On the Equilibrium Without Loss in the Discrete Time Models of Economic Dynamics

The model of economic dynamics with a fixed budget is considered. The conditions are derived under which the model with a fixed budget has an equilibrium state with the equilibrium prices. The necessary and sufficient conditions for the existence of equilibrium prices are found.


Introduction
Let's consider a model defined at the moment by the productive mapping
Definition. The prices 8 = 8 , … , 8 defined in the equilibrium without loss is called an equilibrium prices without loss.
Note that due to the consequence of the lemma 1 [4] we have ; ̅ • = ∅.
First, consider the consumer problem without loss, and then the equilibrium without loss.
Note that the solution of the − ℎ problem without loss has the following property where J ≥ 0 is some constant that is equal to the value of the − ℎ production function in the point ̅ • . Indeed from the conditions ̅ • = ;, ; ̅ • = ∅ we get that Consider the problem of ℎ consumer without loss. To analyze the problem (1) we apply the necessary and sufficient conditions for the extremum, wherein in the point ̅ the maximum is reached if and only if when As is known [2], Then in our case (without loss) we get that the necessary and sufficient optimality conditions for ̅ of the branch take the forms Lemma 1. The number ] ∈ ; , defined in the lemma 4 [4] in the case of without loss ̅ = ; coincides with the maximal growth rate of the total wealth of the − ℎ branch and is equal to Proof. In the case, when = ; , as follows from lemma 4 [4] the number ] is in the form of (4).
From the other hand let ̅ be a maximum point in the problem (1) of the ℎ consumer that is indeed similar to the following relations Then and only then where ℓ V , iU are defined in the lemma 3 [4] for the case ̅ = ;. The proof follows from the consequence of 1 [4], theorem 1 [4] and lemma 1 Remark. By given ] ∈ ; , the equality (3) may be considered as a system of linear equation with respect to variables-coordinates of equilibrium vector of prices 8 without loss where + • = + , … , + , ℓ V = ℓ • # , … , ℓ • # and in contrary by the given prices 8 from the equality (3) are defined uniquely the maximal growth rate ] of the total wealth of the − ℎ branch ∈ ; . Let's consider the following problem. Let b ≥ 0, + > 0 &, ∈ ; be given. By which ℓ > 0, ] > 0 &, ∈ ; there exists the vector 8 = 8 , … , 8 , that for some : , … , : , 3 is an equilibrium prices without loss in the model 1, defined by the set 2ℓ, : , … , : , 3 7.
Note that it follows from the lemma 3 [4] that in the case of lossless ̅ = ; subdifferential iU takes the form [5,6] It can be shown that for each set ] > 0, … , ] > 0 there are weights : , … , : such that the growth rate in the model defined by the weights : , … , : , coincides up to those cells in which : = 0 and the growth rate does not depend on the choice of the equilibrium price 8.
the inequality below is valid Proof. It follows from (7) that Denote the set of the form h X ℓ V + iU by Φ , that considering lemma 3 [4] (in the case ̅ = ;) indeed has a form Since for all ∈ ; due the conditions of the lemma3 [4] (6) is fulfilled we have This system may be written as a system of inequalities Let's introduce the denotations where v is & − ℎ ort in the space .
Rewriting the system (10) in new denotation considering (11) we obtain Then, by the theorem of [7] for the compatibility of the system (12) is necessary and sufficient that for In our case necessary and sufficient conditions have a form . Then it follows from (13) that q ' are such that Substituting q ' -∈ ; into (13) we get (8) where • is Note that where |•| is a determinant of the matrix.
Expanding the determinant of the matrix † ' over the element of the -− ℎ column (t refers the column ‡), we get here • ƒ ' is − 1 × − 1 matrix obtained from the matrix • by removing -− ℎ column and ? − ℎ row.
The proof immediately follows from the lemmas 2 and 3. Note 1. Note that the condition (14), that is necessary and sufficient condition of existence equilibrium prices without loss does not depend on the vector of distributed resources 3. where |•| is a determinant of the 2 × 2 matrix •; b) by = 3 conditions (14) turn to: and vector • -' : Proposition 2. The numbers ] ' > 0 -∈ ; , satisfying (14), exist if and only if when there exists the index Ÿ ∈ ; and ae such that Thus by this way the system of = inequalities (24) is reduced to the system of superlinear inequalities.

Results
In the paper the following results are obtained: -The necessary and sufficient conditions are derived for the optimality of the branch trajectories; -The maximal growth rate is defined for the branches in the without loss case; -The necessary and sufficient condition is derived for the existence of the equilibrium prices without loss; -The form of the superdifferential is given for the utility function of the consumer; -The conditions are defined for the reducing the system of = linear inequalities to the system of superlinear inequalities of the same variables.