industries through capital flow matrix by industry. In virtue of technology limitation, the dynamic input-output model was nowadays merely united in linear programming rather than dynamic programming for making some long-term decisions. Na (2006) and Zhang (2001) made the multi-goal programming on the basis of the dynamic input-output model for economic forecast. All of above models are aimed at maximizing the sum of over the planning term, but simplicity application of linear programming have no consideration the efficient growth theory (i.e“turnpike theorem”); indeed, their objectives and constraints with the time sequence ought to be used the sequential constrained technique in dynamic programming in place of simply linear programming. Using the optimal technology to overcome the unbalance system can be also traced back to the origin of turnpike theorem. The Dorfman, Samuelson, Solow (1958) first combined the linear programming with dynamic input-output model in “Linear Programming and Economic Analysis” and demonstrated that the solution is exactly Leontief trajectory, which is the efficient growth path, so called turnpike. Turnpike theorem is concerned with efficient paths of accumulation of economic expansion. Efficiency is defined with reference to the terminal stocks paths, namely, a path of accumulation lasting N periods is efficient if no other path, starting from the same initial stocks and complying with the technical restrains, reaches larger terminal stocks after N periods. Their concept is contrast to von-Norman (1932), who proofed the existence of a balance, where the price just compensate the cost so as to no profit exist, stimulatingly, the expand of economy activity make the capital stock expanding at maximum rate, meanwhile the optimal model with objective function takes no effect. Samuelson (1949) thought that it is necessary to introduce optimal in the case of economy unbalance. He also first put the concept of turnpike forward from von-Norman model, in which exist a turnpike, and of which the maximum objective function is the scale of capital stock in final term. Put another way, turnpike is the speediest path of capital balance growth, so-called von-Norman equilibrium. Later, Dorfman, Samuelson, Solow (1958) demonstrated the turnpike theorem in the model with two capital goods, and found the efficient capital growth path is consist with Leontief trajectory. The technology advance make the Leontief trajectory became a curve. term
(2) G. Liu (1998, 2009) gives some reverse algorithms of dynamic programming in unconstraint dynamic input-output model, unrelated to turnpike thermo, also another pattern considering price variable. It is shown as follows and has two aspects to improve (3.1) The model can have constraints: the production and investment as dynamic IO model, capacity of resource and environment, and international trade factors. The theoretical foundation of efficient growth path on basis of turnpike theorem determines the reasonable industrial structure of production and investment, and the rational ratio of consumption to output. (3) Turnpike theorem is applied in dynamic input output model by Tusukui & Mutakami (1979) through linear programming with maximizing the capital of terminal year. Three theoretical Tsukui’s turnpike models (1968, 1979) are shown in Table 2. The two types are taken into practice. Their objective function is respectively capital accumulation (a time invariance of final year) and Von Neumann’s balance growth rate (a time independent variance), so these models only require linear programming. The third type is inapplicable by linear programming. The objective function is maximizing accumulated welfare over the planning year, related to time invariance over planning time, so it requires dynamic programming. It will belong to the type model in the following paragraph. TABLE 2. Tsukui’ models and their application. Type of Tsukui model Model pattern Turnpike theorem Algorithm Model of capital accumulation Leontief dynamic model Linear programming; (has algorithm and application) Model of balance growth rate Von Neuman’s Linear programming; (has algorithm application) Model of consumption Remesey’s model Dynamic programming; (no algorithm application) (4) A consumption turnpike model, combined Leontief dynamic input-output model with the dynamic programming, is currently in the blank. Here, a new consumption turnpike, within an environment input-output system, will be developed aiming at accumulated consumptions
generalize algorithm for all kinds of discrete consumption turnpike model. Mixed-Consumption Turnpike (MCT) model is technology advanced dynamic model. The model uses a series of energy-pollutant-economy IO Tables with capital matrix. Comparing to the classic input-output model, it has two distinguish properties. First, it allows change in structure coefficient to embody technology advance. The technology change is embodied in material saving (the input-output coefficient) and capital composition improvement(capital coefficient). Second, it considers the change of energy mixture and corresponding pollutants removal efficiency. Table 4: Comparison of MCT model with other optimal IO models Theorem Method Turnpike theorem Non-Turnpike theorem model objective constraints model objective constraints Static IO model Linear programming - Xia(2010) Minimum Energy consumption IO equations; Constraints of industry structure Fu Xue(2012) maximum GDP in expenditure approach IO equations; Carbon constraints Liu et al. (2009) maximum GDP in income approach (value-added) IO equations; Water use constraints & Minimum cost of water, dual problem Water use constraints; water resource endowment Dynamic programming - Rose (1996) maximum accumulated GDP in expenditure approach IO equations; resource endowment; investment accumulation equation Dynamic IO model Linear programming & Dorfman, et al.[25] Capital in final term; balance growth rate Dynamic IO equations; resource endowment Zhang (2001) maximum accumulated GDP ; minimum pollutants Dynamic IO equations; pollutants constraints; balance growth; final-term condition J. He[1], He at al.[36], He at al.[37] Dynamic IO equations; water resource endowment; final-term condition Na,[7][8] Yu et al.[84] Accumulated value-added Dynamic IO equations 1)