When looking at utility functions, how do you compare preferences? I understand the economic concepts, but the math part is holding me down from understanding a lot in the textbook. (I am not typing this so that someone can 'do my homework for me'). For example, given these three functions, how do I compare the different consumer's preferences for the two goods? I think this is pretty straightforward. I just need a quick explanation to understand how I interpret these numbers. Thanks!
Given that the problem is "compare the preferences of the individuals below":
The first two are Cobb-Douglas utility functions. The general function is: $$ U(x,y)=x^{\alpha}y^{\beta} \space ,\space \alpha,\beta\in (0,1) \space and \space \alpha +\beta=1 $$ In this equation the higher the exponent $\alpha$ (or $\beta$) the more that particular good is important for the consumer.
We can take the $ln(U_{Steve}(n,c))$ and compare the 3 functions.
$$U_{steve}(n,c)=\frac{2}{3}ln(n)+\frac{1}{3}ln(c)$$
We can clearly see that Steve and Patrick have the same preferences. While Jeff shows a utility function which is not the standard cobb-douglas, but if we have to apply the same reasoning even for Jeff, then he prefers good $n$ and $c$ more than the other guys. Plus he has one unit of utility even withouth consuming $n$ or $c$.
