By G. Viennot

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**Extra info for Algèbres de Lie libres et monoïdes libres : Bases des algèbres de Lie libres et factorisations des monoïdes libres**

**Example text**

M. 7). 8). For simplicity, we shall formulate the corresponding result in a somewhat weakened, but symmetric form. 2. 4. 4), respectively, if and only if x∗ ≥ 0, y ∗ ≥ 0, Ax∗ ≤ b, (y ∗ )T [Ax∗ − b] = 0; T ∗ A y ≥ c, (x∗ )T [AT y ∗ − c] = 0. 10) where z ∗ = (x∗ , y ∗ ) ∈ Rn+m , q= c −b , Q= 0 AT −A 0 . Observe that the matrix Q is skew-symmetric. 3 allow us to explain the economic sense of all the components of the dual problem. 4) also indicates the cost value. e. yi∗ determines the sensitivity of the maximal income with respect to small changes of the i-th factor.

The answer to (Q1) is positive. This result can be strengthened under an additional condition. 7. 12) has a positive solution which is defined uniquely up to scalar multiples. 3. CLOSED INPUT-OUTPUT MODEL 27 Proof. 6 it follows that there exists an element π ≥ 0, π = 0, such that (I − A)π = 0. Suppose that πj > 0 for j ∈ J and πj = 0 for j ∈ J ′ and J ′ = ∅. Then J ∩ J ′ = ∅, J ∪ J ′ = {1, . . 12) gives n aij πj = πi = 0 for every i ∈ J ′ . aij πj = j=1 j∈J Since πj > 0 for j ∈ J, it follows that aij = 0 for each pair (i, j) ∈ J ′ × J.

We suppose that the production in the (t + 1)-th period uses only the commodities which were produced in the previous t-th period. 3) is the following xt ≥ Axt+1 , pt+1 ≤ AT pt , xt ≥ 0, for t = 0, 1, . . 4) where the starting values x0 and p0 are given. 4) describes an economic system with non-decreasing outputs and non-increasing prices. Clearly, it always has the trivial solution, hence we are interested in finding nontrivial solutions. Observe that non-increasing prices and the absence of profits do not prevent the existence of such a system for an indefinitely long time while the input-output coeﬃcients are constant.