Economics · 4th Semester · GU CBCS

Single Variable Optimization

Introductory Quantitative Techniques for Economics — Unit 4

Classes: 8
Marks: 20
Type: Calculus-Based

📚 Table of Contents

  1. What is Optimization?
  2. Local vs Global Optima
  3. Geometric Characterization
  4. Calculus: First Order Conditions
  5. Second Order Conditions
  6. Profit Maximization
  7. Cost Minimization
  8. Revenue Maximization
  9. Quick Reference Table
  10. Exam Corner
01

What is Optimization?

Core Idea

Optimization means finding the value of a variable x that makes a function y = f(x) as large (maximum) or as small (minimum) as possible.

🔑 Key Concept
In economics: firms maximize profit/revenue and minimize cost. Optimization provides the mathematical tool to find the optimal output level Q*.

Maximization

Find x such that f(x) is the highest possible value.

Profit Max Revenue Max

Minimization

Find x such that f(x) is the lowest possible value.

Cost Min Loss Min
💡 Memory Trick
"FIND the PEAK or the PIT" — optimization finds either the peak (maximum) or the pit (minimum) of a curve.
02

Local vs Global Optima

🏔️ Local Optimum

A point that is the best in its neighbourhood (nearby region) — but not necessarily the best overall.

Definition f(x*) ≥ f(x) for all x near x*
(local maximum)

Think of it as: best on your street, not necessarily in the city.

🌍 Global Optimum

A point that is the best over the entire domain of the function.

Definition f(x*) ≥ f(x) for ALL x in domain
(global maximum)

Best in the entire city — no other point beats it.

Visual: Local Maximum vs Global Maximum
x y Local Max Global Max Local Min x₁ x*
⚠️ Important Distinction
Every global optimum is also a local optimum. But every local optimum is NOT necessarily a global optimum. In economics, we usually seek the global optimum.
03

Geometric Characterization

What the Tangent Line Tells You

Geometrically, an optimum occurs where the tangent to the curve is horizontal — i.e., slope = 0.

Maximum — ∩ Shape (Concave)
f'(x)=0 x* f'>0 ↗ f'<0 ↘
Minimum — ∪ Shape (Convex)
f'(x)=0 x* f'<0 ↘ f'>0 ↗

Key Geometric Rules

SituationSlope Before x*Slope After x*Type
MaximumPositive (+)Negative (−)Peak / ∩
MinimumNegative (−)Positive (+)Trough / ∪
Inflection PointSame signSame signNeither (saddle)
💡 Memory Trick
MAX = Mountain (+→−)  |  MIN = Valley (−→+)
Slope goes from positive to negative for a mountain peak; negative to positive for a valley bottom.
04

First Order Condition (FOC)

The Necessary Condition

For f(x) to have a local optimum at x = x*, the first derivative must equal zero:

First Order Condition (FOC) f '(x*) = 0

i.e., dy/dx = 0 at x = x*
⚠️ Warning
FOC is necessary but NOT sufficient. f '(x*) = 0 tells us there is a stationary (critical) point — but it could be a max, a min, or an inflection point. We need the Second Order Condition (SOC) to decide.

Steps to Find Critical Points

  1. Write the function: y = f(x)
  2. Differentiate: find f '(x) = dy/dx
  3. Set f '(x) = 0
  4. Solve for x → this gives critical point(s) x*
📖 Example
If f(x) = −2x² + 8x + 5, then f '(x) = −4x + 8.
Setting f '(x) = 0: −4x + 8 = 0 → x* = 2. Check SOC next!
05

Second Order Condition (SOC)

The Sufficient Condition — Tells MAX or MIN

After finding critical point x* from FOC, differentiate again (second derivative) to confirm the nature:

For MAXIMUM f ''(x*) < 0
(Second derivative is NEGATIVE)
⟹ function is concave (∩) at x*
For MINIMUM f ''(x*) > 0
(Second derivative is POSITIVE)
⟹ function is convex (∪) at x*

What if f ''(x*) = 0?

SOC is inconclusive. The point could be a maximum, minimum, or inflection point. Higher-order derivatives or graphical inspection needed.

Second Derivative Intuition
Concave ∩ : f''<0 → MAX ↓ bends down Convex ∪ : f''>0 → MIN ↑ bends up
💡 Memory Trick
NEGATIVE second → MAXIMUM (like a ∩, top is negative/down)
POSITIVE second → MINIMUM (like a ∪, cup holds positivity/up)
Mnemonic: "NM–PM" → Negative=Max, Positive=Min

Complete Test Summary

Conditionf '(x*)f ''(x*)Result
Maximum= 0< 0🏔️ Local Max
Minimum= 0> 0🏞️ Local Min
Inconclusive= 0= 0❓ Need further test
06

Application: Profit Maximization

🟢 Profit Maximization

A firm wants to find the output level Q* that maximizes profit π.

Profit Function π(Q) = TR(Q) − TC(Q)

where TR = Total Revenue, TC = Total Cost
FOC (Necessary Condition) dπ/dQ = 0 ⟹ dTR/dQ − dTC/dQ = 0 ⟹ MR = MC
SOC (Sufficient Condition for Maximum) d²π/dQ² < 0 ⟹ dMR/dQ < dMC/dQ (MC must be rising faster than MR, or MR falling)
🔑 Golden Rule
Profit is maximized when MR = MC, provided the MC curve cuts the MR curve from below (SOC).

Worked Example

Let TR = 50Q − 2Q² and TC = Q³ − 5Q² + 20Q + 10

  1. π = TR − TC = 50Q − 2Q² − Q³ + 5Q² − 20Q − 10 = −Q³ + 3Q² + 30Q − 10
  2. dπ/dQ = −3Q² + 6Q + 30 = 0
  3. Divide by −3: Q² − 2Q − 10 = 0 → Q* ≈ 4.32 (taking positive root)
  4. d²π/dQ² = −6Q + 6; At Q*: negative ✓ → Maximum confirmed
Profit Maximization: MR = MC Rule
Q P/C MR MC Q* MR=MC π increasing (MR>MC) π decreasing (MC>MR)
07

Application: Cost Minimization

🔴 Cost Minimization

A firm wants to produce a given output at the lowest possible total cost, or find the output where average cost is minimized.

Total Cost Function TC = TC(Q) → usually a cubic: TC = aQ³ + bQ² + cQ + d
Average Cost AC = TC / Q
Minimize AC — FOC d(AC)/dQ = 0
SOC — Confirm Minimum d²(AC)/dQ² > 0
🔑 Key Result
AC is minimized where MC = AC (the MC curve crosses the AC curve at its minimum point).
Cost Minimization: MC cuts AC at its Minimum
Q Cost AC MC Q* AC min MC < AC: AC falling MC > AC: AC rising
💡 Memory Trick
"MC is the boss — it pulls AC up when it's above, and pulls it down when below."
AC minimum always occurs where MC = AC.
08

Application: Revenue Maximization

🟣 Revenue Maximization

A firm maximizes Total Revenue TR regardless of profit (common in Baumol's model of firm behaviour).

Revenue Function TR = P × Q = AR × Q

If demand: P = a − bQ, then TR = aQ − bQ²
FOC (Marginal Revenue = 0) dTR/dQ = 0 ⟹ MR = 0
SOC — Confirm Maximum d²TR/dQ² < 0
🔑 Key Result
Total Revenue is maximized when MR = 0 (not when MR = MC!).
Revenue Maximization: TR peaks where MR = 0
Q TR TR Q* MR MR=0 MR > 0 (TR rising) MR < 0 (TR falling)

Worked Example

Demand: P = 100 − 4Q → TR = 100Q − 4Q²

  1. MR = dTR/dQ = 100 − 8Q
  2. Set MR = 0: 100 − 8Q = 0 → Q* = 12.5
  3. d²TR/dQ² = −8 < 0 → Maximum confirmed ✓
  4. TR* = 100(12.5) − 4(12.5)² = 1250 − 625 = 625
09

Quick Reference Table

Application Objective Function FOC SOC Optimum Rule
Profit Max π = TR − TC dπ/dQ = 0 d²π/dQ² < 0 MR = MC
Cost Min (AC) AC = TC/Q d(AC)/dQ = 0 d²AC/dQ² > 0 MC = AC
Revenue Max TR = P × Q dTR/dQ = 0 d²TR/dQ² < 0 MR = 0
General Max y = f(x) f'(x*) = 0 f''(x*) < 0 Concave at x*
General Min y = f(x) f'(x*) = 0 f''(x*) > 0 Convex at x*
Optimization Decision Flowchart
Given f(x) Step 1: Find f'(x) = 0 → get critical point x* Step 2: Find f''(x*) → evaluate second derivative f''(x*) < 0 → MAXIMUM f''(x*) > 0 → MINIMUM f''(x*) = 0 → Inconclusive negative positive = 0 π max / TR max MR=MC or MR=0 Cost min MC = AC

🎯 Exam Corner — Likely Questions (20 Marks)

3M Define local and global maxima. How do they differ geometrically? Draw a diagram.
5M State and explain the first and second order conditions for maximization and minimization of a single variable function.
5M Derive the profit-maximizing output for a firm using calculus. Show that MR = MC is the FOC.
4M Show that Total Revenue is maximized when Marginal Revenue equals zero.
3M Explain geometrically why the average cost curve is U-shaped and where it is minimized.
5M A firm's total cost is TC = Q³ − 6Q² + 15Q + 10. Find the output that minimizes average cost. Verify your answer.
5M Demand is P = 80 − 2Q and TC = Q² + 4Q + 10. Find the profit-maximizing output and price. Calculate maximum profit.
★

Last-Minute Formula Sheet

Critical Point f'(x*) = 0
Maximum Test f'(x*)=0 AND f''(x*) < 0
Minimum Test f'(x*)=0 AND f''(x*) > 0
Profit Max Rule MR = MC (and MC rising)
Revenue Max Rule MR = 0
AC Min Rule MC = AC (at bottom of AC)
Marginal Relationships MR = dTR/dQ | MC = dTC/dQ | AC = TC/Q
MR = AR + Q·(dAR/dQ) | For competitive firm: MR = P = AR