16 Input output model

📊 Input-Output Model: 9 Numerical PYQs (UGC NET / SWAYAM)

✔️ प्रत्येक प्रश्न में पूरा हल (step-by-step) और अंतिम उत्तर दिया गया है।

प्रश्न 1 दो-सेक्टर वाली अर्थव्यवस्था में Total Output की गणना

[UGC NET Paper 2: Economics 3rd Jan 2025 Shift 2]
The input-output matrix for a two-sector economy is given by: \[ A = \begin{bmatrix} 0.25 & 0.40 \\ 0.10 & 0.10 \end{bmatrix} \] If the external demand for the outputs of the two sectors is \( D = \begin{bmatrix} 10 \\ 20 \end{bmatrix} \), what will be the optimal output levels of the two commodities?
Options: (1) 30.20 and 26.40    (2) 28.50 and 20.40    (3) 26.77 and 25.20    (4) 25.50 and 22.20

✅ Solution (Step-by-Step):

Step 1: Formula: \( X = AX + D \) ⇒ \( (I - A)X = D \) ⇒ \( X = (I - A)^{-1} D \)

Step 2: Find \( I - A \): \[ I - A = \begin{bmatrix} 1-0.25 & 0-0.40 \\ 0-0.10 & 1-0.10 \end{bmatrix} = \begin{bmatrix} 0.75 & -0.40 \\ -0.10 & 0.90 \end{bmatrix} \]

Step 3: Determinant: \( |I - A| = (0.75 \times 0.90) - [(-0.40) \times (-0.10)] = 0.675 - 0.04 = 0.635 \)

Step 4: Inverse: \( (I - A)^{-1} = \frac{1}{0.635} \times \begin{bmatrix} 0.90 & 0.40 \\ 0.10 & 0.75 \end{bmatrix} \)

Step 5: Multiply with D:
\( X_1 = \frac{(0.90 \times 10) + (0.40 \times 20)}{0.635} = \frac{9 + 8}{0.635} = \frac{17}{0.635} = 26.77 \)
\( X_2 = \frac{(0.10 \times 10) + (0.75 \times 20)}{0.635} = \frac{1 + 15}{0.635} = \frac{16}{0.635} = 25.20 \)

✅ Final Answer: 26.77 and 25.20 (Option 3)

प्रश्न 2 स्टील और ऑटोमोबाइल अर्थव्यवस्था (Steel-Automobile Economy)

[SWAYAM MOOCS - Mathematical Economics]
An economy consists of two industries - Steel and Automobiles. To produce one-rupee worth of Steel, the Steel industry requires ₹0.2 worth of Steel and ₹0.7 worth of Automobiles. To produce one-rupee worth of Automobiles, the Automobile industry requires ₹0.5 worth of Steel and ₹0.1 worth of Automobiles. The economy has to export ₹15,000 worth of Steel and ₹5,000 worth of Automobiles. How much worth of Steel and Automobiles should be produced to meet the total demand?

✅ Solution (Step-by-Step):

Step 1: Let X = Steel (in thousand ₹), Y = Automobile (in thousand ₹).
\( X = 0.2X + 0.5Y + 15 \)    \( Y = 0.7X + 0.1Y + 5 \)

Step 2: Rearranged: \( 0.8X - 0.5Y = 15 \)
\( -0.7X + 0.9Y = 5 \)

Step 3: Matrix form: \( (I - A) = \begin{bmatrix} 0.8 & -0.5 \\ -0.7 & 0.9 \end{bmatrix} \), \( |I-A| = 0.72 - 0.35 = 0.37 \)

Step 4: \( X = \frac{(0.9 \times 15)+(0.5 \times 5)}{0.37} = \frac{13.5+2.5}{0.37} = \frac{16}{0.37} = 43.235\) thousand ₹
\( Y = \frac{(0.7 \times 15)+(0.8 \times 5)}{0.37} = \frac{10.5+4}{0.37} = \frac{14.5}{0.37} = 39.189\) thousand ₹

✅ Final Answer: Steel = ₹43,235 , Automobiles = ₹39,189

प्रश्न 3 आगत गुणांक मैट्रिक्स (Input-Coefficient Matrix) बनाना

[SWAYAM MOOCS - Mathematical Economics]
Given is an input-output table of an economy (₹ lakhs):

	Banking	Insurance	Education & Research	Total Output
Banking	80000	8000	30000	122000
Insurance	21000	6000	3000	30000
Education & Research	0	0	10000	110000

Convert the input-output table into an input-coefficient matrix.

✅ Solution (Step-by-Step):

Step 1: Formula: \( a_{ij} = x_{ij} / X_j \)

Step 2: Column 1 (Banking): \( a_{11}=80000/122000=0.6557,\ a_{21}=21000/122000=0.1721,\ a_{31}=0 \)

Step 3: Column 2 (Insurance): \( a_{12}=8000/30000=0.2667,\ a_{22}=6000/30000=0.20,\ a_{32}=0 \)

Step 4: Column 3 (Education): \( a_{13}=30000/110000=0.2727,\ a_{23}=3000/110000=0.0273,\ a_{33}=10000/110000=0.0909 \)

✅ Final Answer:
\[ A = \begin{bmatrix} 0.6557 & 0.2667 & 0.2727 \\ 0.1721 & 0.20 & 0.0273 \\ 0 & 0 & 0.0909 \end{bmatrix} \]

प्रश्न 4 किसान का राजस्व (Farmer's Revenue Problem)

[SWAYAM MOOCS - Mathematical Economics]
A farmer sells potatoes (₹30/kg) and tomatoes (₹20/kg). Three customers demand:
Customer 1: 2 kg potatoes + 1 kg tomatoes
Customer 2: 1 kg potatoes + 2 kg tomatoes
Customer 3: 5 kg potatoes + 2 kg tomatoes
Find revenue from each customer and total revenue.

✅ Solution (Step-by-Step):

Step 1: Customer 1: \( (30\times2)+(20\times1)=60+20= ₹80 \)

Step 2: Customer 2: \( (30\times1)+(20\times2)=30+40= ₹70 \)

Step 3: Customer 3: \( (30\times5)+(20\times2)=150+40= ₹190 \)

Step 4: Total Revenue = \( 80+70+190= ₹340 \)

✅ Final Answer: C1: ₹80, C2: ₹70, C3: ₹190, Total = ₹340

प्रश्न 5 चौथा ग्राहक जोड़ना (Fourth Customer Addition)

[SWAYAM MOOCS - Mathematical Economics (Extended from Q4)]
(a) Draw the input-output table for the situation given in Q4 and find revenue.
(b) Formulate a mathematical expression to show the relationship between price, quantity and revenue.
(c) If a fourth customer demands 3 kg potatoes + 2 kg tomatoes, calculate total revenue.

✅ Solution (Step-by-Step):

Step (a): Input-Output Table

Customer	Potatoes(kg)	Tomatoes(kg)	Revenue(₹)
1	2	1	80
2	1	2	70
3	5	2	190
Total	8	5	340

Step (b): Mathematical expression: \( P_x X_1 + P_y X_2 = Y \) where \( P_x=30, P_y=20 \)

Step (c): Fourth customer: \( 30\times3 + 20\times2 = 90+40 = ₹130 \); New total = \( 340+130 = ₹470 \)

✅ Final Answer (c): ₹130, Total = ₹470

प्रश्न 6 Direct and Indirect Requirements (प्रत्यक्ष एवं अप्रत्यक्ष आवश्यकताएँ)

[Based on UGC NET Pattern]
Suppose Leontief input-output coefficient matrix is: \( A = \begin{bmatrix} 0.1 & 0.4 \\ 0.2 & 0.5 \end{bmatrix} \)
Final demand vector \( D = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \). Find the total direct and indirect requirement of the second input to satisfy final demand.

✅ Solution (Step-by-Step):

Step 1: \( I - A = \begin{bmatrix} 0.9 & -0.4 \\ -0.2 & 0.5 \end{bmatrix} \), \( |I-A| = 0.45 - 0.08 = 0.37 \)

Step 2: \( (I-A)^{-1} = \frac{1}{0.37} \begin{bmatrix} 0.5 & 0.4 \\ 0.2 & 0.9 \end{bmatrix} \)

Step 3: \( X = (I-A)^{-1} D = \frac{1}{0.37} \begin{bmatrix} 0.5+0.4 \\ 0.2+0.9 \end{bmatrix} = \frac{1}{0.37} \begin{bmatrix} 0.9 \\ 1.1 \end{bmatrix} = \begin{bmatrix} 2.43 \\ 2.97 \end{bmatrix} \)

✅ Total requirement of second input = 2.97 units

प्रश्न 7 Coal and Steel Economy (कोयला और इस्पात)

[Based on SWAYAM MOOCS Pattern]
An economy produces only Coal (C) and Steel (S). To produce 1 tonne of Steel: needs 0.4 tonne Steel + 0.7 tonne Coal. To produce 1 tonne of Coal: needs 0.1 tonne Steel + 0.6 tonne Coal. Economy needs 100 tonnes Coal and 50 tonnes Steel for export. Find gross output of Coal and Steel.

✅ Solution (Step-by-Step):

Step 1: \( C = 0.6C + 0.7S + 100 \) , \( S = 0.1C + 0.4S + 50 \)

Step 2: Rearrange: \( 0.4C - 0.7S = 100 \) , \( -0.1C + 0.6S = 50 \)

Step 3: \( (I-A) = \begin{bmatrix} 0.4 & -0.7 \\ -0.1 & 0.6 \end{bmatrix} \), \( |I-A| = 0.24 - 0.07 = 0.17 \)

Step 4: \( C = \frac{(0.6 \times 100) + (0.7 \times 50)}{0.17} = \frac{60+35}{0.17}=558.82 \) tonnes, \( S = \frac{(0.4 \times 50)+(0.1 \times 100)}{0.17} = \frac{20+10}{0.17}=176.47 \) tonnes

✅ Final Answer: Coal = 558.82 tonnes, Steel = 176.47 tonnes

प्रश्न 8 Hawkins-Simon Conditions – Numerical Verification

[UGC NET Paper 2: Economics 30th Sept 2020]
Verify whether the following system satisfies Hawkins-Simon conditions: \( A = \begin{bmatrix} 0.5 & 0.3 \\ 0.2 & 0.4 \end{bmatrix} \)

✅ Solution (Step-by-Step):

Step 1: \( I-A = \begin{bmatrix} 0.5 & -0.3 \\ -0.2 & 0.6 \end{bmatrix} \)

Step 2 – Condition 1 (diagonals >0): \( 1-a_{11}=0.5>0 \), \( 1-a_{22}=0.6>0 \) ✓

Step 3 – Condition 2 (determinant >0): \( |I-A| = (0.5 \times 0.6) - [(-0.3)(-0.2)] = 0.30 - 0.06 = 0.24 >0 \) ✓

✅ Final Answer: Both conditions satisfied ⇒ System is viable.

प्रश्न 9 Leontief Production Function – Concept Based

[UGC NET Paper 2: Commerce 17th Oct 2020 Shift 1]
In Leontief production function, L (Labour) is considered as:
(1) Free variable  (2) Slack variable  (3) Binding constraint in the production process  (4) Surplus variable

✅ Solution:

Step 1: Leontief production function = Fixed proportions: \( Q = \min(K/a, L/b) \)

Step 2: Factors are used in fixed ratio, no substitution. The factor that limits output is binding constraint. Labour cannot be replaced by capital → becomes binding constraint.

✅ Final Answer: (3) Binding constraint in the production process

📚 स्रोत: UGC NET Official PYQs, SWAYAM MOOCS Mathematical Economics & Examrace Practice Sets.
✅ सभी प्रश्नों में पूर्ण चरणबद्ध हल (Numerical + Conceptual) दिया गया है।

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