# About the 9 tests on Index number theory to evaluate the efficency of the different types of price indexes

Correct me if I'm wrong, I've seen a section in Wikipedia about Index number theory and I've understand it as "if an index x (Laspeyres, Paasche, etc.) satisfies all of this test, then it's 100% efficient. If satisfies most of them (how many? or which tests?), then it's acceptable".

I've also read the section 2 mentioned in the paper (1993) and I have 3 questions about it.

1. Are this test used nowadays? Is there a more recent paper about it?

2. The conclusion (on page 78) of that paper was that it's best an economic approach rather that the test approach. How is this difference?

3. What's an example of the tests? neither on wiki nor on the paper I could got an example to illustrate the text.

1. Are this test used nowadays? Is there a more recent paper about it?

As far as I know, (theoretical) index number theory is not so popular anymore nowadays. The feeling is probably that we know most about the topic that is to be known about it from a theoretical perspective. Of course this does not mean that no new discoveries could be made, but my feeling is that the topic has fallen a bit 'out of fashion'.

Unfortunately, index number theory is also no longer taught at universities (in great detail), which is somewhat unfortunate as we use price and quantity indices all the time (without being aware of their various limitations.)

My guess is that nowadays most effort related to index numbers is done by (statistical) government agencies that are charged with providing index numbers for policy.

1. The conclusion (on page 78) of that paper was that it's best an economic approach rather that the test approach. How is this difference?

The test approach tries to construct index numbers by having them satisfy as many desirable tests (properties) as possible. You could think of tests in the sense of mathematical axioms. You state some desirable properties that you think price indices should satisfy and then you try to find indices that satisfy these properties. In general, no index can satisfy all of them, but it gives you a clear picture of the trade off between the various indices.

The economic approach to index number theory, on the other hand, constructs index numbers based on micro-economics consumer (and producer) theory.

Consider the standard expenditure minimisation problem: $$c(p,u) = \min_q p'q \text{ s.t. } u(q) \ge \overline{u}.$$ Now fix a utility level $$u$$ and consider a price change from $$p_0$$ to $$p_1$$. Then you can look at the cost of living index: $$\frac{c(p_1,u)}{c(p_0,u)}.$$ This gives the amount of income you would need at prices $$p_1$$ to reach utility level $$u$$, as a fraction of the amount of income you need at prices $$p_0$$ to reach the same utility level $$u$$. If this is larger than one, then life has become more expensive. If it is below one, life has become less expensive.

As such, you can see this as a price index (from period $$0$$ to period $$1$$).

A problem with such cost of living index is that, in general, the index will vary with the utility level $$u$$. So you will have many indices. Two attractive choices are the utility level $$u_0$$ obtained at base year and the utility level $$u_1$$ obtained after the price change. This gives: $$\frac{c(p_1, u_0)}{p_0' q_0} \text{ and } \frac{p_1' q_1}{c(p_0, u_1)}.$$ as $$c(p_1, u_1) = p_1'q_1$$ and $$c(p_0, u_0) = p_0' q_0$$.

Alternatively, if you assume that the utility function is homothetic, then we have that $$c(p,u) = c(p) u$$ where $$c(p) = c(p,1)$$ so: $$\frac{c(p_1,u)}{c(p_0,u)} = \frac{c(p_1)}{c(p_0)},$$ This is now independent of the utility level $$u$$, so there is a unique price index.

For several (homothetic) utility functions, you can actually compute this ratio and therefore obtain an exact price index (i.e. a price index that is a cost of living index for some utility function).

Notice that if preferences are homothetic then $$\frac{p_1' q_1}{p_0' q_0} = \frac{c(p_1,u(q_1)}{c(p_0, u(q_0))} = \frac{c(p_1)}{c(p_0)}\frac{u(q_1)}{u(q_0)}$$ So $$\frac{c(p_1)}{c(p_0)}$$ acts like a price index and $$\frac{u(q_0)}{u(q_1)}$$ is like a quantity index.

1. What's an example of the tests? neither on wiki nor on the paper I could got an example to illustrate the text.

Consider for example the Laspeyres price index (from prices $$p_0$$ to prices $$p_1$$): $$L(p_0, q_0, p_1, q_1) = \frac{\sum_i p_{1,i} q_{0,i}}{\sum_i p_{0,i} q_{0,i}} = \frac{p_1 'q_0}{p_0' q_0}.$$ The Invariance to changes in scale test requires that: $$L(\alpha p_0, \beta q_0, \alpha p_1, \gamma q_1) = L(p_0, q_0, p_1,q_1).$$ This is satisfied for the Laspeyres: $$L(\alpha p_0, \beta q_0, \alpha p_1, \gamma q_1) = \frac{(\alpha p_1)'(\beta q_0)}{(\alpha p_0)' (\beta q_0)} = \frac{p_1' q_0}{p_0'q_0} = L(p_0, q_0, p_1, q_1).$$ The Symmetric treatment of time requires that: $$L(p_0, q_0, p_1,q_1) = \frac{1}{L(p_1, q_1, p_0, q_0)}$$ This condition is (in general) not satisfied for the Laspeyres as: $$\frac{p_1' q_0}{p_0' q_0} \ne \frac{1}{\dfrac{p_0' q_1}{p_1' q_1}} = \frac{p_1' q_1}{p_0' q_1},$$ unless, for example, $$q_1 = q_0$$. (The right hand side is the Paasche index). Similarly, you can show that the Circularity test will also not be satisfied for the Laspeyres.