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I am wondering if it is possible to write down a following CRRA utility function ;

$$u\left(c(t)\right)=a\frac{c\left(t\right)^{1-\sigma}}{1-\sigma}$$ where $a>0$ is a constant scale parameter. I need this $a$ to have some numerical results but I am not sure if $a$ can be justified. In a growh model, I am trying to show the existence of Hopf bifurcation and limit cycles. Then, with a scale parameter like that, I can show that it exists.

Normally, it does not change the usual assumptions on a CRRA utility function, (an increasing concave function)

Is there any way to justify it (are there some examples of this kind ?) or are there any types of utility function with constant scale parameters ?

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  • $\begingroup$ "Numerical results" at what level? Please elaborate. For example, in the usual intertemporal model, growth rates are not affected since the constant cancels out. $\endgroup$ – Alecos Papadopoulos Aug 29 '17 at 20:01
  • $\begingroup$ @AlecosPapadopoulos In fact, I am trying to show that for some parameter set, there is a Hopf bifurcation and limit cycles. I edit the question. You are right that growh model does not change but steady state levels change. $\endgroup$ – optimal control Aug 29 '17 at 20:32
  • $\begingroup$ I trying to understand : if you have a model with a steady state in growth rates (a "balanced growth path") then you cannot have a periodic solution since it would imply a change in the growth rate (and alpha does not affect the growth rate). If instead you have a model with a steady state in levels, certain alpha values transform the long-run constant level-value to a periodic solution? $\endgroup$ – Alecos Papadopoulos Aug 30 '17 at 0:06
  • $\begingroup$ This is exactly what you said. It is not a model with balanced growth path. I have a model with a steady state in levels and certain values are likely to make periodic solution and with the scale parameter bifurcation occurs. $\endgroup$ – optimal control Aug 30 '17 at 6:00
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As it is well-known, preferences (even under risk) are invariant to affine transformations of the utility function. Therefore, adding a 'scale' parameter adds nothing in terms of fundamental preferences (i.e. in terms of comparing streams of consumption). I cannot even imagine what kind of 'numerical results' would be sensitive to it.

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This paper might be of interest to you. This bit is important:

enter image description here

So any linear transformation of $u(c)$ solved the differential equation.

In section 3 the author derives a linear transformation that allows for a balanced growth path under bounded consumption. In particular, the author computes an expression for $a$ above. This is not necessarily what you are interested on, but gives an idea of the values of $a$ given assumptions on $\sigma$, which might give you an idea about how to justify the value of $a$.

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