Question

Suppose that in a certain open economy there are two production sectors: agriculture and manufacturing. Each dollar of production in agriculture requires 25 cents of input from the agriculture sector and 45 cents from manufacturing. Each dollar of production in the manufacturing sector requires 25 cents of input from itself, 45 cents from the agriculture sector. In your answer below keep the columns and rows in alphabetical order (agriculture, then manufacturing). The consumption matrix for this economy is C= Solve the Leontief equation (I−C)X=D to find a level of production in each of the two sectors that will satisfy an external demand from the open sector for 200000 dollars of agricultural output and 150000 of manufacturing output. The required production vector is X=

Answer 1

To find the production levels for agriculture and manufacturing sectors to meet external demand, one must solve the Leontief equation (I-C)X=D, involving calculating the inverse of the matrix (I-C) and multiplying it by the demand vector D.

Step-by-step explanation:

To satisfy an external demand in an open economy with agriculture and manufacturing sectors, we must solve the Leontief equation (I-C)X=D. First, we must define the consumption matrix C:

C =


0.25 0.45


0.45 0.25

Where the rows represent the inputs from agriculture and manufacturing, respectively, to produce one dollar of output in the corresponding sector. We must also set up the identity matrix I and the external demand vector D:

I =
1 0

0 1


D =

200,000

150,000

Now, the Leontief equation can be written as:

(I - C)X = D

Subtracting C from I, we get:

A =


0.75 -0.45


-0.45 0.75

Multiplying the inverse of matrix A by D yields the production vector X:

X = A-1D

We can calculate A-1D using matrix algebra, which will give us the level of production required in both sectors to meet the external demands.