401-t2-10wa - E 401 1 Test 2 Tuesday 2nd March 2010 Answer...

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E frqrplfv 401 1 Test 2 Tuesday 2nd March 2010 Answer all questions. Each is worth 10 marks. 1. Consider a three-good demand system where x 1 = a + bp 1 /p 3 + cp 2 /p 3 + dp 1 p 2 /p 2 3 x 2 = e + fp 1 /p 3 + gp 2 /p 3 + hp 1 p 2 /p 2 3 . where the demand function for the third good follows from the person’s budget constraint. If this system is derived from a well-behaved utility-maximizing problem what are the restrictions on a, b, c, d, e, f, g and h ? Defend your answers carefully. ANSWER If this system is derived from a well-behaved utility-maximizing problem then it must satisfy the budget constraint, the demands must be homogeneous of degree zero in prices and money income and the Slutsky substitution matrix must be symmetric and nsd. The demand for the third good is de fi ned so that the budget constraint holds. Inspection of the equations for x 1 and x 2 shows that scaling prices and income by the same positive number has no e ff ect on the demands, so homogeneity is satis fi ed. Inspection of the demands for the fi rst two goods also shows that they are independent of income so not only are they Marshallian demands but they are also Hicksian demands. Thus S = h 1 p 1 h 1 p 2 S 13 h 2 p 1 h 2 p 2 S 23 S 31 S 32 S 33 = b/p 3 + dp 2 /p 2 3 c/p 3 + dp 1 /p 2 3 S 13 f/p 3 + hp 2 /p 2 3 g/p 3 + hp 1 /p 2 3 S 23 S 31 S 32 S 33 Note that the budget constraint and homogeneity imply the rows and columns of S are linearly dependent so we need check only the upper 2 X 2 submatrix of S. For symmetry we must have c/p 3 + dp 1 /p 2
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