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Lowe and Cobb-Douglas CPIs and their substitution bias

Lowe and Cobb-Douglas CPIs and their substitution bias. Bert M. Balk Statistics Netherlands and Rotterdam School of Management Erasmus University Washington DC, 17 May 2008. Lowe. A Lowe price index is defined as P Lo (p 1 ,p 0 ;q b ) ≡ p 1 ∙q b / p 0 ∙q b

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Lowe and Cobb-Douglas CPIs and their substitution bias

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  1. Lowe and Cobb-Douglas CPIs and their substitution bias Bert M. Balk Statistics Netherlands and Rotterdam School of Management Erasmus University Washington DC, 17 May 2008

  2. Lowe • A Lowe price index is defined as • PLo(p1,p0;qb) ≡ p1∙qb / p0∙qb • where pt (t = 0,1) is a vector of prices and qb is a vector of quantities. Typically • b ≤ 0 < 1.

  3. Cobb-Douglas • A Cobb-Douglas price index is defined as • PCD(p1,p0;sb) ≡Πn(pn1/pn0)snb • where pt (t = 0,1) is a vector of prices and sb is a vector of value shares pnbqnb/ pb∙qb. Typically • b ≤ 0 < 1.

  4. Benchmark Cost-of-Living index • The benchmark Konüs COLI is defined as • PK(p1,p0;qb) ≡ C(p1,U(qb)) / C(p0,U(qb)) • where U(.) is the consumer’s utility function and C(.) the dual cost function. • It is assumed that • C(pb,U(qb)) = pb∙qb .

  5. Second order approximations (1) • Taylor series around pb: • C(p1,U(qb)) = p1∙qb – e1 • C(p0,U(qb)) = p0∙qb – e0 • where et (t = 0,1) are second-order terms which are non-negative.

  6. Balk-Diewert (2003) result • Under systematic long run trends in prices the bias • PK(p1,p0;qb) - PLo(p1,p0;qb) • is negative; depends on the distance between periods b and 0; and is a quadratic function of the distance between periods 0 and 1.

  7. Second order approximations (2) • An other Taylor series around pb: • ln C(pt,U(qb)) = ln C(pb,U(qb)) + • ∑n snb ln (pnt/pnb) + εt • where εt (t = 0,1) are second-order terms which are not necessarily non-negative.

  8. Bias of CD index • Based on the foregoing one obtains • PK(p1,p0;qb) = PCD(p1,p0;sb) exp{ε1 – ε0}. • A priori not much can be said about the bias of a CD CPI.

  9. Lowe and CD • It turns out that • ln PLo(p1,p0;qb) – ln PCD(p1,p0;sb) • is proportional to the covariance between price-adjustment of shares over [b,t] and relative price changes over [0,t]. This is likely to be non-negative.

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