Abstract
The purpose of this chapter is to formulate a linear numerical general equilibrium model. The model is essentially a Leontief type of input–output model, extended with resource constraints. In this chapter the equilibrium model is developed and analysed under conditions of competitive market behaviour. To provide the reader with an understanding of the nature of this model and its link to economic theory, the concept of welfare optimum (Pareto efficiency) and its logical relation to competitive equilibrium is used as a connecting thread between the concept of economic equilibrium and the mathematical programming formulation. The following sections will highlight the major features of the model. At the same time, the assumptions necessary to make the model operational are made explicit.
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- 1.
Generally, a commodity is defined by its physical characteristics, its location, and the date of its delivery. Commodities differing in any of these characteristics will be regarded as different. However, in this model a commodity is synonymous with the industry supplying the commodity (sector classification principle).
- 2.
Thus, there is only use of primary commodities, not production of them.
- 3.
See further Debreu G. (1959), p. 42.
- 4.
In mathematical language, the utility function S, is continuous and increasing, twice continuously differentiable, strictly quasi-concave and its first derivatives are not all simultaneously equal to zero.
- 5.
This forms a matrix with capacity input coefficients in its principal diagonal and zero elements everywhere else. Hence, i = j for all c ij .
- 6.
A commodity is desirable if any increase in its consumption, ceteris paribus, increases utility.
- 7.
Koopmans T.C. (1957), p. 84.
- 8.
Kuhn H. W. and A. W. Tucker (1950). The Kuhn-Tucker theorem for con-strained optimisation tells us that the necessary conditions for the solution of the primal are equivalent to finding the solution of the dual. It does not in itself provide us with a practical solution method for the problem.
- 9.
The shadow prices of the model cannot be considered as “ideal”, because this interpretation would be valid only if the specification of the objective function quantitatively embodied all goals of the economy.
- 10.
These prices carry to each producer and each consumer a summary of information about the supply possibilities, resource availabilities and preferences of all other decision makers.
- 11.
Following Jaffe (1980),: “When Walras defined his entrepreneur as a fourth per-son, entirely distinct from the landowner, the worker and the capitalist, whose role it is to lease land from the landowner, hire personal faculties from the labourer, and borrow capital from the capitalist, in order to combine the three productive services in agriculture, industry and trade.” Thus, then he (Walras) said in a state of equilibrium, les entrepreneurs ne font ni bénéfices ni pertes’ (entrepreneurs make neither profit nor loss), he did not mean that there are no returns to capital in state of equilibrium, but only that there is nothing left over for the entrepreneur, qua entre-preneur, when selling price equal all cost of production including the cost of capital-services for payment is made to capitalists. “See further Jaffe W. and Morishima M. (1980).
- 12.
Assuming that each consumer is on his budget constraint, the system as a whole must satisfy Walras’s Law, i.e. the value of market demands must equal the value of market endowments at all prices.
References
Debreu G (1959) Theory of value, Monograph 17. Cowles Foundation. Yale University Press, New Haven/London
Jaffe W, Morishima M (1980) On interpreting Walras. J Econ Lit XVIII:528–558
Koopmans TC (1957) Three essays on the state of economic science. McGraw-Hill, New York
Kuhn HW, Tucker AW (1950) Non-linear programming. In: Neyman J (ed) Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 481–492
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Norén, R. (2013). The Outlook of the Sovereign Planner: The Linear Activity Model. In: Equilibrium Models in an Applied Framework. Lecture Notes in Economics and Mathematical Systems, vol 667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34994-2_2
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