_{1}

While the routine use of Leontief’s closed model is limited to the case in which the whole income of an economy goes to wages, this paper shows that the model also permits the representation of production programs corresponding to every level of income distribution between wages and profits. In addition, for each of these programs, the model allows calculating the price system and the profit rate when this rate is the same in all industries. Thus, the results obtained in Sraffa’s surplus economy are established following an alternative way, this makes it possible to build a particular standard system for each level of income distribution between wages and profits. Besides, the fact that the model includes the set of households as a particular industrial branch permits to build a balanced-growth path of the economy in which the quantities of work used in each industry as well as the goods consumed by the workers are studied explicitly, unlike what happens in von Neumann’s model. The paper also shows that, under a weak assumption, the balanced-growth rate is independent of the worker’s choice.

In the specialized literature, Leontief’s closed model is an instrument of analysis applied mainly to calculate certain relations between inputs and outputs in an industrial system and also to calculate prices in the particular case when all the income goes to wages (e.g., Berman & Plemmons [

Including this introduction, the paper is divided in 9 sections and an Appendix. Sections 2 and 3 present respectively the open and the closed Leontief’s model. Section 4 studies prices and income distribution in Leontief’s closed model when the profit rate is the same in all industries. Section 5 presents within the model the equality established by von Neumann [

The reference economy is integrated by

beled i or j so that

D-set if it contains D different goods, for any particular D-set,

For each j,

sidered. It is useful to write these quantities in matrix notation defining the column vectors

This permits the representation of the relations between inputs and outputs of the different goods and the relation between each price and its production cost, respectively, by means of the following equation systems.

The Frobenius roots of matrices

Moving along to the topic of viability, a square matrix

Condition (1) implies that in the economy there is at least one good that produces itself either directly or indirectly (see Lemma 1.1 by Seneta [

Equation (3) is an economy or a production program reproducing itself if it produces all the inputs consumed, in which case

If some goods are not produced, it is possible to eliminate from the program the equations corresponding to those goods together with the coefficients corresponding to them in the remaining equations. Then, reassigning the indexes among the goods produced, a new program results where

Given that vector

Proposition 1. In a viable open economy every good either is in the net product or produces at least one good that is in the net product, or both.

Proof. Given any i, consider the D-set consisting of i and all the goods produced by i either directly or indirectly. If

In which E is the square matrix formed by the intersection of the first D columns and the first D rows, H is a

imply the equation

Frobenius root of E is greater than or equal to one, a result allowing to conclude that

The model presented in this section constitutes the basis on which Leontief’s closed model is to be built in the next section. We shall see Proposition 1 allows the establishing of some important properties of the closed model.

In this section, Leontief’s closed model is built by adding to the model presented in the previous section the data from the set of households considered as an industrial branch. For this purpose, we will define first some additional notation.

For each

and, for each

Therefore, for each j,

As explained below, Leontief assumes that:

We can use the information from the program to form the following matrix:

in which A is the matrix of means of production coefficients,

section, but may adopt other values as indicated in the next one, C is the

sumed by the different industries and each column j indicates inputs consumed by industry j. Regarding this, special attention must be paid to column

Let

has a solution

Indeed, it follows from Proposition 1 that each good i such that

Regarding the price system, let

has a solution

If the profit rate (r) is the same in all industries, and if wages are paid at the beginning of production, for each j, the following equation is true:

Hence, it is possible to write (4) as follows:

By measuring prices using the value of the net product, the following equation is satisfied:

In connection to this, let w be the fraction of the value of the net product equivalent to the total wages paid. Multiplying both sides of (20) by w yields:

Dividing both sides of this equation by

Substituting in this equation, for each i, the term in brackets by the left-hand side of (9), and in addition, the right-hand side of the equation by

Furthermore, let

The auxiliary variable

Now, let:

Then, it is possible to write the system formed by Equations (19) and (25) as follows:

The coefficients that are greater than zero in

Proposition 2. There is a continuous monotonic increasing function

This proposition together with condition (6) imply that, for each

Proposition 3. For each

For the reasons given in Proposition 2 and in the paragraph below it, there is a monotonic decreasing function

It follows from the preceding analysis that there is a monotonic decreasing function

coordinate in this vector is equal to the given value of r. For each

Proposition 4. For each

Now, let us consider the matrix

the equation

Proposition 5. There is a monotonic decreasing function

Hence, there is a monotonic decreasing function

due to the fact that

Proposition 6. For each

Dornbush et al. ([

It follows from the preceding analysis that, for each

has a solution q > 0 determined up to a scalar factor. Fixing the magnitude of q by means of the equation:

we get the quantities produced under a program of production using the same amount of work as in system (3).

System (28) can be written as follows:

(30)

(31)

Equations (26) and (31) imply that

Hence, for each i, the product

According to Equations (30) and (31), the ratio between the quantity produced of each good and the amount of the same good consumed is equal to

Furthermore, letting:

we can write Equations (30) and (31) in the following way:

In the system formed by Equations (34) and (35), we can observe that, as a result of the production process, the amount of each good increased at a growth rate equal to g. Therefore, at the end of the production cycle, it is possible to start another cycle investing in each industry (1 + g) times the amount of each good used in the first. If this were to happen, and if, in addition, investments are held similarly at the beginning of each of the following cycles, the economy grows in what is known as the balanced-growth path. In this regard, for each

It is worth adding that, in accordance with what precedes, a hallmark of an economy that is in the balanced-growth path in Leontief’s closed model is the growth of the labor force, while von Neumann’s model considers only the growth of the other industrial branches.

Since matrix A can be decomposable, condition (15) depends on Equation (2) and on the matrix wC. Now, Equation (2) is a feature of the technique used while, in turn, wC can be interpreted in two ways. The first is to consider wC as a bundle of goods actually consumed by workers, which simplifies the analysis. The second is to consider wC as a bundle of goods equivalent to

Nevertheless, the balanced-growth rate does not change if wC is replaced in

For any given level of salary

In this formula, for each i, if i is consumed by workers

Replacing

Given a level of

According to Equations (34), (35) and (36) this is an homothetic system in which the ratio between the quantity produced of each good and the amount of the same good consumed is equal to

Proposition 7. In the balanced-growth path, for each good, the amount of investment and of profit in the industry producing the good are equal to the value, respectively, of the quantity of that good consumed and the surplus of that good produced in the whole industry.

Proof. For each i, multiply by

The left-hand side of each one of these equations is the value of the total consumption of the corresponding good in the system. Now, dividing by

The left-hand side of each one of these equations is the investment made in the corresponding industry which, in the case of industry

This proves the first part of the proposition. To prove the second part, it suffices to multiply the left-hand side of each one of these equations by g and its right-hand side by r.

According to this proposition, in the case of the set of households, the rate of profit measures the growth of the quantity of labor employed in the industrial system. The corresponding profit consists in the increase in the households’ income due to this growth.

For comparative purposes, in this Section, the quantity produced of each good in Equation (3) is used as the unit of measure for the quantities of that good. In this manner,

Let

Equations (9) and (47) imply that

This is the model of surplus economy studied by Sraffa [

However, it should be noted that unlike Sraffa’s model, in which the number of homothetic merchandises is finite (see Benítez Sánchez [

It is worth adding that, for Sraffa ([

This work shows an application of Leontief’s closed model that, as far as I am aware, has not been explored previously. Such application is the study of income distribution between wages and profits when the rate of profit is the same in all industries. The results are consistent with those of Sraffa’s model, except for the fact that in Leontief’s model it is possible to build a standard system for each level of income distribution. This system, except for the scale of production and the units of measure employed, is equal to any whole-industry production process taking place within the balanced-growth path corresponding to Leontief’s closed model for the given level of income distribution. Furthermore, in the balanced-growth path, for each good, the amounts of investment and profit in the industry producing the good are equal to the value, respectively, of the quantity of that good consumed and the surplus of that good produced in the whole industry. For this reason, for the set of households, included in the model as a particular industrial branch, the common profit rate measures the growth of the labor force. Unlike von Neumann’s model, the balanced growth-path corresponding to Leontief’s closed model shows explicitly the quantities of labor used in each industry, the quantities of goods consumed by workers and the growth of the labor force. Under a weak assumption, the growth rate is independent of worker’s choice.

I am grateful to an anonymous referee for helpful comments and suggestions.

Alberto BenítezSánchez, (2016) Income Distribution and Growth in Leontief’s Closed Model. Theoretical Economics Letters,06,7-19. doi: 10.4236/tel.2016.61002

In this Appendix, I consider a system that produces a unit of Good 1 consuming half a unit from the same good, and a unit of work. Then:

I further assume that

Substituting

Thus, the system formed by Equations (30) and (31) can be written in the following way:

Substituting into the first equation

Dividing both sides of the equation by q_{2} and regrouping, yields:

Therefore

Moreover, taking into account Equation (29), we obtain ^{ }

System (27) can be written as follows:

According to Equations (36) and (A.3), in this system

To build a system of type (27) which is in the balanced-growth path for

This system is in a balanced-growth path with

Finally, Sraffa’s system of type (49) that corresponds to this economy is:

According to Equation (48) when

wage and profit yields the net product. Since system (A.6) is homothetic, it is a standard system. Or, system (A.5) is also a standard system determining the same system of relative prices for