Question

Hi, Please help with both questions please!!

1. Two vectors(1,2,4,-6) E R4 and y - (7,4,-9,10) E R4 are given. Compute the following a) The dot product x. y. (b) Weighted averages Xx (1-λ)y for the following values of X i. X - 0.5; iii. A- 0.25 In the definition of the two good budget set {(zî, Χ2) e R.IP1.rı +p2x2 ^ m rn): we argued that we could normalize the price of good 2 to unity. (a) Express the budget set corresponding to pi-7, p2 -3.5, and (b) Repcat the above excrcise for pi -8, p2 -2, and m 12 and 10.5 in such a form. Draw this budget set draw this budget set on the same axes (c) Repeat the above exercise for p 10, p2 -5, and m- 15, and draw this budget set also on the same axes.

Answer 1

Step by step solution is provided in the pic attached here.

x = (1, 2, 4, -6) and y = (7, 4, -9, 10) (a) x middot y = (1, 2, 4, -6) middot (7, 4, -9, 10) = 7 + 8 - 36 - 60 = 15 - 96 = -81 (b) (i) lambda x + (-lambda) y = 0.5 x + 0.5 y = 0.5 (1, 2, 4, -6) + 0.5 (7, 4, -9, 10) = (0.5, 1, 2, -3) + (3.5, 2, -4.5, 5) = (4, 3, -2.5, 2) (ii) lambda x + (1 - lambda) y = 0.75 x + 0.25 y = 0.75 (1, 2, 4, -6) + 0.25 (7, 4, -9, 10) = (0.75, 1.5, 3, -4.5) + (1.75, 1, -2.25, 2.5) = (2.5, 2.5, 0.75, -2) (iii) lambda = 0.25, = 0.25 x + 0.75 y = 0.25 (1, 2, 4, -6) + 0.75 (7, 4, -9, 10) = (1, 0.5, 1, -1.5) + (5.25, 3, -6.75, 7.5) = (6.25, 3.5, -5.75, 6) (a) p_1 = 7

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