By logarithmic differentiation
of (3.4) and after some simplification, we have
Using the identities (2.5) and (2.17) in the logarithmic differentiation
of (2.23) and simplifying the resulting equation, we get
With a focus on procedures for solving problems, rather than detailed explanations, the reader can whiz through chapters that illustrate derivatives, minimum-maximum problems, the chain rule, and logarithmic differentiation
, among other facets of calculus.
of Equation 1 would again yield equations 28 and 37 in PB.
By Shephard's lemma, shadow-cost-minimizing input demand equations are derived by differentiating total shadow cost with respect to shadow prices, and therefore the observed quantities of inputs, [X.sub.i], are homogeneous of degree zero in shadow prices.(1) Similarly, logarithmic differentiation
of the shadow cost function with respect to shadow prices yields the shadow cost shares, [Mathematical Expression Omitted], which are also homogeneous of degree zero in shadow prices.