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I frequently encounter the following two versions of writing CIES or CRRA preferences:

$$u(c_t) = \frac{c_t^{1-\theta}-1}{1 - \theta}$$

...and...

$$u(c_t) = \frac{c_t^{1-\theta}}{1 - \theta}$$

The only difference between the two is the inclusion or omission of the -1 in the numerator.

I cannot for the life of me figure out what the functional difference of these two forms is.

Is there any? What is the reasoning between the different ways of writing it?

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This first function is the ‘original’ CRRA function.

$$u(c_t) = \frac{c_t^{1-\theta}-1}{1 - \theta}$$

The second function is monotonic transformation of the first function. Monotonic transformation of any utility function will still represent the same preferences (monotonic transformation preserves the ordering of preferences).

$$u(c_t) = \frac{c_t^{1-\theta}}{1 - \theta}$$

People will often use monotonic transformation for mathematical convenience. So to save yourself some time and effort you can just use the second question which still represents the same consumer preferences as the first one, but it’s easier work with.

However, as pointed by Michael Greinecker in the comments this is also positive affine transformation so it might not be always appropriate in some stochastic or if you care about having additively separable time preferences.

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  • $\begingroup$ Thank you so much! This is exactly the answer I needed! $\endgroup$
    – maddonis
    Jan 17 at 12:06
  • $\begingroup$ My copy of Ljungqvist & Sargent 2000 Recursive Macroeconomic Theory has all mentions of a CRRA utility functions in the second form, without the $-1$ in the numerator. $\endgroup$
    – VARulle
    Jan 17 at 15:15
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    $\begingroup$ @VARulle my bad, I was using my old macro lecture notes which are not publicly avaiable where I had reference L&S 2000 and I assumed I got it from there but now I think that in my old notes I was just referencing some result there not the utility function without double checking, I removed the reference form the above, thanks for pointing the mistake I will be more careful next time $\endgroup$
    – 1muflon1 ♦
    Jan 17 at 16:07
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    $\begingroup$ It is not just a monotone transformation; it is a positive affine transformation. The difference becomes important when you want additively separable time preferences or want to look at stochastic models. $\endgroup$
    – Michael Greinecker
    Jan 17 at 17:14
  • $\begingroup$ @MichaelGreinecker thanks for the comment I added that caveat to my answer as well $\endgroup$
    – 1muflon1 ♦
    Jan 17 at 22:37

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