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

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

  • $\begingroup$ your equation is hardly readable! $\endgroup$
    – user20105
    Dec 14 '18 at 21:40
  • 1
    $\begingroup$ 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. $\endgroup$
    – 123
    Dec 15 '18 at 15:40
  • 1
    $\begingroup$ Could you provide Euler equation, please? $\endgroup$
    – manifold
    May 3 '21 at 13:07

Have you tried solving the maximization problem? Normally you do this taking into account some market clearing condition or budget constraint and you should be able to solve for equilibrium prices to get future consumption as a function of present consumption. Consumption smoothing will simply show the path of consumption.

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

  • $\begingroup$ I was thinking something more along the lines of EIS or MRS, between C(t) and C(t-1) $\endgroup$ Dec 15 '18 at 10:21
  • $\begingroup$ Either way you to do it you have to solve the maximization problem first. $\endgroup$ Dec 30 '21 at 3:39

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