# How prove that this problem doesn't have a solution?

• ambitionz
In summary, the Leontief system violates the input constraint and cannot have a solution. This can be shown by adding the three equations together, which results in a contradiction. The extra credit question can potentially be explained by the fact that the demand always exceeds the supply in this system, making it impossible for all three equations to have the same solution.

#### ambitionz

How do I prove that this problem doesn't have a solution?

1. Homework Statement

12. Consider the Leontief system

Here the column sums are 1, violating the Leontief input constraint
given in the text. Show that this system cannot have a solution.

Hint: Add the three equations together.

Extra Credit: Try to explain in economic terms why no solution exists.

## Homework Equations

This is some background info : "An understanding of the economy as consisting of linked sectors goes back to the French economist François Quesnay, and was fully developed by Léon Walras in 1874.[4]However, Wassily Leontief was the first to use a matrix representation of a national (or regional) economy. His model depicts inter-industry relationships within an economy, showing how output from one industrial sector may become an input to another industrial sector. In the inter-industry matrix, column entries typically represent inputs to an industrial sector, while row entries represent outputs from a given sector. This format therefore shows how dependent each sector is on every other sector, both as a customer of outputs from other sectors and as a supplier of inputs. Each column of the input–output matrix shows the monetary value of inputs to each sector and each row represents the value of each sector's outputs"

## The Attempt at a Solution

[/B]
I followed the hint and I added up the three equations.

I got x1 + x2+x3

= x1 + x2+x3 + 300

Since each column represents the demand of each industry, the x1 on the right side of the equation represents the aggregate total of all the demands of an industry; the same goes for x2 and 3 . "x1 + x2+x3 + 300(Consumer demand) " represents the total output of the industries. The "x1 + x2+x3 " represents the total amount of supplies. In order for the system to be in equilibrium the supply must equal the demand. Basically, the left hand side must equal the right hand side. In order for that to happen, x1 + x2+x3 must be 300 less than the amount of the supplies(x1 + x2+x3), consequently, "x1 + x2+x3+300"(the demand) will always be greater than the supply.

Also, does anyone have an idea about the extra credit?

Last edited:
For all three equations to have the same solution: x1 + x2 + x3 = x1 + x2 + x3 +300, which means that 300 = 0, which is obviously not correct. So the three equations cannot have the same solution.

AM

## 1. How do you know for sure that a problem doesn't have a solution?

As a scientist, I use the scientific method to prove that a problem does not have a solution. This involves formulating a hypothesis, conducting experiments or observations, and analyzing the results to draw a conclusion. If my experiments consistently show that there is no solution to the problem, then I can confidently say that the problem does not have a solution.

## 2. Can't there be a solution that we just haven't found yet?

It is possible that there could be a solution to a problem that we have not yet discovered. However, through rigorous testing and experimentation, scientists can confidently say that a problem does not have a solution if all evidence points to that conclusion. It is always important to keep an open mind and continue researching, but the current evidence may suggest that a solution does not exist.

## 3. How do you prove a negative, such as the absence of a solution?

Proving a negative, or the absence of something, can be challenging. However, in science, we use empirical evidence and logical reasoning to support our conclusions. This means that we gather data and perform experiments to show that a problem does not have a solution. If all evidence points to the absence of a solution, then we can confidently say that the problem does not have a solution.

## 4. Is it possible that there are multiple solutions to a problem?

Yes, it is possible for a problem to have multiple solutions. However, it is also possible for a problem to have no solutions. In science, we must thoroughly test and analyze a problem to determine the number of solutions, if any, that exist. If all evidence points to the absence of a solution, then we can confidently say that the problem does not have any solutions.

## 5. How does the lack of a solution impact the significance of a problem?

The lack of a solution to a problem does not diminish its significance. In fact, it may make the problem even more important to study and understand. By identifying that a problem does not have a solution, scientists can continue researching and potentially discover new information that could lead to a solution in the future. Every problem, regardless of whether it has a solution or not, can provide valuable insights and contribute to the advancement of knowledge in a particular field.