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I encountered a necessary optimality conidtion involving growth rate of Lagrange multiplier in this note on Ramsey model
$$\frac{\dot \lambda}{\lambda} = \mathcal {v} - (f^\prime(k)-\delta-\xi)$$
in which $\mathcal {v}$ is the time preference rate, $\xi$ is the population growth rate, and $\delta$ is the depreciation rate.
Is there some way to make sense of it in a similar fashion as Euler equation?
The multiplier is equal to marginal utility of consumption. In case of CRRA utility (i.e. $\lambda_t = c_t^{-\gamma}$), its rate of growth is then related to consumption growth:
$$ \frac{\dot \lambda}{\lambda} = \frac{\mathrm{d} \log \lambda}{\mathrm{d} t} = \frac{\mathrm{d} (-\gamma \log c)}{\mathrm{d} t} = -\gamma \frac{\dot c}{c} $$
The condition can be then rewritten as
$$ \frac{\dot c}{c} = \frac{1}{\gamma} \left[ (f'(k) - \delta - \xi) - \nu\right], $$ so that the consumption growth rate is proportional to return on capital (less rate of time preference), which is pretty much the same as Euler equation in discrete time.
