# Lagrangian: How to understand the No-Ponzi Condition

In http://www.uni-hamburg.de/fachbereiche-einrichtungen/fb03/iwwt/makro/slides2.pdf page 8, lagrangian is written as follow: $$L = E_0 \sum_{t=0}^{\infty}\beta^t\{U(C_t,N_t) + \lambda_t(P_tC_t + Q_tB_t - B_{t-1}-W_tN_t+T_t)+\psi_t(\lim_{T \to \infty} B_T)\}$$ where bond $B_t$ has solvency condition $\lim_{T \to \infty} B_T \geq 0$.

In page 9 then all first-order conditions are derived, but I don't see anything related to $\psi_t$ and solvency condition. Why can the first-order condition relating to $\psi_t$ be dropped?

The condition is mostly referred to as the No-Ponzi (-scheme) [NP] condition. It is one additional constraint, that prevents Ponzi-schemes: Paying debt with new higher debt, ad infinitum.

By the way: The NP condition is one condition, hence the associated multiplier should be $$\psi$$ instead of $$\psi_t$$. While certainly nothing is lost repeating the same condition over and over again (for any $$t$$), we don't need it more than once, and it is being imprecise.

Think about optimization for finite $$T$$ periods. Then, you have the condition that $$B_T \geq 0$$. The Lagrangian optimization gives you the local optimization between $$0, 1, 2$$... There are many solutions that are locally optimal, but you will only allow solutions that in the end lead to $$B_T > 0$$.

### A simple example

Your example is much too messy to think about these core issues. Look instead at the problem

$$\max_{\{c_t, a_{t+1}\}_t} \sum_t \beta^t U(c_t) + \lambda_t (a_{t+1} + c_t - Ra_t)$$

That is, a household that choses assets $$a$$ and consumption $$c$$ to maximize his utility. You can summarize the FOC as

$$\beta^t U'(c_t) = \lambda_t \\ \lambda_t = R\lambda_{t+1}\\ \Leftrightarrow U'(c_t) = \beta R U'(c_{t+1})$$

Look for a moment at the special case where $$\beta R = 1$$ (what does that imply?). With most preferences, this necessarily leads to $$c_t = c_{t+1}$$. This is the local optimization that I was referring to, which is what the Lagrangian gives you. There are, however, infinitely many solutions that satisfy $$c_t = c_{t+1}$$. Next, we try to use the budget constraint:

$$a_{t+1} + c_t = R a_t\\ \Leftrightarrow R a_0 = \lim_{T\to\infty}\sum_{t=0}^T \frac{c_t}{R^t} + \frac{a_{T+1}}{R^T}$$

This is as far we get using the (infinite) set of local budget constraints, where I have used forward iteration (hopefully correctly), assuming any start date $$t=0$$.

Now, if the household also has to satisfy the NP condition, this boils down to

$$R a_0 = \lim_{T\to\infty}\sum_{t=0}^T \frac{c_t}{R^t}$$

which, as we showed $$c_t$$ to be constant, we can solve easily and receive a single budget constraint. The unique solution to the problem that satisfies the NP condition is the solution where $$c_t$$ is a constant and this last equation holds.