<p>I have the following inter temporal utility function: </p>

<p><span class="math-container">$U(t)=(\frac{s(t)}{1-\sigma})(c_t/c_{t-1}^\gamma)^{(1-\sigma)} - \chi*h(t)$</span></p>

<p>where <span class="math-container">$h(t)$</span> is the hours worked.
I know that gamma is responsible for consumption smoothing as I have achieved the following results after a positive shock on <span class="math-container">$s(t)$</span>:<a href="https://i.sstatic.net/23Lmr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/23Lmr.png" alt="The consumption after a shock when \gamma =1, is much smoother"></a></p>

<p>The problem is i do not know how to prove mathematically that when gamma=1 consumption is smoother </p>

CC BY-SA 4.0
2018-12-15T15:40:23.747Z
This is a utility function with habit formation where $\gamma$ governs the degree to which your consumption yesterday matters in terms of utility today (relative to current consumption). If you care only about comparing $\gamma = 0,1$ then check that the function when $\gamma=0$ is everywhere differentiable. If not, you are done. If so, then you could make an argument about strict concavity vs. concavity between the two functions. Not sure else how to discuss relative smoothness.
123
CC BY-SA 4.0
2021-05-03T13:07:58.323Z
Could you provide Euler equation, please?
manifold
<consumption><rbc><dynare>
Consumption smoothing in RBC Model
Artur Mukhin
# Consumption smoothing in RBC Model

(Asked by: username_0 on 2018-12-14)

I have the following inter temporal utility function:

$U(t)=(\frac{s(t)}{1-\sigma})(c\_t/c\_{t-1}^\gamma)^{(1-\sigma)} - \chi\*h(t)$

where $h(t)$ is the hours worked.
I know that gamma is responsible for consumption smoothing as I have achieved the following results after a positive shock on $s(t)$:[![The consumption after a shock when \gamma =1, is much smoother](https://i.sstatic.net/23Lmr.png)](https://i.sstatic.net/23Lmr.png)

The problem is i do not know how to prove mathematically that when gamma=1 consumption is smoother

**Tags:** consumption, rbc, dynare
> **Comments:**
> **username_1**: This is a utility function with habit formation where $\gamma$ governs the degree to which your consumption yesterday matters in terms of utility today (relative to current consumption). If you care only about comparing $\gamma = 0,1$ then check that the function when $\gamma=0$ is everywhere differentiable. If not, you are done. If so, then you could make an argument about strict concavity vs. concavity between the two functions. Not sure else how to discuss relative smoothness.  
> **username_2**: Could you provide Euler equation, please?  

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## 0 answers

