Curvatures An S-shaped value function v(·) can be defined by an expression that has two components: one corresponding to the realm of gains and one corresponding to the realm of losses. For example: v (x) = { Using this equation, the value of ±0 is v(± 0) = 0 while the value of + 10 is v (+10) = √10/2 = √5 ≈ 2.24. The difference is v(+10) − v(± 0) ≈ 2.24 − 0 = 2.24. Meanwhile, the value of + 1000 is v (+1000) = √+1000/2 ≈ 22.36 while the value of + 1010 is v (+1010) = √+1010/2 ≈ 22.47. The difference is only v(+1010) − v(1000) ≈ 22.47 – 22.36 = 0.11. The difference between v(± 0) and v(+10) is indeed greater than the difference between v(+1000) and v(+1010).

Because we are dealing with positive numbers throughout, we are in the realm of gains, which means that we are using the upper half of the equation. In the next exercise you will be using the lower half of the equation, since you will be dealing with negative numbers and the realm of losses. By the way, |x| is the absolute value of x: that is, x with the minus sign removed (if there is one).