<p>In the case of a simple utility maximization, the lagrange multiplier <span class="math-container">$\lambda$</span> is the marginal utility of income. The rate of increase in utility as income increases. Check this <a href="http://the%20rate%20of%20increase%20in%20maximized%20utility%20as%20income%20increases." rel="nofollow noreferrer">references</a> for more details.</p>
<p>In the specific case of the Ramsey Cass Koopmans model, The Lagrange multiplier <span class="math-container">$\lambda$</span> measures the marginal value of wealth (or resources) in period
t. if we exogenously give the economy <span class="math-container">$\epsilon$</span> units of the good during period <span class="math-container">$t$</span>, where <span class="math-container">$\epsilon$</span> is small enough, welfare increases by approximately <span class="math-container">$\lambda\epsilon$</span>.<br />
You may refer to the following <a href="https://ocw.mit.edu/courses/economics/14-05-intermediate-macroeconomics-spring-2013/lecture-notes/MIT14_05S13_LecNot_Ramsey.pdf" rel="nofollow noreferrer">lecture notes</a> for more clarity.</p>

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2021-11-24T00:06:53.283Z
2021-11-24T00:06:53.283Z
Macosso
<p>Consider lagrange function for ramsey problem:
<span class="math-container">$L=E_0 \sum_{t=0}^{\infty} \beta^t \{u(c,l)+\gamma_t (s^t)[E_0 \sum_{j=0}^{\infty} \beta^j u_c (s^{t+j}) z(s^{t+j}) -u_c (s^t) b_t(s^{t+1})] \}$</span><br />
where <span class="math-container">$[E_0 \sum_{j=0}^{\infty} \beta^j u_c (s^{t+j}) z(s^{t+j}) -u_c (s^t) b_t(s^{t+1})]$</span> means &quot;net surplus - net debt&quot; of government.</p>
<p>In sargent's book(Recursive macroeconomics) it says<br />
if government is borrowing(bond issue) then lagrange multiplier <span class="math-container">$\gamma_t (s^t)$</span> is positive <br />
and if government accumulates asset then lagrange multiplier <span class="math-container">$\gamma_t (s^t)$</span> is negative.<br />
I don't know how to determine that sign.<br />
What is the meaning of lagrange multiplier in this problem?</p>

<macroeconomics><lagrangian>
What is the meaning of lagrange multiplier (especially in ramsey problem)
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# What is the meaning of lagrange multiplier (especially in ramsey problem)

(Asked by: username_0 on 2021-11-21)

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?

**Tags:** macroeconomics, lagrangian

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## 1 answers

### Answer by username_1 on 2021-11-24. Score: 0 | Accepted answer

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](http://the%20rate%20of%20increase%20in%20maximized%20utility%20as%20income%20increases.) 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](https://ocw.mit.edu/courses/economics/14-05-intermediate-macroeconomics-spring-2013/lecture-notes/MIT14_05S13_LecNot_Ramsey.pdf) for more clarity.


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