Appendix for: Which Workers Get Insurance Within the Firm? David Lagakos∗

˜ Guillermo Ordonez

Arizona State University

Yale University

November 14, 2011

Abstract This appendix contains proofs of all propositions from the model section and additional robustness analysis to complement the regression analysis presented in the paper.

Keywords: insurance within the firm, risk sharing, limited commitment, displacement costs, wage smoothing JEL codes: D21, E32, J24, J41

∗

Corresponding author. Email: [email protected], phone: 480-965-3531, address: Department of Economics, Arizona State University, 501 E Orange St., CPCOM 412A, Tempe, AZ 85287-9801.

A Appendix A.1 Proofs of Propositions Proof of Proposition 1: Wages are Smoothed in the Optimal Contract Fix a state (v, p) and let η be the Langrange multiplier on the promise keeping constraint (6). For the worker and firm self-enforcing constraints (7) and (8) let the multipliers be βαp′ |p λe (p′ ) and βαp′ |p λf (p′ ). The first order conditions for w and each v ′ (p′ ) imply 1 1 = (1 + λf (p′ )) − λe (p′ ) ′ uw (w) uw (w )

∀p

′

(17)

If w ′ = w then it must be true that λf (p′ ) = λe (p′ ) = 0, which implies that v ′ (p′ ) > V (p′ ) and Π(v, p′ ) > Π(p′ ). If w ′ > w then uw (w ′ ) < uw (w) by concavity, which by (17) implies that λe (p′ ) > 0 and hence v ′ (p′ ) = V (p′ ). By a similar argument w ′ < w implies that λf (p′ ) > 0, and hence Π(v ′ , p′ ) = Π(p). 

Set up for Proofs of Propositions 3, 4, 5, and 6 It is useful for the proofs of Propositions 3, 4, 5, and 6 to express the two-state environment in matrix notation. This allows us to express equations (9) and (10) in terms of the primitive parameters and to conduct comparative statics. Define    ¯ u(pH (1 − d))   VH  −1  , =A   u(pL (1 − d)) V¯L 









u(wH )   VH  −1   , and =A   u(wL ) VL

30









p − wH   ΠH  −1  H , =A   pL − w L ΠL where 

 A−1 = inv 



1 − βα





−β(1 − α)  1  1 − βα β(1 − α)  =   |A| −β(1 − α) 1 − βα β(1 − α) 1 − βα

and |A| = (1 − βα)2 − β 2 (1 − α)2 . Define further

aH =

A−1 (1 − βα) 11 ∈ [0.5.1] −1 −1 = (1 − βα) + (β − βα) A11 + A12

aL =

(β − βα) A−1 21 ∈ [0, 0.5]. −1 −1 = (1 − βα) + (β − βα) A21 + A22

and

We can rewrite constraints (9) and (10) as, VH (wH , wL ) − V¯H (pH , pL ) ≥ 0,

(18)

Π(wH , wL , pH , pL ) ≥ 0,

(19)

where the worker’s value of the contract in the high state is

VH (wH , wL ) ≡ u(wH )aH + u(wL )(1 − aH ),

(20)

the worker’s outside option in the high state is V¯H (pH , pL ) ≡ u(pH (1 − d))aH + u(pL (1 − d))(1 − aH ),

31

(21)

and the firm’s value of the contract in the low state is

Π(wH , wL , pH , pL ) ≡ (pH − wH )aL + (pL − wL )(1 − aL ).

(22)

The strategy for the proofs of Propositions 3, 4, 5, and 6 is the following. First, we fix an original wage contract, wL and wH , which solves the firm’s constrained profitmaximization problem some productivity process. Then, we modify the productivity process and ask how the constraints (18) and (19) relax or tighten at those original wages. If one or both constraints relax, we know by Propositions 1 and 2 that the new optimal contract must reduce the difference between wages in the two states, and wage smoothing increases. In contrast if one or both constraints tighten, the new contract must increase the differences in wages in the two states, and wage smoothing decreases.

Proof of Proposition 3: Wage Smoothing is Independent of the Average level of Productivity Let wL and wH be the solutions to the firm’s problem given a productivity process with states pH and pL . Now consider a new process in which productivity changes from pH to p′H = xpH and from pL to p′L = xpL for some x. We claim that the new ′ optimal wages are exactly wH = xwH and wL′ = xwL .

To establish this claim, note that because preferences are CRRA, u(xc) = u(x)u(c). Thus scaling both productivity and wage levels by x does not modify constraint (18), since ′ VH (wH , wL′ ) = u(x)VH (wH , wL )

and V¯H (p′H , p′L ) = u(x)V¯H (pH , pL ). 32

Thus, if the original constraint (18) binds, the new constraint also binds: u(x)[VH (wH , wL ) − V¯H (pH , pL )] = 0.

Scaling both productivity and wage levels by x does not modify constraint (19) either. This is because ′ ΠL (wH , wL′ , p′H , p′L ) = xΠL (wH , wL , pH , pL ),

thus, if the original constraint (19) binds then the new constraint also binds:

xΠL (wH , wL , pH , pL ) = 0.

Hence, smoothing remains identical when productivity and wages are both scaled by x. The elasticity in the new contract is εxw,xp = elasticity in the original contract, εw,p =

(wH −wL ) . (pH −pL )

x(wH −wL ) , x(pH −pL )

and is the same as the



Proof of Proposition 4: Wage Smoothing is Increasing in Displacement Costs Let wL and wH be the solutions to the firm’s problem given some productivity process with displacement costs d. Now consider an increase to higher displacement costs d′. Note that this change lowers the worker’s outside option in the high state, (21). Thus constraint (18) loosens. On the other hand, at the original wages, the firm’s value in or out of the contract in the low state does not change. Thus constraint (19) does not change. Since one of the two constraints loosens at the original wages, the new ′ ′ optimal contract specifies wages wH and wL′ such that wH − wL′ < wH − wL . This

implies that the new elasticity is lower, and hence there is more wage smoothing. 

Proof of Proposition 5: Wage Smoothing is Decreasing in the Autocorrelation of Productivity

33

Let wL and wH be the solutions to the firm’s problem given some productivity process with autocorrelation parameter α. Now imagine a change to a higher parameter α′ . At the original wages, we show that this affects the value of the contract for the worker in the high state (equation 20), the worker’s outside option in the high state (equation 21), and the value of the contract for the firm in the low state (equation 22). The reason is that an increase in α increases aH and reduces aL : β(1 − β) ∂aH = >0 ∂α (1 + β(1 − 2α))2 β(1 − β) ∂aL =− < 0. ∂α (1 + β(1 − 2α))2 This implies both VH (wH , wL |α) and V¯H (pH , pL |α) are higher. However, since pH > wH , VH (wH , wL |α′ ) − V¯H (pH , pL |α′ ) < VH (wH , wL |α) − V¯H (pH , pL |α) which tightens constraint (18). Similarly, a larger α reduces aL and thus tightens constraint (19). Hence the original smoothing given by wH and wL is not sustainable ′ anymore, implying the solution requires wH − wL′ > wH − wL , and hence a larger

elasticity εw,p and less wage smoothing. 

Proof of Proposition 6: Wage Smoothing is Increasing in the Volatility of Productivity if the Discount Factor is Sufficiently High Let wL and wH be the solutions to the firm’s problem given some productivity process pH and pL . Now assume a mean-preserving increase in the volatility of the productivity process, such that p′H = pH + z and p′L = pL − z for some fixed z. We show that volatility (which is proportional to z) affects both the worker’s outside option and the value of the contract for the firm. An increase in z reduces the value of the contract for the firm, since aL ≤ 0.5, and 34

hence Π(wH , wL , p′H , p′L ) − Π(wH , wL , pH , pL ) = −z(1 − 2aL ) ≤ 0. Note that an increase in z tightens constraint (19) for all β < 1 and does not modify it for β = 1. Furthermore, the derivative of the difference with respect to β is negative (since

∂aL ∂β

=

(1−α) (1+β(1−2α))2

> 0). This implies that constraint (19) tightens monotonically

more the lower the β. An increase in z modifies the worker’s outside option as well: V¯H (p′H , p′L ) − V¯H (pH , pL ) = ∆H (z)aH − ∆L (z)(1 − aH )

where ∆H (z) ≡ u(pH + z) − u(pH ) < ∆L (z) ≡ u(pL ) − u(pL − z) (the inequality arises from the concavity of the utility function). Furthermore, the derivative of the difference with respect to β is negative (since

∂aH ∂β

(1−α) = − (1+β(1−2α)) 2 < 0). This implies

that constraint (18) tightens monotonically more the lower the β. Next we consider the extreme cases of β = 1 and β = 0. We claim first that when β = 1, the contract sustains more wage smoothing. To see this, note that when β = 1, aL = aH = 0.5. Hence, under the new productivity process, constraint (18) relaxes because the worker’s outside option decreases, and constraint (19) does not change because the firm’s value of the contract does not change. Second, we claim that when β = 0 the contract sustains less wage smoothing. To see this, note that when β = 0, aL = 0 and aH = 1. Thus, under the new productivity process, constraint (18) tightens because the worker’s outside option increases, and constraint (19) also tightens because the firm’s value in the contract decreases. Given the results for the extremes values of β, and since both constraints (18) and (19) monotonically relax with β, it implies there is a unique β(z) ∈ (0, 1) where constraint (18) is relaxed and (19) is tightened, such that wage smoothing does not change. For 35

all β > β(z) wage smoothing increases and for all β > β(z) wage smoothing reduces in the new optimal contract. 

Proof of Proposition 8: Separation Probability Decreases in Displacement Costs First, from the Envelope Condition of the firm’s maximization problem

∂Π(p,ǫ) ∂V (p,ǫ)

=

− u′ 1(w) < 0, which is negative since by assumption u′w > 0. w

Second, for a given aggregate productivity p and a fixed promised utility v,

∂Π(p,ǫ) ∂ǫ

> 0,

implying that, as the idiosyncratic shock ǫ increases, Π(p) increases. Third, from the definition of the worker’s outside option,

∂V (p) ∂d

<0

Now, fixing (p, ǫ), ǫ∗p is determined by the point at which Π(V (p), ǫ∗p ) = 0. Evaluated at the state (p, ǫ∗p ) where the worker binds (i.e., V (p) = V (p)), ∂Π(p, ǫ∗p ) ∂V (p) ∂Π(p, ǫ∗p ) ∂ǫ∗p + =0 ∂d ∂ǫ∗p ∂d ∂V (p) From the signs of the partial derivatives above ∂ǫ∗p <0 ∂d

Naturally, this means the probability of separation F (ǫ∗p ) decreases. 

Proof of Proposition 9: Wage Smoothing Increases in Displacement Costs; Richer Model We will proceed in three steps. First, considering an initial wage w−1 we will prove that as d increases (say from d1 to d2 > d1 ), in states where wages increase, they increase by less and in states where wages decrease, they also decrease by less, sustaining more smoothing. Second we will show the number of states in which wages 36

do not change is larger. Finally, we show how these properties translate into more wage smoothing by implying a lower elasticity Considering a given state in the previous period (p−1 , ǫ−1 ) with a corresponding wage in the previous period w−1 , we can split all possible current states into four subsets.

1) States (p, ǫ) (such that ǫ < ǫ∗p ) where both the worker and the firm binds. In this case there is endogenous separation (s(p, ǫ) = 1). 2) States (p, ǫ) (such that ǫ > ǫ∗p ) where there is continuation (s(p, ǫ) = 0). We can classify these states in three independent subsets. a) States (p, ǫ) ∈ Ωw p (d) where only the worker binds (v(p, ǫ) = V (p) and Π(p, ǫ) = b ǫ) > Π(p)). Hence w(p, ǫ) = w p,ǫ ≥ w−1 . Π(p,

b) States (p, ǫ) ∈ Ωfp (d) where only the firm binds (v(p, ǫ) = Vb (p, ǫ) > V (p) and

Π(p, ǫ) = Π(p)). Hence w(p, ǫ) = w p,ǫ ≤ w−1 .

c) States (p, ǫ) ∈ Ωnp (d) where none binds (v(p, ǫ) > V (p) and Π(p, ǫ) > Π(p)). Hence w(p, ǫ) = w−1 .

In the first step we show an increase in d (from d1 to d2 ) reduces (weakly) the change in wages in all states, both directly by reducing the outside option and indirectly by reducing the separation probability. The second step shows that an increase in d increase the number of states in which wages remain unchanged. −1 ) ≤ 0 for all (p, ǫ). This is, an increase in d reduces (weakly) wage Step 1: ∂(w(p,ǫ)−w ∂d

changes at each state. a:Direct effect from outside options Starting from a level of displacement costs d1 , we know that in all states in the subset 37

w Ωw p (d1 ), v = V (p) determines w p,ǫ ≤ w−1 . More specifically, in states in Ωp (d1 ).

V (p, ǫ) = u(wp,ǫ) + βEp′,ǫ′ V (p′ , ǫ′ ) = V (p)

where Ep′ ,ǫ′ V ′ (p′ , ǫ′ ) = (

Ep′

"

′ F (ǫ∗p′ )V (p′ ) + (1 − F (ǫ∗p′ )) P r(Ωw p′ )V (p ) +

P r(Ωfp′ )Vb (p′ )

+

Z

V ′ (p′ , ǫ′ )dF (ǫ′ )

ǫ′ ∈Ωn p′

#)

Similarly, in all states in the subset Ωfp (d1 ), Π(p, ǫ) = Π(p) determines w p,ǫ ≥ w−1 . More explicitly, Π(p, ǫ) = p + ǫ − w p,ǫ ) + βEp′,ǫ′ Π(p′ , ǫ′ ) = Π(p) where Ep′ ,ǫ′ Π′ (p′ , ǫ′ ) = (

Ep′

"

f ′ b ′ F (ǫ∗p′ )Π(p′ ) + (1 − F (ǫ∗p′ )) P r(Ωw p′ )Π(p ) + P r(Ωp′ )Π(p ) +

Z

Π′ (p′ , ǫ′ )dF (ǫ′ )

ǫ′ ∈Ωn p′

Assume an increase of displacement costs from d1 to d2 . We focus first in the case in which the separation probability F (ǫ∗p′ ) does not change, hence highlighting the direct effects of d through its impact on outside options. In the next part of the step we discuss how changes in separation reinforce the results. From equation 12,

∂V (p′ ) ∂d

< 0 for all p′ . If d increases but wages do not change in any

state, all equations in the set Ωfp (d1 ) remain binding (since Π(p) do not change for any p) while all equations in the set Ωw p (d1 ) stop binding (since V (p) decrease for all p). Hence, it is optimal for the firm to reduce wages in some states (reducing average wages in the match). The same is true for all future aggregate states p′ in equilibrium. 38

#)

Since V (p′ ) decrease for all p′ and

∂Π(p,ǫ) ∂V (p,ǫ)

b ′ ) increase in all states (p′ , ǫ′ ) ∈ < 0, Π(p

′ ′ ′ Ωw p′ (d1 ). This implies an increase of Ep′ ,ǫ′ Π (p , ǫ ) and then an increase in w p,ǫ for all

states (p, ǫ) ∈ Ωfp (d1 ), where the firm used to bind with d1 . The only way this increase in wages is consistent with a decrease in average wages is that w p,ǫ decrease in some positive mass of states (p, ǫ) ∈ Ωw p (d1 ) where the worker used to bind with d1 .

b:Indirect effect from outside options through separation rates: From Proposition 8, an increase in d reduces F (ǫ∗p ) in all aggregate states p, further increasing (weakly) Ep′ ,ǫ′ Π′ (p′ , ǫ′ ) (since the expected profit from continuation is by construction greater than the expected outside option). This further increases w p,ǫ in all states (p, ǫ) ∈ Ωfp (d1 ) and further reduces wp,ǫ in a positive mass of states (p, ǫ) ∈ Ωw p (d1 ), based on the same argument used in a). Formally these effects can be summarized in

∂(w(p,ǫ)−w−1 ) ∂d

< 0 for all states (p, ǫ) (and strict for a positive mass of

f states) belonging to Ωw p (d1 ) and Ωp (d1 ).

Step 2: Ωnp (d1 ) ⊆ Ωnp (d2 ) for d1 < d2 . This is, an increase in d increases (weakly) the number of states in which wages do not change For this step, we compare the states (p, ǫ) for which the match continues with d2 , which gives the right information for the comparison of wage smoothing. Recall the subset Ωfp (d1 ) is defined by all states in which w−1 > wp,ǫ (d1 ) (since w−1 is not enough to cover the firm’s outside option, it is necessary to reduce wages until she binds). As shown in Step 1, wp,ǫ increases with d in all states (p, ǫ) ∈ Ωfp (d1 ). For a fixed w−1 , this implies Ωfp (d2 ) ⊆ Ωfp (d1 ). Furthermore, since the worker does not bind in (p, ǫ) ∈ Ωfp (d1 ) and V (p|d2 ) < V (p|d1 ), he continues without binding in those states. This implies Ωfp (d1 ) \ Ωfp (d2 ) ⊆ Ωnp (d2 ). A symmetric reasoning can be applied to the subset where the worker binds, such w n w w that Ωw p (d2 ) ⊆ Ωp (d1 ) and Ωp (d1 ) \ Ωp (d2 ) ⊆ Ωp (d2 ). The combination of these two

39

results implies Ωnp (d1 ) ⊆ Ωnp (d2 ) (i.e., there are more states in which none binds and wages do not move).

Step 3:An increase in d increases wage smoothing The previous two steps allows us to characterize how smoothing changes with d. Increasing d reduces the changes from w−1 to w(p, ǫ) in some states where wages increase and from w−1 to w(p, ǫ) in all states where wages decrease. Furthermore there are more states (p, ǫ) in which wages do not change. Since the elasticity of wages to productivity is defined by εw,(p,ǫ) =

∆w , ∆(p,ǫ)

and we proved

(p, ǫ) (strictly for a positive mass of states), then

∂εw,(p,ǫ) ∂d

∂(w(p,ǫ)−w−1 ) ∂d

≤ 0 for all states

< 0, and wage smoothing

increases in displacement costs. 

A.2 Tests of Alternative Hypothesis We now assess several alternative hypotheses about what drives the industry variation in wage smoothing. Table 1 adds several other industry characteristics, one by one, to the regression results of Table 6, to see whether alternative hypotheses are supported by the industry data. Columns 1 and 2 of Table 1 show, as a frame of reference, the regression of the wage elasticity of productivity on (1) just percent college graduate, and then (2) percent college graduate, displacement costs, plus the volatility and autocorrelation of productivity. The first hypothesis we test is that unionization keeps wages from responding to productivity. The idea that workers involved in collective bargaining agreements will have smoother wages has been well studied – see for example Lindbeck and Snower (1988). To test this hypothesis in our data, Column 3 of Table 1 adds the unionization rate of the industry to the set of regressors. The coefficient estimate is negative, meaning that a higher unionization rate leads to smoother wages, as one might expect. 40

Nevertheless, the P-value is 0.315, meaning that the estimate is statistically insignificant at any reasonable confidence level. In addition, the coefficient on displacement costs and the other independent variables maintain their sign and magnitude. This casts doubt on the hypothesis that differences in unionization rates explain much of the cross-industry variation in wage smoothing in our data.17 Next we test whether industry differences in the importance of labor in the production process drive wage smoothing. This could be because workers are better able to bargain for wage increases in industries where they are a more important factor of production. To test this hypothesis, Regression 4 adds the labor share, measured by the average ratio of the industry wage bill to value added, to the set of regressors. The coefficient estimate turns out to be small in magnitude and far from being statistically significant, while leaving the other coefficients largely unchanged. We conclude that the labor share is unlikely to be an important driver of wage smoothing. The third hypothesis is that wage productivity elasticities are explained in part by the distinction between manufacturing industries than other industries due to potential differences in quality of output data. If output is measured less accurately in services, which many have argued (see e.g. Bosworth and Triplett (2004) on the challenges with measuring service output), then we would expect to have very low elasticities in service since output is so noisy. (In the limit, if detrended output were white noise, then the service elasticity would be zero.) Thus, our headline observation of lower elasticities in high skill industries could really be because services industries have high skill, and service elasticities are low because of measurement error. The flip side is that manufacturing industries have lower skill levels, but higher elasticities because output is actually measured with more accuracy. 17

A number of studies have found a role of unionization on wage smoothing in Europe, where unions are arguably more powerful. See e.g. Dickins et al (2007) and Nickell and Wadwhani (1990). Further study of U.S.-European differences in wage smoothing determinants would be interesting, but is beyond the scope of this paper.

41

We test this hypothesis in Regression 5, which includes a manufacturing dummy into the regression. The regressions show that the manufacturing dummy has a coefficient close to zero and a very high P-value, and has little bearing on the size or significance of other regressors. Therefore, the manufacturing/non-manufacturing distinction appears to be an unimportant factor. While measurement error might cloud service output measures, this is not driving our results at all.

(1)

(2)

(3)

(4)

(5)

(6)

-1.122*** (3.01e-05)

-0.581** (0.0110)

-0.638*** (0.00711)

-0.612** (0.0135)

-0.608** (0.0113)

-0.656** (0.0110)

Displacement Costs

-0.610*** (0.00307)

-0.556*** (0.00859)

-0.570** (0.0152)

-0.552** (0.0252)

-0.523** (0.0440)

Volatility of Productivity

-1.935** (0.0195)

-1.689** (0.0491)

-1.909** (0.0231)

-1.903** (0.0234)

-1.682* (0.0558)

Autocorrelation of Productivity

0.363** (0.0159)

0.350** (0.0206)

0.351** (0.0242)

0.310 (0.115)

0.325 (0.105)

Regression # Variable Percent College Graduate

Unionization Rate

-0.145 (0.315)

Labor Share

-0.136 (0.374) -0.060 (0.719)

Manufacturing dummy

Constant

Observations R-squared

-0.020 (0.910) -0.024 (0.669)

-0.009 (0.879)

1.161*** (7.68e-09)

0.699*** (8.82e-05)

0.755*** (6.73e-05)

0.762*** (0.00281)

0.747*** (0.000501)

0.792*** (0.00294)

48

48

48

48

48

48

0.318

0.607

0.616

0.608

0.608

0.617

Note: Dependent variable is the wage-productivity elasticity; observations are U.S. industries. P-values are in parentheses: *** p < 0.01, ** p < 0.05, * p < 0.1

Table 1: Testing Alternative Hypotheses

42