Table 3 The mathematical for the framework and structure of the model

Category

Subcategory

Mathematical formulas

Production structure

Production function

\({Q}_{i}^{l}={\phi }_{i}^{l}\left[{\sum}_{s}{\delta }_{{is}}^{l}{\left({F}_{{is}}^{l}\right)}^{\frac{{\sigma }_{l}^{l}-1}{{\sigma }_{i}^{l}}}\right]\frac{{\sigma }_{l}^{l}}{{\sigma }_{i}^{l}-1}\), \(i\) represents the country, \(l\) represents the industry, and \(s\) represents the factors of production. \({Q}_{i}^{l}\) is the output of country \(i\) industry\(l\), \({F}_{{is}}^{l}\) is the demand for factor s in production of country \(i\) industry \(l\), \({\phi }_{i}^{l}\) is the production scale parameter of country \(i\) industry \(l\), \({\delta }_{{is}}^{l}\) is country \(i\) industry \(l\) The input share parameter of factor s in production, \({\sigma }_{l}^{l}\) is the elasticity of substitution of input factors in the production of industry \(l\) in country \(i\).

Factor demand

\({F}_{{is}}^{l}={\frac{{Q}_{i}^{l}}{{\phi }_{i}^{l}}\left[{\delta }_{{is}}^{l}+(1-{\delta }_{{is}}^{l}){\left(\frac{{\delta }_{{is}}^{l}\left({\sum}_{s}{w}_{{is}}-{w}_{{is}}\right)}{\left(\begin{array}{c}1-{\delta }_{{is}}^{l}\end{array}\right){w}_{{is}}}\right)}^{\left(\begin{array}{c}1-{\sigma }_{i}^{l}\end{array}\right)}\right]}^{\frac{{\sigma }_{i}^{l}}{1-{\sigma }_{i}^{l}}}\), \({w}_{{is}}\) represents the price of factor \(s\) in country \(i\).

Consumption structure

Utility function

\({U}_{i}\left({x}_{{ij}}^{l}\right)={\sum}_{l}{\sum}_{j}{\left[{{\alpha }_{{ij}}^{l}}^{\frac{1}{{\sigma }_{i}}}{\left({x}_{{ij}}^{l}\right)}^{\frac{{\sigma }_{i}-1}{{\sigma }_{i}}}\right]}^{\frac{{\sigma }_{i}}{{\sigma }_{i}-1}}\), \({x}_{{ij}}^{l}\) represents the consumption share parameter of industry \(l\) in country \(i\) for goods produced in country \(j\). \({\sigma }^{i}\) represents the elasticity of substitution in consumption for different products in country \(i\).

Consumer demand

\({x}_{{ij}}^{l}=\frac{{\alpha }_{{ij}}^{l}\left({\sum}_{s}{w}_{{is}}{\bar{F}}_{{is}}\right)}{{\left(p{c}_{{ij}}^{l}\right)}^{{\sigma }_{{sharedi}}}\left[{\sum}_{j}{\sum}_{l}{{\alpha }_{{ij}}^{l}\left({{pc}}_{{ij}}^{l}\right)}^{\left(1-{\sigma }_{i}\right)}\right]}\), \({\alpha }_{{ij}}^{l}\) represents the share parameter of country j’s product consumed by industry \(l\) in country \(i\) as a proportion of total consumption in country \(i\). \({\sigma }_{i}\) represents the elasticity of substitution in consumption for country \(i\). \(p{c}_{{ij}}^{l}\) represents the consumption price of industry \(l^{\prime}\) s product from country j in country \(i\).

Market clearing conditions

Factor market

\({\sum}_{l}{F}_{{is}}^{l}={\bar{F}}_{{is}}\), \({\bar{F}}_{{is}}\) represents the resource endowment of factor \(s\) in country \(i\). Factor demand = Factor supply.

Product market

\({Q}_{i}^{l}={\sum}_{j}{x}_{{ji}}^{l}\), Sector output = Sector consumption.

Global trade

\({\sum}_{i}{Y}_{i}=0\), \({\sum}_{l}{\sum}_{i}{P}_{i}^{l}{X}_{{ij}}^{l}={\sum}_{l}{\sum}_{i}P{C}_{{ij}}^{l}{M}_{{ij}}^{l},i \, \ne \, j\), Trade balance.

Zero-profit condition

\({p}_{i}^{l}{Q}_{i}^{l}={\sum}_{s}{w}_{{is}}^{l}{F}_{{is}}^{l}\), Total income = Total expenditure.