Matrix Model of a Closed Economy That Will Not Balance
A simple two-sector economy has input-output matrix A = [[0.2, 0.4],[0.5, 0.1]], where aᵢⱼ is the input from sector i needed per unit output of sector j. Final demand is d = [30, 20]ᵀ. A student solves (I − A)x = d, obtains x, and reports production levels. They then check that total inputs equal total outputs and find a mismatch of 10 units, concluding the matrix is wrong. The teacher says the model is fine but the check was misconceived.
1. Compute the production vector x and show the working.
Read the worked answer
I − A = [[0.8, −0.4],[−0.5, 0.9]].
Determinant = 0.8·0.9 − (−0.4)(−0.5) = 0.72 − 0.20 = 0.52.
Inverse = (1/0.52)[[0.9, 0.4],[0.5, 0.8]].
x = inverse · [30, 20]ᵀ = (1/0.52)[[0.9·30 + 0.4·20],[0.5·30 + 0.8·20]] = (1/0.52)[35, 31] ≈ [67.31, 59.62]ᵀ.
So sector 1 produces about 67.3 units and sector 2 about 59.6 units.
2. Explain why the student's 'total inputs equals total outputs' check was invalid, and state a correct consistency check.
Read the worked answer
In an open input-output model, total output of each sector equals intermediate demand plus final demand: x = Ax + d. Total inputs across sectors need not equal total outputs in the naive sense the student used, because final demand absorbs the difference and the columns of A need not sum to 1. The mismatch of 10 is simply the final demand not being counted in their check.
A correct check is to substitute back: Ax + d should equal x. Compute Ax = [[0.2·67.31 + 0.4·59.62],[0.5·67.31 + 0.1·59.62]] ≈ [[13.46 + 23.85],[33.66 + 5.96]] = [37.31, 39.62]. Adding d = [30, 20] gives [67.31, 59.62] = x, confirming consistency. The judgement call is knowing what conservation law the model actually encodes.