- Appendix B: Variable Shares in the Index of Real Household Incomes
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APPENDIX B VARIABLE SHARES IN THE INDEX OF REAL HOUSEHOLD INCOMES Expression (6.2 ) i n Chapter 6 could be considered to hold quite accuratel y for predictin g rea l incom e change s fro m on e perio d t t o th e nex t t+1. However, in period t+1 pric e changes will induce shifts in the shares, and hence the expression will progressively lose its accuracy over time. The original expression (6.2) presupposes that the household operates along a productio n possibilit y frontie r o f agricultura l an d nonagricultural activities . Suppose that this frontier is given by the following constan t elasticity o f transformation (CET ) separable expression: (B.l) where A, N are the quantities of agricultura l an d nonagricultural products produced , Q i s the index of total household real product, Lh and Khare the amounts of labo r an d capital supplie d b y th e specifi c household , T is th e elasticit y o f transformation, and a, b are parameters. If Pah and Pn are the prices of A and N faced by the household (notice Pah i s not the same as Pa), the n profit maximization subject to (B.l) implies that the quantities supplied are equal to 196 Appendix B. Variable Shares in the Index of RealHouseholdIncomes19 7 where (B.2) (B.3) (B.4) Furthermore, we assume that A is a CET index of the three classes of agricultural products, while N is a CET index of formal and informal products: (B.5) (B.6) The allocation of A to the .A, 's and N t o the Nt's wil l be done according to the formulas (B.7) where (B.8) (B.9) (B.10) under the nestedCET structure. and the nominal income of the household is equal to 198 Alexander H.Sarris and Rogier van den Brink Log-differentiating expressio n (B.4) , we obtain the term in the first bracket in (6.2) if we assume that Kh , and in particular Lh , stay unchanged, so that the amount of the total product Q does not change. If, however, we assume that there is a short-run aggregate labor supply response of the sort, for instance, (B.ll) then expressio n (6.2 ) woul d hav e t o b e multiplie d b y 1 + X where X is th e elasticity of labor supply with respect to real income. In the empirical part later, the value of X is assumed to be equal to zero. The issue, however, is to derive expressions fo r the next period weights s a, Sat, sn, an d s nt given th e impose d structure . Th e log-chang e i n th e shar e o f household income deriving from agriculture between two periods can be written from the above structure, as follows: (B.12) Utilizing (B.2) and (B.4), this can be written as (B.13) The change in s„ will obtain as a residual from (6.1). The log-change i n sai i s given by the following expressio n in view o f (B.7 ) and (B.9): (B.14) where in accordance with (B.9) (B.15) Similarly, for the nonagricultural shares (B.16) where (B.17) Appendix B. Variable Shares in the Index of RealHousehold Incomes 19 9 The price Pnf i s the formal-sector average wage rate, while Pnu i s the unit labor reward r of the informal sector. The above expressions can be used for period-to-period updating of shares, as follows: The above expressions depend on only three parameters, namely, T, Ta, an d Tn. Give n the lack of empirical information on these, we could examine alternative values. Notice that under the assumptions made, the short-run agricultural supply response can be derived by log-differentiating th e expressions (B.7) and (B.2). Utilizing also the corresponding price expressions (B.3) and (B.9), we can write the following: (B.24) The expression multiplyin g P ai i n (B.24 ) abov e i s th e short-run , own-suppl y price elasticity , whil e th e expression s multiplyin g P aj are th e cross-pric e elasticities. Notice that the analysis up to now concerned the period-to-period changes in the various shares that pertain to the typical household. For the analysis of...

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