# prelims_Micro Prelim ANSWERS August 2008 - Microeconomics...

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Microeconomics Prelim August 2008 A NSWER K EY Question 1. (a) . Let P the set of relevant price-wealth vectors ( p , w ), a subset of 1 L + ++ , and denote by : j xP →ℜ ± a consumer’s Walrasian demand for good j , j = 1,… L , assumed to be strictly positive and differentiable on P . (a).1 . What do we mean when we say that good j is a necessity for the consumer at ( p , w )? Same for luxury and for borderline necessity-luxury . Necessity : (,) 01 j j xpw w wx p w << ± ± . Luxury : 1 j j w p w > ± ± . Borderline Necessity-Luxury : 1 j j w p w = ± ± . (a).2 . Show that the concepts of luxury and borderline necessity-luxury can be characterized by a property of the budget share function j bpw of the good. Can you do the same with the concept of necessity ? Explain. The property is the sign of the derivative ( ) jj j p ww ∂∂ ± , which can be computed as 22 1 1 j j j j p w pp x p w w x p w w ⎡⎤ −= ⎢⎥ ⎣⎦ ±± ± ± ± . Hence, j w p w w >⇔ > ± ± , which is the definition of a luxury at ( p , w ). Similarly for the necessity-luxury borderline. If good j is a necessity, then the previous equality shows that 0 j w < . But if 0 j w < ± , then 0 j w < , but the good is not a necessity in the sense of the above definition. For the rest of this question we consider the indirect utility function 1 () :: ( , ) ( ) ln( / ( )) Fp vP vpw Gp wCp = + , (1)

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2 where C ( p ) >> 0, and the functions C , F and G are such that v ( p , w ) has the properties of an indirect utility function on P . (b) . For j = 1, …, L , obtain the Walrasian demand function (,) j x pw ± and the budget share function j bpw corresponding to (1). By Roy’s identity, / / j j vpw p xpw w ∂∂ =− ± , which if computed from (1) yields 2 2 2 2 2 ln () ln( / ( )) ln 1 ln( / ( )) ln jj j j Fw C wC F pC p Fp G Gp wCp p w C C F w C ⎛⎞ −− ⎜⎟ ⎡⎤ ⎝⎠ −+ + ⎢⎥ ⎣⎦ × ± 2 2 ln ln j C G w F p Cw C p p C F w + = , i. e., 2 l n l n j j wF w wC wG w Fp C Cp Fp C =+ + ± . (2) This in turn yields 2 l n l n j j j j j px p p p C G w wF p C C p F p C ≡= + + ± . (3) (c). Consider first the case of (1) with G ( p ) = 0, all p . Show that if good j is a luxury at some (,) 0 pw >> , then it is a luxury at p w , for all w > 0. If G ( p ) = 0, p , then l n j j j pp C Fp C Cp ≡+ . (4)
3 As seen in (a).2, luxury at (,) p w 0 j bpw w > , which for (4) is equivalent to () 0 j j p Fp > , in which case 0, 0 j w w >∀> . (d). Consider now the general case of (1) where G ( p ) is not always zero. (d).1 . Suppose that good j is a luxury at some (,) 0 pw >> . Does it follow that it is a luxury at p w , for all w > 0? Explain. Now 2ln( ) ln( ) jj j p p FG w w Fp Fp Cp ∂∂ =+ , which may in principle change signs as w varies, in which case good j is a luxury at some w ’s, but at for others.

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prelims_Micro Prelim ANSWERS August 2008 - Microeconomics...

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