Editing Price index (section)

== Theoretical evaluation ==
Price index formulas can be evaluated based on their relation to economic concepts (like cost of living) or on their mathematical properties. Several different tests of such properties have been proposed in index number theory literature, and [[Walter Erwin Diewert|W. Erwin Diewert]] summarized past research in a list of nine such tests:<ref>Diewert (1993), 75–76.</ref>

# Identity test:
#: <math>I(p_{t_m},p_{t_n},\alpha \cdot q_{t_m},\beta\cdot q_{t_n})=1~~\forall (\alpha ,\beta )\in (0,\infty )^2</math>
#: If prices remain the same between periods and quantities are scaled proportionally (each quantity of an item is multiplied by the same factor of either <math>\alpha</math>, for the first period, or <math>\beta</math>, for the later period), then the index should be 1.
# Proportionality test:
#: <math>I(p_{t_m},\alpha \cdot p_{t_n},q_{t_m},q_{t_n})=\alpha \cdot I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})</math>
#: If each price in the later period increases by a factor <math>\alpha</math>, then the index should increase by the factor <math>\alpha</math>.
# Invariance to changes in scale test:
#: <math>I(\alpha \cdot p_{t_m},\alpha \cdot p_{t_n},\beta \cdot q_{t_m}, \gamma \cdot q_{t_n})=I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})~~\forall (\alpha,\beta,\gamma)\in(0,\infty )^3</math>
#: If prices in both periods are scaled by <math>\alpha</math> and quantities by <math>\beta</math> and <math>\gamma</math>, then the index should remain unchanged, meaning the magnitude of prices and quantities shouldn’t affect the result.
# Commensurability test:
#: If units of measurement change (e.g., from kg to lbs), then the index should not be affected.
# Symmetric treatment of time:
#: <math>I(p_{t_n},p_{t_m},q_{t_n},q_{t_m})=\frac{1}{I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})}</math>
#: If the order of time periods is reversed, then the index should be the reciprocal of the original.
# Symmetric treatment of commodities:
#: If the order of commodities is [[Permutation|permuted]], then the index should remain unchanged, ensuring all goods are treated equally.
# Monotonicity test:
#: <math>I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \le I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\Leftarrow~~p_{t_n} \le p_{t_r}</math>
#: If later prices in one period (<math>t_n</math>) are less than or equal to those in another (<math>t_r</math>), then the index for <math>t_n</math> should be less than or equal to that for <math>t_r</math>.
# Mean value test:
#: The overall price relative implied by the index should lie between the smallest and largest price relatives for all commodities.
# Circularity test:
#: <math>I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \cdot I(p_{t_n},p_{t_r},q_{t_n},q_{t_r})=I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\Leftarrow~~t_m \le t_n \le t_r</math>
#: If three ordered periods are considered (<math>t_m</math>, <math>t_n</math>, <math>t_r</math>), then the product of the index from <math>t_m</math> to <math>t_n</math> and from <math>t_n</math> to <math>t_r</math> should equal the index from <math>t_m</math> to <math>t_r</math>.