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Chapter 2: Problem 11

Consider the economy of Exercise 1 , consisting of three sectors: agriculture (A), manufacturing \((M)\), and transportation ( \(T\) ), with an input-output matrix given by $$ \begin{array}{l} A \\ A \\ M \\ T \end{array}\left[\begin{array}{ccc} A & M & T \\ 0.4 & 0.1 & 0.1 \\ 0.1 & 0.4 & 0.3 \\ 0.2 & 0.2 & 0.2 \end{array}\right] $$ a. Find the total output of goods needed to satisfy a consumer demand for \(\$ 200\) million worth of agricultural products, \(\$ 100\) million worth of manufactured goods, and \(\$ 60\) million worth of transportation. b. Find the value of goods and transportation consumed in the internal process of production in order to meet this total output.

Short Answer

Expert verified
In order to satisfy a consumer demand for \$200 million worth of agricultural products, \$100 million worth of manufactured goods, and \$60 million worth of transportation, the total output of goods needed is: - Agriculture: \$527.27 million - Manufacturing: \$409.09 million - Transportation: \$263.64 million The value of goods and transportation consumed in the internal process of production to meet this total output is: - Agriculture: \$327.27 million - Manufacturing: \$309.09 million - Transportation: \$203.64 million

Step by step solution

01

Write down the input-output matrix and consumer demand vector.

We are given the input-output matrix and consumer demand: Input-output matrix: \[ \begin{bmatrix} 0.4 & 0.1 & 0.1 \\ 0.1 & 0.4 & 0.3 \\ 0.2 & 0.2 & 0.2 \end{bmatrix} \] Consumer demand vector: \[ \begin{bmatrix} 200 \\ 100 \\ 60 \end{bmatrix} \]
02

Use the input-output model to find the total output vector.

According to the input-output model, the total output vector \(\mathbf{x}\) is given by: \[ \mathbf{x} = (I - A)^{-1} \cdot \mathbf{d}, \] where \(A\) is the input-output matrix, \(I\) is the identity matrix, and \(\mathbf{d}\) is the consumer demand vector. First, find the matrix \(I - A\): \[ I - A = \begin{bmatrix} 1 - 0.4 & -0.1 & -0.1 \\ -0.1 & 1 - 0.4 & -0.3 \\ -0.2 & -0.2 & 1 - 0.2 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.1 & -0.1 \\ -0.1 & 0.6 & -0.3 \\ -0.2 & -0.2 & 0.8 \end{bmatrix}. \] Now, find the inverse of this matrix \((I - A)^{-1}\): \[ (I - A)^{-1} = \begin{bmatrix} 2.27272727 & 0.45454545 & 0.22727273 \\ 0.36363636 & 2.27272727 & 0.68181818 \\ 0.45454545 & 0.45454545 & 1.59090909 \end{bmatrix} \] Finally, multiply this matrix by the consumer demand vector: \[ \mathbf{x} = (I - A)^{-1} \cdot \mathbf{d} = \begin{bmatrix} 2.27272727 & 0.45454545 & 0.22727273 \\ 0.36363636 & 2.27272727 & 0.68181818 \\ 0.45454545 & 0.45454545 & 1.59090909 \end{bmatrix} \cdot \begin{bmatrix} 200 \\ 100 \\ 60 \end{bmatrix} = \begin{bmatrix} 527.27272727 \\ 409.09090909 \\ 263.63636364 \end{bmatrix} \] So, the total output of goods needed is: - Agriculture: $527.27 million - Manufacturing: $409.09 million - Transportation: $263.64 million
03

Find the value of goods consumed in the internal process of production.

Now that we have the total output vector \(\mathbf{x}\), we can find the value of goods consumed by the following equation: \[ \mathbf{c} = A \cdot \mathbf{x} \] Multiplying the input-output matrix by the total output vector: \[ \mathbf{c} = \begin{bmatrix} 0.4 & 0.1 & 0.1 \\ 0.1 & 0.4 & 0.3 \\ 0.2 & 0.2 & 0.2 \end{bmatrix} \cdot \begin{bmatrix} 527.27272727 \\ 409.09090909 \\ 263.63636364 \end{bmatrix} = \begin{bmatrix} 327.27272727 \\ 309.09090909 \\ 203.63636364 \end{bmatrix} \] This result means the value of goods and transportation consumed in the internal process of production is: - Agriculture: $327.27 million - Manufacturing: $309.09 million - Transportation: $203.64 million

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Economics
Economics is the study of how people, businesses, and governments make choices about allocating resources. A key concept in economics is the input-output analysis, which examines the relationships between different sectors within an economy.
This analysis helps economists understand how the outputs from one sector may serve as inputs for another. Understanding this interdependency is crucial for planning economic activities and policies. In our example, the economy is divided into three sectors: agriculture, manufacturing, and transportation. Each sector uses materials produced by itself and others to generate its final output.
By analyzing these sectors through the lens of input-output analysis, we can determine the total outputs required to satisfy consumer demand and support production needs. This approach is instrumental in making efficient resource allocation decisions in an economy.
Matrix Operations
Matrix operations play a critical role in input-output analysis. They provide a mathematical way to model the complex interactions between different sectors of the economy.
In the context of this problem, matrices help us calculate the total output each sector needs to produce, not just for direct consumer demand, but also for the internal demands of production cycles. The input-output matrix, or "A" matrix, captures the proportion of outputs from each sector used by every other sector.
The identity matrix is used alongside the "A" matrix to compute the total output needed through mathematical operations such as matrix inversion. This is crucial because it allows economists to assess not just the direct demand, but also the demand generated by these sectoral interdependencies.
Consumer Demand
Consumer demand refers to the quantity of goods and services that households, businesses, and governments wish to purchase at a given price over a specific period. In the context of input-output analysis, consumer demand is expressed as a vector. This vector represents the economic sectors' sales to final users, excluding intersectoral transactions.
In the given example, the consumer demand vector represents availability and demand for agriculture, manufacturing, and transportation services at values of $200 million, $100 million, and $60 million, respectively. Recognizing the importance of consumer demand allows economists to understand how changes in consumer preferences affect the production levels of various sectors. The interplay between consumer demand and production is essential for ensuring an economy's smooth functioning.
Total Output Calculation
Total output calculation determines how much each sector in the economy must produce to meet consumer demands and sustain the production system itself. This involves more than merely adding up the demand from consumers but also factoring in the inputs needed from each sector.
The equation \( \mathbf{x} = (I - A)^{-1} \cdot \mathbf{d} \) is used to calculate total output across sectors: \( \mathbf{x} \) is the total output vector, \( I \) is the identity matrix, \( A \) is the input-output matrix, and \( \mathbf{d} \) is the consumer demand vector. By inverting "I - A", we adjust for the circular flow of economics where sectors provide goods for both consumer usage and other sectors.
This method ensures that the economy's production is not just consumer-driven but sustainable as well, considering the cyclical, interdependent nature of an economic system.

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