As the basic component of the national economic construction and development, the production benefits shown by enterprises directly determine the stable operation of the national economy. In recent years, the pace of enterprise economic construction innovation, as it is affected by a variety of factors and restrictions, has been the attention of the whole society. On the basis of understanding the variational calculation of constraint forms, according to the impact of financing constraint on the production efficiency of enterprises, this paper conducts an in-depth discussion on the actual constraint effects by building the corresponding research model and proposes corresponding solutions from the perspective of the development of the new era.

#### Keywords

- constraint form
- variational calculation
- financing
- nonparametric frontier analysis
- parameters
- the weight
- financing policy
- credit monitoring

#### MSC 2010

- 76U05

Based on the analysis of the basic organisational structure of enterprises shown in Figure 1, it can be seen that financing constraints, as the main factor affecting the production efficiency of enterprises, are mainly caused by three aspects: First, information is not symmetrical. Some financial institutions, such as banks, will ask financiers to provide more guaranteed information or interest in order to guarantee their own capital safety, which will not only increase financing costs but also produce new influences on the internal capital allocation of enterprises. Second, the central bank will limit the credit ratio of other banks and financial intermediaries. Generally speaking, state-owned enterprises are more inclined to finance, while the financing level of some small- and medium-sized enterprises is relatively low. Third, at present, China’s financial market development is not perfect, and some small and medium-sized enterprises do not have direct financing channels, so in the practical development of the inevitable phenomenon of financing constraints. The impact path of financing constraints on enterprise production efficiency is mainly shown in Figure 2. Therefore, this paper combined the variational calculation of constraint form to study the constraint effect of production efficiency.

It specifically involves the following points: First, some enterprises produce the same kind of products, so that in the limited market conditions of the non-cooperative game, the ultimate purpose is to maximise their own interests. Second, the market has the law of conservation of energy. Third, the production cost of all enterprises will be affected by the number of products, scientific research investment, factor allocation, pollutant emission and other factors. Fourth, the product price will be limited by the market supply quantity. Fifth, there is a negative correlation between pollution emission loss and research investment.

In this study, _{+} represents a non-negative real number, and Rm refers to a matrix of 1×m, in which all elements are selected values in the range of real numbers.

_{+} _{+}

_{ij}_{i}

At the same time, Qi represents the total amount of products produced by enterprise I, and

According to the analysis of Nash equilibrium definition, if this paper wants to study whether there is a strategic combination (^{*}^{*}^{*}^{*}^{*}

_{i}_{i}_{i}_{i}_{i}_{i}), where

_{1}, _{1}, _{1}, _{1}, _{1}) _{2}, _{2}, _{2}, _{2}, _{2})

_{2}, _{2}, _{2}, _{1}, _{1}), _{1}, _{1}, _{1}, _{1}, _{1}),

Input–output evaluation indicators

Input indicators | Current depreciation of fixed assets | Ten thousand yuan |

Selling expenses | Ten thousand yuan | |

Financial expenses | Ten thousand yuan | |

Overheads | Ten thousand yuan | |

Main business costs | Ten thousand yuan | |

Number of employees | people | |

Output indicators | Total profit | Ten thousand yuan |

Main business income | Ten thousand yuan | |

Gross industrial output | Ten thousand yuan |

According to the various domestic enterprises in recent years production operation analysis, production efficiency the evolution trend of the change trend of adult show continued volatility, and will be affected by the policies of the time and is influenced by social and economic development level and so on are shown in Figure 3, as the evolution of enterprise production efficiency trend chart in recent years, based on the change form constraint variational calculation effect analysis, More valuable information can be found.

Based on the analysis of related concepts of formal variational calculation, it can be seen that if {_{i}_{N}^{N}_{1},..., _{N}_{i}

At this point, the vector fields corresponding to all ^{N}

In particular, under the condition that _{u}_{i}_{g}_{a}_{fga}_{f}_{g}_{[f, g]}, _{g}f_{a}

^{N}

Inner product:

Li derivative: _{a}_{a}d_{a}Q^{q}^{q}_{1}, _{2},...,_{q}_{1},...,_{q}

At this point, Q-shaped infinity is equivalent to 0 and can be regarded as infinity ~ 0. Assuming _{1} ∈ ^{q}_{D}ω_{1}, then we can get

The usual form of the element in Q’(A) is:

If

And we can specify

By constructing the equivalence between the Nash equilibrium model and the problem of a variational inequality, we can get

And let _{1}, _{2},...,_{m}^{*}^{*}^{*}^{*}^{*}_{i}_{i}_{i}_{i}_{i}_{i}

Equivalent to specifying (^{*}^{*}^{*}^{*}
, ^{*}
) ∈

Also, clarify the following requirements for

Then, in _{1}, _{2},...,_{m}_{i}_{i}_{i}_{i}

Model 1 is mainly used to study the effect of financing constraints on the production efficiency, model 2 is mainly used to study the effect of R&D financing constraints on the production efficiency, model 2 is a measure of financing constraints on investment – cash flow sensitivity of the influence on the production efficiency, model 4 is used to evaluate the financing constraint in the act of export influence on production efficiency and model 5 is used to analyse the impact of financing constraints on productivity in employee compensation in Pingliang. Meanwhile, the corresponding control variable set formula is as follows:

Among them, _{1}, _{2}, _{3} and _{1} to _{5}, respectively, represent the influence levels of various control variables on production efficiency. For example, _{1} refers to the impact level of financing constraints on the production efficiency of enterprises in R&D expenditure. If this value is positive, it means that financing constraints will have a negative impact on production efficiency, which requires the market to relax constraints. On the contrary, if the value is negative, the impact of financing constraints on production efficiency is positive. In this case, it is necessary to strengthen the practice of financing constraints so as to improve the actual production efficiency of enterprises. Table 2 shows the indicators and measurement methods to be defined in this study.

Specific indicators and measurement methods

Te | Productivity | DEA model operation |

Fr | Financing constraints | Financing constraint = interest expense/total assets |

Int1 | Financing constraints and R&D expenditure interaction | Interaction term = financing constraint * R&D investment |

Int2 | Interaction item between financing contract and investment behaviour | Interactivity = Financing Constraints = Long-term Investment |

Int3 | Financing constraints and export behaviour interaction item | Interaction term = financing constraint * export delivery value |

Int4 | Interaction term between financing constraints and employee compensation | Interaction term = financing constraint * employee compensation |

Size | The size of the business | Firm size = number of employees take logarithm |

Air | Debt to asset ratio | Asset-liability ratio = total liabilities/total assets |

Den | Asset intensity | Capital intensity = net fixed assets/number of employees |

Sub | Government subsidies | Industrial Enterprise Database Direct Access to |

Tax | Tax burden | Tax burden = tax expenditure/sales |

Assuming that Enterprise 1 and Enterprise 2 only produce the same product, and there is only one demand market, the above analysis results show that the function relationship between the production cost of Enterprise 1 and the total amount of product _{1}, the research fund

The corresponding formula for the functional relationship between various elements of Enterprise 2 is:

According to the analysis of the negative correlation between product price and product quantity, the market selling prices of enterprise 1 and enterprise 2 are set as , _{1}, _{2}) = 5 − _{1} − _{2}while the losses caused by pollutant removal of enterprise 1 and enterprise 2 are, respectively,

The function formula for enterprise 2 to obtain profits is:

Since the four factors of the enterprise studied in this paper have a specified range in the actual development and are not close to infinity, the value range of these variables is assumed to be between 0 and 5, then the constraint set is defined as:

Then, in order to determine whether

By (1), and (6) (7)

Using the variational inequality projection shrinkage algorithm, combined with the MATLAB language software to write the calculation, the final product output of enterprise 1 is

The data selected in this paper are the information stored in the enterprise database from 2018 to 2020, and the selected indicators involve the total assets, R&D expenditure, the number of employees, subsidy income and other contents. It should be noted that in the process of sorting out the relevant data, it must be clearly recognised that there are also problems such as missing or errors in the database reserve information. Therefore, it is necessary to preprocess the relevant data information before the research and detection, and then to conduct an in-depth discussion on the results generated.

Combine in Table 3 shows the financing constraints on production efficiency of regression result analysis shows that the influence of the use of fixed effects regression model and stata software regression study found that financing constraints, the key variables under the condition of 1% of the change is very significant, negative and symbols, proving the financing scale continues to expand the negative effect to enterprise production efficiency. In this process, financing constraints can effectively improve the production efficiency of enterprises, thus verifying Model 1 proposed in the analysis of this paper. However, the output efficiency in a short period of time cannot reach the best, and the output increment should be lower than the input increment, which will inevitably reduce the actual production efficiency. At the same time, due to the excessive investment behaviour of enterprises, the continuous expansion of their production scale, resulting in the actual production efficiency, is not effectively improved. Although the variable of capital intensity presents a positive correlation, it has no significant influence on the composition of production efficiency, and the sign of tax variable is negative, which proves that the increase of tax burden will further reduce the production efficiency of the enterprises.

Analysis of regression results

Fe | −0.1791*** |

(0.0294) | |

Size | −0.0243*** |

(0.0030) | |

Alr | −0.0275*** |

(0.0060) | |

Den | 0.0000*** |

(0.0000) | |

Sub | −0.0056* |

(0.0031) | |

Tax | −0.0080 |

(0.0268) | |

Observations | 223.198 |

R^{2} |
0.5278 |

*, ** and *** indicate significant at the level of 10%, 5% and 1%, respectively. The parentheses indicate robust standard errors.

According to the content and the final result in this paper, analysis shows that although the Nash equilibrium under the enterprise can get optimal value from the perspectives of its own development, the whole industry sees this state is not to maximise value, which requires companies to build good relations of cooperation in the development of the practice, and only in this way, can we truly achieve benefit maximisation. At the same time, enterprise development should also focus on improving the efficiency of factor allocation, so that high-tech enterprises can be concentrated in developed areas, while low-end manufacturing can also be concentrated in backward areas, which plays a positive role in promoting the overall development of the social economy. Therefore, it is necessary to increase research and exploration in this aspect in future development.

In conclusion, under the background of the new era, the production efficiency of enterprises shows a trend of continuous decline. At this time, the level of financing constraint has a profound impact on the production efficiency of enterprises. Therefore, both the national government and the market economy should put forward clear solutions after clarifying the relationship between financing constraints and enterprise production efficiency and cannot directly reduce the degree of financing constraints. At the same time, the corresponding analysis model should be built according to the variational calculation of constraint form, and the degree of influence under different conditions can be obtained accordingly. Then, according to the accumulated experience in the past, the constraints management work in the subsequent development should be analysed. In addition, the enterprise production management needs to foster more high-quality high-level professional talents and reference more modern management ideas and technologies and learn from excellent research literature at home and abroad, in such a way not only to make effective solutions based on different situations but also to further encourage the enterprises to promote the efficiency of production.

#### Specific indicators and measurement methods

Te | Productivity | DEA model operation |

Fr | Financing constraints | Financing constraint = interest expense/total assets |

Int1 | Financing constraints and R&D expenditure interaction | Interaction term = financing constraint * R&D investment |

Int2 | Interaction item between financing contract and investment behaviour | Interactivity = Financing Constraints = Long-term Investment |

Int3 | Financing constraints and export behaviour interaction item | Interaction term = financing constraint * export delivery value |

Int4 | Interaction term between financing constraints and employee compensation | Interaction term = financing constraint * employee compensation |

Size | The size of the business | Firm size = number of employees take logarithm |

Air | Debt to asset ratio | Asset-liability ratio = total liabilities/total assets |

Den | Asset intensity | Capital intensity = net fixed assets/number of employees |

Sub | Government subsidies | Industrial Enterprise Database Direct Access to |

Tax | Tax burden | Tax burden = tax expenditure/sales |

#### Input–output evaluation indicators

Input indicators | Current depreciation of fixed assets | Ten thousand yuan |

Selling expenses | Ten thousand yuan | |

Financial expenses | Ten thousand yuan | |

Overheads | Ten thousand yuan | |

Main business costs | Ten thousand yuan | |

Number of employees | people | |

Output indicators | Total profit | Ten thousand yuan |

Main business income | Ten thousand yuan | |

Gross industrial output | Ten thousand yuan |

#### Analysis of regression results

Fe | −0.1791 |

(0.0294) | |

Size | −0.0243 |

(0.0030) | |

Alr | −0.0275 |

(0.0060) | |

Den | 0.0000 |

(0.0000) | |

Sub | −0.0056 |

(0.0031) | |

Tax | −0.0080 |

(0.0268) | |

Observations | 223.198 |

R^{2} |
0.5278 |

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