Slutsky’s theorem (1915)

Named after its proposer, Soviet economist Eugen (Evgeny) Slutsky (1880-1948), Slutsky’s theorem was later developed by English economists John Hicks (1904-1989) and ROY ALLEN (1906-1983).

In its simplest form:

Price effect = income effect + substitution effect

Slutsky asserted in 1915 that demand theory is based on the concept of ordinal utility.

This idea was developed by Hicks who separated the consumer’s reaction to a price change into income and substitution effects.

Source:
E Slutsky, ‘On the Theory of the Budget of the Consumer’, Readings in Price Theory, K E Bould-ing and G J Stigler, eds (1953)

Statement

Let {\displaystyle X_{n},Y_{n}} be sequences of scalar/vector/matrix random elements. If {\displaystyle X_{n}} converges in distribution to a random element {\displaystyle X} and {\displaystyle Y_{n}} converges in probability to a constant {\displaystyle c}, then

• {\displaystyle X_{n}+Y_{n}\ {\xrightarrow {d}}\ X+c;}
• {\displaystyle X_{n}Y_{n}\ \xrightarrow {d} \ Xc;}
• {\displaystyle X_{n}/Y_{n}\ {\xrightarrow {d}}\ X/c,}   provided that c is invertible,

where {\displaystyle {\xrightarrow {d}}} denotes convergence in distribution.

Notes:

1. The requirement that Y_n converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. For example, let {\displaystyle X_{n}\sim {\rm {Uniform}}(0,1)} and {\displaystyle Y_{n}=-X_{n}}. The sum {\displaystyle X_{n}+Y_{n}=0} for all values of n. Moreover, {\displaystyle Y_{n}{\xrightarrow {d}}{\rm {Uniform}}(-1,0)}, but {\displaystyle X_{n}+Y_{n}} does not converge in distribution to {\displaystyle X+Y}, where {\displaystyle X\sim {\rm {Uniform}}(0,1)}, {\displaystyle Y\sim {\rm {Uniform}}(-1,0)}, and {\displaystyle X} and {\displaystyle Y} are independent.^[4]
2. The theorem remains valid if we replace all convergences in distribution with convergences in probability.

Proof

This theorem follows from the fact that if X_n converges in distribution to X and Y_n converges in probability to a constant c, then the joint vector (X_n, Y_n) converges in distribution to (X, c) (see here).

Next we apply the continuous mapping theorem, recognizing the functions g(x,y) = x + y, g(x,y) = xy, and g(x,y) = x y⁻¹ are continuous (for the last function to be continuous, y has to be invertible).

See also

• Convergence of random variables

References

Further reading

• Casella, George; Berger, Roger L. (2001). Statistical Inference. Pacific Grove: Duxbury. pp. 240–245. ISBN 0-534-24312-6.
• Grimmett, G.; Stirzaker, D. (2001). Probability and Random Processes (3rd ed.). Oxford.
• Hayashi, Fumio (2000). Econometrics. Princeton University Press. pp. 92–93. ISBN 0-691-01018-8.