What is the meaning of lagrange multiplier (especially in ramsey problem)

Question

Consider lagrange function for ramsey problem: $L=E_0 \sum_{t=0}^{\infty} \beta^t \{u(c,l)+\gamma_t (s^t)[E_0 \sum_{j=0}^{\infty} \beta^j u_c (s^{t+j}) z(s^{t+j}) -u_c (s^t) b_t(s^{t+1})] \}$
where $[E_0 \sum_{j=0}^{\infty} \beta^j u_c (s^{t+j}) z(s^{t+j}) -u_c (s^t) b_t(s^{t+1})]$ means "net surplus - net debt" of government.

In sargent's book(Recursive macroeconomics) it says
if government is borrowing(bond issue) then lagrange multiplier $\gamma_t (s^t)$ is positive
and if government accumulates asset then lagrange multiplier $\gamma_t (s^t)$ is negative.
I don't know how to determine that sign.
What is the meaning of lagrange multiplier in this problem?

Answer 1

In the case of a simple utility maximization, the lagrange multiplier $\lambda$ is the marginal utility of income. The rate of increase in utility as income increases. Check this references for more details.

In the specific case of the Ramsey Cass Koopmans model, The Lagrange multiplier $\lambda$ measures the marginal value of wealth (or resources) in period t. if we exogenously give the economy $\epsilon$ units of the good during period $t$, where $\epsilon$ is small enough, welfare increases by approximately $\lambda\epsilon$.
You may refer to the following lecture notes for more clarity.