Input-Output Models - Finite Mathematics

1. What is the external demand for agricultural products from this economy?
   Answer: $320 million The external demand vector $D$ can be calculated using the balance equation $X = AX + D$, which rearranges to $D = X - AX = (I - A)X$. First, calculate $I-A$: $I - A = \begin{pmatrix} 1 - 0.25 & -0.40 \\ -0.10 & 1 - 0.20 \end{pmatrix} = \begin{pmatrix} 0.75 & -0.40 \\ -0.10 & 0.80 \end{pmatrix}$. Now, multiply by the production vector $X$: $D = \begin{pmatrix} 0.75 & -0.40 \\ -0.10 & 0.80 \end{pmatrix} \begin{pmatrix} 800 \\ 500 \end{pmatrix} = \begin{pmatrix} 0.75(800) - 0.40(500) \\ -0.10(800) + 0.80(500) \end{pmatrix} = \begin{pmatrix} 600 - 200 \\ -80 + 400 \end{pmatrix} = \begin{pmatrix} 400 \\ 320 \end{pmatrix}$. The external demand for agricultural products is the second component of the vector $D$, which is $320 million.
2. An economy's input-output analysis reveals that the Leontief inverse matrix $(I-A)^{-1}$ has an entry of 1.25 in row 2, column 3. Which statement correctly interprets this coefficient?
   Answer: A $1 increase in Sector 3's final demand generates $1.25 in total output for Sector 2 The Leontief inverse $(I-A)^{-1}$ gives total output requirements per unit of final demand. Entry (2,3) = 1.25 means that each $1 of final demand for Sector 3's output requires $1.25 of total output from Sector 2 (including direct, indirect, and induced effects). Choice A confuses this with the technology matrix A. Choice C misinterprets the coefficient as a percentage comparison. Choice D incorrectly describes it as additional activity rather than total requirement.
3. As a result of this change in demand, what is the required increase in the total output of the Energy sector?
   Answer: $40 billion The change in production, $\Delta X$, is related to the change in demand, $\Delta D$, by the formula $\Delta X = (I - A)^{-1} \Delta D$. The change in demand is an increase of $55 billion for Manufacturing (sector 2) and no change for Energy (sector 1). So, $\Delta D = \begin{pmatrix} 0 \\ 55 \end{pmatrix}$. First, we find $(I - A)^{-1}$. $I - A = \begin{pmatrix} 1-0.3 & -0.4 \\ -0.2 & 1-0.1 \end{pmatrix} = \begin{pmatrix} 0.7 & -0.4 \\ -0.2 & 0.9 \end{pmatrix}$. $\det(I - A) = (0.7)(0.9) - (-0.4)(-0.2) = 0.63 - 0.08 = 0.55$. $(I - A)^{-1} = \frac{1}{0.55} \begin{pmatrix} 0.9 & 0.4 \\ 0.2 & 0.7 \end{pmatrix}$. Now, we calculate the change in production: $\Delta X = \frac{1}{0.55} \begin{pmatrix} 0.9 & 0.4 \\ 0.2 & 0.7 \end{pmatrix} \begin{pmatrix} 0 \\ 55 \end{pmatrix} = \frac{1}{0.55} \begin{pmatrix} 0.9(0) + 0.4(55) \\ 0.2(0) + 0.7(55) \end{pmatrix} = \frac{1}{0.55} \begin{pmatrix} 22 \\ 38.5 \end{pmatrix} = \begin{pmatrix} 40 \\ 70 \end{pmatrix}$. The vector $\Delta X$ shows the required increase in production for each sector. The increase for the Energy sector (sector 1) is $40 billion.
4. Which of the following statements is the most accurate interpretation of the entry in row R, column T of the matrix $(I - A)^{-1}$?
   Answer: To satisfy $1 of external demand for Technology, a total of $0.50 worth of Real Estate output is required. The entry in row $i$, column $j$ of the Leontief inverse matrix $(I - A)^{-1}$ represents the total value of output from sector $i$ that is required to satisfy $1 of external (final) demand for the products of sector $j$. In this case, the entry in row R (row 2) and column T (column 3) is $0.5$. This means that to satisfy $1 of external demand for Technology, the economy must produce a total of $0.50 worth of output from the Real Estate sector. This total includes the direct inputs to the Technology sector and all indirect inputs to other sectors that support the production for that final demand.
5. What is the total value of the Energy sector's output that is used as input by all three sectors combined?
   Answer: $480 million The total value of a sector's output used as internal input is a component of the internal demand vector, $AX$. The Energy sector is sector 3. We need to calculate the third component of the vector $AX$. This is found by multiplying the third row of $A$ by the vector $X$. Third row of $A$: $[0.2, 0.2, 0.1]$. Calculation: $(AX)_3 = (0.2)(x_1) + (0.2)(x_2) + (0.1)(x_3)$ $(AX)_3 = (0.2)(800) + (0.2)(1000) + (0.1)(1200)$ $(AX)_3 = 160 + 200 + 120 = 480$. So, the total value of the Energy sector's output used as input by all sectors is $480 million.
6. To meet this external demand, what is the required total production for the Biofuels sector?
   Answer: $400 million We need to solve the system $(I - A)X = D$ for $X = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$. First, find $I - A$: $I - A = \begin{pmatrix} 0.8 & -0.1 & -0.3 \\ 0 & 0.6 & -0.2 \\ 0 & 0 & 0.5 \end{pmatrix}$. The system of linear equations is: 1) $0.8x_1 - 0.1x_2 - 0.3x_3 = 290$ 2) $0.6x_2 - 0.2x_3 = 140$ 3) $0.5x_3 = 250$ Since the system is upper triangular, we can solve by back substitution, starting from the last equation. From equation (3): $x_3 = \frac{250}{0.5} = 500$. Substitute $x_3 = 500$ into equation (2) to find $x_2$ (production for Biofuels): $0.6x_2 - 0.2(500) = 140$ $0.6x_2 - 100 = 140$ $0.6x_2 = 240$ $x_2 = \frac{240}{0.6} = 400$. So, the required production for the Biofuels sector is $400 million.
7. Based on this projection, what is the value of Raw Materials that will be consumed by the Manufacturing sector?
   Answer: $200 million The entry $a_{ij}$ of the technology matrix $A$ represents the dollar value of input from sector $i$ required to produce $1 of output for sector $j$. Here, Raw Materials is sector 1 and Manufacturing is sector 2. We want to find the value of input from Raw Materials (row 1) consumed by Manufacturing (column 2). This corresponds to the entry $a_{12}$. From the matrix, $a_{12} = 0.4$. This means $0.40 of Raw Materials are needed for every $1 of Manufacturing output. Since the total production of the Manufacturing sector is $500 million, the value of Raw Materials consumed by this sector is: $$a_{12}$ \times $x_2$ = 0.4 \times $500 \text{ million} = $200 \text{ million}.
8. What is the total value of output from Sector 2 that is consumed internally by all three sectors of the economy?
   Answer: $400 billion The internal demand is given by the vector $AX$. The total production vector $X$ can be found using $X = (I - A)^{-1}D$. After finding $X$, the internal demand can be calculated using the identity $AX = X - D$. Step 1: Calculate $X$. $X = \begin{pmatrix} 1.5 & 0.5 & 0.5 \\ 1.0 & 2.0 & 1.0 \\ 0.5 & 0.5 & 1.5 \end{pmatrix} \begin{pmatrix} 100 \\ 200 \\ 100 \end{pmatrix} = \begin{pmatrix} 1.5(100)+0.5(200)+0.5(100) \\ 1.0(100)+2.0(200)+1.0(100) \\ 0.5(100)+0.5(200)+1.5(100) \end{pmatrix} = \begin{pmatrix} 150+100+50 \\ 100+400+100 \\ 50+100+150 \end{pmatrix} = \begin{pmatrix} 300 \\ 600 \\ 300 \end{pmatrix}$. Step 2: Calculate internal demand $AX$. $AX = X - D = \begin{pmatrix} 300 \\ 600 \\ 300 \end{pmatrix} - \begin{pmatrix} 100 \\ 200 \\ 100 \end{pmatrix} = \begin{pmatrix} 200 \\ 400 \\ 200 \end{pmatrix}$. The second entry of this vector, $400, represents the total value of output from Sector 2 consumed internally.
9. A simplified economy has three sectors: Agriculture (A), Manufacturing (M), and Services (S). The technology matrix $T$ shows the input requirements per dollar of output: $T = \begin{pmatrix} 0.1 & 0.2 & 0.3 \\ 0.4 & 0.2 & 0.1 \\ 0.2 & 0.3 & 0.2 \end{pmatrix}$. If the final demand vector is $d = \begin{pmatrix} 100 \\ 150 \\ 200 \end{pmatrix}$ (in millions of dollars), what is the total output required for the Manufacturing sector?
   Answer: $267.3 million, including all inter-sector dependencies and feedback effects To find total output, we solve $(I - T)x = d$ where $I - T = \begin{pmatrix} 0.9 & -0.2 & -0.3 \\ -0.4 & 0.8 & -0.1 \\ -0.2 & -0.3 & 0.8 \end{pmatrix}$. Using matrix inversion or elimination, we get $x = \begin{pmatrix} 223.1 \\ 267.3 \\ 321.5 \end{pmatrix}$. The Manufacturing sector requires $267.3 million total output. Choice A uses an incorrect inverse calculation. Choice B omits some inter-sector feedback. Choice D ignores the input-output multiplier effects entirely.
10. An economist analyzes an input-output model where the technology matrix has been estimated from data spanning five years. She discovers that one sector's input coefficients have been trending upward due to technological change. What is the most significant implication for policy analysis using this model?
    Answer: The model's predictions become unreliable for long-term structural economic analysis Input-output models rely on a fundamental assumption that technology coefficients remain stable over time. When you encounter questions about changing technology matrices, focus on how this violates the model's core assumptions and affects its reliability for policy decisions. If input coefficients are trending upward over five years, this signals that the technology matrix is not constant—it's evolving. Input-output models assume fixed relationships between inputs and outputs, meaning each sector requires the same amount of inputs from other sectors to produce one unit of output. When these coefficients change systematically, the model can no longer provide reliable predictions about future economic structure or the effects of policy interventions. This makes option A correct: the model becomes unreliable for long-term structural economic analysis because its foundational assumption is violated. Option B incorrectly assumes the Leontief inverse will consistently overestimate—the direction of bias depends on whether coefficients continue rising or stabilize. Option C misses the point entirely; this isn't about current efficiency appearance but about model reliability over time. Option D makes an extreme mathematical claim that's generally false—equilibrium solutions typically exist even with changing coefficients, though they may be less meaningful. The key insight is that input-output analysis requires technological stability. When you see questions about changing technology matrices or evolving input coefficients, immediately think about model assumptions and reliability rather than getting caught up in mathematical details about specific calculations or efficiency measures.
11. Suppose the technology matrix is $A = \begin{pmatrix} 0.1 & 0.6 \\ 0.5 & 0.2 \end{pmatrix}$ and the external demand vector is $D = \begin{pmatrix} 42 \\ 84 \end{pmatrix}$. What is the required total production of the entire economy (Goods and Services combined)?
    Answer: $430 million The relationship between total production $X$, the technology matrix $A$, and external demand $D$ is given by the equation $X = AX + D$. This can be solved for $X$ as $X = (I - A)^{-1}D$. First, calculate $I - A$: $I - A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} 0.1 & 0.6 \\ 0.5 & 0.2 \end{pmatrix} = \begin{pmatrix} 0.9 & -0.6 \\ -0.5 & 0.8 \end{pmatrix}$. Next, find the inverse of $(I - A)$. The determinant is $\det(I - A) = (0.9)(0.8) - (-0.6)(-0.5) = 0.72 - 0.30 = 0.42$. The inverse is $(I - A)^{-1} = \frac{1}{0.42} \begin{pmatrix} 0.8 & 0.6 \\ 0.5 & 0.9 \end{pmatrix}$. Now, calculate $X$: $X = \frac{1}{0.42} \begin{pmatrix} 0.8 & 0.6 \\ 0.5 & 0.9 \end{pmatrix} \begin{pmatrix} 42 \\ 84 \end{pmatrix} = \frac{1}{0.42} \begin{pmatrix} 0.8(42) + 0.6(84) \\ 0.5(42) + 0.9(84) \end{pmatrix} = \frac{1}{0.42} \begin{pmatrix} 33.6 + 50.4 \\ 21 + 75.6 \end{pmatrix} = \frac{1}{0.42} \begin{pmatrix} 84 \\ 96.6 \end{pmatrix} = \begin{pmatrix} 200 \\ 230 \end{pmatrix}$. The total production is the sum of the components of $X$, which is $200 + 230 = 430$.
12. A regional input-output study finds that the 'backward linkage' index for the Construction sector is 1.8, while its 'forward linkage' index is 0.6. Based on this information, what development strategy would be most appropriate for this region?
    Answer: Treat Construction as a final demand sector requiring support from other regional industries A high backward linkage (1.8 > 1) means Construction has above-average demand for inputs from other sectors, while low forward linkage (0.6 < 1) means other sectors have below-average demand for Construction output. This suggests Construction acts more like a final demand sector - it purchases heavily from others but doesn't supply much to intermediate users. Choice A ignores the low forward linkage problem. Choice B is too negative given the positive backward effects. Choice C misses that Construction itself, not its suppliers, should be the focus.
13. In a Leontief input-output model, the technology matrix $A$ has the property that all column sums are less than 1. A policy maker claims this guarantees the economy is 'productive' and that $(I-A)^{-1}$ exists with all positive entries. Which aspect of this claim requires the most careful verification?
    Answer: Whether all entries of $(I-A)^{-1}$ are necessarily positive under this condition When you encounter Leontief input-output models, you're dealing with the fundamental question of whether an economy can produce enough output to meet both intermediate demands (inputs for other sectors) and final demand. The key insight is that column sums less than 1 is a necessary but not sufficient condition for economic productivity. The policy maker's claim contains a critical gap. While column sums less than 1 does guarantee that $(I-A)^{-1}$ exists (since it ensures the spectral radius of $A$ is less than 1), it doesn't automatically guarantee that all entries of $(I-A)^{-1}$ are positive. For economic productivity, you need the stronger condition that $A$ is primitive or that the economy is indecomposable and aperiodic. A matrix can have column sums less than 1 yet still produce some negative entries in $(I-A)^{-1}$, which would be economically meaningless. Option B is incorrect because column sums less than 1 does ensure invertibility of $(I-A)$. Option C is wrong because the column sum condition is necessary for productivity but not sufficient—you need additional conditions. Option D misses the point since the question is about what the given condition guarantees, not what invertibility implies. The most careful verification needed is whether the inverse matrix has all positive entries, since negative entries would violate the economic interpretation that increased final demand should lead to increased production in all sectors. Study tip: Remember that in input-output models, economic conditions are often stronger than pure mathematical conditions. Always check whether mathematical results make economic sense.
14. In a Leontief input-output model, the technology matrix is $A$. If the external demand changes from $D_1$ to $D_2$, the production level must change from $X_1$ to $X_2$ to maintain balance. Which expression correctly represents the new production level, $X_2$?
    Answer: $X_2 = X_1 + (I - A)^{-1}(D_2 - D_1)$ The fundamental equations are $X_1 = (I - A)^{-1}D_1$ and $X_2 = (I - A)^{-1}D_2$. We want to express $X_2$ in terms of $X_1$ and the change in demand. Let $\Delta D = D_2 - D_1$ be the change in demand and $\Delta X = X_2 - X_1$ be the change in production. We have $\Delta X = X_2 - X_1 = (I - A)^{-1}D_2 - (I - A)^{-1}D_1 = (I - A)^{-1}(D_2 - D_1)$. From this, we can solve for $X_2$: $X_2 = X_1 + \Delta X = X_1 + (I - A)^{-1}(D_2 - D_1)$. This shows that the new production level is the old production level plus the change in production required to meet the change in demand.
15. If the external demand for industrial products doubles, what is the resulting percentage increase in the total production of the agricultural sector?
    Answer: 42.9% This is a multi-step problem. First, find the initial production $X_1$. Second, find the new production $X_2$ after the demand changes. Third, calculate the percentage increase for the agricultural sector. Step 1: Find initial production $X_1$. $X_1 = (I - A)^{-1}D_1 = \begin{pmatrix} 2 & 0.75 \\ 1 & 2.25 \end{pmatrix} \begin{pmatrix} 100 \\ 200 \end{pmatrix} = \begin{pmatrix} 2(100) + 0.75(200) \\ 1(100) + 2.25(200) \end{pmatrix} = \begin{pmatrix} 200 + 150 \\ 100 + 450 \end{pmatrix} = \begin{pmatrix} 350 \\ 550 \end{pmatrix}$. The initial agricultural production is $x_{A1} = 350$. Step 2: Find the new demand $D_2$ and new production $X_2$. The demand for industrial products (sector 2) doubles, so $d_{I2} = 2 \times 200 = 400$. The new demand vector is $D_2 = \begin{pmatrix} 100 \\ 400 \end{pmatrix}$. $X_2 = (I - A)^{-1}D_2 = \begin{pmatrix} 2 & 0.75 \\ 1 & 2.25 \end{pmatrix} \begin{pmatrix} 100 \\ 400 \end{pmatrix} = \begin{pmatrix} 2(100) + 0.75(400) \\ 1(100) + 2.25(400) \end{pmatrix} = \begin{pmatrix} 200 + 300 \\ 100 + 900 \end{pmatrix} = \begin{pmatrix} 500 \\ 1000 \end{pmatrix}$. The new agricultural production is $x_{A2} = 500$. Step 3: Calculate the percentage increase. Percentage increase = $\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\% = \frac{500 - 350}{350} \times 100\% = \frac{150}{350} \times 100\% = \frac{3}{7} \times 100\% \approx 42.86\%$. This rounds to $42.9\%$.
16. In a two-sector Leontief model, Sector 1 uses $0.30 of its own output and $0.20 of Sector 2's output per dollar produced. Sector 2 uses $0.10 of Sector 1's output and $0.40 of its own output per dollar produced. If Sector 1 increases its final demand by $50 million while Sector 2's final demand remains constant, what is the induced increase in Sector 2's total output?
    Answer: $24.7 million, incorporating both direct and indirect multiplier effects The technology matrix is $T = \begin{pmatrix} 0.3 & 0.1 \\ 0.2 & 0.4 \end{pmatrix}$, so $I - T = \begin{pmatrix} 0.7 & -0.1 \\ -0.2 & 0.6 \end{pmatrix}$. The inverse is $(I-T)^{-1} = \begin{pmatrix} 1.493 & 0.249 \\ 0.498 & 1.741 \end{pmatrix}$. For demand change $\begin{pmatrix} 50 \\ 0 \end{pmatrix}$, the output change is $\begin{pmatrix} 74.65 \\ 24.9 \end{pmatrix}$. Sector 2's increase is approximately $24.7 million. Choice A uses only first-order effects. Choice C overcalculates the multiplier. Choice D uses only the direct coefficient 0.2 × 50 = 10.
17. What is the value of $a_{12}$, which represents the input from Goods required to produce $1 of Services?
    Answer: $0.10 The governing equation is $X = AX + D$. We can write this out for each sector. The question asks for $a_{12}$, which appears in the equation for the first sector (Goods). $x_1 = a_{11}x_1 + a_{12}x_2 + d_1$. Substitute the given values: $1000 = (0.1)(1000) + a_{12}(2000) + 700$. Now, solve for $a_{12}$: $1000 = 100 + 2000a_{12} + 700$ $1000 = 800 + 2000a_{12}$ $200 = 2000a_{12}$ $a_{12} = \frac{200}{2000} = 0.10$.

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