The expected-cost deterrence response on the log scale, and the discount it implies

The expected cost of an offence under certainty \(p\), severity \(s\) and delay \(c\) with discount rate \(\rho\) is \(p s e^{-\rho c}\), whose logarithm is \(\log p + \log s - \rho c\) (Research.P8.Instrument.logcost_linear). A response linear in the log design \((\log p, \log s, c)\) is this model with a common elasticity, and the ratio of the celerity to the certainty coefficient identifies \(-\rho\) (ratio_is_discount). Identification of the three coefficients is the rank condition of morie_deterrence_design_check on the transformed design.

Usage

morie_deterrence_logcost(p, s, c, y = NULL, rho = NULL)

Arguments

p, s, c

Certainty (in (0, 1]), severity (positive) and delay of the sanction, one observation each.

y

Optional response; when given the log-design least-squares fit is returned.

rho

Optional discount rate at which the expected cost is evaluated.

Value

A list with the log design check (identified, rank), and, with y, coefficients and implied_discount; with rho, expected_cost and log_expected_cost.

Examples

set.seed(2)
p <- runif(40, 0.1, 0.9); s <- runif(40, 1, 10); c <- runif(40, 0, 12)
y <- 2 - 0.8 * log(p) - 0.8 * log(s) + 0.8 * 0.1 * c + rnorm(40, sd = 0.1)
morie_deterrence_logcost(p, s, c, y)$implied_discount
#> [1] 0.09283505