<p>The multiplier is equal to marginal utility of consumption. In case of CRRA utility (i.e. $\lambda_t = c_t^{-\gamma}$), its rate of growth is then related to consumption growth:</p>

<p>$$
\frac{\dot \lambda}{\lambda} = \frac{\mathrm{d} \log \lambda}{\mathrm{d} t} = \frac{\mathrm{d} (-\gamma \log c)}{\mathrm{d} t} =  -\gamma \frac{\dot c}{c}
$$</p>

<p>The condition can be then rewritten as</p>

<p>$$
\frac{\dot c}{c} = \frac{1}{\gamma} \left[ (f'(k) - \delta - \xi) - \nu\right],
$$
so that the consumption growth rate is proportional to return on capital (less rate of time preference), which is pretty much the same as Euler equation in discrete time.</p>

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2015-02-15T04:34:32.720Z
2015-02-15T04:34:32.720Z
ivansml
<p>I encountered a necessary optimality conidtion involving growth rate of Lagrange multiplier in <a href="http://www.econ2.jhu.edu/people/ccarroll/Public/LectureNotes/Growth/HamiltonianVSDiscrete/" rel="nofollow">this note</a> on Ramsey model</p>

<p>$$\frac{\dot \lambda}{\lambda} = \mathcal {v} - (f^\prime(k)-\delta-\xi)$$</p>

<p>in which $\mathcal {v}$    is the time preference rate, $\xi$   is the population growth rate, and $\delta$   is the depreciation rate.</p>

<p>Is there some way to make sense of it in a similar fashion as Euler equation?</p>

<macroeconomics><economic-growth>
How to intuit a necessary optimality conidtion involving growth rate of Lagrange multiplier in Ramsey growth model?
Epicurus
# How to intuit a necessary optimality conidtion involving growth rate of Lagrange multiplier in Ramsey growth model?

(Asked by: username_0 on 2015-02-13)

I encountered a necessary optimality conidtion involving growth rate of Lagrange multiplier in [this note](http://www.econ2.jhu.edu/people/ccarroll/Public/LectureNotes/Growth/HamiltonianVSDiscrete/) on Ramsey model

$$\frac{\dot \lambda}{\lambda} = \mathcal {v} - (f^\prime(k)-\delta-\xi)$$

in which $\mathcal {v}$ is the time preference rate, $\xi$ is the population growth rate, and $\delta$ is the depreciation rate.

Is there some way to make sense of it in a similar fashion as Euler equation?

**Tags:** macroeconomics, economic-growth


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## 1 answers

### Answer by username_1 on 2015-02-15. Score: 3 | Accepted answer

The multiplier is equal to marginal utility of consumption. In case of CRRA utility (i.e. $\lambda\_t = c\_t^{-\gamma}$), its rate of growth is then related to consumption growth:

$$
\frac{\dot \lambda}{\lambda} = \frac{\mathrm{d} \log \lambda}{\mathrm{d} t} = \frac{\mathrm{d} (-\gamma \log c)}{\mathrm{d} t} = -\gamma \frac{\dot c}{c}
$$

The condition can be then rewritten as

$$
\frac{\dot c}{c} = \frac{1}{\gamma} \left[ (f'(k) - \delta - \xi) - \nu\right],
$$
so that the consumption growth rate is proportional to return on capital (less rate of time preference), which is pretty much the same as Euler equation in discrete time.


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