How to Quantify Utility/Pleasure/Pain using the Positive Real Numbers?

Question

I am studying about Cardinal Utility in Economics (or more generally, how to quantify pleasure and pain!)

Intuitively, I assign a positive number to pleasurable experiences, and a negative number to painful experiences:

• pleasurable experience $\rightarrow$ positive number
• painful experience $\rightarrow$ negative number
• greater the magnitude of the number $\rightarrow$ more intense is the pleasure or pain

But... is there anyway to transform this scale into the positive real numbers?

• more painful the experience $\rightarrow$ closer to zero
• more pleasurable the experience $\rightarrow$ larger positive number

That way, I won't need to worry about sign!

Answer 1

The answer mentioned by Good Morning Captain is great! If $U_\text{old}$ was my old utility scale, whose range is the real numbers $\mathbb{R}$, then Captain's transformation:

$$U_\text{new} = f(U_\text{old}) = \begin{cases} \frac{1}{1+|U_\text{old}|} &\text{for } U_\text{old} \leq 0\\ U_\text{old}+1 &\text{for } U_\text{old} > 0 \end{cases}$$

maps negative values of $U_\text{old}$ onto the unit interval $[0,1]$, while mapping the positive values onto $\mathbb{R}^+$.

An even simpler solution would be to just use the exponential transformation:

$$U_\text{new} = \exp(U_\text{old})$$

In both the cases (Captain's function and Exponential function):

• painful experiences $\rightarrow$ negative values in the $U_\text{old}$-scale $\rightarrow$ values in the range $[0,1)$ in the $U_\text{new}$-scale. More negative the experience, more close is the scale to 0.
• neutral experiences $\rightarrow$ 0 in the $U_\text{old}$-scale $\rightarrow$ 1 in the $U_\text{new}$-scale
• pleasurable experiences $\rightarrow$ positive values in the $U_\text{old}$-scale $\rightarrow$ values in the range $(1,+\infty)$ in the $U_\text{new}$-scale