Producer Theory: Production Functions, Short-Run vs. Long-Run, Isoquants, MRTS & Returns to Scale

Earlier material that WILL be on the exam
- Supply–demand curves, elasticities, consumer surplus.
- Consumer theory: indifference curves, tangencies with the budget line, utility maximization ➜ derivation of demand.
- Income vs. substitution effects underpin demand.
Today’s transition: move from the demand side to the supply side (producer theory).
- Similar analytic machinery (curvature + tangency logic).
- Harder conceptually because firms choose the selling price, not take it as given.
- Course will therefore devote ≈2× the time on producer theory compared to consumer theory.

Visualize the firm as a flow-chart / conveyor belt:
- Inputs (factors of production) ⇢ Black box ⇢ Outputs.
Firm’s objective (assumed for now): profit maximization.
- \pi = \text{Revenue} - \text{Cost}.
- Profit maximization ⇢ need for production efficiency.
- Caveat/teaser: later lectures will question if firms really do maximize \pi (e.g., corporate jets & lavish perks), but we take it as given for the model.

Only two inputs are used to generate output q:
1. Labor (L) – hours of work (relatively easy to vary).
2. Capital (K) – “everything else”: machines, land, buildings, tools (harder to vary quickly).
Notation discipline: little q = individual firm’s output; big Q = market-wide output.

Variable input: can be adjusted “easily” (e.g., labor hours).
Fixed input: costly or impossible to change in the short run (e.g., plant size).
Short run (SR): at least one input is fixed (capital fixed, labor variable in our base model).
Long run (LR): all inputs are variable.
- Exact calendar time is context-dependent; concept is theoretical.
Economists sometimes mention “quasi-fixed” factors: inputs not perfectly fixed yet not perfectly variable (e.g., white-collar labor schedules).

With K fixed at \bar{K}, firm chooses L.
Marginal Product of Labor (MPL)
- \text{MPL}=\frac{\Delta q}{\Delta L}\Big|_{K=\bar{K}}.
- Analogous to marginal utility in consumer theory.
Diminishing Marginal Product (DMP) (core assumption)
- Each additional worker adds output, but less than the previous worker.
- Intuition: additional employees share the same fixed capital.
- Example: one shovel, many diggers → 2nd, 3rd,… diggers increase output, but the 6th contributes far less than the 2nd.
- We focus on the realistic interior range where MPL > 0 but falling; we ignore pathological regions where MPL = 0 or negative.

In LR the firm selects both L & K.
Production function: q=f(L,K). Example used in class: q=\sqrt{LK}.
Isoquants: curves showing all (L,K) combos that yield the same q.
- Perfect analogue to indifference curves.
- Properties: farther from origin ⇒ higher output; cannot cross; downward-sloping in standard case.
Marginal Rate of Technical Substitution (MRTS)
- Slope of isoquant: \text{MRTS}{LK}=\frac{\Delta K}{\Delta L}\Big|{q=\bar{q}}.
- Measures how much K the firm can give up for an extra unit of L while keeping output fixed.
- DMP in each input ⇒ MRTS diminishes as one moves down an isoquant (mirrors diminishing MRS in consumer theory).

Perfect Substitutes (linear isoquants)
- Example function: q=L+K. Firm is indifferent between input types; slope is constant.
- Quip: “Harvard undergrad vs. Beanie Baby” — essentially interchangeable inputs.
Perfect Complements / Leontief (right-angle isoquants)
- Function: q=\min{L,K}. Inputs used in fixed proportion (cereal vs. cereal box).
- Extra K without matching L (or vice-versa) contributes nothing.

Ask: what happens if all inputs rise proportionally?
Constant RTS (CRTS): f(2L,2K)=2f(L,K)=2q.
Increasing RTS (IRTS): f(2L,2K)>2q.
Decreasing RTS (DRTS): f(2L,2K)<2q.
Technological / organizational drivers:
- IRTS: specialization inside a large steel mill; fixed plant overhead spread over more units.
- DRTS: limited ore body in mining; managerial diseconomies (firm too complex to coordinate).
Graphical illustrations (text Fig. 8-6):
- Tobacco farming → DRTS (doubling inputs ⇒ < double output).
- Primary metal production → IRTS (doubling inputs ⇒ > double output).
Economist’s prior: mature, competitive firms are usually modeled with DRTS (free-lunch skepticism, capital constraints aside).

Consumer ↔ Producer mapping:
- Utility function ↔ Production function.
- Indifference curve ↔ Isoquant.
- Budget constraint ↔ Isocost (to be introduced in future lecture — determines cost-minimization).
- MRS ↔ MRTS.
Tangency condition in utility maximization (MRS = price ratio) will reappear as cost-minimizing condition (MRTS = input price ratio).

Labor rarely “perfectly variable” day-to-day; capital not 100 % fixed forever (renovations, leasing, IT upgrades, etc.).
- Nonetheless, 2-input, SR/LR dichotomy yields ~80 % predictive accuracy — acceptable abstraction.
Ethical/organizational aside: profit maximization assumption raises questions about corporate jets, executive perks ➜ later lectures on non-profit-max behavior.
Investors & entrepreneurs often pitch IRTS (“scale will solve everything”) ➜ economists remain cautious; need proof beyond marketing hype.

Production function: q=f(L,K).
Short-run MPL: MPL=\frac{\Delta q}{\Delta L}\Big|{K=\bar{K}}.
DMP condition: \frac{\partial^2 q}{\partial L^2}<0 (holding K fixed).
Isoquant equation (example): q=\sqrt{LK} \;\Rightarrow\; K=\frac{q^2}{L}.
MRTS: MRTS{LK}=\frac{MPL}{MPK}=\frac{\Delta K}{\Delta L}\Big|{q=\bar{q}}.
RTS tests:
- CRTS: f(cL,cK)=c\,f(L,K)\;\forall c>0.
- IRTS: f(cL,cK)>c\,f(L,K).
- DRTS: f(cL,cK)<c\,f(L,K).

Next lectures: introduce cost curves, derive the firm’s supply decision, connect to market equilibrium & welfare analysis.
Constant refrain: keep the SR/LR distinction and diminishing marginal product intuition top-of-mind — they anchor virtually all subsequent producer-side results.