(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 turnpike models (1968, 1979) are shown in Table 2. The two types Tsukui’s are taken into practice. Their objective function is respectively capital accumulation (a time invariance of final year) and balance growth rate (a time independent variance), Von Neumann’s 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. models and their application. Tsukui’ Type of Model pattern Turnpike theorem Algorithm Tsukui model Model of Leontief dynamic Linear programming; capital model (has algorithm and accumulation application) Model of Von Linear programming; Neuman’s balance (has algorithm growth rate application) Model of model Dynamic Remesey’s consumption 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
(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
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