- #1
Manfred1999
- 5
- 0
Hi all,
I have a non-economic background and I am currently interested in utility functions and how they are modeled in economics. Some internet research and I found out that there are obviously different ways of model utility. In my own studies of related topics I encountered the following two examples:
$$u(c) = \frac{{c}^{1-n}-1}{1-n}$$ if $$n\ne 1$$
Obviously, in case of n = 1 then it is $$u(c) = \ln\left({c}\right)$$
I know that this is an iso-elastic utility function. However, I do not understand why economists do use this power function to model iso-elastic utilities. What are the mathematical properties that do make this form so beneficial?
A second example is from Mulligan (1997). He uses the following utlity function to model decisions by parents on their own consumption and that of their children:
$$\frac{\sigma}{\sigma-1}{{C}_{t}}^{\frac{\sigma-1}{\sigma}}+\alpha\frac{\sigma}{\sigma-1}E[{{C}_{t+1}}^{\frac{\sigma-1}{\sigma}}]$$
In this example, I do not understand why the author chooses $$\frac{\sigma}{\sigma-1}{{C}_{t}}^{\frac{\sigma-1}{\sigma}}$$ to model utility in this case (he also does not explain it in his book). To me it almost appears to be random. Again, what are the mathematical properties that would make anyone choose this form?
Of course I would appreciate any hints for why one would choose these forms in the examples mentioned above. But I would also appreciate any book/ reference that explains the mathematical properties (and thus the rationale) for these functions. Maybe it is just me but I did not find any useful book that walks the reader for the mathematical justification of the different forms of utility function.
I hope my questions do makes sense. I would be grateful for any help.
Manfred
I have a non-economic background and I am currently interested in utility functions and how they are modeled in economics. Some internet research and I found out that there are obviously different ways of model utility. In my own studies of related topics I encountered the following two examples:
$$u(c) = \frac{{c}^{1-n}-1}{1-n}$$ if $$n\ne 1$$
Obviously, in case of n = 1 then it is $$u(c) = \ln\left({c}\right)$$
I know that this is an iso-elastic utility function. However, I do not understand why economists do use this power function to model iso-elastic utilities. What are the mathematical properties that do make this form so beneficial?
A second example is from Mulligan (1997). He uses the following utlity function to model decisions by parents on their own consumption and that of their children:
$$\frac{\sigma}{\sigma-1}{{C}_{t}}^{\frac{\sigma-1}{\sigma}}+\alpha\frac{\sigma}{\sigma-1}E[{{C}_{t+1}}^{\frac{\sigma-1}{\sigma}}]$$
In this example, I do not understand why the author chooses $$\frac{\sigma}{\sigma-1}{{C}_{t}}^{\frac{\sigma-1}{\sigma}}$$ to model utility in this case (he also does not explain it in his book). To me it almost appears to be random. Again, what are the mathematical properties that would make anyone choose this form?
Of course I would appreciate any hints for why one would choose these forms in the examples mentioned above. But I would also appreciate any book/ reference that explains the mathematical properties (and thus the rationale) for these functions. Maybe it is just me but I did not find any useful book that walks the reader for the mathematical justification of the different forms of utility function.
I hope my questions do makes sense. I would be grateful for any help.
Manfred