5. The Fisher's model

#### 1∘ : Reduction to optimization problem

A special class of the models is formed by the models with fixed budgets. This is the case when each consumer has all commodities in equal quantities: w<sup>i</sup> <sup>j</sup> ¼ λ<sup>i</sup> for all j∈J, and thus, <sup>p</sup>; <sup>w</sup><sup>i</sup> � � <sup>¼</sup> <sup>λ</sup><sup>i</sup> for all <sup>p</sup> <sup>∈</sup>σ. Such a model is known as the Fisher's model. Note that we have this case in the abovementioned example.

The main feature of these models is the potentiality of the mappings G associated with the arising polyhedral complementarity problems.

Let <sup>f</sup> be the function on <sup>R</sup><sup>n</sup> that f pð Þ for <sup>p</sup><sup>∈</sup> <sup>σ</sup> is the optimal value in the transportation problem of the model, and f pð Þ¼�∞ for p∉ σ. This function is piecewise linear and concave. It is natural to define its subdifferential using the subdifferential of convex function ð Þ �f : ∂f pð Þ¼�∂ð Þ �f ð Þp .

Let G be the mentioned associated mapping.

Theorem 1. The subdifferential of the function f has the representation:

$$\partial f(p) = \{ \ln q + te | q \in G(p), t \in \mathbb{R} \},$$

where e ¼ ð Þ 1;…; 1 and ln q ¼ ln q1;…; ln qn � �. (The addend te in this formula arises because it holds P <sup>j</sup><sup>∈</sup> <sup>J</sup> pj ¼ 1 for p ∈σ.)

Consider the convex function h, defining it as follows:

$$h(p) = \begin{cases} (p, \ln p)\_{\prime} & \text{for } p \in \sigma^\*,\\ 0, & \text{for } p \in \partial \sigma, \\\ \qquad -\infty, & \text{for } p \notin \sigma. \end{cases} \right\}.$$

Introduce the function

$$
\varphi(p) = h(p) - f(p) \tag{4}
$$

Proposition 1. For the Fisher's model, the following formula is valid:

programming:

Eisenberg and Gale [6].

: Algorithms

polyhedron.

Let be f gr ¼ L ∩ M.

function ψð Þq on M.

2∘

Proposition 2. For all p, q∈σ<sup>∘</sup> the inequality

or ψð Þq on the affine hull of the current cell.

other one using the Theorem 1 is quite similar [9].

these cells. We need to obtain the point of their intersection rk.

holds. This inequality turns into equality only if p ¼ q.

f ∗ ð Þ¼� ln <sup>q</sup> <sup>X</sup>

Theorem 3. The fixed point of G is the maximum point of the concave function <sup>ψ</sup>ð Þ<sup>q</sup> on <sup>σ</sup><sup>∘</sup> .

Corollary. φð Þ¼ r ψð Þr if and only if the point r is the fixed point of the mapping G.

Consider a couple of two cells Ω ∈ ω, Ξ∈ξ corresponding to each other.

Let L⊃ Ω, M ⊃Ξ be their affine hulls. It will be shown that L ∩ M is singleton.

i∈ I

For the functions φð Þp and ψð Þq , there is a duality relation as for dual programs of linear

φð Þp ≥ψð Þq

Thus, the equilibrium problem for the Fisher's model is reduced to the optimization one on the price simplex. It should be noted that this reduction is different from well-known one given by

The mentioned theorems allow us to propose two finite algorithms for searching fixed points. Algorithmically, they are based on the ideas of suboptimization [11], which were used for minimization quasiconvex functions on a polyhedron. In considered case, we exploit the fact that the complexes ω and ξ define the cells structure on σ<sup>∘</sup> similarly to the faces structure of a

For implementation of the algorithms, we need to get the optimum point of the function φð Þp

Lemma. The point r is the minimum point of the function φð Þp on L and the maximum point of the

Now, we describe the general scheme of the algorithm [8] that is based on Theorem 2. The

On the current k-step of the process, there is a structure B<sup>k</sup> ∈ B. We consider the cells <sup>Ω</sup><sup>k</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>k</sup> ,Ξ<sup>k</sup> <sup>¼</sup> <sup>Ξ</sup>ð Þ <sup>B</sup><sup>k</sup> and have the point qk <sup>∈</sup>Ξk. Let Lk <sup>⊃</sup> <sup>Ω</sup>k, Mk <sup>⊃</sup>Ξ<sup>k</sup> be the affine hulls of

λ<sup>i</sup> max j∈ J ln ci j qj

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(5)

35

Theorem 2. The fixed point of G coincides with the minimum point of the convex function <sup>φ</sup>ð Þ<sup>p</sup> on <sup>σ</sup><sup>∘</sup> .

Another theorem for the problem can be obtained if we take into account that the mapping G and the inverse mapping G�<sup>1</sup> have the same fixed points. For the introduced concave function f , we can consider the conjugate function:

$$f^\*(y) = \inf\_z \left\{ (y, z) - f(z) \right\}.$$

(see [14]) With this function, we associate the function ψð Þ¼ q f ∗ ð Þ ln <sup>q</sup> , which is defined on <sup>σ</sup><sup>∘</sup> . Proposition 1. For the Fisher's model, the following formula is valid:

$$f^\*(\ln q) = -\sum\_{i \in I} \lambda\_i \max\_{j \in I} \ln \frac{c\_j^i}{q\_j} \tag{5}$$

Theorem 3. The fixed point of G is the maximum point of the concave function <sup>ψ</sup>ð Þ<sup>q</sup> on <sup>σ</sup><sup>∘</sup> .

For the functions φð Þp and ψð Þq , there is a duality relation as for dual programs of linear programming:

Proposition 2. For all p, q∈σ<sup>∘</sup> the inequality

$$
\varphi(p) \ge \psi(q)
$$

holds. This inequality turns into equality only if p ¼ q.

Corollary. φð Þ¼ r ψð Þr if and only if the point r is the fixed point of the mapping G.

Thus, the equilibrium problem for the Fisher's model is reduced to the optimization one on the price simplex. It should be noted that this reduction is different from well-known one given by Eisenberg and Gale [6].

#### 2∘ : Algorithms

5. The Fisher's model

34 Optimization Algorithms - Examples

∂f pð Þ¼�∂ð Þ �f ð Þp .

Introduce the function

holds P

: Reduction to optimization problem

case in the abovementioned example.

arising polyhedral complementarity problems.

Let G be the mentioned associated mapping.

where e ¼ ð Þ 1;…; 1 and ln q ¼ ln q1;…; ln qn

f , we can consider the conjugate function:

Consider the convex function h, defining it as follows:

<sup>j</sup><sup>∈</sup> <sup>J</sup> pj ¼ 1 for p ∈σ.)

each consumer has all commodities in equal quantities: w<sup>i</sup>

Theorem 1. The subdifferential of the function f has the representation:

h pð Þ¼

f ∗

(see [14]) With this function, we associate the function ψð Þ¼ q f

ð Þ¼ y inf

8 ><

>:

A special class of the models is formed by the models with fixed budgets. This is the case when

<sup>p</sup>; <sup>w</sup><sup>i</sup> � � <sup>¼</sup> <sup>λ</sup><sup>i</sup> for all <sup>p</sup> <sup>∈</sup>σ. Such a model is known as the Fisher's model. Note that we have this

The main feature of these models is the potentiality of the mappings G associated with the

Let <sup>f</sup> be the function on <sup>R</sup><sup>n</sup> that f pð Þ for <sup>p</sup><sup>∈</sup> <sup>σ</sup> is the optimal value in the transportation problem of the model, and f pð Þ¼�∞ for p∉ σ. This function is piecewise linear and concave. It is natural to define its subdifferential using the subdifferential of convex function ð Þ �f :

∂f pð Þ¼ f g ln q þ tejq∈ G pð Þ; t ∈ R ,

ð Þ <sup>p</sup>; ln <sup>p</sup> , for <sup>p</sup><sup>∈</sup> <sup>σ</sup><sup>∘</sup> , 0, for p ∈∂σ, �∞, for p∉ σ:

Theorem 2. The fixed point of G coincides with the minimum point of the convex function <sup>φ</sup>ð Þ<sup>p</sup> on <sup>σ</sup><sup>∘</sup> . Another theorem for the problem can be obtained if we take into account that the mapping G and the inverse mapping G�<sup>1</sup> have the same fixed points. For the introduced concave function

<sup>z</sup> f g ð Þ� <sup>y</sup>; <sup>z</sup> f zð Þ

� �. (The addend te in this formula arises because it

9 >=

>;

φð Þ¼ p h pð Þ� f pð Þ (4)

∗

ð Þ ln <sup>q</sup> , which is defined on <sup>σ</sup><sup>∘</sup> .

<sup>j</sup> ¼ λ<sup>i</sup> for all j∈J, and thus,

1∘

The mentioned theorems allow us to propose two finite algorithms for searching fixed points.

Algorithmically, they are based on the ideas of suboptimization [11], which were used for minimization quasiconvex functions on a polyhedron. In considered case, we exploit the fact that the complexes ω and ξ define the cells structure on σ<sup>∘</sup> similarly to the faces structure of a polyhedron.

For implementation of the algorithms, we need to get the optimum point of the function φð Þp or ψð Þq on the affine hull of the current cell.

Consider a couple of two cells Ω ∈ ω, Ξ∈ξ corresponding to each other.

Let L⊃ Ω, M ⊃Ξ be their affine hulls. It will be shown that L ∩ M is singleton.

Let be f gr ¼ L ∩ M.

Lemma. The point r is the minimum point of the function φð Þp on L and the maximum point of the function ψð Þq on M.

Now, we describe the general scheme of the algorithm [8] that is based on Theorem 2. The other one using the Theorem 1 is quite similar [9].

On the current k-step of the process, there is a structure B<sup>k</sup> ∈ B. We consider the cells <sup>Ω</sup><sup>k</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ <sup>B</sup><sup>k</sup> ,Ξ<sup>k</sup> <sup>¼</sup> <sup>Ξ</sup>ð Þ <sup>B</sup><sup>k</sup> and have the point qk <sup>∈</sup>Ξk. Let Lk <sup>⊃</sup> <sup>Ω</sup>k, Mk <sup>⊃</sup>Ξ<sup>k</sup> be the affine hulls of these cells. We need to obtain the point of their intersection rk.

Return to the transportation problem of the model and to the descriptions of cells. Consider the graph Γð Þ B<sup>k</sup> . This graph can have more than one connected components. Let τ be number of connected components, and i∈Iν, ð Þ m þ j for j∈J<sup>ν</sup> be the vertices of ν-th component. It is easy to verify that the linear system (3) for Lk is going to be equivalent to this one:

$$\sum\_{j \in I\_v} p\_j = \sum\_{i \in I\_v} \lambda\_{i\nu} \qquad \nu = 1, \dots, \tau. \tag{6}$$

Theorem 4. If the transportation problem of the model is dually nondegenerate, the described

Polyhedral Complementarity Approach to Equilibrium Problem in Linear Exchange Models

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37

Figure 4 illustrates two described cases on one step of the algorithm. The point q∈Ξ is the

We show how the described method works on the Fisher's model example of Section 3.

the structure B<sup>12</sup> ¼ f g ð Þ 1; 2 ;ð Þ 1; 3 ;ð Þ 2; 1 ;ð Þ 2; 2 will be depicted as the matrix

(this is the structure for the cell Ω12). Let us start with the structure

<sup>q</sup><sup>1</sup> <sup>¼</sup> ð Þ <sup>0</sup>:05; <sup>0</sup>:35; <sup>0</sup>:<sup>6</sup> . It is easy to verify that <sup>B</sup><sup>1</sup> <sup>∈</sup> <sup>B</sup> and <sup>q</sup><sup>1</sup> <sup>∈</sup><sup>Ξ</sup> <sup>B</sup><sup>1</sup> � �.

Thus, we have <sup>r</sup><sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup>; <sup>0</sup>; <sup>0</sup> . The cell <sup>Ξ</sup> <sup>B</sup><sup>1</sup> � � is given by the system

move the point <sup>q</sup><sup>1</sup> to the point <sup>r</sup>1. For the moving point q tð Þ it will be:

This point reaches a face of <sup>Ξ</sup> <sup>B</sup><sup>1</sup> � � at <sup>t</sup> <sup>¼</sup> <sup>t</sup>

as equality. We obtain <sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>B</sup><sup>1</sup>

For the start, we need a structure B<sup>1</sup> ∈ B and a point q<sup>1</sup> ∈Ξ B<sup>1</sup> � �: We depict the structures as matrices m � n with elements from f g �; � , and � corresponds to an element of B. For example,

> <sup>B</sup><sup>12</sup> <sup>¼</sup> � �� ��� � �

<sup>B</sup><sup>1</sup> <sup>¼</sup> ��� ��� !

It means that both consumers prefer only first good. Let us choose as q<sup>1</sup> the price vector

p<sup>1</sup> ¼ 1, p<sup>2</sup> ¼ 0, p<sup>3</sup> ¼ 0:

<sup>2</sup> , (7)

<sup>3</sup> , (8)

<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:1111: the inequality (7) for <sup>q</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ is fulfilled

Step 1. The graph Γ B<sup>1</sup> � � has three connected components and the system (6) has the form

q1 <sup>1</sup> <sup>≤</sup> <sup>q</sup><sup>2</sup>

q1 <sup>1</sup> <sup>≤</sup> <sup>q</sup><sup>3</sup>

We have q<sup>1</sup> ∈Ξ B<sup>1</sup> � � and r<sup>1</sup> ∉Ξ B<sup>1</sup> � �. It is the case (i) in the description of algorithm. We have to

q1ðÞ¼ t 0:05 þ 0:95t, q2ðÞ¼ t ð Þ 1 � t 0:35, q3ðÞ¼ t ð Þ 1 � t 0:6:

<sup>∪</sup>f g ð Þ <sup>1</sup>; <sup>2</sup> and <sup>q</sup><sup>2</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ.

suboptimization method leads to an equilibrium price vector in a finite number of steps.

current point of the step.

6. Illustrative example

The linear system (1) for the cell Ξ<sup>k</sup> defines coordinates qj on each connected component up to a positive multiplier:

$$
\mathfrak{q}\_{\mathfrak{j}} = \mathfrak{t}\_{\nu} \mathfrak{q}\_{\mathfrak{j}'}^{k} \qquad \mathfrak{j} \in \mathfrak{J}\_{\nu}.
$$

To obtain the coordinates of the point r<sup>k</sup> , we need to put pj ¼ qj in corresponding Eq. (6), which gives the multiplier tν.

For the obtained point, r<sup>k</sup> can be realized in two cases.

(i) <sup>r</sup><sup>k</sup> <sup>∉</sup>Ξk. Since <sup>r</sup><sup>k</sup> is a maximum point on Mk for the strictly concave function <sup>ψ</sup>ð Þ<sup>q</sup> , the value of the function will increase for the moving point q tðÞ¼ ð Þ <sup>1</sup> � <sup>t</sup> qk <sup>þ</sup> tr<sup>k</sup><sup>Þ</sup> when <sup>t</sup> increases in [0,1]. In considered case, this point reaches a face of Ξ<sup>k</sup> at some t ¼ t <sup>∗</sup> < 1. Some of corresponding inequalities (2) for <sup>p</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ is fulfilled as equality. Choose one of them. Corresponding edge ð Þ l; m þ j will be added to graph. It unites two of connected components. We obtain <sup>B</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>B</sup>k∪f g ð Þ <sup>l</sup>; <sup>j</sup> , accept <sup>q</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> q t<sup>∗</sup> ð Þ and pass to the next step.

It should be noted that the dimension of the cell Ξ reduces. It will certainly be r<sup>k</sup> ∈Ξ<sup>k</sup> when the current cell <sup>Ξ</sup><sup>k</sup> degenerates into a point, and we have <sup>r</sup><sup>k</sup> <sup>¼</sup> <sup>q</sup><sup>k</sup> . But it can occur earlier.

(ii) <sup>r</sup><sup>k</sup> <sup>∈</sup>Ξk. In this case, we can assume qk <sup>¼</sup> <sup>r</sup><sup>k</sup> . Otherwise, we can simply replace qk by r<sup>k</sup> with an increase of the function'<sup>s</sup> <sup>ψ</sup>ð Þ<sup>q</sup> value. We verify qk <sup>∈</sup> <sup>Ω</sup>k? For this, we obtain from the equations of the transportation problem the variables zij, ið Þ ; j ∈ Bk, as linear functions zijð Þp and check zij q<sup>k</sup> � � ≥ 0. If it is true, the point q<sup>k</sup> is the required fixed point. Otherwise, we have zsl <sup>q</sup><sup>k</sup> � � <sup>&</sup>lt; 0. We accept <sup>B</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>B</sup><sup>k</sup> f g ð Þ <sup>s</sup>; <sup>l</sup> , q<sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>q</sup><sup>k</sup> and pass to the next step.

Figure 4. Illustration of one step of the algorithm.

Theorem 4. If the transportation problem of the model is dually nondegenerate, the described suboptimization method leads to an equilibrium price vector in a finite number of steps.

Figure 4 illustrates two described cases on one step of the algorithm. The point q∈Ξ is the current point of the step.
