Proposition 1. If the firm performance function is linear, effort has little effect in large firms. Thus it is optimal to implement an interior level of effort, to avoid exerting excessive costs on the operator. So we assume that the function of firm performance outputs is linear, which can be written as follows:
where 
is the outputs of firm performance, 
is effort level of operator, 
is ability level of operator, 
is an is an endogenous variable,
.
Proposition 2. We assume owner is risk neutral, operator is risk aversion. The utility function of operator is constantly absolute risk averse, which the function form is :
where 
is a constant absolute risk aversion variable that satisfies the condition
, and 
is actual income of operator.
Proposition 3. We assume the function of operator effort cost can be written as follows:
where 
is the effort cost coefficient. The bigger 
is, the more disutility taken by the same effort level
. In other words, the bigger 
is, the little disutility taken by the same effort level 
for operator had different ability levels.
Proposition 4. Owner takes material incentive measures and non-material incentive measures simultaneously. In our setting, the cost of incentives is a function of the operator’s wage, but the benefits of effort are a function of firm value, which is substantially greater. Because maximum effort is always optimal, the efficient contract takes a simple form. Since effort has a percentage effect on both firm value and utility, the percentage change in pay for a percentage firm return is the relevant incentive measure, and it must be sufficiently high to induce maximum effort. Translated into real variables, this measure equals the proportion of total salary that is comprised of material and non-material. Thus, the fraction of pay that must be composed of non-material should be constant across operator of different salaries. We assume that the cost of non-material incentive measures is
, where current non-material incentive measures can make the next issue income of operator increase by
,which 
is the elastic coefficient of the non-material incentive inputs, said the material incentive effect of input and output. If the condition 
holds, the owner will simply adopt the material incentive measures which meet condition that the perceived equivalent income by the operator is equal or greater than the monetary inputs of non-material incentive. The function of operator compensation contract is as follows:
where 
is operator fixed income, 
is coefficient of share profits which reflect the operator share degree of outputs, 
is discount coefficient, and 
is the increased monetary of operator current income by the discounted non-material incentive.
Based on above fundamental assumptions, we get the utility function of firm performance outputs as follows:
,
.
The expected utility function of owner is equal to its net earnings because he is risk neutral, which can be written as follows:
The actual income function of operator is operator is risk aversion neutral in considering efforts costs. Specifically, we use the functional form:
,
.
In the meanwhile, operators have to take risks, the actual income satisfy the condition:
.
Consequently, Agent Certainty Equivalence (ACE) can be written as follows:
if 
is operator reservation utility, then participation constraint is
the formula get an equal sign in the optimal principle. Using the first-order conditions, we obtain the operator’s incentive compatibility constraint as
which also can be written as
.
From the above equation, we can see 
increases with
, which means the bigger operator ability is, the higher its optimal effort level is.
Therefore, the compensation contract can be described as
and substituting the constraint condition into the target function
, we get
:
where 
is operator fixed income, 
is coefficient of share profits which reflect the operator share degree of outputs, 
is ability level of operator, 
is the elastic coefficient of the non-material incentive inputs, said the material incentive effect of input and output, 
is discount coefficient, 
is a constant absolute risk aversion variable that satisfies the condition
, 
is the effort cost coefficient, 
is effort level of operator, 
, 
are constant variables.
We generally follow the solution procedures:
First, a calculating the first derivative of the Lagrange function (1), so as to get the first-order condition of 
respect to 
as Equation (2),
The optimality condition then is given by Equation (3)
w.r.t. Equation (3), we get
simplify the Equation (4), we get
w.r.t.
, and solving for
, we get
Second, substituting 
into participation constraint, when the condition 
takes the equal sign, we get
where
w.r.t.
, and solving for
, we get
Finally, substituting 
and 
into the target function 
in formula (*), we get:
(9).
We can get the first-order results of Equation (6) respect to
, 
, 
, 
, 
as follows:
,
,
since the condition 
holds, the following inequality holds,
,
,
.
since the condition 
holds, the following inequality holds,
,
,
.
since the condition 
holds, the following inequality holds,
,
,
.
Other first-order results of Equations (8) and (9) respect to
, 
, 
, 
, 
will be seen in appendix.
, which means
increases with the decrease of
. Put another way, the smaller the input elasticity coefficients is, the bigger utility of non-material incentives to operator, that is to say the owner should set a larger coefficient of sharing profits to operator.
, which means 
. If the discount coefficient is smaller, indicates that the adverse impact of the future environment on the income is larger, the owner should set a larger coefficient of sharing profits to operator too.
, which means 
increases with the increase of
. The result indicates that if the operator's ability is higher, the owner should set a larger coefficient of sharing profits to operator. All other things being equal, operator who had a higher ability appear willing to take more risk.
, which means
decreases with the increase of
. If the degree of operator’s risk aversion is higher, the owner should set a less coefficient of sharing profits.
, which means 
. the result demonstrates that the bigger operator effort cost coefficient is, the less the coefficient of sharing profits is.
, which means
increases with the decrease of
. The result states clearly that the less utility of non-material incentives to operator, the high level of fixed wage should owner set.
, which means 
. A smaller discount coefficient indicates that the adverse impact of the future environment on the income is larger; the owner should set a high level of fixed wage.
, which means 
increases with the increase of
. The result indicates that if the operator's ability is higher, the owner should set a lower level of fixed wage to operator.
, which means
increases with the increase of
. If the degree of operator’s risk aversion is higher, the owner should set a lower level of fixed wage to operator.
, which means 
. The result demonstrates that the bigger operator effort cost coefficient is, the higher level of fixed wage is.
, which means 
increases with the decrease of
. The result states clearly that the more utility of non-material incentives to operator, the high owner net earnings is.
, which means
increases with the increase of
. A smaller discount coefficient indicates that the adverse impact of the future environment on the income is smaller; the owner net earnings should be larger.
, which means
increases with the increase of
. The result indicates that if the operator’s ability is higher, the owner net earning should be larger.
, which means 
increases with the decrease of
. If the degree of operator’s risk aversion is lower, the owner net earning should be larger.
, which means
increases with the decrease of
. The result demonstrates that the smaller operator’s effort cost coefficient is, the larger the owner net earning is.
in the current period, and operator increased income is 
the next period. In that situation, the non-material incentive with the character of saving cost (when the condition 
holds) is not the optimal incentive in the long run. The degree of saving cost is influenced by the ability of guaranteeing income cashed in of the future environment. Only in the case that the discount coefficient is bigger than elastic coefficient (namely
), the non-material incentive can really save costs for the owner, maximize the utility of operator in the certain total incentive costs or minimize the total incentive cost under the condition of the unchanged operator utility.
since,the condition 
holds, the following inequality holds, 
,
.
,
,
,
.
since,
holds, the following inequality holds, 
,
.
,
,
,