LINEAR CONTRACTS
19.2 OPTIMAL LINEAR CONTRACTS
19.2.4 Brownian Motion Model
HM examine a setting in which the agent controls the drift of a continuous-time Brownian motion, over some fixed unit time interval [0,1]. The significant advantage of this approach is that, under certain conditions, the optimal contract in the dynamic agency problem may be found as the optimal linear contract in the basic agency model with the agent's action representing the mean of a nor- mally distributed performance measure. Not only does this simplify the calcula- tion of an optimal contract, but the dynamic model avoids the Mirrlees Problem with normal distributions. Here we review their model as the limiting case of the repeated binary signal model in Sections 19.2.2 and 19.2.3 as the length of the periods goes to zero. The analysis in this section is based on Hellwig and Schmidt (2002).
One-dimensional Brownian Motion
Let the unit interval be divided into 1/zf time periods each of length zf, and let T = 0, 1,..., 1/zf be the time index. In each period, there is either a good or a bad signal (represented by the numbers y^ and y^ , respectively) and the agent takes an action, which is represented by the probability c/ of obtaining the good sig- nal. Here we appeal to Proposition 19.4, which shows that when periods are in- dependent and the agent has negative exponential utility with no wealth effects, it is optimal to pay the agent period-by-period compensations (each depending on the outcome in that period) such that the same action is implemented in every
period independent of the signal history. Let Tg{T) denote the number of times the good signal has occurred in the first Tperiods. Obviously, Tg{T) is increasing over time. Hence, in order to get to a Brownian motion (that can both increase and decrease) we compare Tg{T) to T (T) = dr, which is the expected number of good signals given some "standard" probability for the good outcome, a.
We assume that the bad signal is negative, i.e., y^ < 0,^^ and fix the standard probability so that the expected performance is zero, i.e.,
ay^ + (1 - a)y^ = 0. (19.6) Let y^ denote the "excess performance" from obtaining the good signal as
opposed to the bad signal, i.e., y^ = y^ - y^ - We now define a "performance account" by
Z^(T) =y^(T^(T)-dT), (19.7) which is simply the aggregate performance up to that date, i.e.,
Z^(r) = j ; ( r / T ) - a r ) - y , ^ ( 7 ; ( r ) - a r )
= y'^T^d) + yt(T - r / r ) ) - (y^ - yt)dT - y^r (19.8)
= y'^T^iT)+y^(T-TlT)).
The performance account is a candidate to be represented by a one-dimensional Brownian motion as the length of the intervals A goes to zero. However, before we can specify this limit, we must specify how the excess performance and the deviation of a^ from the standard probability a depend on the length of the period. This specification is designed so that the expected performance and effort costs over the unit interval are independent of the length of the time inter- vals A if the effort is constant. To achieve this, we assume the excess perform- ance in each period is proportional to A'^\ i.e.,
^^ Note that this is a necessary condition if we want the aggregate performance over the unit interval to be normally distributed. In fact this is not just a matter of subtracting an arbitrary constant from each signal, since the aggregate performance when substituting that constant back in can only be normally distributed if the untransformed signals have some negatives.
The expected performance over the unit interval from choosing (/ instead of the standard probability a is defined to be
/ - -Ic^yt ^ (l-^K^] - \{c^-a)y (19.9)
We want the total expected performance over the unit interval to be independent of the length zf of each time interval we consider. Given the specification of//^
in (19.9), this requires that the agent chooses deviations from the standard of the order of magnitude A'^\ Therefore, the agent's cost over an interval of length A of taking action (/ relative to taking action a is expressed as:
a^ - d^
K\a^) = AK[a + ^—^).
Ifa^ is taken over the entire unit interval, then the total effort cost is
1/cV) - Aa^^^X
A ^ A'^^ '
Hence, if we let a= [a^ - d]/A'^' represent the order of magnitude of the action difference, then the total effort cost over the unit interval depends on a, but is independent of zf.
Note that Tg(T) is generated by a binomial process with Tg(T + l) - TJj) e {0,1 } andE^[r^(T + l) - r^(T)|a^] ^ (/}^ Hence, the expected change in aggre- gate performance takes the following simple form:
E,[Z^(T + 1) - Z\T)\c^]=E,[f^{TJ,T^\) - TJ,T) - a)|a^] = fi'A, and we can write the Z^(T) process as:
[ + (1 - af^)A^' with probability o^,
Z^(T + 1) - Z\T) = /A + ygx\ (19.10) [ - a^A'""' with probability 1 - a^.
Note that the "drift-term," i.e., the expected performance in a time interval of length A is of the order of magnitude zf, while the variation around the mean is
^^ The symbol E^ [ • ] denotes the expected value operator conditional on the information available at date T.
of the order of magnitude zf'^'. This latter characteristic ensures that the variance, i.e.,
Var,[Z^(T + l) -Z^(T)|a^] = j)^V(l-a^)zf,
is also of the order of magnitude A. Hence, neither the drift component nor the variance component of the process for Z^(T) dominates as A goes to zero. It is now relatively straightforward to show that for zf approaching zero, the binomial process for Z^(T) converges to a continuous-time Brownian motion Z{t), t e
[0,1],^^ with instantaneous drift // and diffusion parameter cr, which we formally write as
dZ{t) = judt + adB(t% (19.11) where a = y \/d(l - a),
B(t)is a standard Brownian motion with B(0) = 0, independent increments and B(t') - B(t) ~ N(0, f - t) for t < t',
^^ The limit can be derived from the Central Limit Theorem as the increments in the performance account are identically and independently distributed given a constant action choice.
The Central Limit Theorem (Billingsley 1986, Theorem 27.1)
Suppose that ^4, X^, ...,X^ is an independent sequence of random variables having the same distribution with mean c and finite positive variance ^^. If ^S^ = X^ + X2 + ... + X„, then
S^ - nc
-^ - A^(0,1).
s\fn
Define X^ = Z^(T) - Z ^ C T - I ) ; - c = / z l , s = aA^\
The performance account aXt = \ (r = \/A) is
Z\\/A) = Z\\) - Z^(0) + Z\2) - Z^(l) + ... + Z\\IA) - Z\\IA -1)
= J 4 + X2 + ... + Xy^ = Sy^.
Hence, Z\yA)-VA,^A _ Z\XIA) - ,^ ^ ^^^^^^^
oA'^'^VA ^ or Z\\IA) - N{iiia\
Hence, the account process Z{t) starts at zero, has independent increments, and over any finite interval [t, t'^ the increment in the account is normally distributed with
Z{t') - Z(t) ~ N((^'- t)ju, (f- t)G\
Note that the diffusion parameter a depends neither on time t nor on the agent's choice of action - it depends only on the excess performance over the unit inter- val, y , and the standard probability, a, that gives a zero expected performance.
This is a consequence of the fact that a^ converges to a as the length of each time interval A goes to zero (although it does so at the rate zf'^'). The instanta- neous drift //, on the other hand, depends on the agent's action.
We now derive the compensation contract that implements a constant ac- tion. To do this it is useful to think of the agent choosing (a constant) j / in each period of length zf, which then determines an associated action a^{j/) by (19.9), i.e.,
c^{fi') = a +ju^—- (19.12) Since o^ is a probability, we must restrict the agent's choice of//^ by
yj J A ( i - ^)
- -^— < ju < -^ .
A ^2 ' A'A
However, note that as A goes to zero, the bounds on //^ become trivial, and ju^
can be chosen to be any real number by the agent. Similarly, we can express the agent's cost function in terms of//^,
zf/cV) =
K\AM'))= AK[ay^^'^~^\
Let df denote the period-by-period compensation for obtaining signal / = b,g that gives the agent a certainty equivalent equal to zero given j / . A zero cer- tainty equivalent in each period and incentive compatibility of j / implies that (compare to (19.5))
d'^ - AK{ii^) - Un[l-rKX/)y/'' + ra'KX/)y/j,
dt = AK(JU^) - l l n ( l + ra^K'(/)y^A'^j.
Using a Taylor series expansion of the logarithmic term, the required compen- sations are given by
S^ = AK(M^) - a'K'if/)})/'^ + V2ra'\K'(p')y^fA + 0(A"').
The accumulated compensation "earned" in the first z periods is
Hence, the expected incremental compensation is E,[C^(T+1)
- c\T)\a'] =a'd^ + (i-c/)d^
= AK(/) + V2ra^(l-c^){K\/)y^fA + 0(A^%
and the difference between the actual incremental compensation and the expect- ed incremental compensation is
C\T + 1) - C\T) - E , [ C ^ ( T + 1) - C\T)\a']
[ + (1 - a^)A'^' + 0(A) with probability O^,
= KXM')y^x\
[ - c^A^' + 0(A) with probability 1 - c^.
The variance of the incremental compensation is
Var,[C^(T + l) - C\T)\a'] = c^{\ - c^)[K\fi^)y^fA + 0{A^'\
Since A^'^ goes faster to zero than A, the 0(zf^^^)-terms can be ignored in both the expected incremental compensation and the variance of that incremental. Hence, as A goes to zero, the process for the accumulated compensation C^(T) conver- ges to a continuous-time Brownian motion C(t), ^ e [0,1], on the form
dC{t) = \K(JU) + V2r[K'{id)y^fa{\ -dyjdt
+ K'(ju)yg^d(l-a)dB(t). (19.13) The drift-term has two components. The first component is compensation for
the incurred effort cost and the second component is a risk premium the agent is paid to compensate him for the incentive risk, i.e., the diffusion-term. The key here is that these payments are fixed such that the agent gets a certainty equivalent of zero.
The relation between the compensation and the performance measure fol- lows from substituting the performance account process Z(t) from (19.11) into (19.13):
dC(t) = \K(JU) + y2r{K'(ju)y ) d(l - d) - K'(ju)jujdt + K'(ju)dZ(t).
Hence, the total compensation at ^ = 1 is (by "integrating both sides" and noting thatZ(0) =0)
C(l) = K(JU) + y2r{KXju)yJ^d(l-d) - KXJU)M + I<XM)Z(11
where Z(l) is normally distributed with
Z(l)-N(My%a = y^^fd(r~d).
Since the performance account at ^ = 1 is equal to the aggregate performance at t = I, i.Q.,y = Z(l), we may write the optimal compensation contract as a linear function of a normally distributed aggregate performance measure, i.e.,
c(y)=f^vy, y-N(M,a'l (19.14) where / = w + K(JU) + V2r{K'(ju)a) - K'(JU)JU,
V = K'ifi).
The fixed component of the compensation consists of the agent's reservation wage, a compensation for the incurred effort cost, a risk premium for the incen- tive risk, minus the expected incentive wage. The incentive wage is determined by the agent's marginal cost of providing expected aggregate performance //.
Note that the characterization of the optimal contract in (19.14) is precisely the same as the characterization (19.3) of the optimal linear contract in Section 19.1. Hence, the optimal contract in a setting where the agent continuously con- trols the drift of a Brownian motion for aggregate performance may be found as the optimal linear contract in a static setting, where the agent's action is the mean of a normally distributed performance.^"^
Multi-dimensional Brownian Motion
We now turn to the multi-dimensional Brownian version of the setting with two performance measures considered in Section 19.2.3. In each period of length A there are two binary performance measures j ; • which may take values j ; ^ • and j ; ^ . , / = 1,2. As in Section 19.2.3, there are four possible signals: XJ/Q =
(yLyti)^ ¥t = (yLy^i)^ ¥2 = (y^3^/2) ^ and ^3^ =(y^i,3;^2)- We represent the four signals by numbers also denoted if/f, / = 0,1,2,3.^^
The agent's action is the probability of each of these four performance sig- nals, (pf, / = 0,1,2,3. In the subsequent analysis we assume the agent's action space is the simplex determined by S^ ^/^ = 1, (p^^ > 0. Hence, the agent's action cannot be represented as a single-dimensional choice variable that affects the probabilities of the good signals for both performance measures. The problem with a single-dimensional choice variable is that the compensations for the four different signals cannot be determined from the incentive constraints and the contract acceptance constraint alone. However, that is possible if we represent the set of choices as a simplex, and we can perform the analysis as in the pre- vious section except that the agent now controls a multinomial instead of a binomial process. In particular, we fix a standard probability vector (p as the probabilities that give an expected performance of zero, i.e.,
J2<p]y^1=0, (19.15)
/ = 0
^^ Note that the Brownian motion model avoids the Mirrlees "problem" even though the aggregate performance measure is normally distributed. This has to do with the fact that the agent can effectively avoid the penalties by his action choices as he observes the performance measure continuously. If he only observes the normally distributed performance measures at discrete points in time, the Mirrlees argument applies no matter how small the intervals are between his observations (see Mtiller, 2000, for a formal development of this point). However, it is not important how often the principal observes the aggregate performance measure.
^^ At this point, these numbers are generic representations of the information in each period.
Below we consider the case in which these numbers are linear aggregates of the two performance measures.
and let ^. denote the "excess performance" from obtaining signal / as opposed to the signal 0, i.e., ^. = ^. - ^Q , / = 1,2,3. We can now define the "perfor- mance account" for each of the signals / = 1,2,3 by
zf{T) = xfifiTiT) - ^ > ) , / = 1,2,3, (19.16) where T^{T) is the number of times signal / has occurred in the first T periods.
Note that these accounts are not independent since only one account can change value in each period. If we take the sum of the performance accounts at any date T using (19.14), we get the aggregate performance, i.e.,
Y.ZfiT)=Y.¥fTI,T). (19.17)
i=\ i=0
Again, we let the "excess performance" in each period be proportional to A'^\
i.e.,
anddefine ju- = y / . — ^ ^ — - , i = 1,2,3. (19.18)
An
Note that by using (19.15) the total expected performance over the unit interval is
3 3
^i=0 i=\
such that JU- can be interpreted as the contribution to the aggregate expected performance over the unit interval from shifting probability to signal /, / = 1,2, 3, from signal zero.
Note that each T^(T) is generated by a binomial process with T^(T +1) - T^(T) E {0,1} andE,[r,(r + l) - r,(r)|(p^] = cpf. Hence,
E,[Z/(r + l) -Zf(r)|(p^]
= E,[^f (r,(r + l) - UT) - c^.W] = //fzf, / = 1,2,3, so that we can write the process Zf(T) as follows:
Zf{x^l)-Zfix)
Mi ^ + ¥i
+ (1 - (pf)A'' with prob. (pf
A A'A
(pfA with prob. 1 - ^ / ,
i- 1,2,3. (19.19)
The variance of the change of account / is
Var,[Z/(r + l) - zfix) |(p^] = ^]ff{\ - (pf)A, i = 1,2, 3, and the covariance between any two accounts is
Cov,[Zf(r + l) - Zf(r), zfir^X) - zf{T)W']
= - xfi-xfijCpfcpfA, Uj = 1,2,3; / ^j.
As in the one-dimensional case it is now relatively straightforward to show that for A approaching zero, the process for the three performance accounts con- verges to a three-dimensional continuous-time Brownian motion Z(^), ^ e [0,1 ], with instantaneous drift vector ^ and diffusion matrix S'^' which we formally write as
dZ(t) = yidt + S'/^ JB(0, (19.20) where
B(Ois a standard three-dimensional Brownian motion with B(0) = 0, inde- pendent increments and B{t') - B(0 ~ N(0,(^'- 01) for t < t'.
S = {
[ ^ # l ( l - ^ l ) -WxWi^x^i -^i^3^i^3 1 ^ WiWx^iix Wiiii'^-V^ -WiW^iii^
- ^ 3 XJJ^Cp^Cp^ - ^ 3 ^ 2 ^ 3 ^ 3 ^ 3 ( 1 - ^ 3 ) J ^
Hence, the account process TXi) starts at zero, has independent increments, and over any finite interval \t, t'\ the increments in the accounts are jointly normally distributed with
T{t') - Z(0 ~ N ( ( r - i)\!iXt'- OS).
In order to derive the compensation contract that implements the (constant) drift vector \i we express the agent's cost as a function of ^^ recognizing the relation between the drift rates and the associated action given by (19.18), i.e., as a function /c(^^). Following steps similar to those in the one-dimensional case, it can be shown that the process for accumulated compensation (with a zero cer- tainty equivalent) converges to a (one-dimensional) continuous-time Brownian motion C(0, ^ e [0,1], on the form^^
dC{t) = (/c(^) + 'Ark'diyUk'iii) - k'(ii)ii)dt + k'(ii)dZ(tX where K ^ ^ ) denotes the vector of the partial derivatives of the agent's cost func- tion with respect to the drift rates, //^, in each account. The total compensation at ^ = 1 is
C(l) = Kill) + V2rk'(iiyjlk'(ii) - k'(ii)ii + K X ^ ) Z ( I ) ,
where Z(l) is normally distributed with Z(l) ~ N(fi,S).
Hence, we can write the optimal compensation contract as a constant plus a linear function of the three jointly normally distributed performance accounts at ^ = 1, i.e.,
c(z)=f^y% z~N(^,S), (19.21a) where f=w + /c(^) + 'Ark'diyUk'iii) - k'(ii)ii, (19.21b)
v = k'iii). (19.21c) The fixed component has an interpretation similar to the one-dimensional case,
i.e., it consists of the agent's reservation wage, compensation for the incurred effort cost, and a risk premium for the incentive risk, minus the expected incentive wage. The incentive wage is determined by the agent's vector of mar- ginal costs of providing expected aggregate performance ju^ in each of the dif- ferent performance accounts. Hence, the optimal contract in a setting in which the agent continuously controls the drift vector of a Brownian motion for aggre- gate performance accounts may be found as the optimal linear contract in a
^^ Compare to HM, Theorems 6 and 7, and Hellwig and Schmidt (2002, Theorem 1).
static setting, where the agent's action is the mean vector of jointly normally distributed performance accounts.
It is useful to note that the agent's compensation is independent of the num- bers assigned to the four signals. To illustrate, assume that all numbers are multiplied by a scalar A. This will result in a mean vector \i^ = A^, a covariance matrix E^ = A^ E, a vector of account totals z^ = Az, and a marginal cost vector K^'{\i^) = K'(II)/^' Substituting these relations into (19.21) readily establishes that the new optimal contract is characterized by/^ =/and v^ = v/A, so that
c'iz') =f + v ' V =f+ (vVA)(Az) =f+ \'z = c(z).
Can we express the optimal compensation contract as a linear function of the aggregate performance for the two performance measures j^^ and3;2? ^^ general, the answer is NO!
We can define the numbers that represent the four signals by a linear aggre- gate of the two basic performance measures, i.e.,
for some constants g^ and g2- For example, if y^ and y2 are revenue and cost measures, then ^ is a profit measure if g^ = 1 and g2 " " 1- The aggregate of the performance accounts in (19.17) is then given by
i=\ i=0
which is equal to a linear function of the aggregate performances for each of the two basic performance measures. Hence, in the limiting continuous-time setting the aggregate of the performance accounts is a linear function of the aggregate performances for each of the two basic performance measures, i.e,
E Z.(l) = g,y, + g,y,. (19.22)
i = l
However, the optimal incentive wage is
v'z = Y. ' ^ » 2 , ( l ) , (19.23)
i = \
where K.{\I) is the agent's marginal cost of providing expected aggregate per- formance //^ on performance account /. Hence, unless these marginal costs are the same for all accounts, we cannot express the optimal compensation contract as a linear function of j^^ and3;2- The problem is that with two binary perform- ance measures, the necessary Brownian motion to describe optimal compensa- tion is not two-dimensional, it is three-dimensional. In general, with A^"binary"
performance measures, the Brownian motion must be of dimension A^^ - 1.
HM consider two special cases of the multi-dimensional Brownian motion model in which the optimal compensation is in fact a linear function of the ag- gregate performance for the two performance measures j^^ ^^^yi- The first case is a setting in which the agent's cost function depends only on the expected ag- gregate performance, i.e., the cost function can be written as
^» = ^/[E/^/J
In that setting, the agent's marginal costs are the same for all accounts and the result follows from (19.22) and (19.23). Unfortunately, as demonstrated by Hellwig (2001), this Brownian motion model cannot be derived as the limit of a discrete-time model since the corresponding cost function in discrete time im- plies that it is optimal for the principal only to induce positive probability for two "neighboring signals" in order to reduce the risk premium to the agent.
Hence, the first-best solution can "almost" be implemented. This does not occur in the continuous-time Brownian motion setting, since the covariance matrix can be exogenously specified in that setting.
The second special case examined by HM is a setting in which the principal does not observe the vector of performance accounts Z(0 but only some linear aggregate of those accounts (see HM, Theorem 8), for example the sum of the accounts, i.e.,
Y{t) = Y.m-
i = \
In that case, the optimal contract is a linear function of 7(1) as in (19.21). Of course, contracting on 7is less desirable than contracting on 7J}^ The principal loses if he throws away information. It is more costly to implement a given action vector ^, and only action vectors in which the agent has the same mar- ginal costs for each account can be implemented. Hence, using a linear contract based on the aggregate performance 7 is sub-optimal if more detailed informa- tion is available. Hellwig and Schmidt (2002) also note that this continuous- time Brownian motion setting has no immediate discrete-time analog. The key is that in a discrete-time setting, observing the process for aggregate perform- ance 7^(T) enables the principal to infer the processes of the individual accounts (by observing the size of the increments). Hellwig and Schmidt (2002) develop a discrete-time setting in which "approximately" optimal sharing rules are linear in aggregate performance. They impose two key assumptions: the principal only observes the final aggregate performance 7^(1) and not the time path, and the agent can understate aggregate performance such that the sharing rule has to be non-decreasing.