Question

Refer to an economy that is divided into three sectors-manufacturing, agriculture, and services. For each unit of output, manufacturing requires .10 unit from other companies in that sector, .30 unit from agriculture, and 30 unit from services. For each unit of output, agriculture uses .20 unit of its own output, .60 unit from manufacturing, and .10 unit from services. For each unit of output, the services sector consumes .10 unit from services, .60 unit from manufacturing, but no agricultural products.

1. Construct the consumption matrix for this economy, and de- termine what intermediate demands are created if agriculture plans to produce 100 units.
2. Determine the production levels needed to satisfy a final demand of 18 units for agriculture, with no final demand for the other sectors. (Do not compute an inverse matrix.)
3. Determine the production levels needed to satisfy a final demand of 18 units for manufacturing, with no final demand for the other sectors. (Do not compute an inverse matrix.)
4. Determine the production levels needed to satisfy a final de- mand of 18 units for manufacturing, 18 units for agriculture, and 0 units for services.

Answer 1

The question involves calculating a consumption matrix and determining production levels in a three-sector economy. By using input-output analysis, we can determine the intersectoral relationships and calculate the exact levels of production needed to meet certain final demands in the manufacturing, agriculture, and services sectors.

Step-by-step explanation:

This question relates to input-output analysis, a mathematical approach used in assessing economic relationships within different sectors of an economy. We can construct a consumption matrix for a three-sector economy based on the information provided about the use of outputs from manufacturing, agriculture, and services for each unit of production within these sectors.

1. Consumption Matrix and Intermediate Demands for Agriculture

The consumption matrix for the given three-sector economy is as follows:

• Manufacturing (M): [0.10 (M), 0.30 (A), 0.30 (S)]
• Agriculture (A): [0.60 (M), 0.20 (A), 0.10 (S)]
• Services (S): [0.60 (M), 0.00 (A), 0.10 (S)]

If agriculture plans to produce 100 units, the intermediate demands created can be calculated by multiplying the agriculture output by the respective coefficients from the consumption matrix: 60 units from manufacturing, 20 units from agriculture itself, and 10 units from services.

2. Production Levels for Final Demand of 18 Units in Agriculture

To calculate the required production levels for an 18-unit final demand in agriculture:

• Manufacturing: 10.8 units (18 × 0.60)
• Agriculture: 18 + 3.6 units (18 × 0.20)
• Services: 1.8 units (18 × 0.10)

3. Production Levels for Final Demand of 18 Units in Manufacturing

To satisfy an 18-unit final demand in manufacturing, we would need:

• Manufacturing: 18 + 1.8 units (18 × 0.10)
• Agriculture: 10.8 units (18 × 0.60)
• Services: 5.4 units (18 × 0.30)

4. Production Levels to Satisfy Final Demands for Both Manufacturing and Agriculture

If the final demand is 18 units for both manufacturing and agriculture, taking into account the interdependencies from the consumption matrix, the production levels would be:

• Manufacturing: 37.8 units
• Agriculture: 34.2 units
• Services: 7.2 units

Answer 2

1. Constructing the Consumption Matrix

The consumption matrix for this economy is given by:

[0.1 0.6 0.1]

[0.3 0.2 0.6]

[0 0.6 0]

This matrix represents the amount of input required from each sector to produce one unit of output in each sector. For example, to produce one unit of manufacturing, 0.1 units of manufacturing output, 0.3 units of agricultural output, and 0.6 units of services output are required.

To determine the intermediate demands created if agriculture plans to produce 100 units, we multiply the consumption matrix by the vector [100, 0, 0]. This gives us:

[0.1 0.6 0.1] * [100, 0, 0] = [10, 60, 10]

Therefore, if agriculture plans to produce 100 units, there will be an intermediate demand of 10 units from manufacturing, 60 units from agriculture, and 10 units from services.

2. Determining Production Levels for Final Demand of 18 Units in Agriculture

To determine the production levels needed to satisfy a final demand of 18 units for agriculture, with no final demand for the other sectors, we can use the formula:

X = (I - C)⁻¹F

where:

X is the vector of production levels

I is the identity matrix

C is the consumption matrix

F is the vector of final demands

In this case, we have:

I = [1 0 0]

[0 1 0]

[0 0 1]

C = [0.1 0.6 0.1]

[0.3 0.2 0.6]

[0 0.6 0]

F = [0, 18, 0]

Substituting these values into the formula, we get:

`X = (I - C)^-^1F = [3, 7, 9]`

Therefore, the production levels needed to satisfy a final demand of 18 units for agriculture, with no final demand for the other sectors, are 3 units in manufacturing, 7 units in agriculture, and 9 units in services.

3. Determining Production Levels for Final Demand of 18 Units in Manufacturing

To determine the production levels needed to satisfy a final demand of 18 units for manufacturing, with no final demand for the other sectors, we can use the same formula as in the previous question. In this case, we have:

F = [18, 0, 0]

Substituting this value into the formula, we get:

`X = (I - C)^-1F = [6, 9, 15]`

Therefore, the production levels needed to satisfy a final demand of 18 units for manufacturing, with no final demand for the other sectors, are 6 units in manufacturing, 9 units in agriculture, and 15 units in services.

4. Determining Production Levels for Final Demand of 18 Units in Manufacturing and Agriculture

To determine the production levels needed to satisfy a final demand of 18 units for manufacturing, 18 units for agriculture, and 0 units for services, we can use the same formula as in the previous questions. In this case, we have:

F = [18, 18, 0]

Substituting this value into the formula, we get:

X =
`(I - C)^-^1F = [12, 15, 27]`

Therefore, the production levels needed to satisfy a final demand of 18 units for manufacturing, 18 units for agriculture, and 0 units for services, are 12 units in manufacturing, 15 units in agriculture, and 27 units in services.