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

<p>By the way: The NP condition is <strong>one</strong> condition, hence the associated multiplier should be <span class="math-container">$\psi$</span> instead of <span class="math-container">$\psi_t$</span>. While certainly nothing is lost repeating the same condition over and over again (for any <span class="math-container">$t$</span>), we don't need it more than once, and it is being imprecise.</p>

<p>Think about optimization for finite <span class="math-container">$T$</span> periods. Then, you have the condition that <span class="math-container">$B_T \geq 0$</span>. The Lagrangian optimization gives you the local optimization between <span class="math-container">$0, 1, 2$</span>... There are many solutions that are locally optimal, but you will only allow solutions that in the end lead to <span class="math-container">$B_T &gt; 0$</span>.</p>

<h3>A simple example</h3>

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

<p><span class="math-container">$$ \max_{\{c_t, a_{t+1}\}_t} \sum_t \beta^t U(c_t) + \lambda_t (a_{t+1} + c_t - Ra_t)$$</span></p>

<p>That is, a household that choses assets <span class="math-container">$a$</span> and consumption <span class="math-container">$c$</span> to maximize his utility. You can summarize the FOC as </p>

<p><span class="math-container">$$ \beta^t U'(c_t) = \lambda_t \\
\lambda_t = R\lambda_{t+1}\\
\Leftrightarrow U'(c_t) = \beta R U'(c_{t+1})
$$</span></p>

<p>Look for a moment at the special case where <span class="math-container">$\beta R = 1$</span> (what does that imply?). With most preferences, this necessarily leads to <span class="math-container">$c_t = c_{t+1}$</span>. 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 <span class="math-container">$c_t = c_{t+1}$</span>. Next, we try to use the budget constraint:</p>

<p><span class="math-container">$$ a_{t+1} + c_t = R a_t\\
\Leftrightarrow R a_0 = \lim_{T\to\infty}\sum_{t=0}^T \frac{c_t}{R^t} + \frac{a_{T+1}}{R^T}$$</span></p>

<p>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 <span class="math-container">$t=0$</span>.</p>

<p>Now, <em>if</em> the household also has to satisfy the NP condition, this boils down to </p>

<p><span class="math-container">$$R a_0 = \lim_{T\to\infty}\sum_{t=0}^T \frac{c_t}{R^t}$$</span></p>

<p>which, as we showed <span class="math-container">$c_t$</span> to be constant, we can solve easily and receive a single budget constraint. The <strong>unique</strong> solution to the problem that satisfies the NP condition is the solution where <span class="math-container">$c_t$</span> is a constant and this last equation holds.</p>

CC BY-SA 4.0
2021-02-06T04:14:55.467Z
What happens if there are 2 assets with different returns? Will there be two no ponzi conditions? Would be glad to get some reference on this.
EndLoop
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2021-02-10T10:10:44.797Z
@EndLoop intuitively, you can't go roll forward debt until the end of time. The way you would write it as $\psi(B_T^1 + B_T^2)$, where $1$ and $2$ index the two assets. You could have two conditions separately $\psi^1 B_T^1 + \psi^2 B_t^2$, which would lead to the same result, but it would not be "on the letter" what the NP condition is about.
FooBar
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2021-02-12T01:41:41.887Z
Perhaps I should put a question up for this to continue this discussion. $p_{t}c_{t} + p_{t} b_{t+1} + b_{gt+1} =  y_{t} + b_{gt}(1+i_{t}) + p_{t} b_{t} $ where $b_{t}$ denotes an asset which is traded in two countries and $b_{gt}$ is an asset which is traded in only one country. Now the No-ponzi condition just prevents me from borrowing more than what I could pay in one lifetime. So as you said it is going to be a combination of these assets. So the initial value for these assets for the problem would also be a combination of these two assets?
EndLoop
CC BY-SA 4.0
2021-02-12T03:32:58.850Z
I have posted my question  here - https://economics.stackexchange.com/questions/42524/what-is-the-no-ponzi-condition-for-this-representative-consumer-problem-tags-i
EndLoop
CC BY-SA 4.0
2015-06-08T12:53:09.817Z
2020-03-13T21:55:43.107Z
2020-03-13T21:55:43.107Z
FooBar
<p>In <a href="http://www.uni-hamburg.de/fachbereiche-einrichtungen/fb03/iwwt/makro/slides2.pdf" rel="noreferrer">http://www.uni-hamburg.de/fachbereiche-einrichtungen/fb03/iwwt/makro/slides2.pdf</a> 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 $\lim_{T \to \infty} B_T \geq 0$.</p>

<p>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? </p>

<macroeconomics>
Lagrangian: How to understand the No-Ponzi Condition
nba
# Lagrangian: How to understand the No-Ponzi Condition

(Asked by: username_0 on 2015-06-08)

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?

**Tags:** macroeconomics


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## 1 answers

### Answer by username_1 on 2015-06-08. Score: 9

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.

> **Comments:**
> **username_2**: What happens if there are 2 assets with different returns? Will there be two no ponzi conditions? Would be glad to get some reference on this.  
> **username_1**: @username_2 intuitively, you can't go roll forward debt until the end of time. The way you would write it as $\psi(B_T^1 + B_T^2)$, where $1$ and $2$ index the two assets. You could have two conditions separately $\psi^1 B_T^1 + \psi^2 B_t^2$, which would lead to the same result, but it would not be "on the letter" what the NP condition is about.  
> **username_2**: Perhaps I should put a question up for this to continue this discussion. $p_{t}c_{t} + p_{t} b_{t+1} + b_{gt+1} =  y_{t} + b_{gt}(1+i_{t}) + p_{t} b_{t} $ where $b_{t}$ denotes an asset which is traded in two countries and $b_{gt}$ is an asset which is traded in only one country. Now the No-ponzi condition just prevents me from borrowing more than what I could pay in one lifetime. So as you said it is going to be a combination of these assets. So the initial value for these assets for the problem would also be a combination of these two assets?  
> **username_2**: I have posted my question  here - https://economics.stackexchange.com/questions/42524/what-is-the-no-ponzi-condition-for-this-representative-consumer-problem-tags-i  

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