Optimal Behaviour and the General Equilibrium

The fact that the model is “computable” means that a numerical solution exists (e.g., Arrow-Debreu, 1954; McKenzie, 1959; Ginsburgh and Keyzer, 1997), and “general equilibrium” refers to simultaneously matching demand and supply on all markets. In the example below, note the difference between a partial and a general equilibrium in the traditional way of analysing a market handed down by the Marshall and Walras schools. Let us suppose a Cobb-Douglas two-sector economy with two commodities Xi two sector inputs L1,K1 (labour and capital sectors) and two sector income Yi, with i = 1,2. Then, the partial equilibrium model is defined by the next optimal program:


Introduction
The fact that the model is "computable" means that a numerical solution exists (e.g., Arrow-Debreu, 1954;McKenzie, 1959;Ginsburgh and Keyzer, 1997), and "general equilibrium" refers to simultaneously matching demand and supply on all markets.
In the example below, note the difference between a partial and a general equilibrium in the traditional way of analysing a market handed down by the Marshall and Walras schools. Let us suppose a Cobb-Douglas two-sector economy with two commodities X i two sector inputs L 1 ,K 1 (labour and capital sectors) and two sector income Y i , with i = 1,2. Then, the partial equilibrium model is defined by the next optimal program: Objective: In the case of a general equilibrium, we need to add an income balance restriction to ensure that all inflows and outflows are balanced.

Introduction
Let us now generalize the above formulation and consider a simple economy with m finite number of producers, n finite number of consumers, r commodities, and let us suppose that the Walras hypotheses are fulfilled. Thus, under these conditions, let us present, below, the behavioural functions of economic representative agents and conditions of market equilibrium.
Producer behaviour. Each producer, j (j = 1..m), is confronted with a set of possibilities of production v j , the general element v j of which is a program of production with dimension r, where outputs have a positive sign and inputs a negative sign. The objective of each producer is to select, for a given price p (p = 1..r), an optimal program of profits pv j .
Consumer behaviour. Each consumer i (i = 1..n) is supposed to have an initial endowment of goods w i (outputs or inputs) that the consumer is ready to exchange against remuneration by the producer.
Thus, the consumer is confronted with X i possibilities of consumption of which the general element is x i with dimension r. The consumer is never saturated in consuming X i and his endowment w i allows him to survive. For a given price p of dimension r, consumer i has the objective of maximizing total utility U i (x i ) under his given budgetary constraints: subject to: ) from a decentralized system.
Market clearance. Excess demand for r goods is not positive: Commodities with supply excess, i.e., free commodities, have price zero while other commodities have a positive price: ) guaranteeing that each of the markets will have realizable equilibrium. This, too, is an equilibrium for a decentralized economy since it guarantees compatibility of consumer and producer behaviours (Equations 5.1 and 5.2). This is a competitive equilibrium. The price from Equation (5.3) is imposed on all actors of the market.

Economic Efficiency Prerequisites for a Pareto Optimum
The purpose of this section is to clarify the connection between the general equilibrium model and the optimum Pareto state. This will allow us in the next chapter to go beyond such an equilibrium and to analyse impact on social welfare. We must then check whether or not the three conditions below are fulfilled. a) Equality of marginal rates of technical substitution for different producers.
Let us limit our generalization to an economy with two goods q 1 and q 2 , and two limited inputs x 1 and x 1 , for two respective producers.
q 1 = f 1 (x 11 , x 12 ), for producer 1, q 2 = f 2 (x 21 , x 22 ), for producer 2, This means that x̅ 1 = x 11 + x 21 and x ̅ 1 = x 12 + x 22 Let us maximize the quantity produced of q 1 under restriction of known quantity q̅ 2 . Using the Lagrange multiplier, we have: Finally, we get: TmST TmST The Pareto criterion having been satisfied, it becomes impossible to increase q 1 without decreasing q 2 and vice versa. b) Marginal rate of substitution of products for different consumers. Let U = f(q 1 , q 2 ) be the total utility of any consumer and let q 1 and q 2 be the quantities consumed of two products. Assuming a constant level of total utility, the next relations follow: As for the first condition, limiting our generalization to two consumers and two products which supply them, then one can pose: U 1 = f 1 (q 11 , q 12 ) and U 2 = f 2 (q 21 , q 22 ) U 1 and U 2 represent levels of utilities for the two consumers. The quantities q 1 , q 2 are, respectively, consumed by consumer one and two. Thus, maximizing the utility of consumer 1 under the restriction of a given quantity of consumer 2 and using the Lagrange multiplier, we obtain: L = f 1 (q 11 , q 12 ) + λ[f 2 (q̅ 1 -q 11, q̅ 2 -q 12 ) -U ̅ 2 and finally: