ReviewQuestion06 - this imaginary budget line and the new...

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Solution to review question for prelim 1 Question 1 a. The budget is 10 x 1 + x 2 = 270 . Utility u ( x 1 , x 2 ) = x 1 x 2 2 30 25 20 15 10 5 0 300 250 200 150 100 50 0 x1 x2 b. Note that this is a Cobb-Douglas utility so clearly there is an interior solution. It must satisfy: MRS = p 1 p 2 p 1 x 1 + p 2 x 2 = m. x 2 2 x 1 = 10 1 10 x 1 + x 2 = 270 . Solution is x 2 2 x 1 = 10 x 2 = 20 x 1 10 x 1 + 20 x 1 = 270 x 1 = 270 30 = 9 , x 2 = 180 . c. 1
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30 25 20 15 10 5 0 300 250 200 150 100 50 0 x1 x2 Optimal choice after tax must satisfy: MRS = p 1 p 2 p 1 x 1 + p 2 x 2 = m. x 2 2 x 1 = 10 1 10 x 1 + x 2 = 250 . Solution is x 2 2 x 1 = 10 x 2 = 20 x 1 10 x 1 + 20 x 1 = 250 x 1 = 250 30 = 8 1 3 , x 2 = 500 3 = 166 2 3 . Consumption of both goods decreases, just like one expect from normal good when income decreases. d) With the per unit tax the price of good x 1 is p 1 = 10 + t. MRS = p 1 p 2 p 1 x 1 + p 2 x 2 = m. x 2 2 x 1 = 10 + t (10 + t ) x 1 + x 2 = 270 . 2
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Solution is x 2 2 x 1 = 10 + t x 2 = 2(10 + t ) x 1 (10 + t ) x 1 + 2(10 + t ) x 1 = 270 x 1 = 270 3(10 + t ) = 90 (10 + t ) , x 2 = 180 . To fi nd graphically income and substitution e ff ect we need to fi nd a budget line that has the slope of the new budget p 1 p 2 = 10 + t , but is tangent to the original indi ff erence curve. Because we have a C-D utility, the tangency between
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Unformatted text preview: this imaginary budget line and the new indi f erence curve must be on the same ray from the origin as the no intervention point. (You should add labels to your graph: bold lines show old consumption and budget, this corresponds to a tangency point A, where dashed line tangent to bold indi f erence curve is the point B that. A → B gives the SE. New budget and indi f erence curve are the thin lines. Intersection there is C. B → C is the IE. Finally A → C is the TE. Because this problem has a C-D utility, B and C are on the same ray from the origin (the steeper dotted line). Also A and C are on the same horizontal line x 2 = 180 . 30 25 20 15 10 5 300 250 200 150 100 50 x1 x2 3...
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