Table Of ContentASYMPTOTIC EFFICIENCY OF THE PROPORTIONAL
COMPENSATION SCHEME FOR A LARGE NUMBER OF PRODUCERS
DMITRY B. ROKHLIN AND ANATOLY USOV
Abstract. We consider a manager, who allocates some fixed total payment amount be-
7 tween N rational agents in order to maximize the aggregate production. The profit of i-th
1
agent is the difference between the compensation (reward) obtained from the manager and
0
the production cost. We compare (i) the normative compensation scheme, where the man-
2
agerenforcesthe agentsto followanoptimalcooperativestrategy;(ii)the linear piece rates
n
compensationscheme,where the managerannounces anoptimalrewardper unit good;(iii)
a
J the proportional compensation scheme, where agent’s reward is proportional to his contri-
1 butionto the totaloutput. Denoting the correspondenttotalproductionlevelsby s∗, sˆand
2 s respectively,wherethe lastone is relatedto the unique Nashequilibrium, weexamine the
limitsofthepricesofanarchyAN =s∗/s,AN′ =sˆ/sasN →∞. Theselimitsarecalculated
]
for the cases of identical convex costs with power asymptotics at the origin, and for power
C
costs, corresponding to the Coob-Douglas and generalized CES production functions with
E
decreasing returns to scale. Our results show that asymptotically no performance is lost in
.
n terms of AN′, and in terms of AN the loss does not exceed 31%.
i
f
-
q
[
1. Introduction
1
v
Consider a manager, who allocates some fixed some fixed total payment amount M be-
8
3 tween N producers (agents) in order to maximize the aggregate production. The profit of
0 each agent equals to the difference between the reward obtained from the manager and the
6
production cost ϕ . If the cost functions are known, the manager can determine rewards,
0 i
. stimulating the optimal aggregate production s∗. We call such compensation scheme nor-
1
0 mative. Besides the quite unrealistic assumption that the cost functions are known, this
7
scheme suffers from another drawback: it does not announce any common reward sharing
1
: rules. However, s∗ can serve as a benchmark.
v
Another idea is to use the linear piece rates compensation scheme, announcing a price
i
X
µ of the unit good. So, the reward µx of i-th agent will be linear in his production level
i
r
x . Assuming an individually optimal (rational) agent behaviour, the manager can chose µ
a i
in such a way that the total reward does not exceed M, and the total production sˆ cannot
be improved by another linear reward rule. Clearly, this compensation scheme also requires
the knowledge of production cost functions, although it is easier to assign one parameter
µ, rather than the full set of rewards, as in the normative scheme. Note also that in the
piece rates allocation scheme the total reward requested by irrational agents can exceed M.
Nevertheless, we regard the value sˆ as another benchmark.
The main focus of the present study is the proportional compensation scheme, where the
reward Mx /(x +···+x ) of i-th agent is proportional to his contribution to the aggregate
i 1 N
production. The realization of this scheme requires no information concerning the cost
2010 Mathematics Subject Classification. 91B32,91B40, 91B38.
Keywordsandphrases. Proportionalcompensationscheme,totalproduction,priceofanarchy,asymptotic
efficiency, Tullock contest.
1
2 DMITRYB. ROKHLINANDANATOLYUSOV
functions, and the total reward equals to M irrespective of agent actions (except the trivial
case, where x = 0). So, the manager allows the agents to determine optimal production
levels on their own in the course of a (non-cooperative) game with the payoff functions
x
i
M −ϕ (x ). (1.1)
i i
x +···+x
1 N
Under the assumption that the cost functions ϕ are convex and strictly increasing, the game
i
(1.1) has a unique Nash equilibrium. By s we denote the correspondent total production.
One may argue that the computation of a Nash equilibrium also requires the knowledge
of cost functions. However, such equilibrium also can emerge as a result of agent interaction
in a repeated game through the mechanism of no-regret learning. We recall this concept at
the end of the paper. For each agent the no-regret learning does not require the knowledge
of the cost functions of other agents.
The game (1.1) is a special case of the Cournot oligopoly: [27, 25], and it fits into the
extensively studied theory of contests: see [7, 21, 8, 32] for reviews (an experimental research
is reviewed in [12]). In a contest the payoff function of each player is the difference between
the contest success function (CSF) and the cost of player’s effort. A player’s CSF usually
equals to the expected value of winning an indivisible prize, or, as in our case, to the portion
of the prize, obtained by the player. It depends on the efforts of all players, and it is
increasing in the effort of a selected player and decreasing in the efforts of the other ones.
An account of the CSF’s can be found in [18].
Using the substitution f (y ) = ϕ−1(y ), we can reduce the game (1.1) to a strategically
i i i i
equivalent contest
f (y )
i i
M −y (1.2)
i
f (y )+···+f (y )
1 1 N N
with the CSF of the general-logit form (in the terminology of [32, Chapter 4]). In addition,
in our main example of power costs ϕ = c xα, α ≥ 1, corresponding to generalized CES
i i i
production functions with decreasing returns to scale, the game (1.2) boils down to the
Tullock contest with f (y ) = (y /c )1/α.
i i i i
Contests are used to model conflict situations in rent-seeking, resource allocation, patent
races, sports, advertising, etc. The present paper is related to the analysis of relative perfor-
mance incentive schemes in labour contracts. An active study of such problems was initiated
in 1980s: [23, 14, 26, 24]. We also mention several recent papers with an emphasis on ex-
perimental and empirical studies: [6, 15, 29], where the reader can find a lot of additional
references. The prevailing concept is the rank-order allocation of prizes (rank-order tourna-
ments). However, the proportional prize-contest was promoted by the means of experimental
studies in [3].
The main feature of the present paper is the analysis of the following two versions of the
“price of anarchy”: A = s∗/s, A′ = sˆ/s for a large number N of agents. In line with [22]
N N
the price of anarchy shows how much performance is lost by the lack of coordination. The
study of the prices of anarchy recently became an active area of research. We mention only
a few papers, studying an efficiency of the proportional resource allocation mechanism in
somewhat different models: [19, 5, 2].
In Section 2 we describe three compensation schemes mentioned above. In particular, we
point out that any contest scheme cannot be better than the normative one (Remark 2). In
Section 3 we study the prices of anarchy A , A′ for large N. Our results show that for the
N N
cases of identical convex costs ϕ = ϕ with power asymptotics at the origin (Theorem 1), and
i
ASYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 3
for heterogeneous agents with power costs ϕ (x) = c xα, α > 1 (Theorem 2), asymptotically
i i
no performance is lost in terms of A′ , and in terms of A the loss does not exceed 31%.
N N
These results characterize anasymptotic efficiency of the proportionalcompensation scheme.
We also conjecture that this result remains true for heterogeneous agents with linear cost
functions (α = 1) and i.i.d. marginal costs c .
i
2. Three compensation schemes
Let x be the amount of goodproduced by i-th agent. Denoteby ϕ : R 7→ R therelated
i i + +
production cost. We assume that the functions ϕ are twice continuously differentiable,
i
ϕ (0) = 0, ϕ′(x ) > 0, ϕ′′(x ) ≥ 0, x > 0. It easily follows that ϕ (x ) → +∞, x → +∞.
i i i i i i i i i
(i) Normative compensation scheme. Agent i knows the reward function ψ (x ) ≥ 0 at the
i i
beginning of the production cycle and maximizes his profit:
ψ (x )−ϕ(x ) → max. (2.1)
i i i
xi≥0
Let ψ (0) = 0, and denote by x˜ = x˜ (ψ ) optimal solutions of (2.1), which for simplicity we
i i i i
assume to exist. The manager has M units of capital at his disposal. His aim is to maximize
the total production:
N
x˜ → max
i
i=1
X
over all reward functions ψ , satisfying the conditions
i
N
ψ (x˜ ) ≤ M; ψ ≥ 0, i = 1,...,N.
i i i
i=1
X
Since ψ (x˜ )−ϕ (x˜ ) ≥ 0, we get the estimate
i i i i
N N
ϕ (x˜ ) ≤ ψ (x˜ ) ≤ M.
i i i i
i=1 i=1
X X
Thus, given the budget M, the total production cannot exceed the value
N N
s∗ = sup x : ϕ (x ) ≤ M, x ≥ 0 (2.2)
i i i
( )
i=1 i=1
X X
for any kind of rewards ψ .
i
On the other hand, it is possible to obtain the total production arbitrary close to s∗ by
announcing the rewards
ψi(xi) = ϕi(x∗i)I[x∗i−εi,∞)(xi), (2.3)
where x∗ = (x∗)N is an optimal solution of (2.2) and
i i=1
ε ∈ (0,x∗), if x∗ > 0; ε = 0, if x∗ = 0.
i i i i i
Indeed, in this case the optimal solution of (2.1) is of the form
x∗ −ε , x∗ > 0,
x˜ = i i i
i 0, x∗ = 0
( i
and N x˜ = s∗− N ε , while N ψ (x˜ ) = N ϕ (x∗) ≤ M. Thus, one can regard s∗
i=1 i i=1 i i=1 i i i=1 i i
as the optimal total production amount under the normative compensation scheme.
P P P P
4 DMITRYB. ROKHLINANDANATOLYUSOV
(ii) Linear piece rates compensation scheme. Assume that the cost functions are strictly
convex and the manager tries to choose a best linear reward function ψ (x ) = µx . The
i i i
production levels xˆ (µ) are determined by the problems
i
µx −ϕ (x ) → max. (2.4)
i i i
xi≥0
The functions xˆ (µ) are non-decreasing, and the best choice of µ corresponds to the largest
i
total production which does not violate the budget constraint:
N
µ xˆ = M. (2.5)
i
i=1
X
The aggregate production is given by sˆ= N xˆ .
i=1 i
(iii) Proportional compensation scheme. The left-hand side of (2.5) equals to the total
P
reward. If the agents anticipate that the manager selects µ in this way, then they become
involved in the non-cooperative game with the payoff functions
x
i
H (x) = M −ϕ (x ), x ≥ 0 (2.6)
i N x i i
i=j j
with the convention 0/0 = 0. The totaPl production equals to
N
s = x ,
i
i=1
X
where x is the unique Nash equilibrium (in pure strategies) of the game (2.6):
H (x ,...,x ,...,x ) ≥ H (x ,...,x ,...,x ), j = 1,...,N, x ≥ 0.
i 1 i N i 1 i N i
The existence and uniqueness of a Nash equilibrium (for N ≥ 2) was proved in [30]. The
proof was simplified in [9], see also [10] for an exposition.
It is easy to see that x has at least two positive components. Furthermore, for such x
the functions x 7→ H (x ,...,x ,...,x ) are (strictly) concave. An elementary analysis of
i i 1 i N
the correspondent one-dimensional problems shows that x is characterized by the following
relations
s−x M
ϕ′(x ) = M i, if ϕ′(0) < , (2.7)
i i s2 i s
M
x = 0, if ϕ′(0) ≥ , (2.8)
i i s
N
s = x . (2.9)
j
j=1
X
Following [30], note that for s > 0 the relations
s2ϕ′(z ) = M(s−z ), sϕ′(0) < M, (2.10)
i i i i
z (s) = 0, sϕ′(0) ≥ M
i i
uniquely define continuous functions z (s). Clearly, x is a Nash equilibrium iff x = z(s),
i
where s is a solution of the equation
N
z (s) = s, s > 0. (2.11)
i
i=1
X
ASYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 5
Following [9, 10] let us pass from the replacement functions z to the share functions
i
σ (s) = z (s)/s. The existence and uniqueness of a Nash equilibrium follow from (2.11) in
i i
view of the properties of the share functions (see [10, Proposition 2]): σ are continuous,
i
strictly decreasing where positive,
limσ (s) = 1, lim σ (s) = 0.
i i
s→0 s→∞
So, the equation
N
σ (s) = 1, s > 0, (2.12)
i
i=1
X
which is equivalent to (2.11), has a unique solution.
Remark 1. Introducing the game (2.6), we followed the reasoning of [19]. In their model
(inspired by [20]), the users share a communication link of some given capacity. The link
manager gets payments (bids) from the users and allocates the rates according to the an-
nounced price. The manager adjusts the price in order to allocate the entire link capacity.
If the users are price takers, then the model is referred to as a competitive equilibrium. If
they are price-anticipating, then they are involved in a game, and it is assumed that their
bids correspond to a Nash equilibrium.
The reward functions (2.3), in fact, only tell the agents the production levels x∗ − ε ,
i i
specified for them by the manager. This normative scheme is quite sensible to individual
costfunctionsϕ . Moreover, itdoesnotannounceanycommoncompensationrules. Allthese
i
drawbacks force to seek for more reliable compensation schemes. The following discussion
shows that this task is not trivial.
Weseethat(2.2)isasolvableconvex optimizationproblem, satisfying theSlatercondition.
Hence, x∗ ≥ 0 is an optimal solution of (2.2) iff there exists λ∗ ≥ 0 such that
λ∗ϕ′(x∗) = 1, if x∗ > 0; λ∗ϕ′(0) ≥ 1, if x∗ = 0; (2.13)
j j j j j
N N
λ∗ ϕ (x∗)−M = 0, ϕ (x∗) ≤ M.
i i i i
!
i=1 i=1
X X
Equivalently, x∗ ≥ 0 is an optimal solution of (2.2) iff there exists λ∗ > 0 such that (2.13)
and the equality
N
ϕ (x∗) = M (2.14)
i i
i=1
X
hold true. Furthermore, for given λ∗ > 0 a point x∗ ≥ 0 satisfies (2.13) iff each x∗ is an
i
optimal solution of the problem
x /λ∗ −ϕ (x ) → max (2.15)
i i i
xi≥0
similar to (2.4). Assume for a moment that ϕ are strictly convex. Then (2.14) implies that
i
x∗ is unique and λ∗ is also uniquely defined by (2.13), since at least one component of x∗ is
positive.
It is tempting to try ψ (x ) = x /λ∗ for the role of reward functions. Indeed, by (2.15),
i i i
they stimulate optimal production levels x∗. However, in contrast to the piece rate scheme,
i
6 DMITRYB. ROKHLINANDANATOLYUSOV
ψ (x ) = x /λ∗ are not legal reward functions, since x∗/λ∗ −ϕ (x∗) > 0 for x∗ > 0, and the
i i i i i i i
total reward exceeds the budget M:
N N N
1
ψ (x∗) = x∗ > ϕ (x∗) = M.
i i λ∗ i i i
i=1 i=1 i=1
X X X
Note, that the substitution x = ϕ−1(y ) reduces (2.2) to the following equivalent problem:
i i i
N N
sup U (y ) : y ≤ M, y ≥ 0 , (2.16)
i i i
( )
i=1 i=1
X X
where U (y ) = ϕ−1(y ) are strictly increasing concave functions. This is a customary non-
i i i i
linear resource allocation problem: see, e.g., [28]. If the functions U are strictly concave,
i
then, similarly to the above discussion, there is a unique pair (y∗,µ∗) with y∗ ≥ 0, µ∗ > 0,
satisfying the optimality conditions
N
U′(y∗) = µ∗, if y∗ > 0; U′(0) ≤ µ∗, if y∗ = 0; y∗ = M.
i i i i i i
i=1
X
It follows that the unique optimal solution y∗ of (2.16) can be recovered from the one-
dimensional optimization problems
U (y )−µ∗y → max. (2.17)
i i i
yi≥0
Thus, by selling the resource at price µ∗ (per unit), the manager can stimulate the optimal
plan y∗. But in the present context y = ϕ (x ) correspond to production costs, so the
i i i
optimization problems (2.17) make no economic sense.
Remark 2. Closing this section, we will show that any contest scheme cannot produce
better result than (2.2). Consider a non-cooperative game between N agents with the payoff
functions
H (x) = Ψ (x ,...,x )−ϕ (x), x ≥ 0,
i i 1 N i
whereΨ ≥ 0istherewardofi-thagent, andΨ (0) = 0. Letarandomvector(ξ ,...,ξ ) ≥ 0
i i 1 N
be a Nash equilibrium (in mixed strategies):
E(Ψ (ξ ,...,ξ )−ϕ (ξ )) ≥ E(Ψ (ξ ,...,ξ ,...,ξ )−ϕ (ξ )), ξ ≥ 0.
i 1 n i i i 1 i n i i i
We implicitly assume that all expectations exist. Putting ξ = 0, we infer that
i
E(Ψ (ξ ,...,ξ )−ϕ (ξ )) ≥ 0.
i 1 n i i
If the total reward on average does not exceed M: N EΨ (ξ ,...,ξ ) ≤ M, then
i=1 i 1 n
N P
Eϕ (ξ ) ≤ M.
i i
i=1
X
A fortiori, N ϕ (Eξ ) ≤ M by the Jensen inequality, and from the definition (2.2) of s∗ it
i=1 i i
follows that
P
N
Eξ ≤ s∗.
i
i=1
X
ASYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 7
This negative result is by no means an indication that contest compensation schemes are
useless. The point is that the organization of a contest may not require the knowledge of
production cost functions ϕ .
i
3. The prices of anarchy in case of a large number of producers
In this paper we are interested in the behaviour of the following two versions of the “price
of anarchy”:
A = s∗/s, A′ = sˆ/s
N N
for a large number N of agents. Recall, that the quantities s∗, sˆ, s describe three types of
the aggregate production:
(i) s∗ corresponds to an optimal cooperative strategy, enforced by the reward functions
(2.3): see (2.2) (the normative compensation scheme);
(ii) sˆ is related to the case of “reward-taking” agents: see (2.4), (2.5), where the “best”
common linear reward function is announced by the manager (the linear piece rates
compensation scheme);
(iii) s corresponds to the Nash equilibrium of the game (2.6) for “reward-anticipating”
agents (the proportional compensation scheme).
We will refer to the related problems as (i), (ii) and (iii).
The case of identical cost functions is considered in the following theorem.
Theorem 1. Assume that ϕ = ϕ and
i
ϕ(y) ∼ cyα, ϕ′(y) ∼ αcyα−1, y → +0; c > 0, α ≥ 1. (3.1)
Then
lim A = α1/α, lim A′ = 1.
N N
N→∞ N→∞
Proof. (i) As it was mentioned in Section 2, a vector x∗ is an optimal solution of (2.2) iff it
satisfies (2.14), and there exists λ∗ ≥ 0, satisfying (2.13). It is natural to seek a solution of
the (2.13), (2.14) in the symmetric form: x∗ = y∗ > 0, i = 1,...,N. We have
i
λ∗ϕ′(y∗) = 1, λ∗ ≥ 0; Nϕ(y∗) = M. (3.2)
Clearly, such a pair (y∗,λ∗) exists. From the second equality (3.2) it follows that y∗ → 0,
N → ∞, and using the first condition (3.1), we get
1/α 1/α
M M
y∗ ∼ , s∗ = Ny∗ ∼ N(α−1)/α, N → ∞. (3.3)
cN c
(cid:18) (cid:19) (cid:18) (cid:19)
(ii) From (2.4), (2.5) we see that xˆ are identical: xˆ (µ) = yˆ, and
i i
ϕ′(yˆ) = µ, Nµyˆ= M. (3.4)
Using (3.4) and (3.1), we conclude that yˆ→ 0, N → ∞ and
1/α
M M
µyˆ= = yˆϕ′(yˆ) ∼ αcyˆα; yˆ∼ ,
N αcN
(cid:18) (cid:19)
1/α
M
sˆ= Nyˆ∼ N(α−1)/α, N → ∞. (3.5)
αc
(cid:18) (cid:19)
8 DMITRYB. ROKHLINANDANATOLYUSOV
(iii) We look for a symmetric Nash equilibrium of (2.6): x = y > 0, i = 1,...,N. From
i
(2.7), (2.9) we get
N −1
ϕ′(y) = M .
N2y
Hence, y → 0, N → ∞ and
1/α
N −1 M
M = yϕ′(y) ∼ αcyα; y ∼ ,
N2 αcN
(cid:18) (cid:19)
1/α
M
s = Ny ∼ N(α−1)/α, N → ∞. (3.6)
cα
(cid:18) (cid:19)
(cid:3)
The assertion of the theorem follows from the asymptotic forms (3.3), (3.5), (3.6).
Assume that the agents use the same technology, but obtain resources at different prices.
This situation is natural if the firm has departments in various locations. In this case the
resource prices may depend on the quality of transportation network, the cost of labour,
etc., in a concrete location. Denote by (r ,...,r ) the resource amounts (inputs), and
1 m
by (pi,...,pi ) their prices in i-th location. For the production function F(r ,...,r ) the
1 m 1 m
production cost function is defined by
m
ϕ (x) = inf pir : F(r ,...,r ) ≥ x, r ≥ 0 .
i j j 1 m
( )
j=1
X
For the Cobb-Douglas production function F(r) = A m rβj, A > 0, β > 0 by the
j=1 j j
Lagrange duality (see, e.g., [1]) we have:
Q
m m
ϕ (x) = inf pir : β lnr ≥ ln(x/A), r ≥ 0 = supθ (λ),
i j j j j i
( ) λ≥0
j=1 j=1
X X
where ln0 = −∞ and
m m
θ (λ) = inf pir +λ ln(x/A)− β lnr
i j j j j
r≥0( !)
j=1 j=1
X X
m
= λln(x/A)+ inf {pir −λβ lnr }
j j j j
rj≥0
j=1
X
m
λβ
j
= λln(x/A)+ λβ −λβ ln .
j j pi
j=1 (cid:18) j (cid:19)
X
An elementary calculation shows that
1 1 m pi βj α
ϕ (x) = supθ (λ) = c xα, α = , c = j .
i i i m β i αAα β
λ≥0 j=1 j j=1(cid:18) j(cid:19) !
Y
Similarly, for the generalized CES produPction function (see, e.g., [4, 31]):
m γ/ρ
F(r) = A aρrρ , A,a ,γ > 0, ρ ∈ (0,1)
j j j
!
j=1
X
ASYMPTOTIC EFFICIENCY OF THE PROPORTIONAL COMPENSATION SCHEME 9
we have
m m
x ρ/γ
ϕ (x) = inf pir : aρrρ ≥ , r ≥ 0 = supθ (λ),
i j j j j A i
( ) λ≥0
Xj=1 Xj=1 (cid:16) (cid:17)
where
m m
x ρ/γ
θ (λ) = inf pir +λ − aρrρ
i r≥0( j j A j j!)
Xj=1 (cid:16) (cid:17) Xj=1
m
x ρ/γ
= λ + inf {pir −λaρrρ}
A rj≥0 j j j j
(cid:16) (cid:17) Xj=1
m ρ/(1−ρ)
x ρ/γ a ρ
= λ −(1−ρ) j λ1/(1−ρ).
A pi
(cid:16) (cid:17) Xj=1 (cid:18) j (cid:19)
Maximizing this expression over λ ≥ 0, we get
m ρ/(1−ρ) −(1−ρ)/ρ
1 a
ϕ (x) = c x1/γ, c = j .
i i i A1/γ pi
j=1 (cid:18) j(cid:19) !
X
Thus, the Cobb-Douglas and generalized CES production functions with decreasing re-
turns to scale (that is, with m β ≤ 1 and γ ≤ 1 respectively) correspond to power cost
j=1 j
functions: ϕ (x) = c xα, α ≥ 1. Certainly, this fact is known (see [11, Chapter 5]), and we
i i
P
only recalled it here.
Nowwe have enougheconomic motivationtoconsider a model, representing heterogeneous
agentsbypowercostfunctionswithcommonexponent anddifferent multiplicationconstants.
Theorem 2. Assume that ϕ (x) = c xα, α > 1, c > 0 and
i i i
N 1/(α−1)
min c
1≤k≤N k
lim = ∞. (3.7)
N→∞ i=1 (cid:18) ci (cid:19)
X
Then
lim A = α1/α, lim A′ = 1. (3.8)
N N
N→∞ N→∞
Proof. (i) By the Lagrange duality the value of the problem (2.2) can be represented as
follows:
N N
s∗ = −inf − x : c xα ≤ M, x ≥ 0 = −supθ(λ),
i i i
( ) λ≥0
i=1 i=1
X X
N N N
θ(λ) = inf − x +λ c xα −M = −Mλ+ inf (−x +λc xα)
i i i i i i
x≥0( !) xi≥0
i=1 i=1 i=1
X X X
α N 1
= −Mλ−Bλ−α−11, B = (α−1) 1 α−1 1 α−1 .
α c
(cid:18) (cid:19) i=1 (cid:18) i(cid:19)
X
Maximizing this expression over λ ≥ 0, we get
(α−1)/α
N
1
s∗ = M1/α . (3.9)
1/(α−1)
c !
i=1 i
X
10 DMITRYB. ROKHLINANDANATOLYUSOV
(ii) The optimization problems (2.4) take the form
µx −c xα → max.
i i i
xi≥0
Substituting their optimal solutions xˆ = (µ/(c α))1/(α−1) in (2.5), we get
i i
(α−1)/α
M
µ = α1/α .
N c−1/(α−1)!
i=1 i
Hence, P
N N
µ 1/(α−1) 1
sˆ= xˆ =
i α c1/(α−1)
Xi=1 (cid:16) (cid:17) Xi=1 i
(α−1)/α
1/α N
M 1
= (3.10)
α c1/(α−1)!
(cid:18) (cid:19) j=1 j
X
Comparing with (3.9), we see that sˆ= s∗/α1/α.
(iii) To analyse the proportional compensation scheme consider the equations (2.10),
(2.11):
χ (s,z ) = s2αc zα−1 −Ms+Mz = 0, (3.11)
i i i i i
N
χ(s,z) = z −s = 0.
i
i=1
X
For
1/(α−1)
M
zˆ(s) = , K ∈ (0,1) (3.12)
i
αc s
(cid:18) i (cid:19)
we have χ (s,zˆ(s)) = Mzˆ(s) > 0 and
i i i
χ (s,Kzˆ(s)) = M(Kα−1 −1)s+MKzˆ(s) < 0, for s > s ,
i i i i
(α−1)/α 1/α
K M
s = .
i 1−Kα−1 αc
(cid:18) (cid:19) (cid:18) i(cid:19)
A function χ is strictly increasing in z . Hence, the solution z (s) of (3.11) satisfies the
i i i
inequalities
Kzˆ(s) < z (s) < zˆ(s), s > s . (3.13)
i i i i
Put
N N N
g(s) = K zˆ(s)−s, χ(s) = z (s)−s, h(s) = zˆ(s)−s.
i i i
i=1 i=1 i=1
X X X
From (3.13) we get
g(s) < χ(s) < h(s), s > max s .
i
1≤i≤N
