Adding scaled CRRA utility Suppose our utility function is the usual CRRA utility with $\gamma=2$ so that:
$$u(C) = \frac{C^{1-\gamma}}{1-\gamma} = -\frac{1}{C}$$
Now suppose there are 2 goods, A and B, available for consumption. The consumption of A and B are denoted $C_A, C_B$. How do I compute the final utility?
For example, if I just add the utility, then the summed utility will be less than consuming either one of the goods, because the utility function is negative.
$$u(C_A) + u(C_B) = -\frac{1}{C_A} -\frac{1}{C_B} &lt; u(C_A), u(C_B)$$
Suppose also that we want to make good B &quot;more valuable&quot;. ie. 1 unit of good B should be worth more than 1 unity of good C. How would I reflect this in the utility function? A simple scaling with a value greater than 1 won't work, again because of the negative utility:
$$u(C_A) + u(C_B) = -\frac{1}{C_A} -k \cdot\frac{1}{C_B}$$
where $k&gt;1$
Thanks in advance.
You have to be careful when comparing different utility functions.
For example the function
$$
v(C_A) = - \frac{1}{C_A},
$$
and the function:
$$
u(C_A) + u(C_B) = - \frac{1}{C_A} - \frac{1}{C_B},
$$
cannot simply be compared. The first represents preferences over one good, while the second gives preferences over two goods.
So if you say:
$$
u(C_A) + u(C_B) &lt; v(C_A),
$$
you are in fact comparing two different utility functions.
If you want to know what happens if you only consume one of the goods, you need to compare, for example $u(C_A) + u(C_B)$ with $u(C_A) + u(0)$.
Also if you increase $k$ in the function:
$$
-\frac{1}{C_A} - \frac{k}{C_B},
$$
you are in fact making $C_B$ more valuable. To see this let's compute the marginal utility of $C_B$ relative to the marginal utility of $C_A$:
$$
\frac{MU_B}{MU_A} = \frac{\dfrac{k}{C_B^2}}{\dfrac{1}{C_A^2}} = k \frac{C_A^2}{C_B^2},
$$
which does increase in $k$. So the higher $k$ the higher the value of an additional unit of $B$ compared to an additional unit of $A$.