Consumption smoothing in RBC Model

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

– user20105
Dec 14 '18 at 21:40
• 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
Dec 15 '18 at 15:40
• Could you provide Euler equation, please? May 3 '21 at 13:07

Getting this equation you can show how consumption varies for different levels of $$\gamma$$ (in your case = 0 and = 1)