It is well known for a Cobb-Douglas production function that the elasticity of a factor demand is the inverse of the share of output going to the other factors. Since Cobb-Douglas has a unit elasticity of substitution, the demand elasticity trivially equals the ratio of the elasticity of substitution to the share of output going to the other factor. I show here that this result can be generalized to any constant returns to scale production function. As a result, if a factor is known to be a substitute for (complement of) other factors, the inverse of the share of output going to other factors will be a lower (upper) bound for the factor’s elasticity of demand.
 Blackorby, C. and Russell, R.R. (1989) Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities). American Economic Review, 79, 882-888.
 Krusell, P., Ohanian, L.E., Ríos-Rull, J.-V. and Violante, G.L. (2000) Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis. Econometrica, 68, 1029-1053.