Competitive Credit and Present Bias: A Stochastic Discounting Approach

A prominent theme in behavioural contract theory is the study of present-biased agents represented through quasi-hyperbolic discounting. In a model of competitive credit provision, we study an alternative to this framework in which the agent has a private stochastic discount factor and may overestimate the likelihood of more patient values. Agent preferences, however, are timeconsistent. While a limiting case of our model corresponds to a “fully naive” agent in work on quasi-hyperbolic discounting, another case is where the agent has correct beliefs about future discounting. In equilibrium, the agent selects options with earlier consumption in case of less patient discount factor realisations, but is penalised by receiving worse terms. Our model thus accounts for an important feature of equilibrium contracts identified in Heidhues and Kőszegi (2010). Unlike Heidhues and Kőszegi, our framework often predicts excessively backloaded consumption, including when the agent holds correct beliefs about future discounting.

💡 Research Summary

The paper introduces a novel framework for competitive credit markets in which a single representative agent has time‑consistent preferences but faces a stochastic discount factor that determines how current consumption is valued relative to future consumption. Unlike the standard quasi‑hyperbolic discounting models that generate present bias through a sequence of “selves,” the agent here plans with a single self whose utility function is ordinary (strictly increasing, concave, twice differentiable). The novelty lies in allowing the agent to hold possibly misspecified beliefs about the distribution of future discount factors: the agent may over‑estimate the probability of drawing a more patient (higher) discount factor. Two limiting cases are highlighted. In the “fully naive” limit the agent believes the patient type will occur with certainty, reproducing the preferences of the fully‑naïve quasi‑hyperbolic agent studied in Gottlieb‑Zhang (2021). In the opposite limit the agent’s beliefs are correct, i.e., the perceived distribution matches the true i.i.d. distribution of discount factors.

The market consists of J ≥ 2 banks that compete by offering dynamic lending contracts over T ≥ 3 periods. Each bank faces the same gross interest rate R and can earn profit equal to the present value of the agent’s endowment stream I_T minus the discounted cost of the consumption schedule it finances. After observing his realized discount factor δ₁, the agent sends a message to the chosen mechanism, which then determines the entire consumption path (c₁,…,c_T). The agent evaluates any contract by the expected discounted payoff computed with his (possibly wrong) belief about future discount factors, subject to incentive‑compatibility (the contract must be optimal for the agent given his belief) and a non‑negative profit constraint for the bank, which is evaluated using the bank’s correct beliefs.

The equilibrium analysis yields several key insights. First, when a low‑discount (impatient) factor is realized, the optimal contract front‑loads consumption but imposes harsher terms (e.g., higher effective interest rates). Conversely, when a high‑discount (patient) factor is realized, the contract back‑loads consumption and offers more favorable terms. This pattern mirrors the contract features identified by Heidhues‑Kőszegi (2010) and Gottlieb‑Zhang (2021), yet the underlying preference structure is fundamentally different because the agent’s preferences are time‑consistent.

The authors prove two central propositions. Proposition 2 shows that because the agent over‑weights the likelihood of patient realizations, equilibrium contracts tend to “excessively backload” consumption: the agent receives a larger share of his consumption in later periods than would be efficient. Proposition 3 demonstrates that this back‑loading inefficiency worsens as the horizon T grows, leading to substantial welfare losses in the long run—contrasting sharply with quasi‑hyperbolic models where efficiency is restored for long horizons.

Section 4 extends the analysis to a richer state space with multiple possible discount factors and allows both the agent and the banks to have full support over this set. Lemma 4 establishes that incentive constraints become increasingly binding as the discount factor moves from low to high values, effectively shifting the “utility weight” toward more patient types. Corollary 2 generalizes the earlier result: impatient realizations receive more front‑loaded consumption but on worse contractual terms, while patient realizations enjoy back‑loaded consumption with better terms. Notably, this pattern persists even when the agent’s beliefs are correct, indicating that the contract structure is driven more by the need to satisfy incentive constraints than by belief errors per se.

For welfare evaluation the paper adopts the “firm‑belief” benchmark (following Grubb 2009): the agent’s expected utility is measured using the banks’ correct beliefs, which are assumed to be the true distribution. Under this benchmark, the equilibrium is inefficient because the agent’s consumption is mis‑timed, and a regulator cannot improve welfare without violating either the profit or incentive constraints. The authors also derive inverse Euler equations (Proposition 7) that characterize equilibrium consumption paths. In the special case of logarithmic utility, Corollary 3 shows an intuitive relationship: when the agent over‑estimates patience, expected consumption grows faster than under the efficient benchmark, reflecting the contract’s “pandering” to high‑discount realizations.

Overall, the paper contributes to the literature on dynamic mechanism design and behavioral contract theory by (i) introducing stochastic discount factors as a source of present‑bias‑like behavior without time‑inconsistency, (ii) demonstrating that belief misspecification about future patience can generate contract features previously attributed to quasi‑hyperbolic discounting, (iii) highlighting a novel source of inefficiency—excessive back‑loading of consumption—that persists even with correct beliefs, and (iv) providing a clear welfare comparison that suggests limited scope for traditional regulatory interventions. The work thus offers a fresh perspective on how uncertainty about future discounting shapes credit contract design and the associated welfare outcomes.