# Why is there a Lagrangian Multiplier in the Dynamic Programming Problem of the RBC model?

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.

• I'm hardly an expert on this, but that seems to be some Hamiltonian version of it, i.e. lambda is probably the costate.
– Fizz
May 12 '19 at 14:00