2. THE BASIC EQUATIONS 
We shall use a Leontief type model of the national economy with n sectors. Output is priced by three different systems of prices. In accordance with the terminology generally in use we shall distinguish the following: (a) A system of wholesale prices, which serves in the exchange of products between socialist enterprises. Everything produced receives a wholesale price. (b) A system of retail prices, which serves in the exchange between retail trade enterprises and the consumer. Only the vector of personal consumption from the second quadrant of the inputoutput table is priced in this way. (c) A system of foreign prices which are paid for imported goods bought abroad, or for which our exported goods are sold on foreign markets. This system is used to price only the import and export vectors of the second quadrant of the input—output table. 
A Leontief model in which the output of all sectors would be expressed directly in physical units would have to contain so many sectors that calculation would become very difficult. For practical reasons therefore the output of at least several sectors (which produce nonhomogeneous products) will be expressed in the ‘value of production’ and not directly in physical units. 
The price model we shall describe can be used for the solution of two problems:
Problem No. 1 Finding prices for a given structure of output in physical units.
Problem No. 2 Finding indices of the transformation from the old price system to the new, if the structure of output in the old prices is known.^{6) }
For the sake of simplicity and explicitness we shall in this paper use the terminology of problem 1; but the same model could serve to solve problem 2, if the expressions already introduced are given the following economic interpretation: q is the vector of total output expressed in old prices; x is the vector of total output in new prices. 
Let Q be the inputoutput matrix expressed in physical units and containing the means for simple renewal of productive capacities (depreciation). This matrix can then be defined in the same way as is done by Oscar Lange.^{7)} Let
B be the matrix of output coefficients defined by the following B = q_{diag}^{1}Q. (2) Let
V
be the vector of wages whose elements signify the total volume of wages
paid out in the given sector during a unit of time. Let
Z
be the vector of profits, whose elements express the total volume of
profits made by the given sector during a unit of time. 
If the ‘dual balance equation’ of the Leontief type model (column sums) x = xB + V + Z (3)
is multiplied from the right by the matrix
A = Qq_{diag}^{1
}is the matrix of input coefficients,
V = Vq_{diag}^{1} is the vector of wage cost coefficients,
Z = Zq_{diag}^{1} is the vector of coefficients, expressing the amount of profit in units of output,
then we obtain the following basic equation of the price model: p = pA + v + z, (4)
This equation expresses the dependence of wholesale prices on material costs, wage costs and profits.

We shall further suppose, that retail prices differ from wholesale prices only because of an indirect tax (turnover tax). The trade margin may be considered as the costs and profits of the trade sector. Let us then write:
p as the vector of retail prices; d as the vector of turnover taxes per physical unit of output.^{8) } The relation between the system of wholesale and retail prices is then given by the following equation:
p = p + d. (5)
p = p(I + d) (7)
(a) a system where the turnover tax rate is differentiated, and (b) a system where the turnover tax rate is uniform.

d = (pd_{diag} q_{s}') / pq_{s}' (6a)
where q_{s }is the vector of personal consumption in physical units.
In the second case the numbers expressing the turnover tax rate are the same for all sectors of the economy and are equal to the average rate. For conventional reasons we shall in this case also use the symbol
d to designate the number expressing this uniform turnover tax rate. In a system with a uniform rate of turnover tax it is possible to simplify
p
= p(1 +
d ).
(7a)

We shall now turn to the problem of distinguishing (a) relative prices and (b) price levels under different price systems. By relative prices we understand the relations p_{i}/p_{j} (i,j = 1, 2, . . ., n). It is evident that these relative prices do not change if we multiply the price vector by a scalar. Therefore we
shall say that price vectors p
and p*
for which
holds true (l is a number), express the same relative prices and the same price level if
l = 1, and a different price level if
l ¹ 1. The number  l  1 then signifies the ‘distance’ between price levels. The relation of price levels of two price vectors of different relations can be measured only by a weighted average l = pw'/ p*w' (8)
where w is the vector of weights. This can be a vector of total output, final output, personal consumption, etc. It is quite clear that the distance between price levels of the same price vectors can be different if we choose different weights. 
We shall call a system in which the level of wholesale and retail prices are different a twolevel system of prices, and a system in which the level of wholesale prices is equal to the level of retail prices a onelevel system of prices. Since retail prices only pertain to personal consumption, the only way we can measure the relationship between the levels of wholesale and retail prices is to use the vector of personal consumption
q_{s
}as a vector of weights. We shall use l
for expressing the relationship between the levels of wholesale prices
and retail prices. (Let
l =
p q_{s}/pq_{s}.)

For a twolevel price system it is true that l ¹ 1 . For practical purposes the only relevant case is the one where the level of wholesale prices is lower than the level of retail prices, or to put it differently pq_{s}' < pq_{s}' . In such a case d > 0. The average turnover tax rate in such a case determines the distance between the levels of wholesale and retail prices.

In the case of a uniform rate of turnover tax, the vectors p and p differ only in their levels but contain the same price relations. (This implies l = 1 + d ) We can now sum up several aggregate relationships in the national economy: Let
P = p(q’  q'_{s}) + pq'_{s} be the volume of the gross social product in which all components, with the exception of personal consumption, are priced in wholesale prices and personal consumption is priced in retail prices;
D = d q_{s} is the total volume of the turnover tax. 
From this the following equation holds true:
P = P + D.
(9)
The volume of national income
Y is equal to the gross social product after material costs have been subtracted
Y = P  pAq’ . (10)
Y = vq’ + zq’ + dq_{s }(11) In the case of a onelevel system of prices the whole national income is composed of wages and profits: in a twolevel price system only part of the national income is realised through wholesale prices and the rest is realized by the turnover tax. 



