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Chapter 3: Problem 30

Can an external demand be met by an economy whose technology matrix \(A\) is the identity matrix? Explain.

Short Answer

Expert verified
In conclusion, an economy whose technology matrix A is the identity matrix can only meet an external demand when the demand is 0 (i.e., no external demand). In the case of any non-zero external demand, it cannot be met by such an economy.

Step by step solution

01

Understand the Leontief Input-Output Model

The Leontief Input-Output Model is used to represent the interdependence between different industries in an economy. It is used to calculate production levels for each industry to meet a given external demand. The model equation is: \(X = AX + D\) where: - X is the output vector (unknown) - A is the technology matrix (given, in this case, as the identity matrix) - D is the external demand vector (given) We have to find out if an economy whose technology matrix A is the identity matrix can meet the external demand.
02

Rearrange the model equation to solve for output X

Since we need to find the output vector X, we should rearrange the Leontief model equation to solve for X. \(X - AX = D\) Apply the distributive property: \((I - A)X = D\) where I is the identity matrix.
03

Plug in the given technology matrix A

In our case, A is given as the identity matrix. Therefore, when we plug in A as the identity matrix, the equation becomes: \((I - I)X = D\) which simplifies to: \(0X = D\)
04

Analyze the resulting equation

In the equation we obtained in step 3 (\(0X = D\)), we see that the production level X, regardless of its values, will always result in a product of 0 when constant 0 is multiplied by it. Thus, we have: \(0 = D\)
05

Conclusion

The result we obtained in step 4 (\(0 = D\)) indicates that the only way the external demand can be met by an economy whose technology matrix A is the identity matrix is when the external demand is 0. In other words, such an economy can only meet an external demand when there is no external demand (i.e., D = 0). Hence, in the case of any non-zero external demand, it cannot be met by an economy with a technology matrix as the identity matrix.

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Most popular questions from this chapter

Let \(A\) be the technology matrix \(A=\left[\begin{array}{ll}0.2 & 0.05 \\ 0.8 & 0.01\end{array}\right]\), where Sector 1 is paper and Sector 2 is wood. Fill in the missing quantities. a. units of wood are needed to produce one unit of paper. b. units of paper are used in the production of one unit of paper. c. The production of each unit of wood requires the use of units of paper.

Multiple Choice: If \(A\) and \(B\) are square matrices with \(A B=I\) and \(B A=I\), then (A) \(B\) is the inverse of \(A\). (B) \(A\) and \(B\) must be equal. (C) \(A\) and \(B\) must both be singular. (D) At least one of \(A\) and \(B\) is singular.

You are given a technology matrix \(A\) and an external demand vector \(D .\) Find the corresponding production vector \(X\). \(A=\left[\begin{array}{ll}0.5 & 0.4 \\ 0 & 0.5\end{array}\right], D=\left[\begin{array}{l}10,000 \\ 20,000\end{array}\right]\)

Calculate the expected value of the game with payoff matrix $$ P=\left[\begin{array}{rrrr} 2 & 0 & -1 & 2 \\ -1 & 0 & 0 & -2 \\ -2 & 0 & 0 & 1 \\ 3 & 1 & -1 & 1 \end{array}\right] $$ using the mixed strategies supplied. $$ R=\left[\begin{array}{llll} 0 & 0.5 & 0 & 0.5 \end{array}\right], C=\left[\begin{array}{llll} 0.5 & 0.5 & 0 & 0 \end{array}\right]^{T} $$

More Retail Discount Wars Your Abercrom B men's fashion outlet has a \(30 \%\) chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a \(70 \%\) chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a \(20 \%\) chance that Abercrom A will opt for the empty shirt option. The following payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation: Abercrom \(\mathbf{A}\) \(\begin{array}{ll}\text { Torn Shirts } & \text { Empty Shirts }\end{array}\) Mechanics \(\left[\begin{array}{rr}10 & -40 \\ -30 & 50\end{array}\right]\) Abercrom B \(\quad\) Military What is the expected resulting effect on your customer base?
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