A firm minimizes cost or maximizes profit subject to the constraint implied by a production function. Demand equations for inputs, formulated in terms of changes over time, are described in two steps. The first is the total input decision, which describes the Divisia input volume index in terms of the change in output (for cost minimization) or price changes (for profit maximization). The second is the input allocation decision, which describes the changes in the demand for the individual inputs in terms of the Divisia input volume index and the input price changes. It is shown that the input allocation decision allows a simple transformation so that (1) the change in the demand for each transformed input is independent of the changes in the relative prices of all others and (2) the log-change in output is the sum of certain components, each representing the contribution of one transformed input, in such a way that the interaction of these inputs is confined to terms of the third order of smallness.