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An Analysis of the Substitution Effect and of Revenue Effect in the Case of the Consumer's Theory Provided With a CES Utility Function

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  ACTA UNIVERSITATIS DANUBIUS Vol 8, No. 1/2012 164 An Analysis of the Substitution Effect and of Revenue Effect in the Case of the Consumer’s Theory Provided with a CES Utility Function Catalin Angelo Ioan 1, Gina Ioan 2   Abstract In the consumer’s theory, a crucial problem is to determine the substitution effect and the revenue effect in the case of one good price’s modifing. There exists two theories due to John Richard Hicks and Eugen Slutsky which allocates differents shares of the total change of the consumption to these effects. The paper makes an analysis between the two effects, considering the general case of a CES utility function and introduces three indicators which will characterize these shares. Keywords: CES; substitution; revenue; utility JEL Classification: D11   1. Introduction In the consumer’s theory, a crucial problem is to determine the substitution effect and the revenue effect in the case of one good price’s modifing. The theory due to John Richard Hicks consider after a modifing of a price, first a new allocation of goods preserving the utility, but modifing the revenue and after taking into account that the revenue is the initial one the changing in allocation due to a different utility. The theory of Eugen Slutsky consider a combined displacement of the relative consuming obtained a share of the substitution effect or of revenue effect depending only from the parameters of the utility. The problem is to determine these shares for both methods and to inquire which effect is uppermost. 1  Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, tel: +40372 361 102, fax: +40372 361 290, Corresponding author:   catalin_angelo_ioan@univ-danubius.ro 2  Assistant Professor, PhD in progress, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, tel: +40372 361 102, fax: +40372 361 290, e-mail: gina_ioan@univ-danubius.ro AUDŒ, Vol 8, no 1, pp. 164-175    ŒCONOMICA 165 2. The Analysis Let two goods A and B with the initial prices A p  and B p  and an utility function of a CES type U=  ( )  λ−λ−λ− β+α 1 YXT, α , β >0, λ >0, where X and Y are the consumed quantities in order to obtain an utility U. Let also, at a given time, V – the consumer’s revenue. In order to have the maximum utility for the revenue V it is known that we must have:  +== YpXpV ppUU BABAmBmA  where U mA =  ( ) 111 YXTX  −λ−λ−λ−−λ− β+αα  and U mB =  ( ) 111 YXTY  −λ−λ−λ−−λ− β+αβ  are the marginal utilities corresponding to the two goods A and B respectively. We have now:  +==βα −λ−−λ− YpXpV ppYX BABA11  Let note, in what follows: ϕ = βα , r 1 = BA pp  and: S= 1111 r  +λ−+λλ ϕ+ . We have therefore: XrXppY 1111111AB  +λ+λ −+λ− ϕ=      βα=  XrppV 11111BA      ϕ+=  +λ+λ − =Xprr B111111      ϕ+  +λ+λ −  We obtain now:  ACTA UNIVERSITATIS DANUBIUS Vol 8, No. 1/2012 166 X 1 = A11 SpVr  +λλ , Y 1 = B11 SpV +λ− ϕ  and the corresponding utility is: U 1 = B111 pSTV  λ+λ−λ−λ− ϕβ . Let suppose now that it is a change in the price of one of the goods, let say B, from B p  to B 'p , but the revenue V remains constant. Let note now: r 2 = BB p'p  and, of course: BA 'pp = 21 rr . Let note, also: R= 111211 rr  +λ−+λλ−+λλ ϕ+ , Q= SR . We have, from the upper relations: R-S=      −  +λλ+λλ−+λλ 121211 r1rr   121211 r1SrR +λλ−+λλ−+λ− −−=ϕ  Now: X 3 = A1211 RpVrr  +λλ−+λλ , Y 3 = B211 pRrV +λ− ϕ  and the corresponding utility: U 3 =  λ+λ−λ−λ− ϕβ 1B211 RprTV . We shall apply now the Hicks method for our analysis. At the modify of the price of B, for the same utility: U 1 = B111 pSTV  λ+λ−λ−λ− ϕβ  we shall have: U 1 =  λ+λ−λ−λ− ϕβ 1B211 Rpr'TV    ŒCONOMICA 167therefore: λ+λ−λ−λ−λ+λ−λ−λ− ϕβ=ϕβ 1B211B111 Rpr'TVpSTV  implies that:  λ+λ−λ+λ− = 112 RSVr'V  With the new revenue, we obtain: X 2H = A1111211 pRVSrr λ−λ+λ−+λ+λ λ  Y 2H = B1111 pRVS λ−λ+λ−+λ− ϕ . The substitution effect (which preserves the utility) gives us a difference: ∆ 1H X=X 2H -X 1 =VrSpQQr 11A11112  +λλλ−λ−+λ −   ∆ 1H Y=Y 2H -Y 1 =VSpQQ1 11B11 +λ−λ−λ− ϕ−  The difference caused by the revenue V instead V’ is therefore: ∆ 2H X=X 3 -X 2H = VrrRpQr1 1211A12  +λλ−+λλλ+λ −   ∆ 2H Y=Y 3 -Y 2H =VpRrrQ1 11B221 +λ−λ+λ ϕ−  named the revenue effect.
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