401-t2a-11w

401-t2a-11w - E 401 1 Test 2 Thursday 3rd March 2011 Answer...

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E frqrplfv 401 1 Test 2 Thursday 3rd March 2011 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 certain restrictions must apply to a, b, c, d, e, f, g and h . Impose these restrictions and set p 3 =1 . Now calculate the equivalent variation and the compensating variation measures of welfare change for altering prices from ( p 1 ,p 2 )=(1 , 2) to ( p 1 ,p 2 )=(2 , 1) . 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 f ned so that the budget constraint holds. Inspection of the equations for x 1 and x 2 shows that scaling prices and income bythesamepositivenumberhasnoe f ect on the demands, so homogeneity is satis f ed. Inspection of the demands for the f 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
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E frqrplfv 401 2 c/p 3 + dp 1 /p 2 3 = f/p 3 + hp 2 /p 2 3 , for all p 1 ,p 2 ,p 3 > 0 . This means that d = h =0 and c = f. Then S 11 0 means b 0 and S 22 0 means g 0 .S 11 S 22 S 2 12 0 implies bg c 2 0 . Finally, if we want the demands to be nonnegative for all prices then a, e > 0 . Imposing these restrictions and setting p 3 =1 yields h 1 ( p 1 ,p 2 , 1 ,u 0 )= a + bp 1 + cp 2 h 2 ( p 1 ,p 2 , 1 ,u 0 )= e + cp 1 + gp 2 Since these Hicksian demands are independent of utility, EV = CV. Moreover, with the appropriate restrictions imposed, the Slutsky substitution matrix is symmetric
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401-t2a-11w - E 401 1 Test 2 Thursday 3rd March 2011 Answer...

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