When the own price elasticity is less than one, then cross price elasticity is negative. On the other hand, when cross price elasticity is more than one, then cross price elasticity is positive. In the former case, a rise in the price of a commodity raises the expenditure on this commodity (the commodity being inelastic in demand).
Consequently, the expenditure on the related commodity falls (holding its price and money income of the consumer constant). In other words, its cross price elasticity is negative. Similarly, in the latter case, the cross price elasticity is positive.
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Proof: Budget constrain is written as
Thus, weighted sum of cross price elasticities and own price elasticity equals one. Further, if e11 < 1, then cross price elasticity is negative and the goods one complements. While if e11> 1, then cross price elasticity is positive and the goods are substitutes.