Intro
Intermediate microeconomics students across California—including UCLA, USC, UC Berkeley, UC Irvine, UC Davis, UC Santa Cruz, UC Riverside, and the CSU system—reach cost functions and economies of scale around mid‑semester. This is where production theory becomes graphical and algebraic: SR vs LR costs, envelope curves, and scale economies. These problems appear on every midterm and problem set. For additional support, visit our Microeconomics Tutoring or explore related topics in the Economics Post Hub.
Answer First
The short‑run cost function holds at least one input fixed, while the long‑run cost function allows all inputs to vary. Economies of scale occur when long‑run average cost decreases as output increases. The long‑run cost curve is the lower envelope of all short‑run cost curves.
Problem Setup
Short‑run total cost: \[ STC(q) = wL(q) + r\bar{K} \] Long‑run total cost: \[ LTC(q) = \min_{L,K} \{wL + rK : f(L,K)=q\} \] Average costs: \[ SAC(q) = \frac{STC(q)}{q},\quad LAC(q) = \frac{LTC(q)}{q} \] Economies of scale: \[ \text{If } LAC(q) \downarrow \text{ as } q \uparrow,\ \text{economies of scale} \]
Step-by-Step Explanation
1. Identify which inputs are fixed
Short run: at least one input (usually capital) is fixed. Long run: all inputs are variable.
2. Compute short‑run cost
Plug the fixed input into the production function and solve for the variable input needed to produce q.
3. Compute long‑run cost
Choose the cost‑minimizing combination of inputs for each output level. This uses conditional factor demands from MA1.
4. Compare SAC and LAC
The LAC curve lies below all SAC curves because the firm can adjust all inputs in the long run.
5. Determine economies of scale
Check whether LAC is falling, constant, or rising:
- LAC decreasing → economies of scale
- LAC constant → constant returns to scale
- LAC increasing → diseconomies of scale
Intuition
Short‑run cost curves reflect temporary constraints. Long‑run cost curves reflect full flexibility. Economies of scale arise when spreading fixed costs or using more efficient input combinations lowers average cost.
Common Exam Mistakes
- Confusing returns to scale with economies of scale.
- Thinking SAC must always be above LAC (it can touch at tangency points).
- Using short‑run marginal cost to infer long‑run scale properties.
- Ignoring fixed inputs when computing short‑run cost.
Final Summary
Short‑run cost functions hold at least one input fixed, while long‑run cost functions allow full flexibility. The long‑run cost curve is the lower envelope of short‑run cost curves. Economies of scale occur when long‑run average cost falls as output increases.