1. Consider an economy without production in which there are two goods and two c
ID: 1120999 • Letter: 1
Question
1. Consider an economy without production in which there are two goods and two consumers. The endowment of good l is 1-10 and the endowment of good 2 is 2-20. The preferences of individual 1 are represented by the following utility function: where X1 and x12 are, respectively, her consumption of good 1 and her consumption of good 2. The preferences of individual 2 are represented by the following utility function: Uz(X21, X22) 1X22, where X21 and x22 are, respectively, her consumption of good 1 and her consumption of good 2. The social planner wants to allocate the goods available to the two individuals to maximize the central planner solves the following constrained maximization problem: 27 (1) max 122)Ui(xi1, Xi2) subject to (2) X11 + X21 =Explanation / Answer
Eq 1. x11 + x21 = 20 (endowment of good 1)
Eq 2. x12 + x22 = 10 (endowment of good 2)
Eq 3. sq(x21). x22 = 8000/27
From Eq 3,
sq(x21). (10 - x12) = 8000/27
10 - x12 = 8000/ (27 . sq(x21))
x12 = 10 - (8000 / (27 . sq(x21)))
From Eq 1, x11 = 20 - x21
Therefore, the utility function u1 = x11. sq(x12)
u1 = (20 - x21). sq ( 10 - 8000 / (27 . sq(x21)))
u1 = (20 - x21). sq (270 . sq(x21) - 8000)/ (x21 . sq(27))
To maximise u, we must differentiate it with respect to x21
du1/ dx21 = (1/sq(27)) . (x21 . d((20 - x21). sq (270 . sq(x21) - 8000)) - (20 - x21). sq (270 . sq(x21) - 8000) . d(x21)) / sq(x21)
When the first differential is 0, the function attains maximum or minimum.
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