  Equation (9) is the Ramsey Rule. It is a necessary condition for optimality in Ramsey Mark III and is unarguably the most famous equation in intertemporal welfare economics. The rule is a formal statement of the requirement of \(\{C^*(t)\}\), that the marginal rate of substitution between consumption at two nearby dates (the left-hand-side of eq. 9) equals the marginal rate of transformation between consumption at those same pair of nearby dates (the right-hand-side of eq. (9). It is simple to confirm that equation (9) is invariant under positive affine transformations of the \(U\)-function.