Problem 75 - College Algebra with Modeling and Visualization

Question

Translations (Refer to the discussion in this section about translating a point.) Find a \(3 \times 3\) matrix A that performs the following translation of a point \((x, y)\) represented by \(X .\) Find \(A^{-1}\) and describe what it computes. Leontief Economic Model Suppose that a closed eco nomic region has three industries: service, electrical power and tourism. The service industry uses \(20 \%\) of its own production, \(40 \%\) of the electrical power, and \(80 \%\) of the tourism. The power company use \(40 \%\) of the service indus try, \(20 \%\) of the electrical power, and \(10 \%\) of the tourism The tourism industry uses \(40 \%\) of the service industry \(10 \%\) of the tourism. (A) Let \(S, E,\) and \(T\) be the numbers of units produced by the service, electrical, and tourism industries, respectively. The following system of linear equations can be used to determine the relative number of units each industry needs to produce. (This model assumes that all production is consumed by the region.) $$ \begin{aligned} &0.25+0.4 E+0.8 T=S\\\ &\begin{array}{l} 0.45+0.2 E+0.1 T=E \\ 0.45+0.4 E+0.1 T=T \end{array} \end{aligned} $$ Solve the system and write the solution in terms of \(T\). (B) If tourism produces 60 units, how many units should the service and electrical industries produce?

Step-by-Step Solution

Answer

For (A): \(S = \frac{9}{4}T, E = 4T.\) For (B): 135 units by service, 150 units by electrical industry.

Step 1: Matrix Representation of the Leontief Model

In the given problem, we need to use a system of linear equations derived from the Leontief model of a closed economy to determine the units produced by three industries: Service (S), Electrical Power (E), and Tourism (T). The equations provided are:\[ \begin{align*} 0.20S + 0.40E + 0.80T &= S \0.40S + 0.20E + 0.10T &= E \0.40S + 0E + 0.10T &= T \end{align*} \]Rewriting these equations in terms of a matrix equation, we have:\[ A \begin{bmatrix} S \ E \ T \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix} \]Where matrix \( A \) is:\[ A = \begin{bmatrix} 0.80 & -0.40 & -0.80 \ -0.40 & 0.80 & -0.10 \ -0.40 & 0 & 0.90 \end{bmatrix} \]

Step 2: Solve the System for T

To express the solution in terms of \( T \), substitute \( T \) as some arbitrary value (since every variable will be a multiple of T):Firstly, solve the equation with T explicitly:From the third equation:\[ -0.40S + 0T + 0.10T = 0 \]Re-writes to \[ 0.90T = S + 4T \], thus\[ S = \frac{9}{4}T \].By substituting \( S \) back into other two equations. Provides a solution: \[ E = 2.5T \] (after simplified process) The system solution is: \[ S = \frac{9}{4}T, E = 4T, T = T\]

Step 3: Apply the Solution for T = 60

Now, assuming \( T = 60 \), substitute in:\[ S = \frac{9}{4}(60) = 135 \]\[ E = 4(60) = 150 \]Thus, if the tourism industry produces 60 units, the service industry should produce 135 units, and the electrical power industry should produce 150 units.

Key Concepts

Matrix Representation in Leontief Model

In the Leontief Economic Model, matrix representation plays a crucial role in modeling economic systems. The idea is to express the relationships between industries as a set of linear equations in matrix form. This makes it easier to handle complex economic interactions, like those in a closed economy with interdependent industries.
In the matrix form, the economic input and output relationships of various industries are captured as a matrix equation. For example, if you have three industries, service (S), electrical (E), and tourism (T), you can represent their production interactions in the following manner:

• Firstly, define a matrix \( A \) that holds the coefficients telling how much each industry uses from another.
• This matrix, when multiplied by a vector of production levels \( \begin{bmatrix} S \ E \ T \end{bmatrix} \), will equate to a zero vector if all output is consumed internally.

In our example, the matrix \( A \) encapsulates the consumption relationships between these industries. Finding the solution involves leveraging linear algebra techniques to determine how much each industry should produce under given assumptions.

System of Linear Equations in Economics

A system of linear equations is the backbone of the Leontief Model. It allows us to express complex inter-industry dependencies in simple mathematical terms. Each equation in the system typically states that the total production of an industry equals the sum of its internal consumption and the consumption by other industries.
This approach is effective particularly in closed economies where no outside inputs or outputs affect the internal production statistics. Here's what it involves:

• Each equation corresponds to an industry and calculates how its total output is used by itself and other industries.
• These equations are solved simultaneously to achieve a balance where total output equals total consumption.

The solutions to these systems often come in the form of a vector that specifies the production levels of each industry necessary to meet the consumption needs. For instance, if the tourism industry consumes from both the other industries, this should reflect in tourism's equation in the model.

Understanding Closed Economy

A closed economy is characterized by the absence of trade with outside entities. All goods produced are consumed within the economy, making it an ideal scenario for certain mathematical models like the Leontief Model.
In closed economies:

• There are no imports or exports influencing the production levels. This simplifies the analysis since you only need to focus on internal supply and demand.
• Inter-industry dependencies become crucial as each industry depends on outputs from others to function.

This setting is useful for studying internal economic dynamics without the noise of external factors. It allows economists and mathematicians to test hypotheses regarding industry dependencies and efficiency, using a controlled model. In a Leontief system, this means setting up equations reflecting purely domestic relationships of production and consumption.

Industry Production in Economic Models

Industry production within economic models like Leontief's is determined by the interactions summarized in a system of equations or matrices. This reflects a real-world scenario where industries are interdependent. For example, in the given model:

• The service industry might require outputs from tourism and electrical power sectors to maintain its operations.
• Similarly, the power company could depend on services and tourism for its outputs.

Production levels are adjusted until an equilibrium is reached, where supply throughout the economy satisfies all the demands. The model seeks to balance production according to specified consumption coefficients, which dictate how much one industry's output is used by others. So, in scenarios where tourism suddenly ramps up production, the service and electrical power industries need to adjust their output to cater to increased demands from tourism within this closed economy framework.