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 30’s by Professor Wassily Leontief (picture above) who developed an economic model of the United States economy by dividing it into 500 …show more content…
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 …show more content…
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
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 30’s by Professor Wassily Leontief (picture above) who developed an economic model of the United States economy by dividing it into 500 …show more content…
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 …show more content…
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