Implicit and Explicit Price Indexes Compared

TheoryGuru applied to Chicago Price Theory

Setup

Get["http://economicreasoning.com"]

In the Wolfram Language, x.y refers to the tensor DOT PRODUCT, NOT scalar multiplication.  For TheoryGuru purposes, tensor means vector, so that the result of x.y is a scalar.

Expenditure growth factor

efact=((p2.q2)/(p1.q1));

Inflation factors

pfactLaspeyres=((p2.q1)/(p1.q1));
pfactPaasche=((p2.q2)/(p1.q2));

Quantity growth factors

qfactLaspeyres=((p1.q2)/(p1.q1));
qfactPaasche=((p2.q2)/(p2.q1));

Squared Fisher factors (i.e., Fisher ideal index is a square root)

pfactFishersq=pfactLaspeyres pfactPaasche;
qfactFishersq=qfactLaspeyres qfactPaasche;

Results

The implicit price index ((Expenditure/(Quantity Index))) is the same as the price index if one is Laspeyres and the other is Paasche

efact==pfactLaspeyres qfactPaasche==pfactPaasche qfactLaspeyres

The implicit price index can be different from the price index, in which case the Laspeyres price index differs from the Paasche

TheoryGuru[{efact≠pfactLaspeyres qfactLaspeyres},

pfactLaspeyres≠pfactPaasche]

TheoryGuru[{efact≠pfactPaasche qfactPaasche,qfactPaasche>0},

pfactLaspeyres≠pfactPaasche]

Based on the Fisher ideal index, the implicit price index (its square is ((efact efact)/qfactFishersq)) is the same as the price index

efact efact==pfactFishersq qfactFishersq

Variable interpretations