# **C.1 Setup**

We model an individual’s choice between legitimate business and illicit activities under different conditions—with and without time inconsistency, and with and without financial market imperfections— and assess the predictions for a number of common labor market and crime-reducing interventions: greater punishment, increasing productivity in legitimate business (e.g. through technology or skills improvement), cash or capital transfers, and interventions that shape preferences—either time preferences or personal preferences against illegal behavior.

We use _L_<sup>_b_</sup> and _L_<sup>_c_</sup> to denote time spent in legitimate activities (such as petty business) and illegitimate activities (such as crime). Legitimate business produces revenue according to production function _F_ ( _θ, L_<sup>_b_</sup> _t_<sup>_, Kt_),where</sup><sup>_θ_isproductivityorindividualabilityand</sup><sup>_K_isaccumulatedcapital</sup> used in business. A person’s decision to participate in illegal activity is motivated by the potential gains and costs from such activity. Gains include the expected illegitimate payoff per hour spent in illegal activities, _w_ . Costs include the possibility of apprehension and conviction, which occurs with probability, _ρ_ , and implies a penalty, _fL_<sup>_c_</sup> _t−_ 1<sup>.Thusthe penalty for criminalbehavioris a linear</sup> function of hours spent in criminal activities in the previous period<sup>6</sup> The individual’s total expected earnings from legitimate and illegitimate activities are _yt ≡ F_ ( _θ, L_<sup>_b_</sup> _t_<sup>_, Kt_) +</sup><sup>_wtLc_</sup> _t_<sup>_−ρfLc_</sup> _t−_ 1<sup>..In</sup> addition to investing in business, the individual can also invest or borrow through a riskless asset with constant returns 1 + _r_ . At each period _t_ , the individual decides how much to invest for next period _at_ +1 and reaps interests _rat_ from last period’s investments.

Individuals have utility function _U_ ( _c, l, σL_<sup>_c_</sup> ), where _c_ denotes consumption and _l_ denotes time for leisure. We also allow for individuals to have direct disutility from engaging in crime, as measured by _σL_<sup>_c_</sup> , where _σ >_ 0 implies that implies that illicit work induces some internal penalty such as shame, though in principle it could also reflect social penalties such as a loss of esteem or exclusion from peers and other social networks. We make the standard assumption that _Uc′_<sup>_≥_0,</sup><sup>_U_</sup> _l′_<sup>_≥_0,</sup> _UσL′_<sup>_c≤_0,</sup><sup>_U_</sup> _cc′′_<sup>_<_0,</sup><sup>_U_</sup> _ll′′_<sup>_<_0,</sup><sup>_∂_2</sup><sup>_U/∂L_2</sup> _c_<sup>_≤_0and</sup><sup>_F_</sup> _θ′_<sup>_≥_0,</sup><sup>_F_</sup> _L′_<sup>_≥_0,</sup><sup>_F_</sup> _K′_<sup>_≥_0,</sup><sup>_F_</sup> _θθ′′_<sup>_<_0,</sup><sup>_F_</sup> _LL′′_<sup>_<_0,</sup> _F ′′_<sup>0,and</sup><sup>_F_</sup> _′′_<sup>_F_</sup> _′′_<sup>_F_</sup> _′′_<sup>Weallowfortheindividualtohavequasi-hyperbolic</sup> _KK_<sup>_<_</sup> _θL_<sup>_≥_0,</sup> _θK_<sup>_≥_0,</sup> _LK_<sup>_≥_0.7</sup> ( _β, δ_ ) preferences.

We first consider the case without any uncertainty. The individual’s problem is:

> 6 One reason for this modeling choice is because we want to explore the role that quasi-hyperbolic preferences play in the decision to commit crimes when the punishment is in the future not the present.

> 7For ease of analysis, we also assume that the marginal return to capital is infinity for the first unit of capital invested in business, and that as long _′_ as there is positive capital input, marginal _′_ product of labor for the first unit of labor will be infinity, i.e. _lim K_<sup>(</sup><sup>_θ, Lb, K_)=+</sup><sup>_∞_forall</sup><sup>_Lb_and</sup><sup>_lim_</sup> _K_<sup>(</sup><sup>_θ, Lb, K_)=+</sup><sup>_∞_aslongas</sup><sup>_K>_0.This</sup> _K↓_ 0<sup>_F_</sup> _L_<sup>_b_</sup> _↓_ 0<sup>_F_</sup>

> assumption guarantees that investments and hours in business will always be positive.

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