Paper: Lochner (2007)

"Individual Perceptions of the Criminal Justice System" (2007) AER.
Note: this paper frames everything as in arrest rates. However, Paper: Becker (1968) emphasizes conviction. While arrest likely comes with its own disutility, it is not the full answer: is there a substantial effect of the probability of conviction and the probability of types of punishment (e.g. incarceration) over and above the effect on conviction?
== Summary ==
=== 0. Introduction ===
Empirical estimates of deterrence are based on (1) actual/official, or (2) perceived measures of certainty and severity of punishment.Interesting: "(...) comparatively little attention has been given to examining the origins of risk perceptions and their connection to actual sanction policy." Contributions of this paper to the perceptions-based deterrence literature:
# Examines which factors influence individuals' self-reported perceptions about the probability of arrest: 
#* Correlation with: (a) demographic characteristics, (b) local official arrest rates, (c) neighborhood conditions,
#* Analyzes (d) the role of new information and belief updating.
#* Findings: individuals update their beliefs rationally.What does this mean here? In a way wouldn't it more more rational to not update given that the additional info is n=1? Avoiding arrest reduces its perceived probability, and being arrested increases the perceived probability.
# Examine the effect of beliefs about arrest probabilities on criminal behavior.
=== I. Evolution of Crime and Beliefs ===
Sketch a simple model of belief updating and criminal participation to (1) guide empirical analysis, and (2) highlight few important policy implications.
* B_{it}: benefits from committing a crime, known ex-ante
* J_{it} \geq 0: punishment, known ex-ante
* \pi_{it}: probability of arrest, unknow ex-ante but have some beliefs given by the information set H_i^t
* \mathbb{E}(\pi|H_i^t) : expected probability of arrest conditional on information at time t
The individual will commit a crime iff: B_{it} \geq \mathbb{E}(\pi|H_i^t) J_{it}, absent intertemporal effects of arrest or criminal behavior, except through beliefs (so ignores incentive to commit crime to learn about the probability of arrest).In Paper: Becker (1968) notation: B > pf . He doesn't have an explicit term for the 'benefit' from a crime. He also doesn't have notation for the outside option, the utility when not committing a crime. Could change his notation slightly to let U(Y)be utility when not committing a crime, U(Y+r)the utility when committing a crime and not being caught, and U(Y+r-f)the utility of committing a crime and being caught. Then, adjusting the expected utility expression in fn 16, p.177:\begin{align}
pU(Y+r-f) + (1-p)U(Y+r) & > U(Y) \\
p(U(Y+r) - U(Y+r-f)) & < U(Y+r) - U(Y) \\
pf & < B 
\end{align}
Where the LHS gives the utility cost of punishment, and the RHS gives the benefit of committing a crime (which doesn't depend on being caught or not).
Information accumulates according to H_i^t = (H_i^{t-1}, c_{it}, A_{it}, \tilde c_{it}, \tilde A_{it}, Z_{it}), and let a general rule for belief updating be: \mathbb{E}[\pi|H_i^t] = g(\mathbb{E}[\pi|H_i^{t-1}]), c_{i,t}, A_{i,t}, \tilde{c}_{i,t}, \tilde{A}_{i,t}, Z_{i,t})where: 
* c_{it} \in \{0,1\}: dummy if an individual commits a crime between last and current (survey) dates
* A_{it} \in \{0,1\}: dummy if arrested during the same time period
* \tilde c_{it}, \tilde A_{it}: vectors of these same indicators for individuals that person i associates with
* Z_{it}: other new information about the local environment
The paper uses this model as a benchmark, and evaluates empirically each of:
# g_1 \geq 0: current perceived probability of arrest is increasing in past expected probability of arrest
# g_2, g_4 \leq 0: perceived probability of arrest is decreasing in the number of crimes committed by the person or their associates
# g_2 + g_3, g_4 + g_5 \geq 0: the total effect of committing a crime and being arrested increases the perceived probability of arrest
# Are beliefs increasing in actual local arrest rates?
# Does the 'state' of local communities (rundown buildings, broken windows, etc) affect the perceived probability of arrest?
last ~page of this section about some theoretical implications of this model: (1) criminal profiles will be determined in part by randomness associated with arrest, (2) criminals will on average underestimate arrest probabilities and non-criminals will overestimate, (3) [point abt heterogeneity, don't understand], and (4) if beliefs depend only on policy-invariant priors and crime/arrest histories, any policy will not have an immediate effect on crime but only a lagged one that works through beliefs.
=== II. Crime and Perceptions ===
==== II.A Crime and Beliefs in the NLSY97 ====
NLSY97 sample: 9,022 individuals of age 12-26 in 1997. Data on: family background, test scores, neighborhood characteristics, criminal behavior, and perceptions about the probability of arrest for car theft. People report arrest for some crime in a year, but unfortunately not what crime they were arrested for. Report some descriptives, which allows a simple arrest rate calculation: #arrests/#reported thefts of > $50, this is ~9% for the sample.
They note that the average perceived probability of arrest for auto theft decreases as they subset to 'increasingly criminal' subsets, but that it remains high: 45% for those who report having stolen a car. [PROBLEM]: They then note that this is 'surprisingly' much higher than the general arrest rate, but this seems a strange comparison: they don't in any way observe the actual arrest rate for car theft so can't say that it contradicts the theory (which predicts perceived rate < actual rate for offenders), its possible that car theft arrest rates are >45%, no? Next, he mentions this "discrepancy" may be due to people misinterpreting the Q. [For table 2, they use reported crime data? If arrests are measured accurately, and crimes underreported then this should be an upper bound on the arrest rate for auto theft? Why do they not seem to use it this way?]The OLS in table 2 combines some official crime data to calculate county-lvl arrest rates (fn 14), and survey data for the rest of the variables? It's unclear to me.
Adding all the info they have to do an OLS of perceived probability of arrest for car theft on county-lvl arrest rates and a host of other variables, they find a positive but insignificant effect of county-level arrest rates, and only an R2 of 0.03. 
[OVERALL, IT IS NOT CLEAR WHAT THE TAKEAWAY IS FROM THIS. WHAT ARE ACTUALLY THE FINDINGS?]
==== II.B Crime and Beliefs in the NYS ====
NYS sample: 1,725 individuals of age 11-17 in 1976. Focus on perceptions and criminal behavior as reported by 918 men in the 1983 and 1986 surveys (thus, aged 18-27). Less than 4% report stealing something >$50 or breaking into a building/vehicle (that is < 36 people sample!). #property crimes/#reported break-ins and thefts >$50 = property crime arrest rate of ~5%.
Reported probabilities of arrest in the NYS are (2) 58% for stealing something >$50, (3) 62% for breaking into a building/vehicle, and (4) 72% for attacking someone to hurt of kill them. For the subset of people who committed the type of crime, and weighted by number of times, this goes down to (2) 47%, (3) 45%, and (4) 53%. This contrasts with official arrest rates of (~2) 5.5% for larceny-theft, (~3) 7.4% for burglary, and (~4) 20.4% for assault. 
Note that again in an OLS regression of perceived probability on a range of vars, the R2 is tiny at a maximum of 0.039 for stealing something <$5.
[I GUESS TAKEAWAY IS THAT PERCEPTIONS ARE MUCH HIGHER THAN OFFICIAL ARREST RATES, EVEN FOR THOSE WHO COMMIT THE CRIME?]
=== III. Information-Based Belief Updating ===
=== IV. The Influence of Perceptions on Criminal Behavior ===
=== V. Conclusions ===