  Damage cost curves -- relating the typical damage of a natural hazard to its
physical magnitude -- represent an indispensable ingredient necessary for
climate change impact assessments. Combining such curves with the occurrence
probability of the considered natural hazard, expected damage and related risk
can be estimated. Here we study recently published city scale damage cost
curves for coastal flooding and demonstrate which insights can be gained from
the functions only. Therefore, we include protection cost curves -- relating
the typical investment costs necessary to protect a city against a natural
hazard of certain magnitude -- which are analogous to and consistent with the
above mentioned damage cost curves. Specifically, we motivate log-logistic
functions, which exhibit a power-law increase at the lower end, and fit them to
the cost curves. As expected, cities with large maximum potential loss
(typically large cities) are also more costly to protect. Moreover, we study
the idealized case of continuous adaptation, i.e.\ increasing protection levels
in the same pace as sea-level rise, and compare the associated costs with
residual damage from extreme events exceeding the protection. Based on the
fitted exponents we find that in almost all cities the residual damage rises
faster than the protection costs. Raising coastal protection can lead to lull
oneself in a deceptive safety.
