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Suppose that the household faces the following problem:

$\underset{ c_t , k_{t+1}, n_t } \max \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t \ln c_t + \ln (1 - n_t)$

subjected to

$ k_{t+1} = A_t k_t ^{\alpha} n_t ^{1- \alpha} - c_t $

Usually, in my macroeconomic course, we would formulate the Bellman equation as the following:

$ V(k, A) = \underset {c, k', n} \max \ln c + \ln (1- n) + \beta \mathbb{E} [V (k', A')] $

However, in his lecture notes, he formulated the Bellman equation as the following:

$ V(k, A) = \underset {c, k', n} \max \ln c + \ln (1- n) + \beta \mathbb{E} [V (k', A')] + \lambda [ A k^{\alpha} n_t ^{1- \alpha} - c - k' ]$

What I don't understand is where the expression with $\lambda$ comes from? The method that we used to derive the Bellman equation does not give us the Bellman equation with the $\lambda$, but instead gives us the one without the $\lambda$ expression.

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  • $\begingroup$ I'm hardly an expert on this, but that seems to be some Hamiltonian version of it, i.e. lambda is probably the costate. $\endgroup$ – Fizz May 12 at 14:00

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