Optimization Analysis for Innovative Inputs under the Objective Discrepancy between Government and Enterprise

. Aiming at the improvement of innovation efciency after enterprise obtaining subsidies, this paper constructs two-stage innovation beneft model about research and development (R&D) and transformation and achieves Nash equilibrium of innovative inputs to solve the objective discrepancy of innovation between government and enterprise. Te main conclusions are as follows: there are three kinds of resource allocation structure in the way of achieving Nash equilibrium. Te allocation structure is determined by the sensitivity of benefts (diferentiated by social benefts and enterprise benefts) to R&D and transformation. After obtaining subsidies, enterprise optimizes resource allocation and results in crowding out efect, which is the inevitable choice for enterprise to seek benefts. Relative to the enterprise budget, when the proportion of government subsidies is few, the way of subsidies does not afect benefts. When the government invests more subsidies, which are designated for R&D, there is the possibility of dual losses of social benefts and enterprise benefts. Te conclusion defnes the proportion of subsidies to enterprise budgets so as to diferentiate the allocation structure of innovative inputs. Te practical signifcance is to provide a precise method of resource allocation from the microlevel of enterprise project, which alleviates the objective discrepancy between the government and enterprise.


Introduction
Industrialized countries have tried to foster innovation for many years. Guiding enterprises innovation through fscal and tax policies is an important means to seek leading position in innovation and technological advantages [1]. Governments therefore set up public support programs such as R&D subsidies, subsidized loans, and R&D tax credits. Input-driven policy is typically designed to support R&D and contribute to the enhancement of innovation competences in individual frms [2] because they drive technological change by bringing radically new innovations to market [3]. Under such background, China has developed a set of national innovation policies, such as the Outline of the National Medium and Long term Science and Technology Development Plan, the National High tech Research and Development Plan (863 Plan), and the National Key Basic Research and Development Plan (973 Plan). In America, the Small Business Innovation Research (SBIR) was established under the Small Business Innovation Development Act of 1982, which was a set-aside program for domestic small business to engage in R&D activities. All across Europe, scientifc institutions and innovative companies face a challenging new research framework: HORI-ZON 2020, which has led to a strong focus on the implementation of research and innovation within European societies. Summarizing the policies of these countries or regions, it can be seen that the government's guidance on innovation attaches importance to R&D activities. From innovation project application, fund expenditure, project evaluation, and so on, it emphasizes "focusing on supporting strategic, cutting-edge, and forward-looking high-tech R&D activities related to long-term national development and national security, and preventing decentralized use." At the same time, there exists a danger of these large public investments descending into a "subsidy race," and as a result, science and technology ecosystems could be damaged irreparably if enterprises are attracted by generous subsidies ofered elsewhere and uproot their operations [4].
Because the innovation targets of the government and enterprises are not completely consistent, the enterprises innovation under the guidance of the government also has market failure. From the perspective of the government, the targets are to promote economic development and realize social benefts. Scientifc and technological progresses are the decisive factors to ensure sustainable economic growth; therefore, subsidies will be designated for R&D to form knowledge accumulation and technology upgrading. Different from government's concern about social benefts, enterprises are the main body of innovative activities, with the target of maximizing their own benefts. Te innovative activities of enterprises are to obtain benefts through market. In order to maximize the enterprises benefts, resources are invested in R&D, transformation, and other businesses in a balanced manner, rather than a single R&D. Such discrepancy results in less R&D investment from enterprises, while more R&D investment from government subsidies leads to imbalance in the allocation of innovation resource.
For solving the mismatched allocation of innovative inputs between government and enterprises, it is necessary to regulate the government guidance, balance social benefts, and enterprises benefts and especially give consideration to enterprises benefts. In China, where the local environment for innovation is vastly heterogeneous, it is important for the government to consider diferences, ensuring policies differentiation [5]. How to implement precise subsidies is an urgent problem. At the microlevel of enterprises innovation, this paper analyzes the allocation of innovative inputs to improve the subsidies efciency. Te main works are as follows: (1) based on the existing research results, it constructs two-stage innovation beneft model about R&D and transformation. Under the condition of limited innovation resources, it allocates innovative inputs between R&D and transformation according to the principle of equal marginal revenue. (2) Since innovation has invested from a single enterprises input to a double inputs of government and enterprises, the government pays attention to social benefts, while the enterprises care about their own benefts. Te discrepancy between government and enterprises makes the allocation of innovation input inconsistent. Based on the Static Game of Complete Information Teory, the "iterated elimination of strictly dominated strategy" method is adopted to achieve the Nash equilibrium of innovative inputs between the government and enterprises. (3) We use computer program to simulate the allocation of innovative inputs. From the perspectives of social benefts and enterprises benefts, the simulation compares with the two popular government subsidy modes and analyzes the applicable conditions and the advantages of achieving Nash equilibrium of innovative inputs. Tis paper proposes an allocation scheme of innovative inputs at the microlevel to eliminate the hidden trouble of innovative resource mismatch before the implementation of enterprises R&D activities. Te innovative inputs to achieve Nash equilibrium are a precise government subsidy mode, which conducts the resource allocation combined with government guidance and market-oriented operation.

Literature Review
Under diferent economic backgrounds, scholars have formed theoretical basis for the enterprises innovation with government guidance, such as the Teory of Endogenous Growth and National System of Innovation. Te Teory of Endogenous Growth was gradually put forward in the 1980s to describe macroeconomic development. Romer, one of the representative scholars, introduced four elements including capital, labor force, human capital, and knowledge into the endogenous economic growth model, in which knowledge is the driving force of economic growth [6]. Schultz added the variable of human capital in Cobb Douglas production function and observed that human capital not only afected individual productivity but also improved the productivity of society [7]. Te Teory of Endogenous Growth and subsequent empirical studies showed that technological progress was the endogenous factor of sustainable economic growth, and R&D activities were the source of technological progress. However, there was market failure in the R&D activities led by enterprises, and the national economic development needs to be guided by government policies. Under such backgrounds, on the basis of inheriting innovation theory, National System of Innovation, represented by Christopher Freeman and Richard R. Nelson, proposed that technological innovation was not only enterprises activities but also be driven by the national innovation system, which afected the allocation and efciency of innovation resources [8,9].
With the above theoretical backgrounds, this paper involves two topics including the policies efect of government subsidies and the allocation of innovation resources. Te two topics are so interrelated that subsidy policies regulate the allocation of innovation resources, which refects the efectiveness of subsidy policies. On the one hand, the policies efects of government subsidies on innovation are controversial, with both positive and negative efects. Based on the fact that negative efects occur frequently, it is necessary to re-examine subsidy policies from the perspective of innovative inputs. On the other hand, innovative inputs belong to the allocation of resources, involving the R&D capital, scientifc and technological talents, and other elements. At the same time, the allocation of innovation resources has its particularity. Te high cost, high risk, and the spillover of R&D lead to market failure, so the R&D investment by enterprises is low. Subsidies, as innovative inputs other than enterprises, encourage enterprises to make innovation decisions. It is necessary to review the allocation of innovation resources under government subsidies.

Negative Efects Caused by Government Subsidies.
According to the innovation performance under the government subsidies, the efect of subsidies is controversial in existing literature, including three types of conclusions: the incentive performance of promoting innovation, the negative performance of inhibiting innovation, and the mixed performance of "inverted U" [10,11] or "double-edged sword" [12]. In the negative efects, it is found that subsidies lead to mismatch in the allocation of innovation resources.
One consequence of the mismatch is crowding out efect, for enterprises to reduce their own R&D investment after receiving subsidies. Te higher subsidy level has a crowding out efect on the R&D investment, which is largely attributable to the managerial myopia of enterprises [10]. Based on dataset of Chinese electronic manufacturing industry, Empirical evidence confrms that it will inhibit innovation when there are too excessive subsidies [13]. With the increasing of government subsidies, the crowding in efect weakens gradually. Some scholars further revealed the infuencing factors. When considering enterprises ownership, scale, industry and regional factors, and so on, government subsidies have a signifcant negative efect on stateowned enterprises and large enterprises [14]. Moreover, some fndings indicated that the efect of subsidies is highly heterogeneous across diferent industry [15].
If dividing enterprises innovation into substantive innovation and strategic innovation [16], another consequence of the mismatch is strategic R&D behavior. Enterprises R&D are no longer market-oriented but cater to government preferences. Although the number of patent applications and scientifc citation index (SCI) has increased gradually, enterprises R&D emphasize quantity over quality so much, that they ignore the transformation of R&D achievements in the market. Using Korean pharmaceutical industry data, the empirical evidence that the R&D subsidy program stimulated the biotechnology venture frms to expand their new product R&D activities is found to be rather weak [17]. Plagiarism, falsifcation, and fabrication have become a signifcant problem, so this behavior has led to a tendency to seek quick success and instant beneft [18]. If the crowding out subsidies can be applied to other economic felds, it can make the best use of it while strategic R&D is waste of resources. Strategic R&D not only afects economic benefts but also misleads literature conclusions. Some scholars take patents as dependent variable to verify the subsidies efect, but these R&D achievements have not been tested by the market and cannot be converted into social benefts through commercial operation. Te fndings overall are ambivalent, and the existing literature as a whole is subject to the criticism that the investigators envisage is not adequately specifed [19].
Te efect of subsidies on enterprises innovation depends not only on the amount of subsidies but also on the rational allocation of innovation resources. Te above literatures reveal the negative efects caused by resource mismatch. Te reason for the mismatch is the imbalance of innovative inputs under the guidance of the government subsidies. Innovation is associated with uncertainty and risks [20,21], and at the same time, the knowledge generated by R&D activities has the attribute of public goods [22], and such characteristics determine that the allocation of innovation resources is diferent from the allocation of resources in the production feld. Te chronological alignment of innovation includes elementary R&D, the preparatory phase of production, production, marketing, and fnal sale [23], and therefore, resource allocation can not only emphasizes R&D activities. Government subsidies should be tailored to actual situations, combining conditions of enterprises to formulate schemes. It is necessary to implement precise and diferentiated subsidy policies at the microlevel of innovation projects.

Te Allocation of Innovation Resource.
With further understanding of innovation, studies about the allocation of innovation resource have been accumulated. It has been gradually realized that innovation is not only R&D investment but also the allocation, integration, and reuse of innovation resources. Innovation resources performance is an important driver of global competitiveness of nations [8]. By constructing a performance evaluation system for the allocation of innovation resources, it was found that there were obvious disparities in the ability and efciency in various regions [9]. For example, investment in knowledgebased capital (KBC) has implications for innovation and productivity growth; however, the returns to investing in KBC difer signifcantly across countries [24]. On the whole, the efciency of innovation resources among diferent countries, and among diferent regions in a same country, as well as among diferent industries in a same region, shows obvious heterogeneity [25,26]. In China, there are obvious spatial autocorrelation agglomeration characteristics of regional innovation output and R&D inputs [26]. Te performance of innovation resource in industrial enterprises had a diferent spatial distribution, with high in eastern and central region and low in western region [8,27].
In addition to the disparities among diferent regions and industries, another characteristic of innovation resource allocation is low efciency. Although the overall efciency of the innovation resources increased, the scale efciency was relatively low [28]. For example, the empirical research shows that the efciency of innovation resource allocation in the aerospace industry is generally at the lower middle level, and the efciency improvement in recent years is not obvious [29]. One reason for the low efciency is the mismatched allocation, which can lower aggregate total factor productivity (TFP) [30]. Another reason is the low transformation of innovation achievements, for that the optimization of overall efciency is restricted by lower efciency of innovation achievement transformation [31].
Te above literatures show that the allocation of innovation resources is disparate and inefcient among regions. Terefore, the national innovation strategy is not only to provide subsidies but also to improve the efciency of subsidies and rationally allocate innovation resources. However, previous studies of innovation policies show that there is a mismatch between the underlying assumptions of these policies and the reality of how frms involved in innovation operation [32]. Most of the subsidies are distributed to state-owned enterprises, but the state-owned enterprises have signifcantly lower R&D and productive Complexity efciency than the nonstate enterprises [33]. Te phenomenon of inefcient subsidies is not unique to China. Using microlevel data from French frms, uniform R&D subsidies can accentuate the dynamic misallocation in the economy by over subsidizing applied research activities [34]. For the subsidy, allocation mechanisms in Germany are not associated with relatively higher innovative performance of the subsidized young highly innovative enterprises [35].
Te above literatures have made some analyses on the allocation of innovation resources, providing three contributions. First of all, it objectively measured the efciency of innovation under government subsidies and clarifed the current problems of innovation development. Secondly, it analyzed the infuencing factors and discussed the mechanism on the allocation of innovation resources. Finally, it pointed out the development diferences between regions and industries and put forward improvement measures and directions. Based on these researches, there are still some aspects that should be improved. By collecting enterprises innovation data, some scholars use data envelopment analysis (DEA), stochastic Frontier approach (SFA), or spatial econometric model to conduct empirical researches. Te conclusions were limited due to diferent industries, regions, data time spans, models, and research methods. Moreover, these studies focused on the efciency of innovation resources at the macro level, rather than the allocation structure at the microlevel. Efciency refects the relationship between input and output, while allocation structure focuses on the quantity and collocation relationship among innovative inputs. Particularly, the innovation of enterprises under government subsidies is dual inputs. Te government pays attention to social benefts, and the enterprises only care about their own benefts. Te discrepancy in objectives leads to diferent inputs. Terefore, it is necessary to analyze the equilibrium of innovative inputs between government and enterprises from the microlevel of innovation projects.

Teoretical Basis.
Innovation is a process of converting creative ideas into products, services, or other business operations. Innovation is not an isolated activity but rather an entire process or even sequence of processes. With true innovation, every partial process should be successfully completed [36]. As a process, it should attend to the way in which innovation has been organized so that benefts can come to fruition; this includes an overall research process and a new product development process [37]. Joseph A Schumpeter pointed out that entrepreneurs innovate not just by fguring out how to use inventions but also by introducing new means of production, new products, and new forms of organization [38]. To review the basic concept of innovation, it is mainly to explain that innovation is not a technical concept but an economic concept. Innovation is strictly diferent from R&D, and it is to introduce technology into economic organizations to form new economic capabilities. Te success of innovation must be tested by the market to bring about benefts and achieve sales revenue or proft growth. In the traditional linear model, innovation is represented by a pipeline of sequential processes, which starts at pure scientifc research and ends with commercial applications. Berkhout et al. replace the traditional chain concept to describe the innovation regime by a circle of change including scientifc insights, technological capabilities, product design and manufacturing, and markets [39]. According to the value chain theory, in order to obtain innovative benefts, it is necessary to have R&D investment and corresponding large-scale production, marketing, and other transformational investment. Trough transformational investment, it can put new technologies, methods, and processes into practice to achieve benefts [40,41]. According to the basic concept of innovation and the above related research, this paper divides innovative activities into two stages: R&D and market transformation.
R&D Functions: Griliches was the frst scholar to propose a quantitative model of knowledge production to estimate returns to R&D [42]. Romer, a representative scholar of the New Economic Growth Teory, expressed knowledge production in the form of Cobb Douglas function [6]. Assuming that the enterprises' knowledge stock is A 0 (all the description of parameters is shown in Table 1), the R&D inputs are mainly capital and technological talents, and the capital is used to purchase instruments, equipment, and laboratories. With reference to the above, we defne the R&D function as A � A 0 K x 1 L y 1 + ε. ε is exogenous random variable, which obeys the law of normal distribution N(0, σ 2 ). After R&D stage, the expected value of new knowledge is

Lemma 1.
When the R&D budget and the price of inputs are determined, the inputs quantity of R&D stage are Proof. When the R&D budget and the price of inputs are determined, the objective is to choose K 1 and L 1 , so as to maximize E(A), the problem is described as follows: It can get a unique solution K 1 � (xI e /u(x + y)) L 1 � (yI e /v(x + y)). Substituting

Complexity
Tis completes the proof.
□ Proposition 2. If defning ϕ � (x/u(x + y)) x (y/v(x + y)) y and α � x + y, the expected R&D output from lemma1 can be simplifed as follows: 3.1.1. Transform Function. Te technology and knowledge generated by R&D are transformed into productivity through production and operation activities to realize innovative benefts. Te transformational benefts are expressed as R e � AK s 2 L t 2 . Te new knowledge A generated in the R&D stage becomes the transformational input. Lemma 3. When the transformational budget and the price of inputs are determined, the inputs quantity of transformational stage are K 2 � (sP e /u(s + t)), L 2 � (tP e /w (s + t)), and the transformational benefits is R e � A(s/u (s + t)) s (t/w(s + t)) t P s+t e Te main idea of proof is the same as Lemma 1.

Te Expected Innovative Benefts Function.
R&D and transformation complement each other and work together to generate innovative benefts. Based on the above analysis, the R&D function (2) is substituted into the transformation function equation (2) to get the expected innovative benefts, as shown in the following equation: In the innovative planning stage, enterprises evaluate the innovative benefts and allocate budgets in R&D and transformation, referring to (4) to calculate the quantity of inputs according to the principle of equal marginal benefts.
Proof. When the innovative budgets are determined, the objective is to choose I e and P e , so as to maximize E(R e ), the problem is described as follows: Using It can get a unique solution I e � (αM e /α + β), P e � (βM e /α + β). Tis completes the proof. Modern economic growth shows that R&D activities promote the generation of new technologies and knowledge, and technological progresses and knowledge accumulations are the core of economic growth [43,44]. Te new knowledge generated by R&D has positive externalities through difusion [45], which drives the technological upgrading of the industrial chain. Te knowledge spillover also has an incentive efect on the technological upgrading of the competitive enterprises, making the social benefts of innovation greater than the enterprises benefts [46]. In this paper, the social benefts of innovation are described as (6). Parameter c represents the sensitivity of social benefts to R&D inputs. c > α. □

Innovative Benefts Function under Government
Subsidies. When the government subsidizes enterprises innovation, the social benefts and enterprises benefts of innovation are shown in the following equation: Te objectives of the government and enterprises are not completely consistent. Te government pays attention to social benefts while the enterprises care about their own benefts. Government and enterprises allocate innovative inputs from the perspective of their benefts respectively, so the innovative inputs of both sides are not consistent.

Lemma 6.
From the perspective of social benefts, the optimal R&D input I * g and transformational input P * g are shown in equation (6).
Proof. To maximize social benefts, the innovation optimization is described as follows: s.t. I g + P g � M g , I e + P e � M e .
Solving the above optimization, the government's expected innovative inputs are I g + I e � (c(M g + M e )/c + β) P g + P e � (β(M g + M e )/c + β) . Since the government cannot directly interfere with the enterprise inputs, the optimal R&D input I * g and transformational input P * g are shown in the following equation: Tis completes the proof. □ Lemma 7. From the perspective of enterprises benefts, the optimal R&D input I * e and transformational input P * e are shown in equation (7).
Te main idea of proof is the same as Lemma 6.
From the above (9) and (10), it can be seen that the innovative inputs of government and enterprises are diferent. Te problems faced by the government and enterprises are how to adjust the inputs to maximize their expected benefts, respectively. According to the process of policy issuance and enterprises subsidy application, the following part of the paper uses static game of complete information to solve the problem of the allocation of innovative inputs.

Innovative Inputs Achieving the Nash Equilibrium
According to the subsidies distribution process, the government frst issues subsidy policy, indicating the amount, mode of subsidy, application conditions, and acceptance criteria after completion, etc. According to the subsidy policy requirements, the enterprises shall fll in the application statement, including technical basis, R&D progress, and expected benefts. During the whole process, the information of both sides is so complete, that the government and enterprises understand each other's objectives and budgets. Te objective of the government is to maximize the social benefts, and the enterprises pursue the maximization of its own benefts. In this case, the solution of innovative inputs is consistent with the static game of complete information. Te government and enterprises can reach a Nash equilibrium agreement that they both abide by to achieve a balanced inputs. Based on the Static Game of Complete Information theory, the Nash equilibrium of innovative inputs can be achieved. According to (9), that is, the function of the government's optimal R&D inputs, draw a straight line gg ′ , where the Y-axis is the government's R&D inputs, and the X-axis is the enterprises' R&D inputs. In turn, according to (10), the optimal R&D inputs line of the enterprises is ee ′ . On the Y-axis, the intercept of line gg ′ is larger than that of line ee ′ , as shown in Figure 1. Corresponding to Figure 1, gg ′ and ee ′ in Figure 2 represent the optimal transformational inputs of the government and enterprises, respectively, and the intercept of line ee ′ is larger than that of line gg ′ .
Since subsidies and enterprises budgets are determined, the government and enterprises allocate R&D inputs and transformational inputs to achieve Nash equilibrium, which is divided into three confguration structures.

Complexity
Theorem . When government subsidies and enterprises budgets reach condition (M g /M e ) ≥ (c/β), innovative inputs tend to conform to government preference, and the confguration structure is I g � (c(M g + M e )/c + β), P g � (βM g − cM e /c + β), I e � 0, P e � M e .

Solution Idea. When
According to the R&D inputs line in Figure 1, the above confguration structure can be verifed by the method of "iterated elimination of strictly dominated strategy." (1) Enterprises R&D inputs will not exceed [0, e ′ ], so this information means that for the government, I g ∈ [a, g] is better than I g ∈ [0, a). the government R&D inputs have exceeded the enterprises' optimal R&D inputs (α(M g + M e )/ α + β), so the enterprises choose R&D inputs I e � 0. In this case, the optimal government R&D inputs are point g in Figure 1, that is, I g � c(M g + M e )/c + β.
Trough the method of "iterated elimination of strictly dominated strategy," the fnal R&D inputs portfolio is (I g � c(M g + M e )/c + β, I e � 0). Te government's remaining subsidies are used for transformational inputs P g � M g − I g � (βM g − cM e /c + β), while the enterprises will use all the budgets for transformational inputs P e � M e .
According to the transformational inputs line in Fig From the above analysis, the government provides all R&D inputs and uses the remaining subsidies for transformation. Te enterprises will use all budgets for transformation.
Te sum of R&D inputs is I g + I e � (c(M g + M e )/c + β), and the sum of transformational inputs is P g + P e � (β(M g + M e )/c + β). Innovative inputs tend to conform to government preference, while from the perspective of enterprises, transformational inputs are relatively insufcient. Tis confguration structure is determined by c and β. Te more c is, the more R&D inputs are.
Teorem 8 shows that although the purpose of subsidies is to stimulate R&D, when the proportion of subsidies to enterprises budgets is large, it is inappropriate to use all subsidies for R&D. From the perspective of social benefts, a part of subsidies can be used to transformational inputs, which can achieve balanced resource allocation. When the subsidies remain unchanged, R&D inputs increase with more enterprises budgets. However, if the enterprises continue to increase budgets, which change the proportion of (M g /M e ) ≥ (c/β), it will enter into the second confguration structure.
Theorem 9. When government subsidies and enterprises budgets satisfy condition (α/β) ≤ (M g /M e ) < (c/β), innovative inputs tend to between government preference and enterprises preference, and the confguration structure is I g � M g , P g � 0, I e � 0, P e � M e .

Solution Idea. When
Te subsidies are less than the government's expected R&D inputs but more than the enterprises' expected R&D inputs. Tis situation corresponds to Part B between point e and point g on the Y-axis of Figure 1. From the perspective of government, even if all subsidies are used for R&D, it cannot reach the expected R&D inputs, so the limited subsidies will not be used for transformation. From the perspective of enterprises, the government's R&D inputs have exceeded expectations, and it is the best choice for enterprises to use all budgets for transformation. Figure 1 corresponds to Part B in Figure 2. Te enterprises budgets are less than the expected transformational inputs but more than the expected transformational inputs of government. Te enterprises will use the limited budgets for transformation, and the government will use subsidies for R&D. Te same conclusion can be drawn from the R&D line in Figure 1 and the transformation line in Figure 2.
In this case, the total R&D inputs are I g + I e � M g , the total transformational inputs are P g + P e � M e , and the innovative inputs are between the government preference and the enterprises preference. From the perspective of the government, R&D inputs are relatively less. If the government increases subsidies at this time, it will continue to be used for R&D. From the perspective of enterprises, the transformational inputs are relatively less. If the enterprises increase the budgets, it will continue to be used for transformation. However, if the enterprises continue to increase budgets, which change the proportion of (M g /M e ) < (α/β), it will enter into the third confguration structure. Theorem 10. When government subsidies and enterprises budgets satisfy condition (M g /M e ) < (α/β), innovative inputs tend to conform to enterprises preference, and the confguration structure is I g � M g , P g � 0, I e � (αM e − βM g / α + β), P e � β(M e − βM g /α + β).

Solution Idea.
When (M g /M e ) < (α/β), it is equivalent to M g < (α(M g + M e )/α + β). Subsidies are less than the R&D inputs expected by the government and also less than the R&D inputs expected by enterprises. Tis corresponds to part C between point 0 and point e on the Yaxis of Figure 1. When M g � a, the government's R&D inputs will not exceed point a. At this time, the enterprises' corresponding R&D inputs are point d, that is, I e � (α(M g + M e )/α + β) − M g � (αM e − βM g /α + β), and the enterprises' remaining budgets are used for transformation. Will the government use all the subsidies for R&D inputs? When the enterprises R&D inputs are d, the optimal R&D inputs of the government are point f on the Yaxis. Because the government subsidies are insufcient, even if all the subsidies are used for R&D, it cannot satisfy point f. At this time, the government will not use the limited subsidies for transformational inputs. M g < (α(M g + M e )/ α + β) is equivalent to M e > (β(M g + M e )/α + β). Part C in Figure 1 corresponds to Part C in Figure 2. Enterprises use the remaining budgets for R&D on the premise of satisfying the transformational inputs. Te same conclusion can be drawn from the R&D line in Figure 1 and the transformation line in Figure 2.
In this case, comparing with the enterprises budgets, government subsidies are insufcient. Te government will use all subsidies for R&D. Te enterprises use part of the budgets for R&D to make up for insufcient subsidies and use the other part of the budgets for transformation. Te sum of R&D inputs is I g + I e � M g + (αM e − βM g /α + β) � (α (M g + M e )/α + β), and the sum of transformational inputs is P g + P e � (β(M g + M e )/α + β). Innovative inputs tend to conform to enterprises preference, while from the perspective of government, R&D inputs are relatively insufcient. Tis confguration structure is determined by α and β. Te more α is, the more R&D inputs are.
Te innovative inputs allocation to achieve Nash equilibrium is a precise subsidy policy, which is to allocate resources according to the principle of maximizing social benefts and enterprises benefts. Te confguration structures are oriented by innovative benefts. It not only gives play to the guidance of subsidies on enterprises innovation but also takes in account the independent choice of enterprises.

Numerical Simulation and Discussion
Te above analysis shows that the allocation of innovative inputs needs to adopt the corresponding confguration structure according to the proportion of government subsidies to enterprises budgets. Te computer simulation can visually display innovative inputs. Te simulation program is developed with Java language, and the simulation data is stored in the Access Database. Te innovative benefts have a decreasing scale efect on the inputs [47,48], so the simulation parameter is set as α + β ≤ 1. From the social perspective, referring to the Teory of Endogenous Growth, economic growth comes from scientifc and technological progress [6], so the simulation parameter is set as c + β > 1.

Example of Confguration Structures to Achieve Nash
Equilibrium. Figure 3 shows the confguration structure under diferent enterprises budgets when the subsidies are unchanged. When M e � 0, the government invests in innovation independently and allocates subsidies according to the principle of maximizing social benefts; at this time, I g � (cM g /c + β) and P g � (βM g /c + β). In the simulation, 8 Complexity assuming β is greater than c, so the transformational inputs are slightly larger than the R&D inputs. If the enterprises budgets are in the part A, and that is, (M g /M e ) < (c/β), the confguration structure complies with Teorem 8. In this range, as the enterprises budgets increase, the enterprises will use all the budgets for transformation, and the government will gradually decrease the transformational inputs for increasing R&D inputs. For each additional unit of the enterprises budget, R&D inputs increase by (c/β + c) and transformational inputs increase by (β/α + β). If the enterprises budgets are increased to part B, it is equivalent to (α/β) ≤ (M g /M e ) < (c/β), and the confguration structure complies with Teorem 9. Figure 3 assumes that the government subsidies are unchanged, so at this range, all subsidies are used for R&D, and R&D inputs present a straight line. Every additional unit of the enterprises budgets are used for transformation. If the enterprises budgets are increased to part C, it is equivalent to (M g /M e ) < (α/β), and the confguration structure conforms to Teorem 10. All subsidies are used for R&D, and the enterprises budgets are invested in R&D and transformation in proportion. For each additional unit of the enterprises budget, R&D inputs increase by (α/α + β), and transformational inputs increase by (β/α + β). Figure 4 shows the confguration structure under different subsidies when the enterprises budgets are unchanged. Te confguration structure of part A in Figures 1-4 is the same, so are part B and part C. Figure 4 shows the crowding out efect of subsidies on enterprises R&D inputs. When there is no subsidy, i.e., the X-axis origin, the enterprises allocate the budgets according to the principle of maximizing their own benefts I e � (αM g α+ β/P e ), � (βM g /α + β). If the subsidies are within part C, it is equivalent to (M g /M e ) < (α/β). For each additional unit of subsidy, the enterprises' R&D inputs will decrease by (β/α + β), and the reduced R&D inputs will be used for transformation. If the subsidies are increased to (M g /M e ) < (α/β), that is, part B and part A in Figure 4, the R&D inputs of enterprises are all crowed out for transformation.

Comparative Analysis of Tree Subsidies Modes.
Te confguration structure to achieve Nash equilibrium is derived from theoretical analysis. Whether it is feasible in practice needs further analysis. Tere are two allocating modes in the existing government subsidies. One is that the government designates subsidies for R&D inputs, such as the "Technological Innovation Fund for Small and Medium sized Technological Enterprises," which stipulates that subsidies are used for research, development, and pilot scale test of technological innovation. Te other is that the government does not designate the subsidies for R&D or transformation but depending on the enterprises to fll in the budget according to the subsidies policy. Some special innovation projects adopt this mode such as "intelligent manufacturing integrated standardization and new application project." Although the government does not designate the use of subsidies, it will formulate evaluation criteria in advance and organize professional institutions to audit that subsidies are used for project implementation. Terefore, the computer program simulates three subsidy modes. Te frst is that government led the subsidies for R&D. Te second is that enterprises led the subsidies for R&D or transformation, aiming at maximizing their own benefts according to Lemma (3). Based on the analysis of Teorems 8-10, the third is achieving Nash equilibrium between the government and enterprises to adopt the corresponding confguration structure according to the proportion of government subsidies to enterprises budgets.
In simulation 1, the proportion of government subsidies to enterprises budgets reaches (M g /M e ) < (α/β), and the innovative benefts are shown in Figure 5. In terms of social benefts, achieving Nash equilibrium between government and enterprises is the largest, as shown in part c. At the initial stage on the Y-axis, social benefts under the government led are greater than the enterprises led. However, when subsidies increase to a certain extent along the Y-axis, social benefts under the enterprises led exceed the government led. Te growth rate of social benefts among three modes is    exceed a certain proportion, both social benefts and enterprises benefts are the least. In simulation 2, the proportion of government subsidies to enterprises budgets reaches (α/β) ≤ (M g /M e ) < (c/β), and the innovative benefts are shown in Figure 6. Te benefts under government led and the benefts under achieving Nash equilibrium have the same result, the social benefts are shown in part a, and the enterprises benefts are shown in part c. In accordance with Teorem 9, both modes use subsidies for R&D and enterprises budgets for transformation. Te mode under enterprises led has the least social benefts as shown in part b; however, it has the largest enterprises benefts as shown in part d.
In simulation 3, the proportion of government subsidies to enterprises budgets reaches (M g /M e ) < (α/β). In this case, the innovative benefts under three subsidies modes are the same, so for simplicity, the fgure has been omitted. Because the proportion of government subsidies is few, no matter which mode is adopted, enterprises can adjust innovative inputs to maximize their own benefts.
Comparing the above simulation experiments, when the proportion of government subsidies is relatively few, within (M g /M e ) < (α/β), subsidies mode does not afect the innovative benefts. When the proportion of government subsidies is moderate (α/β) ≤ (M g /M e ) < (c/β), both the innovative inputs under government led and the innovative inputs under achieving Nash equilibrium are benefcial to social benefts. When the proportion of government subsidies is relatively large, within (M g /M e ) ≥ (c/β), innovative inputs under achieving Nash equilibrium are benefcial to social benefts, and innovative inputs under enterprises led are benefcial to enterprises benefts. Under government led, subsidies designated for R&D may cause dual losses of social benefts and enterprises benefts.

Conclusions
With government subsidies, innovation is changed from single enterprises investment to joint investments of government and enterprises. Based on the Static Game of Complete Information theory, the "iterated elimination of strictly dominated strategy" method is adopted to achieve the Nash equilibrium of innovative inputs between the government and enterprises. It is a precise subsidy mode to inhibit mismatch of innovative inputs from the microlevel. Te main conclusions are as follows: (1) Tere are three kinds of resource allocation structure in the way of achieving Nash equilibrium. When the proportion of government subsidies to enterprises budgets reaches (M g /M e ) ≥ (c/β), innovative inputs tend to conform to government preference. When the proportion of subsidies to enterprises budgets is moderate, within (α/β) ≤ (M g /M e ) < (c/β), innovative inputs tend to between government preference and enterprises preference. When the proportion of government subsidies reaches (M g /M e ) < (α/β), innovative inputs tend to conform to enterprises preference. Both the sensitivity of innovative benefts to R&D inputs and the sensitivity of innovative benefts to transformational inputs determine the allocation structure. If the social sensitivity of innovative benefts to R&D inputs is far more than enterprises' sensitivity to R&D inputs, R&D inputs from enterprise are insufcient. It points out the direction for government subsidies. To increase fnancial subsidies for national defense and public infrastructure, innovative projects can meet with necessary R&D inputs and achieve necessary social benefts. (2) Achieving Nash equilibrium in innovative inputs is a precise subsidy mode, which not only realizes the guidance of subsidies on innovation but also takes into account the enterprises benefts. In order to maximize their own benefts, enterprises should allocate innovative inputs according to the principle of equal marginal benefts. If the subsidies are designated for R&D inputs, the enterprises reduce its R&D inputs for other stage of innovation, such crowding out is an inevitable choice for the enterprises to seek their own benefts. Damrich et al. confrmed that the crowding out would occur when government deprives the private sector of the means to exploit new knowledge [49]. From the perspective of resource allocation, this paper shows that additional subsidies used for R&D make the marginal benefts of R&D less than the marginal benefts of transformation. Enterprises reduce their R&D inputs, and the crowding out efect is an inevitable choice for enterprises to seek their own benefts. (3) By comparing three subsidy modes, it makes clear the applicable conditions of precise subsidies to achieve Nash equilibrium. When the proportion of government subsidies to enterprises budget is relatively less, the subsidy mode does not afect innovative benefts. When the proportion of government subsidies to enterprises budget is relatively more, and subsidies are designated for R&D, it may cause dual losses of social benefts and enterprises benefts. Government subsidies reduce enterprises' economic performance despite promoting indigenous innovation in the higher technology industries. If the subsidies are designated for R&D, when the subsidy exceeds a certain proportion, enterprises cannot reach the ideal resource allocation. If the inputs in one stage exceed the reasonable limit and disturb inputs proportion with other stages, it will inevitably distort the resources allocation and reduce the overall innovation efciency.
How to avoid such situation? Prokop et al. advocated that a suitable R&D subsidy mix enables avoiding misallocation [50]. Tis paper shows that rational allocation of innovative inputs can avoid inefciency. By building information infrastructure, including information technology innovation and even digital innovation, government can enhance information transparency to facilitate rational allocation of Complexity innovative inputs. Trough the creation or adoption, and exploitation of an inherently unbounded, valueadding novelty (e.g., product, service, process, or business model) through the incorporation of digital technology [51], digital integration is conducive to improvements innovation performance. It has a signifcant efect on the improvement of strategic fexibility and dynamic capabilities [52]. Regarding the allocation of innovative resources, it promotes the decision-making between government and enterprises to be more scientifc, reasonable, and effective and then improves the allocation of innovation resources.
By achieving the Nash equilibrium of innovative inputs, the government and enterprises can balance each other to maximize their respective benefts, which is of great practical signifcance to improve the efciency of government subsidies and to mobilize the enthusiasm of enterprises. Te policy implication lies in that, in order to improve the effciency of government subsidies, we should attach importance to the market in regulating resources allocation. "Science and technology is the frst productive force," R&D has promoted scientifc and technological progress and knowledge accumulation, and the market has transformed knowledge into productive forces. So the technology and knowledge must be tested by the market to generate benefts in production and operation stages. It is a market behavior to invest resources in R&D, production, and operation stages according to the principle of equal marginal benefts. Excessive R&D inputs and neglect of resource allocation will not efectively promote enterprises innovation.
Based on the classical literatures, this paper constructs two-stage innovation beneft model. But it cannot fully ft the complex reality of innovative process. Moreover, the conclusions drawn from theoretical derivation and numerical simulation need to be further verifed by empirical research. Tese problems will be improved in the follow-up study. Tis paper emphasizes the balanced allocation of innovative resources in R&D and transformation, and the conclusions are not suitable for basic scientifc research, which is theoretical work to obtain the basic principles, not for the purpose of direct market application.

Data Availability
Te data supporting the current study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that there are no conficts of interest.