<p>The constraint is part of the objective function in each period. So, naturally it is subject to discounting, and so it must "be multiplied by beta".  </p>

<p>So the fist setup above is wrong (or just a typo: one should check whether in the specific publication the solution given follows the above image or in reality it implies that the constraint is multiplied by beta).</p>

<p>The "third case" the OP mentions in his post is a different matter. There we "compact" the constraint into one, so here they are outside the sum and the intertemporal discounting, with the understanding that the single multiplier is a present-value multiplier in this case. This is ok, it is done for example on certain macroeconomic models related to optimal taxation.</p>

CC BY-SA 4.0
2022-02-17T10:40:59.773Z
I had the same question today. I have thought about this and concluded that both formulations are identical. See this for reference: http://www-personal.umich.edu/~chouse/macII_2003/syllabus/notespacket.pdf
Rumi
CC BY-SA 4.0
2022-09-02T18:53:58.673Z
So @Rumi- do we need to multiply the budget constraint with beta or not? It seems in the notes you posted above, there is no multiplication.
Kwame Brown
CC BY-SA 4.0
2018-08-13T19:58:39.350Z
2018-08-13T19:58:39.350Z
Alecos Papadopoulos
<p>I am getting a bit confused about setting up the Lagrangian in intertemporal constrained optimization problems. The confusion is as to when is the one-period budget constraint multiplied by the discount factor Beta and/or when is it included in the summation? Here is what I mean. In this first example:</p>

<p><a href="https://i.sstatic.net/iicII.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iicII.png" alt="Example 1"></a></p>

<p>the constraint is not multiplied by Beta, nor part of the summation. Now, in example 2:</p>

<p><a href="https://i.sstatic.net/AxeZD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/AxeZD.png" alt="Example 2"></a></p>

<p>the constraint is multiplied by the discount factor Beta and part of the intertemporal summation. Why this difference? Aren't the two cases pretty much equal, both maximizing subject to one-period constraints? The only difference I see is when the interest rate on the assets is realized, but that should not matter I believe(?)
I also found a third case (will upload an image later if I find it) in which the one-period constraint is summed up intertemporally, but not discounted by Beta.
So to conclude: what is the rule here? What am I missing? </p>

<p>EDIT:</p>

<p>By looking at the FOCs that follow example 1, it becomes clear that the author means the Lagrange Multiplier to be part of the summation, but not discounted by Beta. This is clear from the following:</p>

<p><a href="https://i.sstatic.net/6vFUf.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6vFUf.png" alt="Focs_Example1"></a></p>

<p>So this really looks to me like a different formulation to the second example. The "funny" thing, however, is that in both cases you get the same result (i.e. the same Euler Equation).</p>

CC BY-SA 4.0
2018-08-13T18:29:54.980Z
Hi, Welcome to Economics SE! we have a policy regarding homework questions where users must demonstrate effort using text, markdown syntax and/or images space provided in the text of his question submission for the site. Using only images as proof of work is insufficient. for more details see:https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions
EconJohn
CC BY-SA 4.0
2018-08-13T22:42:50.973Z
I'm fairly certain that in the first example, the constraint should be part of the summation. There's enough ambiguity in the notation for you to interpret what's written to mean this. Of course, it's better to be explicit by putting in parenthesis, as in your last example.
Theoretical Economist
<macroeconomics><utility>
Lagrangian: when to discount budget constraint?
gitcanzo
# Lagrangian: when to discount budget constraint?

(Asked by: username_0 on 2018-08-13)

I am getting a bit confused about setting up the Lagrangian in intertemporal constrained optimization problems. The confusion is as to when is the one-period budget constraint multiplied by the discount factor Beta and/or when is it included in the summation? Here is what I mean. In this first example:

[![Example 1](https://i.sstatic.net/iicII.png)](https://i.sstatic.net/iicII.png)

the constraint is not multiplied by Beta, nor part of the summation. Now, in example 2:

[![Example 2](https://i.sstatic.net/AxeZD.png)](https://i.sstatic.net/AxeZD.png)

the constraint is multiplied by the discount factor Beta and part of the intertemporal summation. Why this difference? Aren't the two cases pretty much equal, both maximizing subject to one-period constraints? The only difference I see is when the interest rate on the assets is realized, but that should not matter I believe(?)
I also found a third case (will upload an image later if I find it) in which the one-period constraint is summed up intertemporally, but not discounted by Beta.
So to conclude: what is the rule here? What am I missing?

EDIT:

By looking at the FOCs that follow example 1, it becomes clear that the author means the Lagrange Multiplier to be part of the summation, but not discounted by Beta. This is clear from the following:

[![Focs_Example1](https://i.sstatic.net/6vFUf.png)](https://i.sstatic.net/6vFUf.png)

So this really looks to me like a different formulation to the second example. The "funny" thing, however, is that in both cases you get the same result (i.e. the same Euler Equation).

**Tags:** macroeconomics, utility

> **Comments:**
> **username_1**: Hi, Welcome to Economics SE! we have a policy regarding homework questions where users must demonstrate effort using text, markdown syntax and/or images space provided in the text of his question submission for the site. Using only images as proof of work is insufficient. for more details see:https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions  
> **username_2**: I'm fairly certain that in the first example, the constraint should be part of the summation. There's enough ambiguity in the notation for you to interpret what's written to mean this. Of course, it's better to be explicit by putting in parenthesis, as in your last example.  

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## 1 answers

### Answer by username_3 on 2018-08-13. Score: 1

The constraint is part of the objective function in each period. So, naturally it is subject to discounting, and so it must "be multiplied by beta".

So the fist setup above is wrong (or just a typo: one should check whether in the specific publication the solution given follows the above image or in reality it implies that the constraint is multiplied by beta).

The "third case" the OP mentions in his post is a different matter. There we "compact" the constraint into one, so here they are outside the sum and the intertemporal discounting, with the understanding that the single multiplier is a present-value multiplier in this case. This is ok, it is done for example on certain macroeconomic models related to optimal taxation.

> **Comments:**
> **username_4**: I had the same question today. I have thought about this and concluded that both formulations are identical. See this for reference: http://www-personal.umich.edu/~chouse/macII_2003/syllabus/notespacket.pdf  
> **username_5**: So @username_4- do we need to multiply the budget constraint with beta or not? It seems in the notes you posted above, there is no multiplication.  

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