Finding the production Matrix

Question

Assume that the production of 1 unit of steel requires 0.8 units of steel and 0.2 units of lumber, and that the production of 1 unit of lumber requires 0.1 units of steel and 0.4 units of lumber.

Find the production schedule that satisfies an external demand for 10 units of steel and 60 units of lumber.

 

X= []

Answer 1

This is a Leontief input–output problem, and the one idea that makes the whole thing click is this: the economy has to produce enough not just for the outside world, but for itself. Making steel consumes steel. That self-consumption is what the matrix bookkeeping handles for you.

Step 1: build the technology matrix

The trick that trips people up is which way the recipes go. Each column of A is the recipe for one unit of one good.

1 unit of steel needs 0.8 steel and 0.2 lumber → that is the first column.

1 unit of lumber needs 0.1 steel and 0.4 lumber → that is the second column.

A = [ 0.8  0.1 ; 0.2  0.4 ]

Reading it row-wise, row 1 says how much steel each good consumes, row 2 how much lumber. If you build A with the recipes as rows instead, everything downstream comes out wrong, so it is worth pausing to check this.

Step 2: set up the equation

Let X = [x₁ ; x₂] be total production of steel and lumber, and D = [10 ; 60] the external demand.

Total output = what the industries consume internally + what goes to the outside:

X = AX + D

Solve for X the way you would in ordinary algebra, but carefully — you cannot write (1 − A) with a scalar 1:

X − AX = D → (I − A)X = D → X = (I − A)⁻¹ D

Step 3: compute

I − A = [ 0.2  −0.1 ; −0.2  0.6 ]

For a 2×2 matrix [a b ; c d], the inverse is (1/(ad − bc)) [ d  −b ; −c  a ]. Here the determinant is (0.2)(0.6) − (−0.1)(−0.2) = 0.12 − 0.02 = 0.10.

(I − A)⁻¹ = (1/0.10) [ 0.6  0.1 ; 0.2  0.2 ] = [ 6  1 ; 2  2 ]

Then

X = [ 6  1 ; 2  2 ] [ 10 ; 60 ] = [ 60 + 60 ; 20 + 120 ] = [ 120 ; 140 ]

So the production schedule is 120 units of steel and 140 units of lumber.

Step 4: check it, because this one checks beautifully

AX = [ 0.8(120) + 0.1(140) ; 0.2(120) + 0.4(140) ] = [ 96 + 14 ; 24 + 56 ] = [ 110 ; 80 ]

X − AX = [ 120 − 110 ; 140 − 80 ] = [ 10 ; 60 ]  ✓

That is exactly the demand. Notice what the numbers are telling you: of the 120 steel produced, 110 is swallowed by the two industries themselves and only 10 reaches the outside. The internal consumption dwarfs the external demand, which is the whole reason you cannot simply produce 10 and 60 and call it done.

One thing worth noticing

Every entry of (I − A)⁻¹ came out positive. That is not luck — it is the signal that this economy is productive, meaning it can meet any nonnegative demand with a sensible nonnegative production plan. If you ever solve one of these and get a negative quantity, you have either built A transposed or been handed an economy that consumes more than it makes. Either way, a negative answer here is a red flag, not something to round away.