This economics question tests your understanding of economic models and analysis. The step-by-step answer below applies the relevant framework and explains the reasoning.
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You're on a roll — here's the solution for question 12. Question 12: Using 1994 as origin, obtain a straight-line trend by the method of least square. Also find the value for year 1997. The straight-line trend equation is given by Y_c = a + bX, where Y_c is the trend value, X is the deviation from the origin, and a and b are constants. The normal equations for the method of least squares are: 1. Y = n a + b X 2. XY = a X + b X^2 Given data: | Years | Income (Y) | | :---- | :--------- | | 1990 | 140 | | 1991 | 144 | | 1992 | 160 | | 1993 | 152 | | 1994 | 168 | | 1995 | 176 | | 1996 | 180 | Origin = 1994. Let X = Year - 1994. Step 1: Create a table to calculate Y, X, XY, and X^2. |c|c|c|c|c| Year & Income (Y) & X = Year - 1994 & XY & X^2 \\ 1990 & 140 & -4 & -560 & 16 \\ 1991 & 144 & -3 & -432 & 9 \\ 1992 & 160 & -2 & -320 & 4 \\ 1993 & 152 & -1 & -152 & 1 \\ 1994 & 168 & 0 & 0 & 0 \\ 1995 & 176 & 1 & 176 & 1 \\ 1996 & 180 & 2 & 360 & 4 \\ Total & 1120 & -7 & -928 & 35 \\ Here, n = 7. Y = 1120 X = -7 XY = -928 X^2 = 35 Step 2: Substitute the values into the normal equations. 1. 1120 = 7a + (-7)b 1120 = 7a - 7b 2. -928 = (-7)a + 35b -928 = -7a + 35b Step 3: Solve the system of equations for a and b. Add equation (1) and equation (2): (1120) + (-928) = (7a - 7b) + (-7a + 35b) 192 = 28b b = (192)/(28) = (48)/(7) ≈ 6.857 Substitute the value of b into equation (1): 1120 = 7a - 7 ((48)/(7)) 1120 = 7a - 48 1120 + 48 = 7a 1168 = 7a a = (1168)/(7) ≈ 166.857 Step 4: Write the straight-line trend equation. Y_c = a + bX Y_c = 166.857 + 6.857X Step 5: Find the value for the year 1997. For the year 1997, X = 1997 - 1994 = 3. Substitute X=3 into the trend equation: Y_c = 166.857 + 6.857(3) Y_c = 166.857 + 20.571 Y_c = 187.428 The straight-line trend equation is Y_c = 166.857 + 6.857X. The trend value for the year 1997 is 187.428. What's next?