1 ). Then, condition (16) implies that after a success at job 2, job 2 is still optimal, given that PH (φ2 , θ2 ) > PL (φ2 , θ2 ) with αf 2 > βf 2 , and PL (φ2 , θ2 ) ≥ ρ2 by the first condition in (15). Next, observe that after a failure at job 2, the prior is updated to PL (φ2 , θ2 ), and job 2 is again optimal by the first condition in (15). To understand W1 (φ2 ), note that under the plan associated with it, in period 1 a worker is assigned to job 1. After success, the prior is updated to PH (φ2 , θ1 ), at which condition (16) and the fact that PH (φ2 , θ1 ) ≥ ρ2 since φ2 ≥ ρ2 imply that job 2 is optimal. After failure, the prior is updated to PL (φ2 , θ1 ), in which case the worker is again assigned to job 1. (If not, then by (16) a worker who starts in job 1 at φ2 works in job 2 in the second period regardless of success or failure. Given that W2 (φ2 ) = y(φ2 , 2), this would imply φ2 = ρ2 and PL (φ2 , θ2 ) < ρ2 since αf 2 > βf 2 , which is inconsistent with the first condition in (15).) The dynamic cutoff φ3 satisfies W2 (φ3 ) = W3 (φ3 ), where W2 (φ3 ) = (1 − δ)y(φ3 , 2) + δ{r(φ3 , θ2 )y(PH (φ3 , θ2 ), 3) + [1 − r(φ3 , θ2 )]y(PL (φ3 , θ2 ), 2)}

(18)

and W3 (φ3 ) = y(φ3 , 3). To understand W3 (φ3 ), note first that after a success at job 3, job 3 is still optimal, since PH (φ3 , θ3 ) > PL (φ3 , θ3 ) with αf 3 > βf 3 , and PL (φ3 , θ3 ) ≥ ρ3 by the second condition in (15). Next, note that after a failure at job 3, the prior is updated to PL (φ3 , θ3 ), and job 3 is still optimal, by the second inequality in (15). To understand W2 (φ3 ), note that after a success at job 2, the prior is PH (φ3 , θ2 ). Now, PH (φ3 , θ2 ) ≥ PH (φ3 , θ3 ) since job 2 is more informative than job 3, PH (φ3 , θ3 ) > PL (φ3 , θ3 ) since αf 3 > βf 3 , and PL (φ3 , θ3 ) ≥ ρ3 by the second condition in (15). Thus, PH (φ3 , θ2 ) ≥ ρ3 and working in job 3 is optimal. After a failure at job 2, the prior is PL (φ3 , θ2 ), in which case the worker is again assigned to job 2. (Analogously to the above, if not, then φ3 = ρ3 and PL (φ3 , θ3 ) < ρ3 , which is inconsistent with the second condition in (15).)

E

Human Capital Acquisition

One dimension of the data the model may not well capture is that individual wages on average increase with time in the labor market, even for individuals who do not experience changes in job or employing firm. Specifically, in the framework considered in the paper, two workers with the same prior receive the same wage regardless of their experience, measured by the number of periods both have been in the labor market. This feature can be easily remedied by allowing for human capital acquisition. For simplicity, suppose that a worker can acquire new productive 11

skills, that is, human capital, just based on labor market experience t, so that now expected output is yf (pt , k, t) = yf (pt , k) + ηt, η ≥ 0. In this case, the equilibrium assignment policy is unchanged whereas wages are simply augmented by ηt. Proposition E1. When workers can acquire human capital, the optimal assignment rule coincides with that of the general economy. Equilibrium wages are given by w(pt , θf k , t) = w(pt , θf k ) + ηt, with w(p, θf k ) determined as in Section 3 in the paper. Proof : The proof that the assignment rules for the two economies coincide follows immediately from the observation that the objective function in the planning problem for the human capital economy is the sum of the objective function for that problem in the original economy plus terms that are independent of the allocations of workers to jobs and firms. Hence, the two solutions coincide. The proof that wages are given by w(pt , θf k , t) = w(pt , θf k ) + ηt follows from the same logic that was used to derive w(pt , θf k ) in the original economy. Here, I have considered the simplest case in which human capital is produced and used in the same way in all jobs. Of course, once human capital is allowed to be differentially acquired in some jobs and differentially productive in other jobs, the job assignment rule in the economy with information and human capital acquisition must be dynamically designed to balance the accumulation of both information and human capital. Hence, the job assignment rules will differ from the rules considered so far in the obvious ways.

References Kim, T.H., and H. White (2003): “On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index,” mimeo.

12

Figure A.1. Promotion and Demotion Rungs A. Promotion Rungs

promotion after successes

promotion after 2 successes

promotion after 1 success

B. Demotion Rungs

k

qk1 demotion after 1 failure

qk 2 demotion after 2 failures

k 1

q k nk demotion after n k failures

Figure A.2. Heterogeneous Priors and Promotions

mass g ( pL ) mass g ( pH )

0

pL

PH ( pL )

pH

2 PH  PH ( pL ) 

PH ( pH )

3

Note: Measure g ( pL ) of workers have prior pL and measure g ( pH ) of workers have prior pH .

1