Adding scaled CRRA utility

Question

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. $$u(C_A) + u(C_B) = -\frac{1}{C_A} -\frac{1}{C_B} < u(C_A), u(C_B)$$

Suppose also that we want to make good B "more valuable". 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 $k>1$

Thanks in advance.

Answer 1

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: $$ u(C_A) + u(C_B) < v(C_A), $$ 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 $A$.