Shepard's Lemma

thm. Shepard’s Lemma (Expenditure Minimization). Relates Hicksian Demand and Expenditure Function

$$\begin{array}{r}{h}_1(p_1,p_2,\bar{u})=\frac{\partial E}{\partial p_1}\\ h_2(p_1,p_2,\bar{u})=\frac{\partial E}{\partial p_2}\end{array}$$

thm. Shepard’s Lemma (Cost Minimization). Related conditional Input Demand and the Cost Function

$$\begin{array}{rl}{l}_c(w,r,\bar{x})& =\frac{\partial C}{\partial w}\\ k_c(w,r,\bar{x})& =\frac{\partial C}{\partial r}\end{array}$$