Lesson Plan: Use Disney+ EMEA Promotions to Teach Optimization and Resource Allocation

Hook: Turn student frustration with abstract optimization into a career-ready, hands-on simulation

Many students struggle to connect constrained optimization and Lagrange multipliers to real-world decision-making. Teachers report that abstract examples feel irrelevant and students forget the mechanics. This lesson plan uses a timely, concrete career-scenario—students act as Disney+ EMEA content executives allocating limited resources (budget, talent, localization) to maximize audience reach—to teach optimization, resource allocation, and the practical use of Lagrange multipliers. By 2026, streaming platforms have sharpened focus on regional strategies, ad tiers, and AI-driven personalization—making this activity both current and career-relevant.

Why this matters in 2026: industry context and classroom relevance

Streaming competition and market nuance in EMEA (Europe, Middle East, Africa) have intensified through late 2025 and early 2026. Content chiefs like Disney+'s EMEA leadership are reorganizing teams and redirecting spend towards local originals, ad-supported offerings, and short-form funnels. These real-world pressures create natural constraints—limited budgets, finite star talent, and localization costs—that mirror the constrained optimization problems students study in calculus and economics.

"Use an authentic career scenario and students instantly ask better questions about trade-offs, utility and constraints." — classroom-tested insight

Learning objectives

• Translate a business problem into a mathematical optimization model.
• Apply Lagrange multipliers to solve a constrained optimization problem algebraically and numerically.
• Use spreadsheet and visualization tools to run sensitivity analysis and present trade-offs.
• Develop teamwork, negotiation, and decision-making skills in a career-oriented role-play.

Overview: the Disney+ EMEA promotions scenario (classroom-staged)

Set the scene: senior content restructuring in Disney+ EMEA (inspired by real promotions and strategy shifts in 2024–2026) leads to a fresh commissioning round. Students are split into content teams. Each team receives a fixed budget and must allocate funds between:

• Advertising & Marketing (B) — spend to drive reach across many markets
• Talent & Production Quality (T) — investment in cast, creators, and production value that increases per-view engagement
• Optional advanced constraints: localization (subtitles/dubbing), short-form promotion, ad-tier experiments

Core constraints and objective

Teams seek to maximize reach R(B,T) subject to a budget constraint B + T = C. For the math model we use a Cobb–Douglas-style production function familiar to students of economics and calculus:

R(B,T) = B^α · T^β, where α and β are positive with α + β = 1 (constant returns to scale assumption for simplicity).

Worked example: algebraic solution using Lagrange multipliers

Provide this step-by-step derivation in class and on worksheets. Use α = 0.6 and β = 0.4 to represent the idea that ad spend marginally contributes more to initial reach than a single talent unit in a crowded market.

Problem: Maximize R(B,T) = B^0.6 · T^0.4 subject to B + T = C (total budget C).

1. Form the Lagrangian: L(B,T,λ) = B^0.6 T^0.4 + λ(C − B − T).
2. Take partial derivatives and set them to zero:
   • ∂L/∂B = 0.6 B^(−0.4) T^0.4 − λ = 0
   • ∂L/∂T = 0.4 B^0.6 T^(−0.6) − λ = 0
   • ∂L/∂λ = C − B − T = 0
3. Equate the first two expressions for λ:

   0.6 B^(−0.4) T^0.4 = 0.4 B^0.6 T^(−0.6)

   Rearrange to find B/T = 0.6/0.4 = 3/2 = 1.5

4. Using B + T = C and B = (1.5)T, solve: (1.5T) + T = C ⇒ 2.5T = C ⇒ T = 0.4C, B = 0.6C.

Interpretation: With these elasticities, the optimal resource split is 60% ad/marketing and 40% talent/production. If C = $1,000,000, then B = $600,000 and T = $400,000.

Why this result is pedagogically valuable

• It connects calculus tools (Lagrange multipliers) to a transparent allocation rule: elasticities determine budget shares.
• Students can interpret α and β as marginal returns and tweak them to reflect different market conditions (e.g., star-driven vs. ad-driven markets).
• The model extends easily to multiple constraints (localization minima, talent caps) using multi-variable Lagrangians or KKT conditions for inequality constraints.

Classroom activity plan (90–120 minutes)

Materials

• Project brief handout (scenario, constraints, market assumptions)
• Worksheets with the Lagrange derivation and blank spaces for student work
• Shared spreadsheet template (Google Sheets) with R(B,T) models and sensitivity sliders
• Calculator or laptop; optional Python/Colab notebook for advanced classes

Roles

• Content Executive (lead negotiator)
• Data Analyst (runs models and sensitivity tests)
• Creative Lead (argues for talent spend and localization)
• Marketing Lead (argues for ad/marketing spend and user acquisition)

Step-by-step flow

1. 5–10 min: Introduce industry context—mention Disney+ EMEA leadership changes and 2026 trends (local originals, ad tiers, AI personalization).
2. 10 min: Present the mathematical model and worked example; walk through Lagrange multipliers.
3. 30–40 min: Team work—students choose α and β for their market assumptions (e.g., local-language drama might have higher β). They solve analytically (if able) or use the spreadsheet solver to find optimal B and T.
4. 15–20 min: Negotiation phase—teams present allocations and justify with math and market reasoning. Other teams act as board members and can challenge assumptions.
5. 10–15 min: Sensitivity analysis and extension—change α, β, or add a localization constraint (minimum spend on dubbing) and re-optimize.
6. 5–10 min: Debrief—link back to calculus, decision-making, and career skills.

Extensions and advanced options (AP/University level)

• Multiple constraints: add B + T + L = C with L for localization, solve with a three-variable Lagrangian.
• Discrete talent units: treat T as integer number of star attachments with fixed cost; introduce mixed-integer programming and use spreadsheets or simple heuristics.
• Ad-tier effects: model reach as R = a·B^α·T^β + s·S where S is spend on short-form/social and s is an effectiveness coefficient. Optimize across three variables.
• Stochastic modeling: include uncertainty in α and β; teach expected-value optimization and risk sensitivity (robust optimization).
• Machine learning tie-in: show how recommender systems change marginal returns and can be modeled through changing α over time.

Assessment and rubric

Evaluate teams on quantitative correctness, reasoning, and communication. Example rubric (out of 20 points):

• Mathematical correctness (8 pts): correct setup and solution using Lagrange multipliers or numerical solver.
• Market reasoning (6 pts): justified choice of α, β and assumptions about EMEA market segments.
• Presentation and negotiation (4 pts): clarity, teamwork, ability to interrogate trade-offs.
• Extension work or sensitivity analysis (2 pts): extra credit for exploring inequalities or stochastic scenarios.

Sample practice problems (with increasing difficulty)

Problem A (Intro)

Maximize R = B^0.5 · T^0.5 subject to B + T = 200,000. Find B and T.

Problem B (Intermediate)

R = B^0.7 · T^0.3, B + T = 500,000. Compute the optimal allocation and explain why ad spend gets a larger share.

Problem C (Advanced)

Maximize R = B^0.6 T^0.3 L^0.1 with B + T + 1.2L = 1,200,000 where L is localization spend and localization is slightly more expensive per effective unit (factor 1.2). Solve using a Lagrange multiplier for three variables and interpret results.

Teaching tips: active learning and assessment alignment

• Make assumptions explicit. Have students write α and β on the whiteboard and defend them with evidence (market size, language reach, star power).
• Use real numbers but normalize where needed. If costs per talent unit are extreme, normalize variables to avoid numeric pitfalls.
• Encourage alternate objective functions: maximize engagement per dollar, maximize diversity-weighted reach, or maximize long-term subscriber retention.
• Incorporate cross-disciplinary skills: negotiation (soft skills), spreadsheets (technical), calculus (math), and communications (presentation).

2026 trends to weave into class discussion

• Ad-supported tiers and dynamic pricing have changed marginal returns—advertising can amplify reach cheaply, lowering α in some markets.
• AI personalization increases the effectiveness of smaller, targeted campaigns—show how effective α and β may evolve across cohorts.
• Localization costs vary drastically by language and region; discuss how inequality constraints reflect minimum spends for regulatory or cultural reasons.
• Executive reorganizations in major platforms (e.g., reshuffles in Disney+ EMEA leadership in recent years) underscore real-world career pathways for students interested in content and product roles.

Common student misconceptions and how to address them

• "Bigger budget always wins." Counter with diminishing returns (explain concavity in Cobb–Douglas functions).
• "Talent is just one unit; more is always better." Show marginal returns and integer constraints—adding another star may cost more than the gain in reach.
• "Lagrange multipliers are magic." Demystify by stepping through each derivative and interpreting the multiplier as the marginal value of relaxing the budget constraint.

Digital resources and templates

Provide students with a Google Sheets template containing:

• Cells for α, β, C editable by students
• Analytical solutions for the simple two-variable case (B = αC, T = βC for α+β=1)
• Sensitivity sliders and charts to show how reach changes with α and β
• Optional Python/Colab notebook for solving multi-constraint cases numerically

Career-scenarios and real-world pathways

Role-playing as content executives gives students insight into careers in OTT platforms, distribution, analytics, and product strategy. Tie the activity to job roles: data analyst (modeling returns), content buyer (managing talent deals), and marketing manager (channel optimization). Reference recent industry moves—leadership changes and commissioning strategy shifts in 2024–2026—to show how math skills map to real decisions.

Final takeaway: why constrained optimization matters beyond the classroom

This lesson demonstrates that the same calculus tools used to solve abstract textbook problems directly inform high-impact business choices—how to split a budget, when to hire a star, and how to prioritize localization in a fragmented market. Students learn to quantify trade-offs, communicate numerical arguments, and adapt models as market assumptions change—a set of skills employers value in 2026.

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Want the full lesson kit (worksheets, Google Sheets template, rubric, and Colab notebook)? Download our ready-to-teach pack at studyphysics.online/lesson-packs or sign up for a free workshop where we run this simulation with teachers. Equip your students to solve real constrained optimization problems and explore career paths that blend math, strategy, and media.

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