Corrigendum to "Managerial Incentive Problems: A Dynamic Perspective"
aa r X i v : . [ ec on . T H ] N ov CORRIGENDUM TO “MANAGERIAL INCENTIVE PROBLEMS: ADYNAMIC PERSPECTIVE”
SANDER HEINSALU
Abstract.
This paper corrects some mathematical errors in Holmstr¨om (1999) and clarifies theassumptions that are sufficient for the results of Holmstr¨om (1999). The results remain qualitativelythe same.Keywords: Career concerns, symmetric incomplete information, dynamic games.JEL classification: C72, C73, D83. Introduction
This note corrects some mathematical errors in the career concerns model with normally dis-tributed type and signal in Holmstr¨om (1999) Sections 2.1–2.3. The corrigendum is organized asfollows. Section 2 introduces the model and the additional assumptions on the cost function thatare sufficient for some of the results in Holmstr¨om (1999). Section 2.1 derives the equilibriumstrategy in detail, correcting some errors in the derivation of Holmstr¨om (1999). Section 2.2 firstcorrects an inconsistency between Holmstr¨om (1999) Sections 2.1 and 2.2. Second, it introducesan assumption necessary and sufficient for the equilibrium labor supply to converge to zero as thetype becomes persistent or the random component of the output large. Third, it discusses ways toresolve the indeterminacy of equilibrium in the undiscounted case. Section 2.3 presents two waysto modify the definitions in Holmstr¨om (1999) Section 2.3, both of which correct an error in theproof of Proposition 2. 2.
The model
The notation and the model follow Holmstr¨om (1999). The players are a manager and a com-petitive market. Time is discrete, indexed by t ∈ N , and the horizon is infinite. The managerinitially has ability η , which is symmetric incomplete information, and commonly believed to bedrawn from the prior probability distribution N ( m , /h ), where h ∈ (0 , ∞ ) is the precision (theinverse of the variance). Ability evolves according to η t +1 = η t + δ t , where δ t ∼ N (0 , /h δ ) i.i.d.over time. The basic model of Holmstr¨om (1999) assumes h δ = ∞ , which is later relaxed. Thiscorrigendum treats the cases h δ = ∞ and h δ ∈ (0 , ∞ ) together. Date : November 5, 2018.Research School of Economics, Australian National University. HW Arndt Building, 25a Kingsley St, ActonACT 2601, Australia. Email: [email protected], website: https://sanderheinsalu.com/ . Each period t , the manager chooses labor ˆ a t ∈ [0 , ∞ ], which generates public output y t = η t + ˆ a t + ǫ t , where ǫ t ∼ N (0 , /h ǫ ) i.i.d. over time, h ǫ ∈ (0 , ∞ ), and ǫ t is independent of η , δ s forany t, s . The competitive and risk-neutral market observes y t − = ( y , . . . , y t − ) ∈ R t − at thestart of period t and pays the manager a wage w t ( y t − ) ∈ [ −∞ , ∞ ] equal to the manager’s expectedoutput. The market’s wage rule w = ( w t ) ∞ t =1 is a sequence of functions w t : R t − → [ −∞ , ∞ ].The manager’s cost of choosing labor ˆ a is g (ˆ a ), with g continuously differentiable on R + , in-creasing, convex, g ′ (0) = 0 and lim ˆ a →∞ g ′ (ˆ a ) = ∞ . Holmstr¨om (1999) only assumes that g isincreasing and convex. Sufficient for the maximizer of the manager’s utility function (1) to existis that g is continuous, because the action set [0 , ∞ ] is compact. For the first order approach usedin Holmstr¨om (1999) to be valid, a convex and continuously differentiable g is sufficient.The manager’s public strategy a = ( a t ) ∞ t =1 is a sequence of functions a t : R t − → [0 , ∞ ], where a t maps the output history y t − to the action ˆ a t . Public strategies (even mixed) are w.l.o.g. whenthe game has a product structure , as shown in Mailath and Samuelson (2006) p. 330. The gamehere has only one strategic player, so trivially a product structure. Hereafter, a public strategy issimply called strategy.The manager’s discount factor is β ∈ [0 , β = 1 is discussed atthe end of Section 2.2. The manager’s ex post utility is U ( w, a ) = ∞ X t =1 β t − [ w t − g ( a t )] . (1)A perfect public equilibrium consists of the manager’s strategy a ∗ and the market’s wage rule w such that for all t and y t − , a ∗ t ( y t − ) ∈ arg max ˆ a t ( w t ( y t − ) − g (ˆ a t ) + ∞ X τ = t +1 β τ − t E (cid:2) w τ ( y τ − ) − g ( a ∗ τ ( y τ − )) (cid:12)(cid:12) y t − (cid:3)) , (2) w t ( y t − ) = E (cid:2) y t | y t − (cid:3) = E (cid:2) η t | y t − (cid:3) + a ∗ t ( y t − ) . Hereafter, a pure perfect public equilibrium is simply called equilibrium. Restricting attention topure strategies is w.l.o.g., because g is convex and any E [ y t + k | y t − ] that the manager can generatewith mixed actions can be generated by pure actions.2.1. Results for the basic model.
The market is Bayesian and conjectures the manager’s strat-egy a ∗ t ( y t − ), so de-biases the output y t to the signal z t := y t − a ∗ t ( y t − ), which in equilibrium equals η t + ǫ t . From the manager’s perspective, z t = η t + ǫ t + ˆ a t − a ∗ t ( y t − ). The mean of the market’sbelief at the start of period t (after t − m t and the precision h t . The precisionof the market’s belief after seeing y t but before taking into account the shock δ t added to η t is The independence of ǫ t , η , δ s for all t, s is implicit in Holmstr¨om (1999). A product structure means that the informativeness of the public signal about the actions of the other strategicplayers is independent of a player’s own action.
ORRIGENDUM TO “MANAGERIAL INCENTIVE PROBLEMS: A DYNAMIC PERSPECTIVE” 3 h t + h ǫ . Bayesian updating implies η t | z t − ∼ N (cid:18) m t , h t (cid:19) = N (cid:18) h t − m t − + h ǫ z t − h t − + h ǫ , (cid:30) ( h t − + h ǫ ) h δ h t − + h ǫ + h δ (cid:19) . (3)The market sets the wage w t ( y t − ) = m t ( z t − ) + a ∗ t ( y t − ), where z t = ( z , . . . , z t ). The commonbelief precision h t evolves deterministically, which is used to take it outside the expectations below.The manager’s time- t expectation E [ w τ ( y τ − ) | y t − ] of the future wage w τ , τ > t given y t − andˆ a t is E (cid:2) w τ ( y τ − ) | y t − (cid:3) = E " h τ − h τ − m τ − + h ǫ z τ − h τ − + h ǫ + h ǫ z τ − h τ − + h ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y t − + E (cid:2) a ∗ τ ( y τ − ) | y t − (cid:3) = E (cid:20) h τ − h τ − m τ − + h τ − h ǫ z τ − ( h τ − + h ǫ )( h τ − + h ǫ ) + h ǫ z τ − h τ − + h ǫ (cid:12)(cid:12)(cid:12)(cid:12) y t − (cid:21) + E (cid:2) a ∗ τ ( y τ − ) | y t − (cid:3) = E (cid:20) h τ − h τ − ( h τ − m τ − + h ǫ z τ − )( h τ − + h ǫ )( h τ − + h ǫ )( h τ − + h ǫ ) + h τ − h ǫ z τ − ( h τ − + h ǫ )( h τ − + h ǫ ) + h ǫ z τ − h τ − + h ǫ + a ∗ τ ( y τ − ) | y t − (cid:21) = m t τ − Y i = t h i h i + h ǫ + h ǫ h τ − + h ǫ τ − X i = t E (cid:2) z i (cid:12)(cid:12) y t − (cid:3) τ − Y j = i h j +1 h j + h ǫ + E (cid:2) a ∗ τ ( y τ − ) | y t − (cid:3) , (4)with the notational convention P ts = t +1 x s = 0 and Q t − j = t x s = 1 for any x s . The manager’s ex-pectation of the time- t signal z t is E [ η t + ǫ t + ˆ a t − a ∗ t ( y t − ) | y t − ] = m t + ˆ a t − a ∗ t ( y t − ). Whenforming expectations about future signals z i , i > t , the manager expects her future selves tofollow the equilibrium strategy, therefore E [ z i | y t − ] = E [ η i + ǫ i + a ∗ i ( y i − ) − a ∗ i ( y i − ) | y t − ] = m t .Substituting the expected signals into (4) results in E (cid:2) w τ ( y τ − ) | y t − (cid:3) = m t τ − Y j = t h j h j + h ǫ + h ǫ h τ − + h ǫ [ m t + ˆ a t − a ∗ t ( y t − )] τ − Y j = t h j +1 h j + h ǫ (5)+ h ǫ h τ − + h ǫ m t τ − X i = t +1 τ − Y j = i h j +1 h j + h ǫ + E (cid:2) a ∗ τ ( y τ − ) | y t − (cid:3) . Substituting (5) into (2) yields the manager’s objective function m t + a ∗ t ( y t − ) − g (ˆ a t ) + m t ∞ X τ = t +1 β τ − t τ − Y j = t h j h j + h ǫ + [ˆ a t − a ∗ t ( y t − )] ∞ X τ = t +1 β τ − t h ǫ h τ − + h ǫ τ − Y j = t h j +1 h j + h ǫ + m t ∞ X τ = t +1 β τ − t h ǫ h τ − + h ǫ τ − X i = t τ − Y j = i h j +1 h j + h ǫ + ∞ X τ = t +1 β τ − t E (cid:2) a ∗ τ ( y τ − ) − g ( a ∗ τ ( y τ − )) | y t − (cid:3) (6) To derive h t +1 , the market first updates η t (which has precision h t ) in response to the signal z t (precision h ǫ ),obtaining precision h t + h ǫ . Then the market adds the variance h δ of the normally distributed shock δ t − to thevariance h t − + h ǫ of the normally distributed updated η t − . If h δ = ∞ , then h t + h ǫ = h t +1 for all t .Holmstr¨om (1999) defines ˆ h t := h t + h ǫ , but interprets it on p. 173 as “the precision on η t +1 before observing y t +1 ” without clarifying that ˆ h t is the precision after observing y t and before adding the shock δ t . SANDER HEINSALU and taking the first order condition (FOC) w.r.t. ˆ a t results in ∞ X s = t +1 β s − t h ǫ h s − + h ǫ s − Y j = t h j +1 h j + h ǫ − g ′ (ˆ a t ) = 0 . (7)The manager’s marginal benefit of ˆ a t is the discounted sum of the responses of future wages toˆ a t . Because the wage is paid in advance, period- t effort does not affect the period- t wage. TheSOC holds, because g is convex. When h δ = ∞ , the FOC in Holmstr¨om (1999) is Equation (10),reproduced in (H10): ∞ X s = t β s − t h ǫ h s = g ′ ( a ∗ t ) . (H10)It corresponds to the current paper’s FOC (7) with h δ = ∞ , in which case h s − + h ǫ = h s ,but Holmstr¨om erroneously starts the sum from index t , not t + 1. Holmstr¨om’s Equation (21),reproduced below in (H21), corresponds to the current paper’s (7) for general h δ and starts thesum correctly from t + 1, but has a different error. Holmstr¨om (1999) defines µ t := h t h t + h ǫ , and hisFOC is (1 − µ t ) ∞ X s = t +1 β s − t s Y i = t +1 µ i = g ′ ( a ∗ t ) . (H21)Taking h δ = ∞ (so that h i + h ǫ = h i +1 ), the LHS in (H21) becomes h ǫ h t + h ǫ P ∞ s = t +1 β s − t Q si = t +1 h i h i + h ǫ = P ∞ s = t +1 β s − t h ǫ h t +1 Q si = t +1 h i h i +1 = P ∞ s = t +1 β s − t h ǫ h s +1 , which is inconsistent with (H10) where the sub-script of h s +1 is s . To make (H21) consistent with (H10) when h δ = ∞ and the sum in (H10) hasbeen corrected to start at t + 1, the product in (H21) should end at s −
1, not s . Thus (H21)should be written (1 − µ t ) ∞ X s = t +1 β s − t s − Y i = t +1 µ i = g ′ ( a ∗ t ) . (8)The LHS of (8) is P ∞ s = t +1 β s − t h ǫ h t + h ǫ Q s − i = t +1 h i h i + h ǫ = P ∞ s = t +1 β s − t h ǫ h s − + h ǫ Q s − i = t +1 h i h i − + h ǫ , same as themarginal benefit in (7). The intuition for the correct FOC is that increasing the manager’s laborˆ a t by one unit increases the market’s mean belief m t +1 next period by h ǫ h t + h ǫ units, but given m t +1 ,does not directly affect any m τ , τ = t + 1. For any n ∈ N , increasing m n by one unit raises m n +1 by h n h n + h ǫ units, but does not directly affect any m τ , τ / ∈ { n, n + 1 } . Therefore increasing ˆ a t by oneunit raises m t + k , k ≥ h ǫ h t + h ǫ Q k − i =1 h t + i h t + i + h ǫ units. The expected wage E [ w t + k | y t − ] increases onefor one with m t + k , because a ∗ n is independent of m ℓ for any n, ℓ , thus there is no multiplier effectfrom ˆ a t to m n + k via a ∗ n . A unit increase in w t + k is worth β k at time t .2.2. The stationary case. A stationary equilibrium requires h δ < ∞ and features a constantbelief precision h t +1 = h t =: h . In terms of µ := hh + h ǫ , the steady state marginal benefit of the ORRIGENDUM TO “MANAGERIAL INCENTIVE PROBLEMS: A DYNAMIC PERSPECTIVE” 5 manager’s action (the LHS of (8)) is(1 − µ ) ∞ X s =1 β s s − Y i =1 µ = (1 − µ ) ∞ X s =1 β s µ s − = β (1 − µ )1 − βµ . (9)This matches Holmstr¨om (1999) Equation (22), reproduced below in (H22), but is not derivedfrom (H21). The steady state version of (H21) is instead (1 − µ ) P ∞ s =1 β s Q si =1 µ = βµ (1 − µ )1 − βµ , whichhas different comparative statics from those in Holmstr¨om (1999) Proposition 1.For the stationary labor supply a ∗ to satisfy Holmstr¨om’s Equation (22) β (1 − µ )1 − βµ = g ′ ( a ∗ ) , (H22)the assumptions g ∈ C , g ′ (0) = 0 and lim ˆ a →∞ g ′ (ˆ a ) > a ∗ is close to zero when β < h δ is large relative to h ǫ , which is true iff theadditional assumption that g ′ ( a ) = 0 ⇒ a = 0 holds. If this assumption is violated, i.e. if g ′ ( a ) = 0 for all a ∈ [0 , k ] for some k >
0, then as the marginal benefit of labor (the LHS of (H22))converges to zero, the manager’s labor supply a ∗ converges to k , not zero.Holmstr¨om (1999) p. 174 compares the manager’s labor supply under h δ = ∞ and h δ < ∞ , bothwhen β < β = 1. The comparison is indeterminate if β = 1, because in this case, oneof the summands in the manager’s utility (6) is infinite , so the maximizer of (6) is undefined. If h δ < ∞ and lim ˆ a →∞ g ′ (ˆ a ) = ∞ , then the FOC (8) still has a finite solution, which may be definedas the manager’s choice of labor when (6) is infinite. In that case the rest of the analysis remainsvalid.If β = 1 and h δ = ∞ , then the marginal benefit of labor in the FOC (8) diverges, so the solutionof the FOC is a ∗ t = ∞ . This infinite labor presents a problem for Bayesian updating and byextension the derivation of the FOC itself. If the market expects a ∗ t ( y t − ) = ∞ , then any ˆ a t < ∞ leads to y t < ∞ , which is off the equilibrium path, so Bayes’ rule does not apply. One way toresolve the updating problem is to use belief threats to deter the manager from deviating, e.g. set m t +1 = −∞ after y t < ∞ . If the manager chooses ˆ a t = ∞ , then the output is y t = ∞ for any η t ,thus uninformative about the type. By Bayes’ rule, the market’s mean belief is m t +1 = m t after y t = ∞ .2.3. Transient effects.
On p. 174, Holmstr¨om (1999) defines b s ( µ ) := (1 − µ ) µ · · · µ s withoutclarifying whether the µ i , i > µ recursively definedin Holmstr¨om’s Equation (17): µ t +1 = 12 + h ǫ /h δ − µ t . (H17) Recall that Holmstr¨om (1999) only assumes that g is increasing and convex. Proof: h j +1 h j + h ǫ ≥ min n h t +1 h t + h ǫ , hh + h ǫ o =: ι > j ≥ t , so P τ − i = t Q τ − j = i h j +1 h j + h ǫ ≥ − ι τ − t − ι > P ∞ τ = t +1 h ǫ h τ − + h ǫ P τ − i = t Q τ − j = i h j +1 h j + h ǫ = ∞ . SANDER HEINSALU
This distinction becomes important in the equation b s +1 ( µ ) = 1 − µ − µ µ b s ( µ ) = 1 − µ r − µ b s ( µ ) , (10)which is unlabeled in Holmstr¨om (1999). If all µ i are treated as independent variables, then b s ( µ ) = (1 − µ ) µ Q si =3 µ i , but if each µ i is understood as a function of the argument of b s , then b s ( µ ) = (1 − µ ) Q si =3 µ i , because each µ i is then obtained from µ by i − µ i are independent variables and µ = µ s +1 , then b s +1 ( µ ) = (1 − µ ) µ s +1 s Y i =2 µ i = 1 − µ − µ µ b s ( µ ) = (1 − µ ) µ s Y i =3 µ i . If each µ i is treated as a function of the argument of b s , and µ s +1 = 1, then(1 − µ ) µ s +1 s Y i =2 µ i = 1 − µ − µ µ b s ( µ ) = (1 − µ ) s Y i =2 µ i . From now on, treat each µ i as a function of the argument of b s . A modification to (10) thatmakes the equalities in it hold is to replace b s with b s +1 , obtaining b s +1 ( µ ) = − µ − µ µ b s +1 ( µ ) = − µ r − µ b s +1 ( µ ). Another way to make (10) valid is to define b s ( µ ) as (1 − µ ) µ · · · µ s +1 , whichsuggests the general definition b s ( µ t ) := (1 − µ t ) t + s − Y i = t +1 µ i for any s ≥ ≤ t ≤ s. (11)Either way to make (10) hold ensures that the inductive proof on p. 174 of Holmstr¨om (1999) iscorrect, i.e. that γ := (1 − µ ) P ∞ s =2 β s − Q si =2 µ i (the LHS of (H21) at t = 1) decreases in µ . Thesame proof shows that the LHS of (8) at t = 1 decreases in µ . Using (11) in (10) and replacing µ , µ with µ t , µ t +1 respectively shows that for any t , the LHS-s of (H21) and (8) decrease in µ t .This proves Proposition 2 in Holmstr¨om (1999).If the definition of b s is not altered to (11), but remains b s ( µ t ) = (1 − µ t ) Q si = t +1 µ i , then to provethat for any t , the LHS of (H21) decreases in µ t , the equation (10) should be modified separatelyfor each t . Specifically, µ and µ should be replaced with µ t and µ t +1 respectively, and both b s +1 and b s should be replaced with b s + t . ReferencesHolmstr¨om, B. (1999): “Managerial incentive problems: A dynamic perspective,”
The Reviewof Economic Studies , 66, 169–182.
Mailath, G. J. and L. Samuelson (2006):