# 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 Yn converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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 Xn converges in distribution to X and Yn converges in probability to a constant c, then the joint vector (XnYn) converges in distribution to (Xc) (see here).

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

• Convergence of random variables

## References

1. ^ Goldberger, Arthur S. (1964). Econometric Theory. New York: Wiley. pp. 117–120.
2. ^ Slutsky, E. (1925). “Über stochastische Asymptoten und Grenzwerte”. Metron (in German). 5 (3): 3–89. JFM 51.0380.03.
3. ^ Slutsky’s theorem is also called Cramér’s theorem according to Remark 11.1 (page 249) of Gut, Allan (2005). Probability: a graduate course. Springer-Verlag. ISBN 0-387-22833-0.
4. ^ See Zeng, Donglin (Fall 2018). “Large Sample Theory of Random Variables (lecture slides)” (PDF)Advanced Probability and Statistical Inference I (BIOS 760). University of North Carolina at Chapel Hill. Slide 59.