I recently learned about the neoclassical growth model in both discrete and continuous time. The intuitive meaning of the shadow price for both cases is that it represents the value of one additional unit of capital. However, in discrete time, this value is the discounted value of the marginal utility at time t consumption level. Whereas, in continuous time, the shadow price is the discounted value of marginal utility at time 0 consumption level. I was wondering why is there such a difference when changing from discrete to continuous time.
Moreover, I feel that the discrete-time seems more meaningful as the value of capital at time t should depend on the marginal utility at time t and not at time 0.