Discussion Papers In Economics
And Business

Hedonic prices and multidimensional incentives
Masaki Nakabayashi

Discussion Paper 05-32-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHedonic prices and multidimensional incentives
Masaki Nakabayashi

Discussion Paper 05-32-Rev.2

May 2006

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHedonic prices and multidimensional incentives∗
Masaki Nakabayashi†
Osaka University
May, 2006

Abstract
Human tasks are often multidimensional. Holmstrom and Milgrom (1991) concluded
that “high-powered” incentives cannot work unless all dimensions of these tasks are observable in the firm. However, as this study shows, if the firm can observe the price vector
of its products in the market, distinguish each dimension of the price vector, and connect
the information with signals from workers in the firm, then the use of multidimensional“high-powered” incentives becomes feasible. Product differentiation with committed quality satisfies those conditions, which has been practiced by the Japanese, but not by theWestern, manufacturing for a century.
Key words: multidimensional incentives, multitask incentives, high-powered incentives,
hedonic prices, Japanese manufacturing.
JEL: L22, D23, N65.

∗

This is a revised version of Discussion Paper 05-32, December 2005. The author is deeply thankful to PaulR. Milgrom, Masahiko Aoki, Minjung Park, Shingo Ishiguro and Toshihiro Matsumura for their path-breaking
suggestions and continuing kind supports. He also appreciates helpful comments from Michihiro Kandori, Akihiko Matsui, Felix Reichling, Andrew R. Nigrinis, Tetsuji Okazaki, Paul A. David, Avner Greif, Hideshi Itoh,
Dan Sasaki, Luz Mariana Arias, Hirotake Hibari, Noriyuki Yanagawa, Kazumi Hori, Yasuyuki Sawada, Takashi
Kurosaki, Yasuhiro Omori, Keijiro Otsuka, Gavin Wright, Masamitsu Ohnishi, Naofumi Nakamura, Yoshiyuki
Takeuchi, Takenobu Yuki, Yasuo Takatsuki, Ryo Kambayashi and the participants of seminars at University ofTübingen, Osaka University, University of Tokyo, Hitotsubashi University, and Stanford University. This study
has been supported by JSPS Postdoctoral Fellowships for Research Abroad (2002-2004, No. 559) and by JSPS
Grant-in-Aid for Scientific Research (No. 18203024, No. 17330048, No. 17330077).
†
Corresponding address: Graduate School of Economics, Osaka University, Machikaneyama 1-7, Osaka 5600043, Japan. phone: +81-6-6850-5240. e-mail address: masaki@econ.osaka-u.ac.jp

I. Introduction: hedonic prices and effort vectors
A. “Low-powered” incentives in the Western manufacturing and theory
Stock prices are often thought to be good signals of executives’ performance, given that profits
are their “products.” Then, the price of products seems to be similarly used as a signal to evaluate
the performance of workers. This paper analyzes the conditions under which “high-powered”
incentives,1 connected with information about the product price, could be efficient.
Many tasks are multidimensional especially where multiple dimensions of product quality
are critical. In that case it could be beneficial to control relevant dimensions of workers’ task tokeep the optimal vector of quality.
However, a common style of business in the American manufacturing has been the mass
production of goods by unskilled workers. At this extreme, the benefit from multidimensional
incentives is relatively small because the premium derived from better quality is small. On theother hand, Europe has a tradition of manufacturing luxury goods by skilled workers. In thiscase, a multidimensional incentive becomes less manageable because it is harder to evaluate theartistically high quality in separate and standardized terms. At both extremes, multidimensional
incentives for workers do not seem to work well. Production of standardized or luxury goods
with “low-powered” incentives for workers is an equilibrium in the Western manufacturing.
This fits the argument forwarded by Ronald Coase and elaborated by Oliver Williamson.
Coase thought a firm replaces the price mechanism with a planned coordination of inputs toavoid the cost of pricing.2 Here a firm is supposed to be intrinsically reluctant to price inputs, or
equivalently, provide incentives. Williamson represented this idea using a clear concept: “lowpowered” incentives are generally used in the firm while “high-powered” incentives, which
explicitly price inputs, exist in the market.3 They believed the main role of modern firms liesin minimizing transaction costs rather than providing incentives with workers, and this views
coincided with practices in the Western manufacturing for the last one century.

B. Linking “hedonic prices” with incentives
However, as mentioned by Alchian and Demsetz (1972) and Rosen (1988), the centralized
pricing of tasks through a better-organized flow of information within the firm could be more
efficient than decentralized pricing in the market, if it is too costly to observe the market priceof each task because of jointed production, although the signals related to the performance ofeach task are observable in the firm. This in-house pricing that composes the internal labormarket can be used for career concerns as long-term evaluation, combined with “low-powered”
incentives for short-term evaluation, as shown in Williamson, Wachter and Harris (1975), andformalized as the “career concerns” model by Holmstrom (1999) and Dewatripont, Jewitt and
1“High powered” incentives reflect a payment explicitly based on observed performance and “low-powered”
on a payment based on the opportunity cost of labor. Hence a “low-powered” incentive can be interpreted as acompensation scheme that only satisfies the participation constraint. Williamson (1985), pp. 131-162.
2
Coase (1937), p .391.
3
Williamson (1985), pp. 131-162.

1

Tirole (1999). But in theory the centralized pricing of tasks could also work for short-term
evaluation, i.e., provide “high-powered” incentives. If a firm can infer implicit relative pricesof each dimension of tasks evaluated in the market, then the firm can adjust its “high-powered”
incentive as an in-house pricing to maximize its profit. Observed multidimensional prices called“hedonic prices” could thus be useful instruments to give “high-powered” incentives.
As Rosen (1974) pointed out, “hedonic prices” of differentiated products are more observable than those of generic goods. But at the same time, the mystical value of luxuries might bedifficult to decompose into several dimensions of a hedonic price. In contrast, hedonic pricesof differentiated goods consisting of several decomposable factors of quality in the middlerange market seem easier to identify. Then it is feasible for firms in the middle range market toconnect the information about the hedonic prices of their products with the multidimensionaltasks required in production to provide explicit multidimensional, or, multitask “high-powered”
with workers and to optimally control their multidimensional efforts. Hence the production for
middle range market with the “high-powered” incentives for workers is another possible equilibrium, with other equilibria of “low-powered incentives” with standardized or luxury goods
seen in the Western manufacturing.
The Japanese manufacturing linked hedonic prices to incentives and moved to this equilibrium from the beginning of its modernization since the middle 1880s, and the earliest andthe most important example was the modernization of silk reeling from the middle 1880s to the
1900s.4 In the industry, which was one of the most important driving force of the Japanese economy due to its huge exports to the US before World War II, multidimensional wage schemes
had been developed by the early 1900s. Interestingly this was the time when firms tried to differentiate their products by establishing their own brands in the New York market, which was
the largest in the world. While the quality of raw silk consisted of various aspects, Japanesemanufacturers focused on only a few critical and well-observed dimensions of quality, committed to them under their brand names, and provided workers with incentives to focus on these
dimensions of quality. Japanese raw silk acquired a vast share in the US market, although thehigh-end of the market was continuously held by Italian raw silk, and the low-end was occupied
by Chinese.

C. Multiple signals of multiple tasks
This study is motivated by Holmstrom and Milgrom (1991). An insight of them was in theirfocus on the optimization of the “direction” of vector-valued effort rather than just its “scalar.”5
Most work done by human beings are multidimensional while work done by engines are measured by a single scalar, such as horse-power or torque. For instance, individual workers often
have to perform along at least two dimensions: increasing productivity and retaining quality. As
easily imagined, it is more difficult to make individuals work both fast and carefully, rather than4

The silk reeling industry is an industry that produces raw silk threads from cocoons. Raw silk is used asa material for luxury clothes. For descriptive research on the silk reeling industry of Japan, see Nakabayashi(2003, forthcoming).
5
Hence, a “direction” mentions exactly the “direction” of a vector in m where m ≥ 2, rather than just + or
− in .

RR

2

just work fast or just carefully. Indeed the result of Holmstrom and Milgrom (1991) emphasizes
it is difficult for multidimensional “high-powered” incentives to work in a firm.
However, as implied by Holmstrom and Milgrom (1991), it could become easier to use ifsome signals of each dimension of effort vector become observable by the firm. One importantsignal that the firm could observe is the price vector of its products; the hedonic price. As
shown later, implementing “high-powered” incentives become feasible if the firm can observe
the hedonic price of its products. Intuitively, The more signals, the better it is for the firm
to provide incentives, and this intuition is easily evoked by the “sufficient statistic” result inHolmstrom (1979). To be utilized for multidimensional incentives, however, multiple signals
have to satisfy another condition: each dimension of each signal must be distinguishable to theprincipal. We can summarize main features of the conditions for multidimensional incentives
to work on the models in Holmstrom and Milgrom (1991) and in this study as follows:
Multidimensional incentives based on one signal (Holmstrom and Milgrom (1991))
Each dimension of the vector-valued signal must be observable, but each dimension of
the error vector of the vector-valued signal can be dependent to each other.
Multidimensional incentives based on more than one signals (this study)
Some dimensions of some vector-valued signals can be unobservable, but each dimensionof the error vector of all vector-valued signals must be independent to each other.
While those conditions will be analyzed in detail in the next section, predictions from thisstudy are:If a firm knows the multidimensional price vector of its products in the market,
distinguishes each dimension of the price vector, and establishes a production organization where the information is preserved and utilized to control production,then the firm can optimize the effort vectors of its workers by introduction of multidimensional incentives.
Hence, this study inquires conditions under which the constraints on multidimensional“high-powered” incentives are relaxed in a sense, but are strengthened in another sense. If a
firm can observe the price vector of its products, the dependance on the signal in the firm could
be reduced. However, for the price vector to be useful information to give incentives, it is necessary that each dimension of the price vector can be distinguished. This condition seems to
be satisfied in the middle range market of the products, but neither in the low-end nor in thehigh-end market.
After Holmstrom and Milgrom (1991) presented their theoretical prediction about the difficulty of applying multidimensional incentives within a firm, the conditions under which theiruse becomes feasible have never been considered.6 This research tries to close this gap in theliterature.
6

An empirical study by Margaret Slade, which applies the Holmstrom and Milgrom (1991) model, showed
how their model exactly fits transactions in the market, not within a firm, of the Western world (Slade (1996)).
In the Holmstrom and Milgrom (1991) model, the risk aversion of the agent plays a critical role, and so anothertheoretical prediction by them was that multidimensional/multitask incentives could be optimal for executives whobear more risk than employees, which was considered by Feltham and Xie (1994) and Preyra and Pink (2001).

3

In the section II, some theoretical predictions are deduced from the model under hedonic
prices. While Holmstrom and Milgrom (1991) analyzed the function of incentives within an
organization, this paper is also interested in the stream of information that flows from the market
into an organization. Inquired first will be the condition where more than two signals can beutilized to provide incentives, considered then will be a case where two signals – observed
performance in the firm and observed price vector of the products in the market – are utilized.
Section III estimates the compensation scheme in a silk reeling factory, and depicts itschanges from the 1890s to 1910s. It will be shown that a 4-dimensional wage scheme was built
in the early 1900s when the firm established its own brand name to differentiate its products.
Section IV evaluates how each dimension of the workers’ effort was optimized.
Section V sums up the results and discusses further related topics.

II. Theoretical predictions
A. The standard model of multidimensional incentives
The model from Holmstrom and Milgrom (1991) is first outlined here. Consider a multidimensional task. An agent has a particular amount of attention as her/his endowment, and allots
her/his attention to each dimension, which composes an effort vector tT = (t1 , t2 , ..., tm ).
Suppose that there are two dimensions t1 and t2 in the task, and that the principal observes
signals x1 and x2 of t1 and t2 , and relates these signals to the wage. If the two dimensions
are substitutes and one of them is unobservable, then the agent tries to get a higher wage byincreasing her/his effort for the observable one and decreasing effort for the unobservable one.
In this case, it is necessary to use “low-powered” instead of “high-powered” incentives in orderto have workers pay attention to both of t1 and t2 . We can summarize the discussion on “lowpowered” incentives in Holmstrom and Milgrom (1991) as follows: Suppose that inputs to eachdimension of the task are substitutes. Then, unless every dimension of the task is sufficiently
observable, multidimensional “high-powered” incentives are generally difficult to apply, and“low-powered” incentives are more efficient.7

B. Hedonic prices in the market and multidimensional tasks in the firm
Since Waugh (1928) and Court (1939), the quality of products has been thought to be intrinsically multidimensional.8 Suppose that consumers in the market have a multidimensional utility
function that is concave with respect to every term of the product quality, and they assign anl-list of quality magnitude (q1 , q2 ..., ql ) to amount Q of a product they purchase; i.e., supposethat the market has a hedonic price function p(q ′). Hence, the hedonic price p(q ′) is a mappingfrom a vector q ′, whose coordinates specify the quality magnitude of the product to a specificamount of money p. By marginally changing the relevant quality qi , where i = 1, ..., l, the firm
7

Holmstrom and Milgrom (1991), pp. 34-35.
Also see Court (1941a, 1941b), Griliches (1961), Lancaster (1966), Baumol (1967), Rosen (1974), and Epple(1987).
84

accordingly observes marginal changes of price, thus can approximate the shape of hedonic
price function p(q ′) in the neighborhood of the amount Q of the product.
Then, the firm can infer the (l + 1)-dimensional price vector of the product p(q)T =p(p1 (q1 ), ..., pl (ql ), pl+1(ql+1 )), where ql+1 specifies the quantity, i.e., ql+1 = Q.

C. The model of multidimensional incentives with multiple signals
Now consider a case where a risk-neutral principal utilizes n signals of a risk-averse agent’s
effort vector t. Put m = l + 1, which is the number of dimension of the price vector of product,and suppose,t ≫ 0: k-dimensional effort vector generated by the agent.
C(t): private cost function of the agent, which is strictly convex.
B(t): gross benefit to the principal, realized by the agent’s effort.

RRµj (t) : k → m+ where j = 1, ..., n: m-dimensional outcome realized by the k-dimensional
effort vector t.
xj (t) = µj (t) + ǫj : m-dimensional signal vector of m-dimensional outcome vector µj (t),observed by the principal. ǫj stands for the measurement error vector of the outcome
vector µj . ǫ1 , ..., ǫn are independently distributed with ǫj ∼ N(0, Σj ).
u (w − C(t)) = −e−r[w−C(t)] : Constant Absolute Risk-Averse utility function of the agent.9
α: m-dimensional incentive vector.
β: transfer of total surplus for allocation.
Suppose k = m, and the i-th coordinate of µj , µji, is a one-to-one mapping of the i-thcoordinate ti of t. Then the principal can use the signal xj , j = 1, ..., n, to construct thecompensation scheme, which extends the model of Holmstrom and Milgrom (1991) for multiple
signals, such that" n#X(1) w = αT [Γ1 x1 (t) + Γ2 x2 (t)+, ..., +Γn xn (t)] + β = αTΓh [µh (t) + ǫh ] + β,h=1where Γj is a k × k matrix. Then,
"

−r αT

u (CE) = E [u (w − C(t))] = −ePn

h=1 Γh µh (t)+β−C(t)−so that the agent’s Certainty Equivalent is
9

Holmstrom and Milgrom (1987), p. 307.

5#r T Pnα [ h=1 Γh Σh Γh ]α2,

CE = αTand the Total Surplus is

" n
X

#

" n
#
r T X
Γh µh (t) + β − C(t) − αΓh Σh Γh α,
2
h=1
h=1

" n
#
r T X
TS = B(t) − C(t) − αΓh Σh Γh α.
2
h=1Then the contract (t, α, β, Γj ), j = 1, ..., n, must solve

(2)

(3)

" n
#
r T X
max B(t) − C(t) − αΓh Σh Γh α, j = 1, ..., n,Γj , α2h=1subject to t = argmax αT
t′

" n
X

#

Γh µh (t′) + β − C(t′).

h=1

(IC)

∗
The
Pn principal must choose the optimal weight of signals Γj such that the variance of wage
α [ h=1 Γh Σh Γh ] α is minimized. Here we have to note there exists a condition to be satisfied by each covariance matrix Σj , for the uniqueness of this problem’s solution.
T

Proposition 1.
For j = 1, ..., n, suppose that each dimension of the measurement error ǫj of the signal xj isnonzero so that the variance of each dimension of ǫj is positive. Then, the optimal contract
which solves (2) is unique only if all dimensions of ǫj are independent from each other, so thatthe covariance matrix Σj of ǫj is diagonal.
Proof. See the Appendix.
Proposition 1 says each dimension of the signals must be clearly distinguishable from otherdimensions by the principal. If some dimensions of ǫj are not independent, then the signal xj
cannot be utilized as a signal of the same dimensions of the effort vector t with other signals
x1 , ..., xj−1, xj+1, ..., xn in the compensation scheme (1). If the principal nevertheless wants to
use such xj , he should use it as a signal of another aspect of the agent’s effort.
Therefore, if the principal wants to use multiple signals by the procedure shown above, eachdimension of any signal must be distinguishable from each other, i.e., respective elements ofthe signals must be easy to find. If the firm, for instance, wants to use the hedonic price of its
products, each factor of the product quality must be observed by consumers easily, so that eachdimension of the product’s hedonic price is separately observed by the firm. In other word,
the hedonic price cannot be used as a signal if the composition of the quality product is very
complicated so that respective factor of the quality is not well distinguishable to consumers, as
happens for very luxury goods.
6Focusing on the situation where multiple signals are used by the firm, suppose for the reminder of this paper that each dimension of the measurement error ǫj is independent to eachother so that Σj is diagonal, hence suppose that optimal Γj is a diagonal matrix
Pn whose diagonal
entries are γj ii , for j = 1, ..., n and i = 1, ...k. Also suppose that 0 < γj ii , h=1 γh ii = 1, forj = 1, ..., n, P
i = 1, ..., k, and normalize outcomes such that µ1 (t) = µ2 (t) =, ..., = µn (t) = t.
Then, since nh=1 Γh = I, (2) and (3) are reduced to
(4)

" n
#
r T X
max B(t) − C(t) − αΓh Σh Γh α, j = 1, ..., n,Γj , α2h=1(5)subject to t = argmax αT t′ + β − C(t′).
t′Denoting Cij as the Hessian of C(t) with respect to t, the first-order condition of (5) gives


∂C(t)
α=
∂t

(6)

T

,

∂α
∂t
= [Cij ] ,= [Cij ]−1 .
∂t∂αThe first order condition of (4) with respect to Γj is

Γj = Σj−1

(7)

" n
X
g=1

Σg−1

#−1

,

which minimizes the risk to the risk-averse agent, i.e., the variance of wage. 10
The first-order condition of (3) with respect to α is


α=

(8)

"

∂B(t)
I +r
∂t

" n
X
h=1

#

Γh Σh Γh [Cij ]#−1 T

Combining (7) with (8) gives .

2
Put σj,ii, j = 1, ..., n, i = 1, ..., k, as the diagonal entry on the i-th row of the covariance matrix Σj of theerror vector ǫj of signal xj . Then, the row vector (γ1,ii , ..., γj,ii , ..., γn,ii ) consisting of diagonal entries on i-throws of Γj , j = 1, ..., n, is given as the “minimum variance portfolio”
10T  2
σ1,ii
1
 ..   ..
∗∗(γ1,ii , ..., γn,ii ) =  .   .


1

0

−1
 T  2
,
...
0
σ1,ii
1
 ..   ..
.. ..
 .   .
.
. 
2. . . σn,ii
10∗∗Then, Γj∗ whose diagonal entries are γj,11, ..., γj,kk
above is equal to (7).

7

−1  ...
01..   ..  .
..
.
.   . 2
. . . σn,ii
1

(9)



α=





n
X



∂B(t) 
Σh−1
I +r
∂t
h=1

" n
X
g=1

−1 T#−2 
 [Cij ] 
Σg−1

The following is a description of this procedure by the principal: The principal
1. observes the signals xj (t), for j = 1, 2, ..., n,
2. assigns the information weight matrix Γj to xj (t), such that the variance of the compensation, which is the risk to the agent, is minimized,
3. and then chooses an incentive vector α such that the total surplus should be maximized.
Now suppose n = 2, that is, suppose that the firm uses two signals of the effort vector
t of the agent; the quality and quantity of the product generated in the firm µ1 (t) ≡ µ(t),and the price vector of the product in the market µ2 (t) ≡ p(t). Let x(t) = µ(t) + ǫ be theperformance of the worker observed with some noise in the firm, and let p̃(t) = p(t) + ε bethe observed price vector with some noise in the market. The compensation scheme is given
by w = αT [Γµx(t) + Γpp̃(t)] + β. Normalize µ(t) and p(t) such that µ(t) = p(t) = t. In
addition, consider the case where k = 2, such that




 2

 2

t1
α1
σ1 0
ς1 0
t=
, α=
, Σµ =, Σp =
,
t2
α2
0 σ22
0 ς22
where t1 and t2 respectively stand for the quality and the quantity of the product.
Then, from (8), we have the optimal α such that,(10) 
(σ12 + ς12 ) [(σ22 + ς22 + rσ22 ς22 C22 ) B1 − rσ22 ς22 B2 C12 ]α=×(σ22 + ς22 ) [(σ12 + ς12 + rσ12 ς12 C11 ) B2 − rσ12 ς12 B1 C12 ]




 −1
rσ12 ς12 C11 rσ22 ς22 C22
2 2 2 2 2
2
2
2
2
2
+ 2
(σ1 + ς1 )(σ2 + ς2 ) 1 + 2+ r σ1 ς1 σ2 ς2 C11 C22 − C12.
σ1 + ς12
σ2 + ς22

An interpretation of the result can be summarized as Proposition 2 and Proposition 3 below.
Proposition 2.
(a) If any dimension of the performance vector µ(t) of workers in the firm or the price vectorp(t) of the product in the market is neither perfectly unobservable, nor perfectly observable(i.e., 0 < σi2 < +∞ and 0 < ςi2 < +∞, for i = 1, 2), then it is optimal to use information aboutboth µ(t) and p(t) for each dimension of the effort vector t in order to provide an incentive.
(b) For the i-th dimension of t, if µ(t) or p(t) becomes perfectly unobservable (σi2 → +∞or ςi2 → +∞), then it is optimal to ignore the signal from the unobservable one and use
information only about the observable one to provide an incentive. For the i-th dimension oft, if µ(t) or p(t) becomes perfectly observable (σi2 → 0 or ςi2 → 0), then it is optimal to use
information only about the perfectly observable one.
8

Proof. See the Appendix.
Consider the conditional joint distribution for a given effort ti , fx (xi |ti )fp̃ (p̃i |xi , ti ) =f (xi , p̃i |ti ). Suppose 0 < σi2 < +∞, and let thi and tli denote high and low achievement ofdimension i of t, respectively. Then the likelihood ratio f (xi , p̃i |thi )/f (xi , p̃i|tli ) depends on p̃iif and only if 0 < ςi2 < +∞. Hence Proposition 2 shows that the “sufficient statistic” result ofHolmstrom (1979) holds for this mechanism.
Proposition 3.
Suppose 0 < σi2 < +∞ and 0 < ςi2 < +∞.
(a)If the i-th dimension of the performance vector µ(t) in the firm or the price vector p(t) in themarket becomes less observable (σi2 or ςi2 increases), then it is optimal to weaken the incentive
for that dimension.
(b) If the both of two signals of a dimension become less/more observable, then the ratio of
marginal decreases/increases of incentive on the dimension due to the marginally increasing/decreasing noises of the signals must depend on the reverse of the ratio of noises of thesignals.
Proof. See the Appendix.
As Proposition 3 indicates, when the i-th dimension of µ(t) becomes less observable, thefirm can keep must re-weight i-th dimension of p(t) by adjusting γii accordingly, and cankeep the incentive α instead of giving up the whole incentive. Also, the adjustment to theincreased noise of a signal must depend on the noisiness of the other signal. This shows thatthe conditions for the multidimensional “high-powered” incentive to work can be relaxed if thefirm uses information about p(t) as well as µ(t).
In business practice, this point could be important. A wage scheme depending only on asignal is naturally sensitive to the noise of signal, so that the risk might be too large to be burdened by the agent. This is often an important reason to give up the “high-powered” incentives.
Adjustment among signals and reduction of risk solve this problem to some extent.
As shown by Holmstrom and Milgrom (1991), as long as the firm uses only information
within the organization, the conditions for the multidimensional “high-powered” incentives towork are strict. However, if each dimension of the price vector p(t) is observed and distinguishable from each other as Proposition 1 requires, and if the firm can preserve the stream of
information about the price vector, then a multidimensional incentive becomes easier to use, as
Proposition 2 and Proposition 3 indicate.
Related to this point, another remark is also necessary.
Remark.
Since p(t) shows the aggregate performance of all workers of the firm while x(t) is individually generated, information about p(t) can be utilized to provide an incentive only for aggregate
performance, not for individual ones.
Now that the benefit from the market-oriented multidimensional incentives looks rather
9straightforward, the next question is under what conditions the price vector can be observed andeach dimension of it can be distinguishable by the firm. The best-case scenario is a monopoly,
where the firm can easily observe the change of each coordinate of p(t) by increasing eachcoordinate of q(t), without any noise from the pricing of other sellers’ products. The worstcase scenario is that a firm just sells a generic product at the same price as other sellers in themarket do, where the firm is never able to observe a price increase from their efforts to enhancea dimension of quality.
However, if information about quality of a product is not perfectly observed by buyers at
the time when they buy it in the market, then some firms will try to establish their own brands
that guarantee some quality in order to differentiate its product and receive a quality premiumfrom buyers.11 Consider the establishment of a brand a little more carefully. In practice, there
are two kinds of brands. One is a brand of luxury goods such as the European fashion brands.
The quality of these products is hard to evaluate at the time of purchase, so consumers choose aproduct by relying on the established image of its brand. The other type is typically a brand ofelectric appliances or automobiles for the middle-range market. The quality of these products
can be seen in catalogues or as samples, and firms whose catalogue specs and samples are
credible find it optimal to establish their brands to reflect this. The quality of Japanese electric
appliances and cars can be seen in catalogues and as samples at shops, and their catalogues
and samples are believed to almost exactly show the real quality of these products so that the
reputation of their brands are kept.
The condition of Proposition 1 might not be satisfied in the first type of brands; luxury
brands. However, the second type of brands of differentiated products in the middle-range
market could satisfy the condition of Proposition 1. Hence, if the second type of brand isestablished, then the price vector is observable to the supplier when consumers differentiate themultidimensional quality of the product and then buy it. Even if the market for such brands iscompetitive among suppliers that have established their brands so that suppliers are price takers,
each supplier can observe the price vector of her/his products.
Then the firm may try to use the information about the price vector of its brand to control
incentives within its production organizations, which indeed happened in the Japanese manufacturing a century ago. Before studying the details, let’s summarize the theoretical predictions.
Prediction 1.
If a firm can observe the price vector of its products by differentiating them through the establishment of a brand name, and if the firm can construct a production organization where theinformation is preserved and utilized to control production, then the firm can introduce multidimensional incentives and optimize the effort vector of workers (Proposition 1 and Proposition
2).
Prediction 2.
If a firm can observe the price vector by establishing its brand so that it can use the information
about the price vector as well as the outcome observed in the firm to provide incentives, then
11

Klein, Crawford and Alchian (1978), p. 306. Klein and Leffler (1981).

10

these incentives will be given for the aggregate performance of all workers as well as individual
performance, since the price vector is a signal of aggregate performance (Remark).
Prediction 3.
Consider a product whose brand is established in the market. If the relative observability of adimension of quality changes in the market, then the multidimensional incentive must change
accordingly (Proposition 3).

III. The wage scheme in early 20th century Japan
A. Establishment of the brand
The modern silk reeling industry in Japan grew in the middle 1880s and increased exports to
the U.S. dramatically. The Japanese share of the U.S. market had reached 50 percent by the endof the 1880s, 70 percent by 1910, and 80 percent in the 1920s, overwhelming both the Italian
and the Chinese silk reeling industries. This was the first case where a competitive exportindustry led Japan’s economic development, which has been repeated by various manufacturingindustries since then.12
When raw silk was traded in the market, it was priced according to several factors of quality.
However, it had been almost impossible for silk reeling manufacturers to acquire information
about this price vector in the New York market until the early 1880s. Prior to the 1880s, theWestern trading companies put their trademarks, or “private chops,” on the raw silk after theyinspected and re-packed it, before exporting it to Europe and the U.S. Thus, the quality premiumand the information about the price belonged to the Western trading companies that owned the“private chops.”
In the middle 1880s, however, leading manufacturers organized cooperatives for joint inspection and shipment, and they put their trademarks, or “original chops,”13 on their products.
The New York raw silk market was a spot market where raw silk was traded by sample. Thus
the establishment of a brand meant that its samples were recognized as credible, and the producer of the brand was able to observe the price vector of her/his products. By establishingproducers’ brands, the quality premium and the information about price thus came to belong tothose manufacturers’ cooperatives, not to Western trading companies. It meant, however, thatthe most important information about the price and effort vectors was still unknown to eachmember manufacturer of a cooperative. Only the cooperatives’ headquarters, which conducted
the joint inspection of products, recorded the performance of workers, and guaranteed the quality of the brands to the market, possessed the information necessary to evaluate the performanceof workers.
Hence, major manufacturers withdrew from cooperatives, built large factories that included
inspection processes from the late 1890s to the early 1900s, and established their own brands.
This allowed respective silk reeling manufacturers to finally grasp the stream of information
12

Nakabayashi (2003), pp. 1-59.
Trademarks of trading companies were called “private chops” and those of silk reeling manufacturers werecalled “original chops.” Duran (1913), pp. 105-106.
13

11

about the price vector in the market. Then multidimensional incentives were introduced by thosemanufacturers. Usually, Labor productivity, Material productivity,14 Evenness of threads15 andLuster of threads of raw silk were monitored to provide incentives.

B. Determination of the wage
Now let us inquire about the real process of optimizing the effort vector. One example here isthat of the Kasahara Factory, in Suwa County, Nagano Prefecture, which was the central districtof the industry in Japan. In the silk reeling industry in Suwa County, all workers were young andfemale, all of them lived in dormitories of the firms they worked for, and all their living expenses
were paid by the firms. They were not unionized in the relevant period, so that obstacles to theintroduction of new management practices were small. Employment contracts were usually
one-year and turnover rate in a factory between two consecutive years was generally high.16
Also the technology of reeling raw silk did not require workers to literally cooperate. In generalwages were determined by the ex post relative evaluation by which the effects of common
exogenous shocks were excluded; hence the incentives of risk-averse workers were enhanced.17
Workers received by lump-sum payments at the end of year, in addition to the living expenses.18
The Kasahara Factory followed these common practices.
Until the early 1900s, the Kasahara Factory had belonged to a cooperative for joint inspection and shipment under the cooperative’s brand. Those days, only Labor productivity andMaterial productivity had been systematically recorded in the Kasahara Factory to determine
wages, not dimensions related to the quality of the products. Raw silk that did not satisfy a specific level of quality was excluded from shipment, but the results of inspections were not used
for determining wages. The Kasahara Factory stopped cooperative inspections and shipments
at the end of 1903, and began to inspect raw silk independently and ship it under its own brandin 1904.
Since 1904, in order to determine wages, 1) Labor productivity, 2) Material productivity,3) Evenness of threads, and 4) Luster of threads were systematically recorded at the Kasahara
Factory, where Luster and Evenness were critical dimensions of quality. These dimensions
were recorded for all produced raw silk everyday during the inspection process before shipment.
During the year under the contracts, workers’ performances were recorded every day, relative
performances were calculated every half month, and workers’ efforts followed the overseers’
guidance based on the recent half-month performance. 19 Then wages were paid at lump sum
according to the relative performance of each worker at the end of the year. This practice spreadthroughout Suwa County in the 1900s, a case of which was the Kasahara Factory.
14

Material productivity was the amount of a product over a unit of material (cocoon), which revealed theperformance of economizing on the raw material: cocoons.
15
Evenness of threads was the most important factor of quality in the U.S. market, where power looms for mass
production prevailed.
16
Thus the career concerns were not relevant.
17
Holmstrom (1982a).
18
Hunter (2003), pp. 144-189. Nakabayashi (2003), pp. 241-255.
19
Therefore the practice satisfied the condition in which a linear compensation scheme can be assumed (Holmstrom and Milgrom (1987), pp. 316-322 ).

12

Table 1 shows the wage distribution that does not include the living expense paid by thefirm. Large variance in the table indicates a feature of the “high-powered” incentive of thiswage system.

C. Construction of the wage scheme
In the Kasahara Factory, the wage scheme had been two dimensional until the early 1900s, as
shown in Table 2. 20
Scheme 1897
w: wage. x1 : Labor productivity. x2 : Material productivity.
w = α1 x1 + α2 x2 + β, α1 , α2 > 0.

(11)

However, since 1904, the wage scheme has had four dimensions as follows:
Scheme 1904
w: wage. x1 : Labor productivity. x2 : Material productivity. x3 : Evenness of threads. x4 :Luster of threads.
(12)

w = α1 x1 + α2 x2 + α3 x3 + α4 x4 + β, α1 , α2 , α3 , α4 > 0.

The regressions of (12) are in Table 2. In 1904, the Kasahara Factory incorporated an inspection process into its own factory and began to sell its raw silk under its own brand. With thisorganizational change, the Kasahara Factory could acquire the signal x(t) of the effort vector tthat included quality dimensions, and recorded it in the daily inspections. Furthermore, it was
20

The coefficients α3 and α4 of x3 and x4 are unstable in some years on Table 2. It is supposedly from two
reasons. One is that t3 : Evenness of threads and t4 : Luster of threads, both of which were for cleaner threads,
presumably interacted with each other. Indeed, the coefficient of the interaction term of a standardized regression
through the origin with year dummies Y19XX is significant as follows:w1904-1913 =1.387x1904-1913+ 2.506x1904-1913+ 0.018x1904-1913+ 0.041x1904-1913+ 0.025(x3 x4 )1904-1913
12
3
4

p value for t

0.000

0.000

0.001

0.000

0.000

− 1.085Y1904 − 1.116Y1905 − 1.026Y1906 − 1.301Y1907 − 0.708Y1908
0.000

0.000

0.000

0.000

0.000

− 0.798Y1909 − 0.809Y1910 − 0.809Y1911 − 0.460Y1912 − 0.780Y1913
0.000

0.000

0.000

0.000

0.000

p value for F : 0.000. The number of samples: 2, 235.
The other reason was a discrepancy between ex ante incentive and ex post performance. If all of workers
were on the path of incentive mechanism and performed exactly as guided, then their performance should haveconcentrated on the optimal point so that their performance should be the same, hence a regression of observed
ex post differences of their performance to the wage should not have fit. For instance, the sign of α4 in 1910 is
opposite in Table 2, but it was not because workers did not follow the incentive for t4 , rather because they followed
it very well as shown on Table 3-c (discussed in the next section).
The wage scheme in Table 1 estimates the final result of calculating a wage. The actual practice leading to theresult was a little more complicated. See Nakabayashi (2003), pp. 256-268.

13able to acquire information about the price vector p̃(t) of its own products, i.e., another signalof t. Prediction 1 suggests at this point the Kasahara Factory could introduce an explicitly
multidimensional “high-powered” incentive scheme, and it was exactly what happened.
The aggregate performance, given the price in the market, can be approximated by the returnon sales, i.e., profits over sales. Indeed a regression of the real wage to the return on sales
through 1904-1913 shows a significant relation.21 This indicates wages, at least to some degree,depended on the overall performance of the firm. In other words, that portion of the wage wasdetermined as a reward for the aggregate performance of the whole factory, a result Prediction
2 suggests.
Since it was not necessary for workers to literally cooperate, a reward for aggregate performance was just for aggregated effort, not cooperative activity. However, a contemporary
observer in a local newspaper pointed out that rewards for aggregate performance were also
useful to keep up workers’ morale.22
Next, we will check whether the effort vector was really optimized during the process.

IV. Optimization of the effort vector
A. Substitute dimensions
In the silk reeling industry, Material productivity t2 , Evenness of threads t3 , and Luster of
threads t4 , all of which needed careful processing, were obviously substitutes for Labor productivity t1 . If the relative price of labor and material changes, and/or the relative “price” of eachdimension of the quality changes, the effort vector that maximizes profits changes accordingly.

B. Controlling the direction of the effort vector
The firm’s target was some optimal effort vector t∗T = (t∗1 , t∗2 , t∗3 , t∗4 ) = t∗1 (1, t∗2 /t∗1 , t∗3 /t∗1 , t∗4 /t∗1 ).
Neither Material productivity t2 , Evenness of threads t3 , nor Luster of threads t4 needed to
be enhanced infinitely. Rather, given the relative price to clear the product market, the labormarket, and the raw material market, optimal levels of the quality of product and the material
productivity were decided such that profit was maximized.
21

A standardized regression is as follows:
RW : real wage. ROS: return on sales of the Kasahara Factory.
RW

1904-1913

p value for t= 1.067x1904-1913− 0.108x1904-1913+ 0.052x1904-1913+ 0.024x1904-1913+ 0.025ROS 1904-1913 .
1
2
3
4
0.000

0.000

0.000

0.000

0.000

p value for F : 0.000. The sample numbers:2, 235.
The sign of x2 is negative probably because x2 heavily depended on the quality of material cocoon, which changed
due to the weather of each year, so that the coefficient of pooled data of x2 for multiple years do not necessarily
depend on the performance of workers. It is indicated by that the coefficient is significantly positive with year
dummies in the note 25.
22
Haizanbo, “Kawagishimura no ichinich (5)” (One day in the Kawagishi Village (5)), Shinano Mainichi Shimbun (Shinano daily), Nagano, July 28, 1903.

14

Therefore, if workers’ ability was sufficiently high, their attention had to be allotted to eachof Material productivity t2 , Evenness of threads t3 , and Luster of threads t4 such that the optimallevels of them were satisfied, and the rest amount of workers’ attention, if they had, had to be
allotted to Labor productivity t1 , at the optimum. However, if some workers’ ability was not
sufficient so that their attention was not enough to satisfy the optimal levels of the t2 , t3 , and t4 ,the firm had to instruct them to enhance all of the t1 , t2 , t3 , and t4 .
Thus, according to the distribution of ability of workers, the firm was supposed to instruct
less able workers to enhance all dimensions t1 , t2 , t3 , and t4 with some weights, and instructabler workers to fix dimensions t2 , t3 , and t4 at the optimal level and to enhance t1 .
Hence, on the (t1 , ti ) plane (i = 2, 3, 4), plots of performance (t1 , ti ) of workers were
supposed to follow an increasing and bounded-above function g(t1 ) if the effort vector t was
optimized. Then, the image of the signal x(t) on the (x1 , xi ) plane (i = 2, 3, 4) was also
supposed to converge an increasing and bounded-above function according to the optimizationof the effort vector t.
Motivated by a Gaussian kernel regression of Labor productivity x1 and Material productivity x2 (Figure 1), we take a first approximation of such a function with(13)

xi = η1 /x1 + η2 , 0 < x1 ; η1 < 0; i = 2, 3, 4.

C. The optimized effort vector and information from the market
The image of x(t) on the Labor productivity x1 - Material productivity x2 plane had not been
optimized at all in 1897 or 1901 (Table 3-a), even though Material productivity x2 had beenrecorded (Table 2). The inspection process had been carried out by the headquarters of thecooperative before 1904, not the Kasahara Factory, so that even information about Material
productivity had probably not been handed to Kasahara on a daily basis. Hence the Kasahara
Factory could neither monitor nor instruct the effort vector t of workers on daily basis. Consequently, workers increased Labor productivity x1 by decreasing Material productivity x2 below
the optimal level. It was optimized after 1904 (Table 3-a).
The process of optimization can also be seen on the Labor productivity x1 - Evenness of
threads x3 plane since 1904 (Table 3-b).
For Labor productivity t1 , Material productivityt2, and Evenness of threads t3 , the effort
vector t has been well controlled since 1904, when the Kasahara Factory started independent
inspection and made its original brand. On the other hand, the image of t on the Labor productivity t1 - Luster of threads t4 plane was optimized as late as 1908 (Table 3-c).
Interestingly, some changes occurred in the US market at the beginning of the 1908 season.
The Silk Association of America, the industrial body consisting of silk manufacturers and merchants, suggested a method of classification for Japanese brands of raw silk, using a standard
brand as a measure. Using this standardized measure allowed buyers to more easily differentiate
the quality of raw silk sold by different brands.23 At the same time it enabled manufacturers tobetter observe the price vector from the purchasing behavior of buyers in the New York market.
23“Classifications of raw silks,” The American Silk Journal, New York, vol.27, no.7, 1908, p. 23.

15

Indeed, Luster of threads became more effective for the determination of wages in 1908-1913
than it had been in 1904-1907. 24
As Prediction 3 claims, faced with a changing market, the firm observed the Luster dimension p̃4 of p̃(t) more clearly, and enhanced the incentive α4 for dimension x4 of the observed
performance x(t), both of which were signals of t4 of the effort vector t. Then, the image ofx(t) on the (x1 , x4 ) plane converged to the optimal curve (Table 3-c, Figure 2), which reflects
the optimization of t on the (t1 , t4 ) plane.
The incentive scheme worked on the basis of information from the product market, as well as
information in the firm, as Prediction 1 mentions. Moral hazard by workers had been a serious
problem until the early 1900s, but after 1904 this problem had been almost completely solved.
After 1904, the Kasahara Factory established its own brand as an instrument to capture thestream of information about the price vector of its products, utilized it to control the incentivesof workers, and explicitly optimized the multidimensional effort vector of workers.

V. Discussion: a viewpoint of comparative analysis
A. A tradition of multidimensional incentives
As predicted by the model, the establishment of a brand that enabled firms to observe the pricevectors of their products made the introduction of “high-powered” multidimensional incentives
for workers possible.
In the Japanese silk reeling industry, manufacturers established their own brands to acquire
quality premiums that had belonged to trading companies. The establishment of brands accompanied the construction of an organization to inspect the quality. Within this organization, themultidimensional performance of workers had begun to be monitored and recorded. As a resultof these organizational changes, silk reeling manufacturers acquired information about the pricevector in the market and about performance of their workers in their factories on a daily basis.
By taking advantage of this condition, they were able to optimize the effort vectors of workers
by connecting information about the price vector with that of the effort vector. The information stream of the price vector from the product market was efficiently utilized for controllingincentives in the firm.
24Standardized regressions through the origin with year dummies Y19XX are below.

w1904-1907 =1.334x1904-1907+ 1.115x1904-1907+ 0.078x1904-1907+ 0.040x1904-1907+ 0.021(x3 x4 )1904-1907
12
3
4

p value for t

0.000

0.009

0.001

0.012

0.147

− 0.677Y1904 − 0.706Y1905 − 0.721Y1906 − 0.953Y1907.
0.001

0.001

0.000

0.000

p value for F : 0.000. The number of samples: 768.
w1908-1913 =1.371x1908-1913+ 5.139x1908-1913+ 0.018x1908-1913+ 0.021x1908-1913+ 0.028(x3 x4 )1908-1913
12
3
4

p value for t

0.000

0.000

0.004

0.000

0.000

− 2.162Y1908 − 2.487Y1909 − 2.481Y1910 − 2.488Y1911 − 1.322Y1912 − 2.424Y1913.
0.000

0.000

0.000

p value for F : 0.000. The number of samples: 1, 467.
The coefficient of x4 was more significant in 1908-1913.

16

0.000

0.000

0.000

B. Segmented quality of Japanese cars
This parable of a historical experiment contains some implication to understand the contemporary difference between the Japanese and the Western manufacturing.
For two decades, Japanese cars have had a good reputation because of their quality, and, in
2006, Consumer Reports25 finally ranked “Japanese models as its top choice in all 10 vehicle
categories.” However, what is this good quality? The annual car-buying guide of Consumer
Reports “is based on tests of more than 200 models, covering performance, comfort, safety andfuel economy, among other factors.”26 Thus the kind of market reports usually evaluates carsalong several categories, such as engine troubles within a few years after purchase, or driver satisfaction in handling, and so forth. Then those reports classify cars by summing up the points
given to respective categories or terms. Therefore, they implicitly assume that consumers’ evaluations, or utility functions, are additively separable across standardized categories. In otherwords, the fact that Japanese cars achieve high quality ratings means the additively separable
dimension of their quality components are high. Japanese car manufacturers likely optimize
the quality components of their cars subject to the multidimensional price that reflect the additively separable and standardized benefits their customers receive from the quality. On the other
hand, the benefits of owning a Corvette or Porsche seem to be hard to separate and calculate intheir separated components, and in such a high-end market, Japanese cars have been relatively
unsuccessful. Therefore the Japanese auto industry is typically good at hedonic approaches toquality control in the middle range market, as the Japanese silk reeling industry was a century
ago. This is exactly a point directly related to Prediction 1. The Japanese manufacturing has
focused on the middle range of the market, where each dimension of the quality can be easily
distinguished from the others, as Prediction 1 requires.
Another particular feature of the Japanese manufacturing is that multidimensional evaluations for wage and promotion schemes are imposed on blue-collar workers as well as whitecollar workers to keep product quality high.27 Combined with career concerns, these compensation schemes consist of multidimensional incentives.
Given those casual observations with the analysis of this research, the multidimensional“high-powered” incentives on shop floors seem to work better, if the quality control is conductedin well-defined multidimensional terms.
That story also induces us to return to the understanding of the “borders of firms” implied
by Alchian and Demsetz (1972). A firm tries to exclude intermediary players and incorporatetransactions if it can acquire a quality premium with smaller transaction costs. At the sametime, the firm can increase total surplus if it can preserve the information stream from theproduct market to optimize the effort vectors of workers in the production organization. Hence,
the borders of the firm can also be decided by the benefits from quality and incentive controls,
as well as the transaction cost, as seen in the Japanese manufacturing more than one century
ago.
25

Consumer Reports New Car Buying Guide 2006, Consumer Reports Books, June 12, 2006.
Financial Times, March 2, 2006, p. 15.
27
Aoki (1988), pp. 49-98.
26

17

C. For a comparative analysis
It has been shown that a multidimensional “high-powered” incentives can be efficient undersome conditions. Then why are similar organizations rare in the West? In the US, Ford style ofthe “contractually fixed wage” became dominant for blue-collar workers in the auto industry in
the 1920-30s, ironically when they discovered the concept of “hedonic prices” (Waugh (1928),
Court (1939)). European auto industries had followed suit by the 1970s. A reason why the fixed
wage was taken there were with some combination of the new technology for mass production
and the new management affiliated with it. However, another point was that incentive wages
had to face union conflicts, and fixed wages have been a part of agreements between firms andunions in the Western manufacturing industries until now. In the auto industry, for instance,
while many Japanese practices were introduced in the 1990s, the industry’s standard pay structure based on an agreement between the Big Three and the UAW remained, and the Japanesestyle of an individualized and “capability-based” wage systems have been rejected.28
The inertia of American industrial relations can be traced back to the very beginning ofindustrialization in the US.29 Hence an important factor is probably the historical path of theindustrialization. The modern textile industry was a newly implemented industry in 19th century Japan, and young female workers did not have a history of guilds. However, in the Westernworld, especially male workers had a tradition of guild-like unions and sometimes they wereagainst firms that tried to control production.30 This seems to be a reason why multidimensional
incentives are not offered to blue-collar workers in the West but they are in Japan.
This difference could also be a part of explanation of why the Western manufacturing is
generally competitive in the production for the low-end or the high-end market, but not for the
middle-range market. Production of generic or luxury goods does not require or suit multidimensional incentives. Instead, implicit incentives like guaranteed positions and career considerations would be suitable to the production of luxury goods. On the other hand, consumersof middle-range products can be recognized across standardized terms, hence multidimensional
incentives work well for production of middle-range differentiated products.

Appendix: proofs of the propositions
Proposition 1.
Proof.
∂αT Γj Σj Γj α
= ΣjT ΓjT ααT + ααT ΓjT ΣjT .
∂ΓjSince the rank of ααT is 1, the solution Γj∗ of ΣjT ΓjT ααT + ααT ΓjT ΣjT = 0 is uniqueonly if each row of Γj contains only one nonzero entry. Such a Γj∗ minimizes αT Γj Σj Γj α
28Jürgens, Malsch and Dohse (2003), pp. 215-280. Wheeler (1993), pp. 71-75. Weinstein and Kochan (1995),
pp. 12-15. Adler, Kochan, MacDuffie, Pil and Rubinstein (1997), pp. 68-69. Sloane and Witney (2004), pp.
280-282. For an overview on the industrial relations in Japan after World War II, see Nakamura and Nitta (1995).
29
Kochan and Katz (2000), pp. 17-48.
30
Clark (1984).

18

only if each row of Σj contains only one nonzero entry. Since each dimension of ǫj is positive,
it means Σj must be diagonal, i.e., each dimension of ǫj must be independent to each other.
Proposition 2.
Proof. (a) By (7), if 0 < σi2 < +∞, and if 0 < ςi2 < +∞, then 0 < γii < 1, so that information
both about the performance observed in the firm xi and the price observed in the market p̃i is
utilized to determine the wage w.
(b) For the observed performance in the firm x(t), as σi2 → +∞, γii → 0, where information
only about the price p̃i is utilized. As σi2 → 0, γii → 1, where information only about theperformance in the firm xi is utilized. For the price vector p(t), the proof is analogous.
Proposition 3.
Proof. (a) By (10),α1 =[(σ22 + ς22 + rσ22 ς22 C22 ) B1 − rσ22 ς22 B2 C12 ]
.
2 2σ12 ς122 2
2 2 σ1 ς1
2
2
2
2
C11 + rσ2 ς2 C22 + r σ2 ς2 2
(C11 C22 − C12 )(σ2 + ς2 ) 1 + r 2
σ1 + ς12
σ1 + ς122
Since C(t) is strictly convex (C11 C22 − C12> 0) and σ12 ς12 /(σ22 + ς12 ) is strictly increasing in
σ12 and in ς12 , α1 decreases if σ12 and/or ς12 increases.
(b) By (10),∂α1 . ∂α1∂(α1 /α2 ) . ∂(α1 /α2 )ς3== 13 ,∂σ1 ∂ς1
∂σ1
∂ς1
σ1

and

∂α1 . ∂α1∂(α1 /α2 ) . ∂(α1 /α2 )ς3== 23 .
∂σ2 ∂ς2
∂σ2
∂ς2
σ2

19

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University of Tokyo Press, 2003.

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Historical document“Kasaharagumi shiryo (Documents of Kasahara group),” held by Okaya Sanshi Hakubutsukan(The Silk Museum of Okaya), Okaya, Nagano Prefecture, Japan.
22

Table 1 Wage in the Kasahara Fusakichi Factory, Suwa County, Nagano Prefecture.
year Number Distributionofaverage maximum minimum median variance kurtosis skewness
samplessen per sen per sen per sen per
workday workday workday workday
1897
138 18.067 32.042
7.413 17.997 29.547
0.003
0.362
1901
163 20.898 44.098
0.638 21.999 61.410
0.377 -0.034
1904
191 22.521 53.983
6.667 23.104 74.095
0.389
0.367
1905
199 20.040 46.415
3.625 20.652 60.918
0.088
0.268
1906
150 23.196 49.549
1.735 23.994 92.764 -0.341
0.034
1907
228 23.191 52.140
0.736 22.852 92.683
0.499
0.254
1908
251 22.931 59.780
1.774 24.001 99.890
0.178
0.309
1909
351 22.682 58.235
1.264 22.999 112.989 -0.229
0.405
1910
351 26.083 60.229
1.364 25.000 130.243 -0.258
0.247
1911
343 23.450 66.998
0.000 22.998 146.768 -0.001
0.405
1912
88 23.779 47.996
2.478 25.494 130.575 -1.062 -0.023
1913
83 23.209 60.000
3.080 20.000 159.526
0.011
0.794
Source : Fusakichi Kasahara, "Seishi keisan bo" (Book for evaluation of silkreeling). Kasaharagumishiryo (Documents of Kasahara goroup).
Notes: "Wage" does not contain supplemental payment, which amounts to 5-10%
of wage. 1 sen (0.01 yen) was approximately 0.5 cent of U.S. dollar. The numberof samples is small in 1906, 1912, and 1913, because some books have been lostfor those years. However, there is not any bias in distribution of performance
depending on each book, so that this loss does not affect the rsult of estimation.

23

Table 2 Wage and observed performancew : Wage. Į = (Į 1, Į 2, Į 3, Į 4): incentive vector. x = (x 1, x 2, x 3, x 4): signal of effort
vector t in the firm, where x 1: Labor productivity . x 2: Material productivity .
x 3: Evenness of threads . x 4: Luster of threads . ȕ : transfer of surplus.
w = Į 1x 1 + Į 2x 2 + Į 3x 3 + Į 4x 4 + ȕ.
Į1

Į2

Į3

Į4

ȕ

2

standardR
p value
error
for F
1897
0.777
0.209

 0.000
0.031
0.681
0.000
p value for t
0.000
0.000
(0.000) 0.000
0.038
0.766
0.000
1901
0.867
0.191
p value for t
0.000
0.000
(0.000)
1904
0.838
0.190
0.042
0.093
0.000
0.038
0.809
0.000
p value for t
0.000
0.000
0.209
0.005 (0.000)
1905
0.836
0.099
0.116
0.144
0.000
0.031
0.841
0.000
p value for t
0.000
0.001
0.000
0.000 (0.000)
1906
0.799
0.150
0.052
0.164
0.000
0.044
0.803
0.000
p value for t
0.000
0.000
0.173
0.000 (0.000)
1907
0.848
0.019
0.010
0.155
0.000
0.047
0.762
0.000
p value for t
0.000
0.637
0.788
0.000 (0.208)
1908
0.794
0.143
0.114 -0.022
0.000
0.048
0.777
0.000
p value for t
0.000
0.000
0.001
0.492 (0.000)
1909
0.860
0.073
0.173
0.016
0.000
0.037
0.883
0.000
p value for t
0.000
0.001
0.000
0.436 (0.000)
1910
0.826
0.191
0.088 -0.046
0.000
0.049
0.821
0.000
p value for t
0.000
0.000
0.001
0.075 (0.000)
1911
0.826
0.192
0.084
0.035
0.000
0.047
0.850
0.000
p value for t
0.000
0.000
0.000
0.129 (0.000)
1912
0.754
0.176 -0.007
0.033
0.000
0.058
0.762
0.000
p value for t
0.000
0.006
0.912
0.619 (0.002)
1913
0.780
0.202
0.030
0.079
0.000
0.051
0.846
0.000
p value for t
0.000
0.000
0.572
0.160 (0.000)
Source : Fusakichi Kasahara, "Seishi keisan bo."Notes : Coeffecients are the results of a standadized linear regression so that thetransfer ȕ is normalized as 0. The p value for t of ȕ is from an
unstandardized regression. The number of samples is the same as on Table 1.
year

24

Table 3-a Optimization of effort vector t (Labor
productivity-Material productivity plane)
x 1: Labor productivity . x 2: Material productivity .
x 2 = Ș 1/x 1 + Ș 2.
Ș1

Ș2

R2
p valuestandard
error
for F
1897
-0.116
0.000
0.404
0.013
0.177
p value for t
0.177 (0.000)
1901
0.035
0.000
0.467
0.001
0.661
p value for t
0.661 (0.000)
1904-0.212
0.000
0.446
0.045
0.003
p value for t
0.003 (0.000)
1905-0.299
0.000
0.403
0.089
0.000
p value for t
0.000 (0.000)
1906-0.539
0.000
0.365
0.290
0.000
p value for t
0.000 (0.000)
1907-0.713
0.000
0.700
0.509
0.000
p value for t
0.000 (0.000)
1908-0.471
0.000
0.261
0.221
0.000
p value for t
0.000 (0.000)
1909-0.468
0.000
0.257
0.219
0.000
p value for t
0.000 (0.000)
1910-0.386
0.000
0.224
0.149
0.000
p value for t
0.000 (0.000)
1911-0.359
0.000
0.270
0.129
0.000
p value for t
0.000 (0.000)
1912-0.563
0.000
0.249
0.317
0.000
p value for t
0.000 (0.000)
1913-0.338
0.000
0.400
0.114
0.002
p value for t
0.002 (0.000)
Source : Fusakichi Kasahara, "Seishi keisan bo."Notes : Coefficients are the results of standardized linear
regression while the p value for t of Ș 2 is from an
unstandardized regression. The number of samples is
the same as on Table 1.
year

25

Table 3-b Optimization of effort vector t (Labor
productivity-Evenness of threads plane)
x 1: Labor productivity . x 3: Evenness of threads .
x 3 = Ș 1/x 1 + Ș 2.
Ș1

Ș2

2

R
p valuestandard
error
for F
1904
-0.156
0.000
0.000
0.024
0.032
p value for t
0.032 (0.000)
1905-0.383
0.000
0.000
0.147
0.000
p value for t
0.000 (0.000)
1906-0.270
0.000
0.000
0.073
0.001
p value for t
0.001 (0.000)
1907-0.589
0.000
0.000
0.347
0.000
p value for t
0.000 (0.000)
1908-0.150
0.000
0.000
0.023
0.017
p value for t
0.017 (0.000)
1909-0.216
0.000
0.000
0.047
0.000
p value for t
0.000 (0.000)
1910-0.266
0.000
0.000
0.071
0.000
p value for t
0.000 (0.000)
1911-0.132
0.000
0.000
0.017
0.015
p value for t
0.015 (0.000)
1912-0.094
0.000
0.000
0.009
0.384
p value for t
0.384 (0.014)
1913-0.071
0.000
0.000
0.005
0.523
p value for t
0.523 (0.207)
Source : Fusakichi Kasahara, "Seishi keisan bo."Notes : Coefficients are the results of standardized linear
regression while the p value for t of Ș 2 is from an
unstandardized regression. The number of samples is thesame as on Table 1.
Year

26

Table 3-c Optimization of effort vector t (Labor
productivity-Luster of threads plane)
x 1: Labor productivity . x 4: Luster of threads .
x 4 = Ș 1/x 1 + Ș 2.
Ș1

Ș2

2

R
p valuestandard
error
for F
1904
-0.003
0.000
0.012
0.000
0.970
p value for t
0.970 (0.103)
1905
0.017
0.000
0.015
0.000
0.816
p value for t
0.816 (0.000)
1906
0.035
0.000
0.008
0.001
0.675
p value for t
0.675 (0.027)
1907
0.027
0.000
0.011
0.001
0.682
p value for t
0.682 (0.267)
1908-0.154
0.000
0.014
0.024
0.015
p value for t
0.015 (0.174)
1909-0.106
0.000
0.014
0.011
0.047
p value for t
0.047 (0.001)
1910-0.339
0.000
0.012
0.115
0.000
p value for t
0.000 (0.000)
1911-0.214
0.000
0.014
0.046
0.000
p value for t
0.000 (0.852)
1912-0.431
0.000
0.013
0.186
0.000
p value for t
0.000 (0.029)
1913-0.492
0.000
0.010
0.242
0.000
p value for t
0.000 (0.000)
Source : Fusakichi Kasahara, "Seishi keisan bo."Notes : Coefficients are the results of standardized linear
regression while the p value for t of Ș 2 is from an
unstandardized regression. The number of sumples is thesame as on Table 1.
year

27

39.1039.05

Material productivity x 2

39.00

38.95
Gaussian kernel, bindwidth 10.
38.90

38.85

38.80

38.75

38.70
30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00Labor productivity x 1

Figure 1: Gaussian kernel regression (Labor productivity-Material productivity): Kasahara Factory, 1904.
Source: “Seishi keisan bo”.
Notes: Material productivity: produced raw silk per 4 shou (7,216 liters) of cocoon. Labor
productivity: momme(3.75 grams)s of raw silk per workday.
0.03
0.02
0.01

Luster of Threads x 4

0
-0.01
-0.02
-0.03
-0.04
Unstandaridized estimated curve-0.05

Unstandardized estimated curve: x 4 0.940x 10.009. R : 0.115. p value for F : 0.000.
p value for t 0.000
0.000
2

-0.06
-0.07
10.00

Observed

30.00

50.00

70.00

90.00

110.00130.00150.00Labor Productivity x 1

Figure 2: Optimization of effort vector t (Labor productivity-Luster of threads): Kasahara
Factory, 1910.
Source: “Seishi keisan bo”.
Notes: Luster of threads: points of Luster per 1 momme (3.75 grams) of raw silk. Labor
productivity: mommes of raw silk per workday.

28