Final answer:
To find the production levels to meet a final demand in a dual sector economy, we use the Leontief input-output model and the consumption matrix. Agriculture's plan to produce 50 units requires solving a matrix equation to determine the intermediate demands for manufacturing.
Step-by-step explanation:
The question concerns the calculation of production levels in a dual sector economy, divided into manufacturing and agriculture, to meet a specific final demand. To determine the production levels needed, we must calculate the intermediate demands created by each sector using a consumption matrix.
Let's denote the manufacturing production level as M and the agriculture production level as A. According to the question, the consumption matrix is an array where manufacturing requires 0 units from itself but 0.6 units from agriculture, while agriculture requires 0.5 units from manufacturing and 0.2 units from itself for each unit of output produced. So, the consumption matrix C can be written as:
C = [[0, 0.6], [0.5, 0.2]]
The final demand is set to 50 units for manufacturing and 30 units for agriculture. The Leontief input-output model can then be used to calculate the total production levels needed to meet this final demand. The model uses the formula (I - C)^-1 * D, where I is the identity matrix and D is the vector of final demand. After calculating the inverse of (I - C), we can multiply it by D to find the total production vector.
In the case where agriculture plans to produce 50 units, we are asked to find the intermediate demands. We substitute 50 for A in the matrix equation A = CA + D to find the intermediate demands for manufacturing (M).
After performing the matrix multiplication and solving the resulting equations, we can obtain values for M and A that satisfy the given conditions.