Application to Leontief input-output model





Introduction In order to understand and be able to manipulate the economy of a country or a region, one needs to come up with a certain model based on the various sectors of this economy. The Leontief model is an attempt in this direction. Based on the assumption that each industry in the economy has two types of demands: external demand (from outside the system) and internal demand (demand placed on one industry by another in the same system), the Leontief model represents the economy as a system of linear equations. The Leontief model was invented in the 30s by Professor Wassily Leontief (picture above) who developed an economic model of the United States economy by dividing it into 500 economic sectors. On October 18, 1973, Professor Leontief was awarded the Nobel Prize in economy for his effort.


1)      The Leontief closed Model Consider an economy consisting of n interdependent industries (or sectors) S1,,Sn. That means that each industry consumes some of the goods produced by the other industries, including itself (for example, a power-generating plant uses some of its own power for production). We say that such an economy is closed if it satisfies its own needs; that is, no goods leave or enter the system. Let mij be the number of units produced by industry Si and necessary to produce one unit of industry Sj. If pk is the production level of industry Sk, then mij pj represents the number of units produced by industry Si and consumed by industry Sj . Then the total number of units produced by industry Si is given by:





In order to have a balanced economy, the total production of each industry must be equal to its total consumption. This gives the linear system:










then the above system can be written as AP=P, where







A is called the input-output matrix.


We are then looking for a vector P satisfying AP=P and with nonnegative components, at least one of which is positive.


Example Suppose that the economy of a certain region depends on three industries: service, electricity and oil production. Monitoring the operations of these three industries over a period of one year, we were able to come up with the following observations:

1.      To produce 1 unit worth of service, the service industry must consume 0.3 units of its own production, 0.3 units of electricity and 0.3 units of oil to run its operations.

2.      To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of service, 0.1 units of its own production, and 0.5 units of oil.

3.      Finally, the oil production company requires 0.3 units of service, 0.6 units of electricity and 0.2 units of its own production to produce 1 unit of oil.


Find the production level of each of these industries in order to satisfy the external and the internal demands assuming that the above model is closed, that is, no goods leave or enter the system.


Solution Consider the following variables:

1.      p1= production level for the service industry

2.      p2= production level for the power-generating plant (electricity)

3.      p3= production level for the oil production company


Since the model is closed, the total consumption of each industry must equal its total production. This gives the following linear system:





The input-output matrix is






and the above system can be written as (A-I)P=0. Note that this homogeneous system has infinitely many solutions (and consequently a nontrivial solution) since each column in the coefficient matrix sums to 1. The augmented matrix of this homogeneous system is





which can be reduced to






To solve the system, we let p3=t (a parameter), then the general solution is




and as we mentioned above, the values of the variables in this system must be nonnegative in order for the model to make sense; in other words, t≥0. Taking t=100 for example would give the solution








2)      The Leontief open Model The first Leontief model treats the case where no goods leave or enter the economy, but in reality this does not happen very often. Usually, a certain economy has to satisfy an outside demand, for example, from bodies like the government agencies. In this case, let di be the demand from the ith outside industry, pi, and mij be as in the closed model above, then




for each i. This gives the following linear system (written in a matrix form):



where P and A are as above and



is the demand vector.


One way to solve this linear system is




Of course, we require here that the matrix I-A be invertible, which might not be always the case. If, in addition, (I-A)-1 has nonnegative entries, then the components of the vector P are nonnegative and therefore they are acceptable as solutions for this model. We say in this case that the matrix A is productive.


Example Consider an open economy with three industries: coal-mining operation, electricity-generating plant and an auto-manufacturing plant. To produce $1 of coal, the mining operation must purchase $0.1 of its own production, $0.30 of electricity and $0.1 worth of automobile for its transportation. To produce $1 of electricity, it takes $0.25 of coal, $0.4 of electricity and $0.15 of automobile. Finally, to produce $1 worth of automobile, the auto-manufacturing plant must purchase $0.2 of coal, $0.5 of electricity and consume $0.1 of automobile. Assume also that during a period of one week, the economy has an exterior demand of $50,000 worth of coal, $75,000 worth of electricity, and $125,000 worth of autos. Find the production level of each of the three industries in that period of one week in order to exactly satisfy both the internal and the external demands.


Solution The input-output matrix of this economy is



and the demand vector is






By equation (*) above,












Using the Gaussian-elimination technique (or the formula B-1=(1/det(B))adj(B)), we find that




which gives



So, the total output of the coal-mining operation must be $229921.59, the total output for the electricity-generating plant is $437795.27 and the total output for the auto-manufacturing plant is $237401.57.





If you like to know more about the subject and the life and achievements of W. Leontielf, check the following links:

        Wassily Leontief (life and achievements of W. Leontielf)