Behavioral economics is a subfield of economics that blends insights from psychology and economics to better understand how individuals make decisions and how these decisions deviate from those predicted by classical economic theories. Traditional economic models often assume that individuals act rationally, maximizing their utility based on stable preferences and full information. However, empirical evidence frequently shows that actual human behavior deviates from these assumptions due to various cognitive biases, emotions, and social influences.

Core Concepts in Behavioral Economics:

1. Bounded Rationality:
   Proposed by Herbert Simon, bounded rationality suggests that while individuals aim to make rational choices, their cognitive limitations and the complexity of the environment often constrain their decision-making processes. They thus use heuristics or mental shortcuts, which can sometimes lead to suboptimal or irrational choices.

2. Prospect Theory:
   Developed by Daniel Kahneman and Amos Tversky, this theory posits that people evaluate potential losses and gains differently, leading to decision-making that deviates from expected utility theory. Prospect theory introduces the concept of loss aversion, which suggests that losses loom larger than gains of the same magnitude. Mathematically, the value function \(v(x)\) in prospect theory is generally concave for gains (\(x > 0\)) and convex for losses (\(x < 0\)), and steeper for losses than for gains.

   \[
   v(x) = \begin{cases}
   x^\alpha & \text{if } x \geq 0 \\
   -\lambda (-x)^\beta & \text{if } x < 0 \\
   \end{cases}
   \]
   where \( \alpha \) and \( \beta \) are typically less than 1, representing diminishing sensitivity, and \( \lambda > 1 \) represents loss aversion.

3. Nudge Theory:
   Introduced by Richard Thaler and Cass Sunstein, nudge theory explores how indirect suggestions and positive reinforcement can affect behavior and decision-making. By designing choice architectures (the way options are presented), policymakers and businesses can ‘nudge’ individuals towards better decisions without restricting their freedom of choice. An example is automatically enrolling employees in a retirement savings plan but allowing them to opt-out.

4. Mental Accounting:
   Proposed by Richard Thaler, mental accounting refers to the tendency of individuals to categorize and treat money differently based on subjective criteria, such as the source of the money or its intended use, rather than considering it as fungible. This can lead to irrational behaviors, such as saving money at low interest rates while carrying high-interest debt.

5. Hyperbolic Discounting:
   Hyperbolic discounting describes the tendency for people to prefer smaller, immediate rewards over larger, delayed rewards more than traditional exponential discounting would predict. This can be mathematically expressed as:

   \[
   D(t) = \frac{1}{1 + \alpha t}
   \]
   where \( D(t) \) is the discount factor at time \( t \), and \( \alpha \) is a parameter indicating the degree of present bias. As \( t \) increases, the value diminishes rapidly, explaining why individuals often overvalue immediate rewards at the expense of future benefits.

Behavioral economics provides valuable insights into various economic phenomena, such as consumer behavior, market anomalies, and public policy design. By incorporating a more accurate representation of human behavior, this field helps in shaping strategies and interventions that can lead to improved economic outcomes both for individuals and society at large.