Define short run production functions.

A short-run production function describes the relationship between variable inputs (typically labour) and output, holding at least one input (usually capital) fixed. It is represented mathematically as:

$$Q=f(L,\bar{K})$$

where:

• $Q$ = Total output
• $L$ = Variable input (e.g., labour)
• $\bar{K}$ = Fixed input (e.g., machinery, land)

Key features:

• Law of Variable Proportions: As more variable inputs are added to fixed inputs, output initially increases at an increasing rate (increasing returns), then at a decreasing rate (diminishing returns), and finally may decline (negative returns).
• Assumptions: Time period is too short to change fixed inputs, technology is constant, and other factors (e.g., management) remain unchanged.
• Example: If $Q=\frac{L^2}{4}$ with fixed capital $K=2$, output depends solely on labour ($L$) in the short run.