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 P Lo (p 1,p 0 ;q b ) ≡ p 1 ∙q b / p 0 ∙q b where p t (t = 0,1) is a vector of prices and q b is a vector of quantities. Typically b ≤ 0 < 1.

3. Cobb-Douglas A Cobb-Douglas price index is defined as P CD (p 1,p 0 ;s b ) ≡ Π n (p n 1 /p n 0 ) s n b where p t (t = 0,1) is a vector of prices and s b is a vector of value shares p n b q n b / p b ∙q b. Typically b ≤ 0 < 1.

4. Benchmark Cost-of-Living index The benchmark Konüs COLI is defined as P K (p 1,p 0 ;q b ) ≡ C(p 1,U(q b )) / C(p 0,U(q b )) where U(.) is the consumer’s utility function and C(.) the dual cost function. It is assumed that C(p b,U(q b )) = p b ∙q b.

5. Second order approximations (1) Taylor series around p b : C(p 1,U(q b )) = p 1 ∙q b – e 1 C(p 0,U(q b )) = p 0 ∙q b – e 0 where e t (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 P K (p 1,p 0 ;q b ) - P Lo (p 1,p 0 ;q b ) 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 p b : ln C(p t,U(q b )) = ln C(p b,U(q b )) + ∑ n s n b ln (p n t /p n b ) + ε t where ε t (t = 0,1) are second-order terms which are not necessarily non-negative.

8. Bias of CD index Based on the foregoing one obtains P K (p 1,p 0 ;q b ) = P CD (p 1,p 0 ;s b ) exp{ε 1 – ε 0 }. A priori not much can be said about the bias of a CD CPI.

9. Lowe and CD It turns out that ln P Lo (p 1,p 0 ;q b ) – ln P CD (p 1,p 0 ;s b ) 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.