Specify the Lucas Supply Function. What are its implications? In what respects is it different from the classical aggregate supply function?

The Lucas Supply Function, formulated by Robert Lucas in the 1970s, is a significant contribution to macroeconomic theory, particularly in the context of aggregate supply. It was developed as part of the Lucas critique, which emphasizes that macroeconomic relationships must account for the fact that agents in the economy form expectations based on the overall economic model, rather than simply reacting to past data. The Lucas Supply Function offers an alternative to traditional aggregate supply (AS) functions, which assumed a direct, fixed relationship between the price level and output in the short run.

### Lucas Supply Function: Specification

The Lucas Supply Function is based on the assumption of rational expectations, which means that economic agents (such as firms and households) form their expectations of future variables, including the price level, in a manner consistent with the underlying economic model. Lucas built on the New Classical framework, incorporating the idea that agents adjust their behavior not only based on current conditions but also in anticipation of future economic events. 

Mathematically, the Lucas Supply Function can be represented as:

\[
Y = Y^n + \alpha (P - P^e)
\]

Where:
- \( Y \) is the actual output in the economy,
- \( Y^n \) is the natural level of output (also referred to as potential output or the long-run output),
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