Lowe and Cobb-Douglas CPIs and their substitution bias

Bert M. BalkStatistics Netherlands

andRotterdam School of Management

Erasmus University

Washington DC, 17 May 2008

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.

Cobb-Douglas

A Cobb-Douglas price index is defined as

PCD(p1,p0;sb) ≡ Πn(pn1/pn

0)snb

where pt (t = 0,1) is a vector of prices and sb is a vector of value shares pn

bqnb/ pb∙qb. Typically

b ≤ 0 < 1.

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 .

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.

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.

Second order approximations (2)

An other Taylor series around pb:

ln C(pt,U(qb)) = ln C(pb,U(qb)) +

∑n snb ln (pn

t/pnb) + εt

where εt (t = 0,1) are second-order terms which are not necessarily non-negative.

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.

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.