An alternative approach to the estimation of functional specifications from physical input data, involves estimation of factor share equations from cost data. This is done by using the duality that exists between the production function and the , cost function along the expansion path. The basic dual input definition of elasticity of substitution renders a number of alternative concepts possible. The Allen, Morishima and Shadow measures are particularly useful. Although specific production functions frequently embody assumptions that may have important disadvantages with respect to the substitutability of inputs, the translog specification represents relaxations of these maintained hypotheses. The translog functional specification is thus the production function of choice for empirical estimation of elasticities of substitution between input pairs when little information about the production process other than cost data, is available.