Forschungsgruppe ORCOS
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Economics of Crime: DNS Thresholds

Skiba (1978) studied a one-sector optimal growth model with a convex-concave production function. He established a critical cut-off capital stock Ks. If the initial value of capital is below Ks, it does not pay to start accumulating more capital because the marginal product of capital is too low. On the other hand, if K>Ks, capital is accumulated up to the Golden Rule limit. Hence the outcome of the optimal policy depends on the initial state, specifically the magnitude of the initial state relative to some critical threshold. An even earlier example of such a threshold phenomenon has been delivered in a marketing example by Gould (1970) in which the goodwill stock grows logistically. A brief introduction to such thresholds, which are called 'Skiba points', is given in Feichtinger and Hartl (1986, p. 116-119, 325-335).

A first sound proof of the existence of a threshold separating two basins of attraction in a one-state dynamic optimization model was given by Dechert and Nishimira (1983). Since that time many papers have been written dealing with multiple steady states and history dependence in one-dimensional optimal control models (see, e.g.Long et al., 1997; and Wirl and Feichtinger, 1999, for survey).

In the economics of crime context, multiple steady states and hence threshold policies arise fairly frequently. Feichtinger et al. (1997) describe a dynamic extension of Becker's (1968) general approach to crime and punishment, where the law enforcement agency determines the optimal dynamic trade-off between damages caused by offenders, law enforcement expenditures and cost of imprisonment.Tragler et al. (1997) and Caulkins et al. (2000) apply dynamic optimisation techniques more specifically to the control of illicit drug consumption. In all these models, there exists a Skiba threshold implying that above the Skiba point the optimal trade-off between social costs implies a steady state with a high level of offenders / drug users, while below the threshold the optimal law enforcement should eradicate crime / drug use. Due to the formal analogies between crime / drug control and, e.g., pollution models (similar control instruments: prevention, treatment, etc.), one can expect that these threshold policies also arise in a variety of other socio-economic contexts, e.g. in environment planning - see also Feichtinger (1999)