When Should Governments Reveal Criminal Histories?

Appendix – Proofs

The publicly known potential offender optimally commits the crime if θ≥θb≡δˉτ, which is increasing in the probability of detection and penalty if caught. □

After some rearranging, we find that the unknown potential offender optimally commits the crime if θ>θg≡δ_τ+βδ_Vg−Vb. Of course, this expression involves two endogenous variables, θ_g and V_g. Since criminal opportunities are distributed uniformly on (0,1) we get E[θ| θ >θ_g] = (1 + θ_g)/2. I write lifetime expected utility as

Some algebra, and the definition of θ_g above, gives

Substituting this into the expression for θ_g gives an equation in one unknown

Employers will pay a wage equal to the worker’s expected productivity, given that his rating is “good.” Let ξ_g be the fraction of workers with r = g in equilibrium. Then lSg=yP+ξgθgP+ξg. To solve for ξ_g, note that potential criminals with r = g commit crimes with probability 1–θ_g and are caught with probability δ_. These agents die with probability 1–ß and are replaced by an equal mass of r = g agents. In the steady state, ξ_g solves

The comparative statics are easily obtained using the Implicit Function Theorem to compute the derivatives after noting that

for all θ_g and P given Assumption 2. □

where V₁ is the expected lifetime utility of an agent with one conviction under the lenient system. Again using techniques similar to those used in the proof of Lemma 2 gives V1=lLg1−β+1−θ1221−β. Substituting this into (2) above allows the system of equations to be written

Again, employers pay a wage equal to the worker’s expected productivity, given that his rating is “good.” Let ξ₀ and ξ₁ be the fraction of workers with zero and one convictions, respectively, in equilibrium. Then lLg=yP+ξ0θ0+ξ1θ1P+ξ0+ξ1. Using techniques similar to those used to solve for ξ_g in the proof of Lemma 2, solving for ξ₀ and ξ₁ gives

The system of equations in θ₀ and θ₁. can now be solved to find the optimal cutoff values.

The comparative static results are found using differentiation via the Implicit Function Theorem on the system in (4). Differentiating the system with respect to Θ = (θ₀, θ₁.) gives the Jacobian

The determinant of the above matrix is always positive, and so its inverse, DΘ−1, exists. It is possible to solve for the inverse matrix, but it is unwieldy and hence suppressed. I note, however, that each component of the inverse matrix is negative.

Differentiating the system in (4) with respect to δ_ gives

Both of the above terms are positive. The Implicit Function Theorem states that dΘdδ=−DΘ−1×Dδ, and hence dθdδ_>0 and dθ1dδ_>0.

The proof that the optimal cutoff opportunities increase in δ₁, τ, y, and β proceeds in a similar manner. □

Subtracting the equations characterizing cutoff values θ₁ and θ_g, given in (3) and (1) respectively, yields

Similarly, subtracting the equations characterizing θ_g and θ₀, given in (1) and (2), yields

Adding together equations 5 and 6 gives an expression for θ₁–θ₀. Solving that expression for θ₀, which will be useful momentarily, gives

Note that the expressions in (5), (6), and ultimately (7), are derived by assuming that l^L(g) = l^S (g). I next derive an implicit expression relating θ₀ and θ₁ by substituting the expression for V₁ into the equation characterizing θ₀. In contrast to the above equations, the following relationship must hold in all equilibria, not just when l^L(g) = l^S(g). Substituting the expression for V1=lLg1−β+1−θ1221−β into (2) gives

Then setting the equations for θ₀ in (7) and (8) equal to one another gives

Recall that θ1=δ1τ+βδ1l1−β+1−θ1221−β−Vb, and so the θ₁ term on the left hand side can be canceled, giving

Canceling the δ_τ+βδ_1−θ0221−β terms from both sides and some rearranging gives

Noting that V1=l1−β+1−θ0221−β, the above can be re-written as

But this is a contradiction because V₁ >V_b and δ_ < 1 so the left hand side is strictly negative.

This proves that wages for agents with good ratings cannot be equal across notification systems; that is, l^S(g) ≠ l^L(g). I exploit the continuity of the wages to show that wages for agents with good ratings are always greater under the strict system than under the lenient system. In the numerical simulations discussed in the proof of Proposition 1, the wage for agents with a good public rating is larger under the strict policy. Since equilibrium wages are continuous in cutoff opportunities, l^S(g) >l^L(g) in all environments satisfying Assumptions 1 and 2. □

Then to see that θ₀ < θ₁, fix the wage at the equilibrium level l^L(g) and consider the system of equations defining the cutoff opportunities

If δ_=δ1, then θ₀ < θ₁ because if some cutoff opportunity x solves the bottom equation, then the left hand side of the top equation is greater than the right hand side evaluated at x because V₁ >V₂. Decreasing the cutoff opportunity from x decreases the left hand side and increases the right hand, thereby restoring equality and showing that θ₀ < θ₁. If δ_<δ1, this effect is only reinforced.

To see that θ₀ < θ_g, define W_x as the value attained by agents with no convictions under the lenient system when using x as their cutoff opportunity. Then

Define Wg=Wx|x=θg. That is, W_g is the expected value agents with no convictions under the lenient system attain when acting like agents with no convictions under the strict system. As a preliminary step, I first show that W_g –V₁ < V_g –V_b. Subtracting V₁ from the above expression for W_x evaluated at x = θ_g gives

Solving for W_g –V₁ gives

Similar operations give

Then W_g – V₁ < V_g–V_b because l^S(g) > l^L(g) and V₁ > V_b.

Then differentiating (9) with respect to x gives

Evaluating this derivative at x = θ_g, gives

The inequality holds because W_g – V₁ < V_g – V_b, and the equality holds because θ_g, is optimal for agents without any convictions under the strict system and the First Order Condition gives θg=δ_τ+βδ_Vg−Vb.

This shows that the lifetime expected utility of h = 0 under L decreases in the cutoff opportunity at x = θ_g. So an agent with no convictions under the lenient system could do better than mimicking the agent with no convictions under the strict system by using a lower cutoff. But this is a local argument – it is possible some y >θ_g actually maximizes utility. Assume, towards a contradiction, that some such y >θ_g exists. Since lifetime expected utility is continuous and differentiable in the cutoff opportunity, there must be some θ′ >θ_g at which lifetime expected utility attains a local minimum, and the slope at θ′ is necessarily zero. But the Second Order Condition holds whenever the First Order Condition holds, and so there cannot be any θ′ >θ_g that is a local minimum. □

Consider as a benchmark the following constellation of parameters:

In this environment, the optimal cutoff opportunities under the strict and lenient systems are θ_g = 0.65, θ_b = 0.72, and θ₀ = 0.57, θ₁ = 0.92, θ₂ = 0.72, respectively. Let ρS=ξg1−θg+ξb1−θb and ρL=ξ01−θ0+ξ11−θ1+ξ21−θb be the steady state crime rates implemented by the strict and lenient notification systems, respectively. Then the crime rates implemented by the strict and lenient systems are ρ^S = 0.32 and ρ^L = 0.29. The crime rate under the strict notification system is roughly 10% larger.

To illustrate the importance of the probability of detection, consider what happens when δ₁ falls to 0.14 and all other parameters are held constant. Since nothing is affected under the strict system, θ_g and θ_b are unchanged, and hence so is ρ^S. The fall in δ₁ leads to a decrease in d₁ to 0.78. The fact that agents with one conviction commit more crimes (and are therefore less productive) with the lower probability of detection puts downward pressure on the wage, which also leads to a decrease in unknown criminals’ cutoff opportunity to θ₀ = 0.55, further increasing the crime rate. The crime rate then increases to ρ^L = 0.36, which becomes greater than ρ^S = 0.32.

Next, consider what happens when ß falls to 0.85 and all other parameters are as in the benchmark above. This example illustrates how β directly affects both cutoff opportunities and the steady state composition of criminal types. The decrease in the probability of living to next period decreases the cutoff opportunities for agents with good ratings to θ_g = 0.50, θ₀ = 0.47, and θ₁ = 0.76. These changes decrease ξ₁, the steady state mass of agents with 1 conviction, from 0.36 in the benchmark to 0.19 after β falls. The corresponding increase in the steady state mass of unknown criminal types, ξ₀, is from 0.55 to 0.77. With this lower β, the crime rates are ρ^S = 0.45 and ρ^L = 0.47.

To see that productivity is maximized when the crime rate is minimized, let πS=P+ξgθg+ξbθb and πL=P+ξ0θ0+ξ1θ1+ξ2θ2 be the steady state productivity levels implemented by the strict and lenient notification systems, respectively. Note that, since ξg+ξb=ξ0+ξ1+ξ2=1, the crime rates can be re-written as ρS=1−ξgθg−ξbθb and ρL=1−ξ0θ0−ξ1θ1−ξ2θb. Then since ρ^S = P+1 – π^S and ρ^L = P+1 – π^L we have

□