Predatory mortgage lending

In our model, a critical assumption is that the existing lender possesses more information than the borrower. Initially, we focus on the most extreme scenario where, at time 1, the lender can perfectly predict the borrower's income at time 2, while the borrower only knows that her likelihood of having a high income at time 2 is p. Essentially, the lender has complete foresight. However, we now relax this assumption and instead depict the lender as receiving a signal about the borrower's income at time 2 that isn't perfect but still informative. The results we derive remain consistent in nature. Moreover, diminishing the lender's level of information has two opposing influences on the prevalence of predatory lending. On one hand, the lender's limited foresight results in a decreased expectation of a negative impact on the borrower's well-being, given the possibility that she might have a higher income at time 2 and can thus repay the loan. This effect mitigates predatory practices. On the other hand, the lender's reduced foresight prompts a greater willingness to lend to borrowers with unfavorable signals, as there's still some chance they could have higher incomes at time 2. This increased lending propensity could lead to more predatory behavior.

In a more formal context, we suppose that the current lender receives a signal σ ∈ {I, K}. Borrowers for whom the lender observes σ = I (respectively, σ = K) are labeled as good-signal borrowers (respectively, bad-signal borrowers). We assume that the probability of σ = I given y2 = I is Pr(σ = I|y2 = I) = Pr(σ = K|y2 = K) = 1 − ε, where ε ∈ [0, 1/2). The case where ε = 0 aligns with the primary model, while higher values of ε correspond to a reduction in the lender's informational advantage. We define θ ≡ Pr(y2 = K|σ = K) = (1−p)(1−ε) / ((1−p)(1−ε)+pε) and φ ≡ Pr(y2 = I|σ = I) = p(1−ε) / (p(1−ε)+(1−p)ε) to represent the lender's adjusted beliefs.