Price systems computable
from input-output coefficients

B. SEKERKA, O. KYN and L. HEJL

 from: Contributions to Input-Output Analysis, A. P. Carter and A. Brody (eds.), 

 North-Holland Publishing Company, 1969

Reprinted in  Heinz D. Kurz, Erik Dietzenbacher, Christian Lager: Input Output-Analysis, volume 3, Part II  Price Models,  Elgar Refeence Collection,  International Library of Critical Writings in Economics,  1998

1. Assumptions of the model

Consider an input-output model in physical terms in very detailed breakdown.⁽¹⁾

Let

X = the column vector of total production in physical terms;

Y = the column vector of final production in physical terms;

A = the matrix of input-output coefficients;

B = the matrix of capital-output coefficients;

C = the matrix of coefficients of workers' consumption

Vectors X, Y and matrices A, B, C, are defined so that the following equations hold

AX + Y = X                              (1)

BX^{^} = F                                           (2)

CX^{^} = D                                   (3)

X^{^}  is a diagonal matrix formed from vector X;

F    is a matrix whose element f_ij represents the stock of capital of kind i in branch j of production;

D   is a matrix whose element d_ij represents the total volume of products i consumed by those working in branch j of production.

Matrices A, B, C are the three basic characteristics of the economic system from which we shall derive various price systems. Note that these matrices represent the requirements for various productive factors for a unit of production in various branches: matrix A represents requirements for intermediate products, matrix B for the stock of physical capital, and matrix C requirements for labor, expressed in subsistence (real) wages. We shall assume that matrices A, B, C are non-negative, that they have positive maximum characteristic roots, and that the maximum characteristic root of matrix A is less than one. These are the only assumptions laid down. In a normal economy, these assumptions are always fulfilled.

2. Some macroeconomic indicators

P(P, X) =  P'X      =  gross social product;                            (4)

N(P, X) =  P'A X  =  total material costs;                              (5)

Y(P, X) =  P'Y     =  national income                                    (6)

W(P,X) =  P'CX   =  total wage fund;                                    (7)

F(P, X) =  P'BX   =  total stock of capital;                             (8)

M(P,X) =  P'(I-A-C)X  = total profits or surplus value           (9)

We shall examine the following aggregate relationships:

a(P,X) = N(P,X)/Y(P,X) =  material costs-income ratio;             (10)

b(P,X) = F(P,X)/Y(P,X) =  capital-income ratio;                        (11)

g(P, X) = W(P,X)/Y(P,X) =  wages-income ratio;                           (12)

y(P, X) = M(P,X)/W(P,X) = the rate of surplus value                (13)

f(P, X) = M(P,X)/[N(P,X) + W(P,X)] = the profits-cost ratio     (14)

d(P, X) = M(P,X)/F(P,X) = the average rate of profit                    (15)

3. A general three-channel type of price system

Values of a(P,X) ,b(P,X), and g(P, X), defined in § 2, depend on the price vector P and the production vector X. From definitions (4)-(12), it follows that, for any arbitrary, non-zero constants h, w, 

a(hP,wX) = a(P,X), 

b(hP,wX) = b(P,X)

g(hP,wX) = g(P,X)

Thus, we see that these values depend on relative prices and relative quantities, that is, on relations p_i/p_j,  x_i/x_j (i, j = 1, 2, ...n) where n is the order of matrix A. They do not, however, depend on the price or production levels.

Let us now ask whether it is possible to find a vector of production X, such that the values of a, b, and  g or some linear combinations of them are independent of prices; and so find a price vector P, such that the values of a, b, and  g  or linear combinations of them are independent of X. 

Consider the matrix

H(m,n,r) = nA + rB + mC                                            (16)

where m,n,r are non-negative parameters, at least one of which is not zero. Under these assumptions,  H(m,n,r)(I - A)^-1 ³ 0. 

Thus, there exists⁽²⁾ a characteristic root   l₀(m,n,r) > 0 and a characteristic vector Y₀ (m,n,r) ³ 0 such that

H(m,n,r)(I - A)^-1Y₀  =   l₀Y₀                                                             (17)    

 From eq. (1)        X₀ = (I - A)^-1 Y₀,

and 

                 H(m,n,r)X₀  =   l₀Y₀ .                                           (18)

Then, for any vector  P ³ 0 such that   P'Y  > 0,

na(P,X₀) + rb(P,X₀) + mg(P, X₀) =  l₀ .                                  (19)

Therefore, for any three non-negative parameters m,n,r of which at least one is greater than zero, there exists a vector of total production X₀ ³ 0 such that eq. (19) holds independently of the price system chosen, provided that P' Y₀ > 0.

Note that if the matrix H(m,n,r)(I-A)^-1 is irreducible, then the vector X₀ is positive and uniquely determined.

For the characteristic root  l₀, there exists a characteristic vector P₀ = P₀(m,n,r) ³ 0 (in case of irreducibility P₀ > 0), such that

(I - A')^-1H'(m,n,r)P₀ =   l₀P₀                                               (20)

This means that for any vector Y ³ 0, such that Y'P₀ > 0, 

na(P₀,X) + rb(P₀,X) + mg(P₀, X) =  l₀ .                               (21)

For each group of three non-negative parameters m,n,r with at least one greater than zero, there exists a non-negative price vector, such that eq. (21) holds independently of the production vector, provided that P'₀Y>0. 

P₀ is the characteristic vector of the matrix (I-A')^-1H'(m,n,r)  and if this matrix is irreducible, it is positive and unique. Because we have assumed l₀ > 0, we can define the following parameters

m* = l₀^-1m,       n* = l₀^-1n,      r* = l₀^-1r                    (22)

From eq. (20), then

P₀    =   (1 +  n*)A'P₀  +  r*B'P₀  + m*C'P₀                          (23)

Clearly, A'P₀ , B'P₀, and C'P₀ are material costs, capital, and wages necessary for one unit of production at P₀ prices. Thus, prices P₀ cover costs of materials and form income as a sum of three components: the first is proportional to cost of materials, the second to capital, and the third to wages. Prices defined by eq. (23) are therefore called three-channel prices.

For parameters  m, n, r we have assumed only non-negativity, and that at least one of them is positive. Otherwise, they may be chosen arbitrarily. Parameters m*,n*,r* are, however, mutually bound by the condition that the maximum characteristic root of the matrix (1+n*)A'+r*B'+m*C' is unity. This means that the values of all three parameters n*, m* and  r* cannot be chosen quite arbitrarily. First of all, the following limits exist for these parameters:

0 £ n* £ ((1/l_a) - 1), 0 £ r* £ (1/l_b), 0 £ m* £ (1/l_c);                 (24)

where l_a is the maximum characteristic root of matrix A; l_b is the maximum characteristic root of matrix B(I-A)^-1; and l_c is the maximum characteristic root of matrix C(I-A)^-1. If two of the three parameters are chosen, then the value of the third is given by eq. (23). In specifying  n*, m* and  r* we can:

a) choose the values of  m, n, r arbitrarily in the desired relations;

(b) find the maximum characteristic root l₀, and on the basis of eq. (22), transform    m, n, r to m*, n*, r*. They will then be within the limits given by condition (24).

Alternatively, we can choose two of them and compute the third from eq. (23). The fact that the two chosen parameters lie within the limits given by eq. (24) is not, however, a sufficient condition for the non-negativity of the third parameter. We shall return to this problem later. 

Note, finally, that if  l_a < 1, then at least one of the parameters m*, n*, r* must be greater than zero. This follows from the inequality

A'  £  (1+n*)A'  +  r*B'  +  m*C'

and the condition that the maximum characteristic root of the matrix  
(1+n*)A'+r*B'+m*C' is unity.

4 Three basic types of prices

4.1. The N-income price system

Let us examine the value of  a(P,X)  defined by eq. (10). As shown in § 3, there exists a vector X₀ ³ 0, such that the ratio of aggregate material costs to income, that is a(P,X), is independent of the price vector; and analogously that there exists a vector P₀³ 0, such that a(P,X)  is independent of the vector of total production X. In other words, the relation between total material costs and national income expressed in prices P₀ will not change with any changes in the vector of production X. For proof, choose  n > 0, m = r = 0 in matrix  H(m,n,r)(I - A)^-1 . If we introduce values of the parameters into eq. (23), we obtain the following price relation:

(1 + n*)A'P₀ =  P₀                                                                     (25)

The price system given by equation (25) is called an N-income price system.⁽³⁾N-income prices are proportional to material costs. In other words, the relation between income and material costs is the same for all prices and is equal to the parameter n*. The parameter n* is uniquely defined by eq. (25).

n* = (1/l_a) - 1                                                     

where l_a is the maximum characteristic root of matrix A. The vector of  N-income prices, P₀  is the characteristic vector of matrix A'. Note that the uniform relation of income to material costs n* depends only on matrix A and, therefore, cannot be influenced by changes in production vector X. From eqs. (25) and (26) it also follows that, at N-income prices, a(P₀,X)   =   l/n*. Here, if matrix A is irreducible, P₀  is positive and uniquely defined.

4.2. The F-income price system

Let us now examine value of b(P,X)  defined by eq. (11). From § 3 it follows that there exists a vector X₀ > 0 such that the aggregate capital-output ratio b(P,X₀)  is independent of the vector of prices P, and also that there exists a vector of prices P₀ > 0 such that  b(P₀, X) is independent of the production vector X.To prove this, set  m = n = 0 and  r>0 for matrix H(m,n,r)(I-A)^-1 Since, by our assumptions, the matrix (I-A)^-1B'  has a positive maximum characteristic root l_b  we obtain from eq. (23), a price equation with m = n  = 0 .

A’P₀   +  r*B' P₀     =  P₀                                                             (26)

 where     r*   =   1/l_b

A price system given by eq. (26) is called an F-income price system. As is now evident, the capital-income ratio is the same for all prices in an F-income price system. It depends only on the matrix of complex capital-output ratios and cannot therefore be influenced by changes in the production vector. 

From eqs. (21) and (22), it also follows that, in this case, the aggregate capital-income ratio is equal to the maximum characteristic root of the matrix of complex capital-income ratios; that is, 

b(P₀, X) = l_b  =  1/ r*.

The vector of  F-income prices is a characteristic vector of the 
matrix (I-A)^-1B' . If this matrix is irreducible, then P₀ > 0  is uniquely defined.

4.3. The value price system

Analogously to the argument just presented, there exists a price vector   P₀³ 0, such that  g(P₀, X) is independent of production vector X.  Thus, the matrix of workers’ total consumption coefficients C(I - A)^-1 would have a positive maximum characteristic root l_c. Setting  r = n  = 0 , we obtain the price formula from eq. (23):

A’P₀   +  m*C' P₀     =  P₀                               (27)

 

m*  =   1/l_c

The price system of eq. (27) is called a value system of prices in the labor theory of value sense of the expression. Value prices are proportional to total wage-costs and the income component of every price is proportional to direct wage-costs. The parameter  m*  now becomes the reciprocal of the maximum characteristic root of the matrix of workers’ total consumption coefficients. The vector of value prices is the characteristic vector of the matrix (I-A)^-1C' For value prices:

g(P₀, X)   =  l_c   =  1/ m* 

and   

y(P₀, X)   =  m* -  1   

where y is the rate of surplus value as defined by eq. (13).

All three types of prices are special limiting cases of a general three-channel price system that can be computed by setting two of the three parameters m*,n*,r* equal to zero, that is, the lower limit of condition (24), and the third parameter to its upper limit.

5. Two-channel prices

If, in the equation for three-channel prices (23), we set one of three parameters m*,n*,r* equal to zero, we obtain a two-channel type of price. We distinguish three variants of two-channel prices.

5.1. N-two-channel prices

N-two-channel prices are obtained if we set  r* = 0.

P₀  = (1 + n*) A’P₀ + m*C’P₀ .                              (28)

If we choose  n* arbitrarily from the interval [0; (l/l_a) - 1), we can write

P₀    =  m*[I -  (1 + n*) A’]^-1 C’P₀                          (29)

The parameter m* is thus uniquely defined as the reciprocal of the maximum characteristic root, and the vector of prices P₀ as the characteristic vector of the matrix  [I-(1+n*)A’]^-1C’. It is thus possible  to consider m* as a function of n*. From the theory of non-negative matrices it follows that dm*/dn*  £ 0, and in cases of irreducibility, dm*/dn* <0.

Similarly, it can be shown that the choice of parameter m* from the interval [0; 1/l_c) uniquely defines n*.

Value prices and N-income prices are evidently limiting cases of N-two-channel prices. A certain ‘middle case’ is also of some interest. If we write m* = 1 + n* we obtain the following price formula:

P₀ = (1 + n*)(A’ + C’)P₀.                                       (30)

These prices cover material and wage costs and profit proportional to costs. We shall call them cost prices. Note that n* > 0, that is, there exists positive profit only if the maximum characteristic root of matrix  A’ + C’ is less than unity. With cost prices P₀, given by eq. (30), 

f(P₀, X) =  n*

where f is the ratio of total profits to total costs as defined by (14), and 1 + n*  is the reciprocal of the maximum characteristic root of matrix    A’ + C’.

5.2. F-two-channel prices          

F-two-channel prices are obtained as follows: in eq. (23), set n* = 0.

P₀    =   A'P₀  + m*C'P₀  +   r*B'P₀                                             (31)

Choose r* arbitrarily from the interval  [0; 1/l_b ). We can then write

P₀    =  m*[I -  r* (I - A)^-1B' ]^-1  (I - A)^-1C' P₀                            (32)

The parameter m* is thus uniquely defined as the reciprocal of the maximum characteristic root, and the vector P₀  as the characteristic vector, of the matrix

[I -  r* (I - A)^-1B' ]^-1  (I - A)^-1C' .

Now m* is a function of the parameter  r* such that   dm*/dr*  £  0  and in case of  irreducibility dm*/dr* < 0. Similarly, the reverse can be shown; by choice of  m* from the interval   [0; 1/l_c ) r* is uniquely defined.

It is evident that value prices and F-income prices are limiting cases of the F-two-channel type. We obtain an interesting ‘middle case’ if we set  m* = 1. If the characteristic root of the matrix A + C is less than unity we can write

P₀    =   r*[I - (A' + C')']^-1 B'P₀                                                    (33)

These prices cover material and wage cost and contain profit in proportion to capital. We call them production prices.

Note in equation (31) that if the maximum characteristic root of matrix  A+C  is unity and m* = 1, then r* = 0 and we would have value prices. Then prices would just equal the sum of material and wage costs and profit would be zero. It can easily be shown that. in a system of production prices P₀   defined by equation (33),

d(P₀, X) =  r*

  [I - (A' + C')']^-1 B'.

5.3. D-two-channel prices

D-two-channel prices are obtained if we set  m* = 0 in eq. (23).

P₀    =   (1 +  n*)A'P₀  +  r*B'P₀                                             (34)

This type of prices is less interesting from the point of view of economic interpretation. However, it can be shown that  n*   is a function of  r*  such that  dn*/dr*  £  0 

6. Determining the level of prices

All the formulae thus far determine only relative prices and not the price level. To determine the level of prices, a normalizing condition is necessary. We shall introduce it in the following form:

V’P = 1,                                                                             (35)

where V is an arbitrary vector. We call it a group numeraire. Vector V can be chosen in various ways.

(1) Choose V as a unit vector with all components zero, but with unity for the rth element. Thus, we have chosen the price of the rth commodity to be one. In other words, we express the prices of all other commodities in units of the rth commodity. This is the case of a simple numeraire. If this rth commodity is gold we have the classical theory of gold as a measure of value.

(2) State that the money value of the gross social product is f monetary units. Then it follows that P = P’X = f. This gives a group numeraire of

 V = (l/f)X.

(3) Analogously, if we write V = (l/y)Y, the volume of national income will be y monetary units.

(4) Vector V can be chosen in various other ways — for instance, so as to maintain the given volume of personal consumption in money terms.

7. Prices and price indices

Thus far we have assumed that the elements of vector P express the number of monetary units for each physical unit of production. The same equations are also used for finding indices of transformation from the prices currently prevailing in the national economy to computed prices. Let us call P° current (empirically given) prices, P¹ calculated prices, and Pⁱ  price indices. If P° > 0, then Pⁱ can be defined:

Pⁱ   =  ( P^{^}°)^-1 P¹                                                                (36)

Let A° be the matrix A expressed in prices P°, that is,

A°      =     P^{^}°A( P^{^}°)^-1                                                       (37)

consider N-income prices

P¹   =  (1 + n*)A'P¹

This equation can be written

( P^{^}°)^-1P¹   =  (1 + n*)( P^{^}°)^-1A'P^{^}°( P^{^}°)^-1P¹

Considering eqs. (36) and (37) we can write

Pⁱ   =  (1 + n*)A'Pⁱ 

If we use matrices expressed in prices P°, instead of those expressed in physical terms, we can calculate price indices Pⁱ for all types of prices by this same method. Note that a different interpretation of the normalizing condition (35) is required when we calculate price indices. We require, for instance, that the volume of the gross social product .~‘, expressed in prices P°, be the same as in prices P¹:

XP¹ = XP⁰.

If we define

V  =  XP^{^}°/ XP° 

we can, again, write this condition

VPⁱ = 1.

8. On the matrix of worker’s consumption

To simplify our formulae, let us assume that matrices A, B and C are expressed in original prices and P are price indices. It is empirically difficult to estimate a matrix of workers’ consumption C. To do so, we introduce the simplifying assumption that the structure of consumption expenditures of workers in all branches of the economy has the same proportions as the vector of personal consumption in the second quadrant of the input-output table. Let Y_c  be the vector of personal consumption. Now define

G =    Y_c/J'Y_c      J' = (1, 1, ..., 1).                                    (38)

Let U be the vector of wage funds (the sum of wages paid during the year in the various branches). We define the vector of wage coefficients

W  =  X^{^-1} U.                                                                    (39)

Let

C = GW’.                                                                            (40)

This matrix C is the matrix of coefficients of workers’ consumption, assuming that workers in all branches spend their wages in the same proportions as total personal consumption. 

Introducing eq. (40) into eq. (23), we obtain

P = (1 + n*) A’P + r*B' P + m*WG’P

If G is the group numeraire, G’P = 1, the simplified price formula becomes

P = (1 + n*)A'P + r*B'P + m*W,                                       (41)

with 

G’P = 1.                                                                             (42)

P is a vector of price indices, designated as Pⁱ above. Given eq. (38), we conclude that eq. (42) is equivalent to the requirement that the total volume of personal consumption in the newly computed prices be the same as in original prices. 

Thus, all types of prices can be calculated using a vector of wage coefficients W in place of matrix C, and with a nurneraire chosen so as to keep total personal consumption constant. Conditions governing m*,n*, and r* are unchanged, provided that both (41) and (42) hold.

The terms n* and r* must both lie within the limits given by eq. (24). Now m* has the following limits:

0 £ m* £ l/[G’ (I - A’)^-1 W]  - 1.                                               (43)

If C is defined by eq. (40), limits for m* given by eq. (43) are identical with those of eq. (24). It is not possible to derive any upper bound for m* from eq. (41) alone, i.e., without the normalizing condition (42).

However, the assumption m* ³  0 assigns the upper bounds of eq. (24) to n* and r*. Even when we fix two of the three parameters  m*,n*, r* in eq. (41), we do not uniquely predetermine the third without eq. (42).

9. Empirical calculations

Tables 1 and 2 present price indices Pⁱ calculated according to eqs. (41) and (42) for various values of  m*,n*, and r*. The calculations were based on an input-output table for the Czechoslovak economy for the year 1966 aggregated to 13 branches. This table was prepared by updating the statistical input-output table for 1962 to production and price conditions valid in 1966. 

TABLE 1.

F-two-channel-prices 

Table 1 contains F-two-channel prices, calculated as follows. In eq. (41) we set n* = 0, and varied r* from 0 to 0.05, 0.10, etc. We calculated m* from eq. (42). The price vectors in table 1 include three special cases: value prices (r* = 0 and m* = 2.71); production prices (m* = 1 and r* = 0.228); and F-income prices (to be more exact, ‘nearly F-income prices’ with parameters r* = 0.42 and m* = 0.021).

At first glance it may seem very strange to permit the values of m* to fall below unity. However, this case has a reasonable economic interpretation. Multiplying matrix C by m* < 1 lowers real consumption of workers proportionally in all branches. Thus, the dependence of  r* on m* can be interpreted as dependence of the rate of profit on the level of real wages. A similar approach is taken in SRAFFA (1963). Another possible interpretation is to assume that part of personal consumption is subsidized from the state budget; (1 - m*) is then something like a negative turnover tax rate.

The changes in price relations in table 1 show that the growth of the value of r* from 0 to 42 per cent is associated with a rise in prices in some branches and a drop in others. The direction of price change depends on the complex capital-output and complex labor-output ratios in a given branch.

Table 2 contains N-two-channel prices, calculated from (41), with  r* = 0 and various values of n*. The value of m* was determined by eq. (42). Among the price vectors in table 2 there are two special cases: value prices (n* = 0; m* =  2.71) and cost prices (n* = m* - 1 = 0.02862). The lower limit of the range of n* is not covered, and hence N-income prices were not calculated.

Fig. 1.

Fig. 2.

On the basis of the calculated F-two-channel prices, we derive a diagram of the function m* = f(r*), where n* = 0. This diagram is shown as fig. 1. Fig. 2 is a diagram of the function m* = g(n*), where r* = 0, derived for an N-two-channel price system. Figs. 1 and 2 clearly show the limits that must be respected in the choice of parameters n*, m* and r*. These limits depend on the sequence in which the parameters are chosen. First, we choose r* arbitrarily from the interval [0; 1/l_b] and designate the chosen value as r*• Given r*, the value of f(r*) can be found, where f is the function represented in fig. 1. Parameter m* can be chosen arbitrarily within the limits [0; f(r*)]. The value of n* is then uniquely determined and n* ³ 0. Alternatively, we can choose parameters in a different sequence. For instance, we can choose m* arbitrarily in the interval [0;1/l_a] and n* in the interval   [0;g^-1(m*)]  where g is the function represented in fig. 2. The value of r* is then uniquely defined.

10. Macroeconomic indicators

On the basis of the prices in tables 1 and 2, we can show how the values of the various macroeconomic aggregates defined by eqs. (4) through (12) vary with changes in n*, m* and r*. This is shown in table 3. Remember that with assumption (42) the value of personal consumption and total wages paid are the only constant values.

TABLE 3

Macroeconomic indicators (in billions of crowns)

11. A note on practical applications

The present price model can be used in various ways.

(1) It can be used for an analysis of the economy. It is usually very difficult to compare economic indicators since they are influenced by price relations. This is especially true for international comparisons. The present model shows that some of these difficulties are side-stepped, if we express the various indicators in specially calculated price systems. Maximum characteristic roots l_a, l_b, l_c are the only aggregate characteristics of the economic system that are independent of relative prices and production structures. Two-channel prices for various values of the parameters n* and r*  show the relative capital and labor intensiveness of sectors, as illustrated in tables 1 and 2.

(2) The model can be used by a central agency for the calculation of mutually balanced price changes. A model of this type was used for this purpose in the general reform of wholesale prices that went into effect in Czechoslovakia on January 1, 1967. With the help of computers, the price indices of an F-two-channel price system were calculated (with r* = 0.06 and m* = 0.22) for 25000 groups of products. Such a large task, of course, presented a number of methodological, as well as practical computational, problems. Some of these problems are described in SEKERKA (1967), SEKERKA and MRENICA (1967), and SEKERKA and TYPOLT (1968).

(3) The model can be used for forecasting future price developments. Given expected technological changes and changes in the distribution of national income, we estimate future matrices A, B, C and parameters m*, n*, r*. We can then calculate expected changes in relative prices. In Czechoslovakia, several alternative forecasts of prices for 1970 and 1975 were calculated on the basis of this model. For further information about these calculations, see HEJL et al. (1966b), HEJL et al. (1967b) and KYN et al. (1966).

Appendix

Characteristic roots and vectors

Since, in this paper, price systems are expressed as characteristic vectors of non-negative matrices, it is useful to state the following theorems, which have been proven in the theory of matrices:

Definition: The characteristic root l₀ of a square matrix A is called a maximum characteristic root if, for all the remaining characteristic roots, l_i of matrix A,  |l_i|  £  |l₀|   holds true.

Theorem (1)

A non-negative matrix A ³  0 always has a non-negative maximum characteristic root l₀, and a non-negative characteristic vector X₀.

For any arbitrarily chosen l > l₀,  (lI - A)^-1 ³ 0;   and(lI - A)^-1 /dl  £ 0.

Note: If  l₀ is a maximum characteristic root of matrix A, it is also the maximum characteristic root of matrix A’.

Theorem (2)

Let A ³ 0 be irreducible. Then there exists a positive maximum characteristic root l₀ and accompanying it a positive characteristic vector X₀. No other positive characteristic vector of matrix A that is linearly independent of X₀ exists.

For any arbitrarily chosen l > l₀,  (lI - A)^-1 > 0;   and(lI - A)^-1 /dl  < 0.

Theorem (3)

Given two non-negative matrices A and A* with maximum characteristic roots l₀ and l^*₀, A <A* (A ¹ A*), then  l₀ £ l^*₀. If A is an irreducible matrix then l₀ < l^*₀. 

Proof of these theorems can be found, for instance, in GANTMACHER (1966).

FOOTNOTES

(1) In the practical computations described in § 9, however, we applied an aggregated model using value data. Instead of prices, we obtained indices of transformation from the original prices to the new ones.   (back)

(2) The appendix contains theorems concerning characteristic roots and vectors of non-negative matrices.   (back)

(3)

Symbol N- and later F-are used to distinguish two different price Systems:

income prices and two-channel prices. N- is taken from the Czech word 'naklady' (cost) and F- from the Czech word 'fondy' (capital). (back)

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