Film Series Five: Differentiation and Integration
The use of differentiation can help us make sense of cost decisions that are being made daily in industries worldwide. In this series we ask a number of questions, such as; would it be cheaper to educate students if universities were larger? What happens to the costs of providing floor space as hotel buildings increase in size? Why does the mark-up on supermarket products vary so much between the different goods they sell?
In the second part of the film we think about how partial differentiation helps us understand the relationship between price, output and profit in multi-product firms; where pricing decisions can be more complex than in single product firms.
In the final section we discover how integration enables us to throw light on one of the most famous paradoxes in economics: the diamond/water paradox.
These video clips and animations can be viewed in isolation, or you can use our learning pathway and follow links to related material in our Question Bank.
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5.01 Introduction to Differentiation - How large should a University be?
Making rational decisions often means thinking about small changes, and many of them revolve around the idea of the margin - the marginal change. A useful way of analysing marginal changes is by differentiation. (7 minutes 21 seconds)
View or comment on YouTube Permalink Download (51 MB) | Related MathsEG material Profit, average and marginal functions |
5.02 Conclusion to film 5.01
Concludes the clip for differentiation and optimal University size. (2 minutes 13 seconds)
View or comment on YouTube Permalink Download (28 MB) | Related MathsEG material Profit, average and marginal functions |
5.03 Finding the optimal number of floors in hotel construction - part one
Up to a certain point, the higher the building, the lower the cost per floor. However at a certain point fixed costs rise dramatically to take into account the cost of deeper foundations and a stronger construction. In this film we introduce calculus to optimise spend and determine the appropriate number of floors to build. (5 minutes 18 seconds)
View or comment on YouTube Permalink Download (41 MB) | Related MathsEG material Optimisation - single variable |
5.04 Finding the optimal number of floors in hotel construction - part two
(5 minutes)
View or comment on YouTube Permalink Download (25 MB) | Related MathsEG material Optimisation - single variable |
5.05 Finding the optimal number of floors in hotel construction - Conclusion
Concluding film. (1 minute 17 seconds)
View or comment on YouTube Permalink Download (17 MB) | Related MathsEG material Optimisation - single variable |
5.06 Differentiation - Prices in monopolistic markets - part one
In this section we look at supermarket pricing, using differentiation to show that equilibrium price in monopolistic markets in higher than in competitive markets. (7 minutes 50 seconds)
View or comment on YouTube Permalink Download (57 MB) | Related MathsEG material Differentiation of Polynomials |
5.07 Differentiation - Prices in monopolistic markets - part two
This film examines the relationship between monopoly power and elasticity, and applies this principle in order to understand why supermarkets mark up some goods much more than they mark up others. (4 minutes 44 seconds)
View or comment on YouTube Permalink Download (28 MB) | Related MathsEG material Elasticity of Demand |
5.08 Differentiation - Prices in monopolistic markets - part three
Supermarkets have a much higher mark up on impulse items such as sweets. In this set we rearrange our formula and work out the elasticity implied in the mark up on any good. (4 minutes 24 seconds)
View or comment on YouTube Permalink Download (24 MB) | Related MathsEG material Elasticity of Demand |
5.09 Differentiation - Prices in monopolistic markets - Conclusion
Differentiation has enabled us to understand why prices in monopolistic markets tend to be higher than in competitive ones. But an understanding of demand elasticity is essential to understanding price determination, not only in the supermarket, but also in markets generally. (4 minutes 26 seconds)
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5.10 Partial Differentiation - Complements and Substitutes
In this section we use partial differentiation to determine the change in demand of one product when the price of another is changed. (10 minutes 37 seconds)
View or comment on YouTube Permalink Download (66 MB) | Related MathsEG material Partial elasticity of demand Theory of the firm/ definitions |
5.11 Partial Differentiation - The Multiproduct Firm - part one
In this section we use the technique of partial differentiation to examine the pricing decisions in diversified organisations. (2 minutes 37 seconds)
View or comment on YouTube: 1 / 2 Permalink Download (71 MB) | Related MathsEG material Unconstrained optimisation |
5.12 Partial Differentiation - Multiproduct pricing - part two
Using partial differentiation to understand pricing decisions in diversified organisations. (5 minutes 38 seconds)
View or comment on YouTube Permalink Download (26 MB) | Related MathsEG material Unconstrained optimisation |
5.13 Partial Differentiation - Conclusion
Concluding clip. (41 seconds)
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5.14 The Diamond Water Paradox - part one
What determines the price of a commodity is its marginal utility, not its total utility, and the price of all units of a good is set by its marginal valuation. This section uses integration to explore this concept. (6 minutes 44 seconds)
View or comment on YouTube Permalink Download (60 MB) | Related MathsEG material Integration of polynomials Consumer and producer surplus |
5.15 The Diamond Water Paradox - part two
Integration is used to determine total utility of water versus diamonds. (6 minutes 9 seconds)
View or comment on YouTube Permalink Download (29 MB) | Related MathsEG material Integration of polynomials Integrating algebraic functions Consumer and producer surplus |
5.16 Series Conclusion
Mathematics provides a hugely constructive way of gaining insights into economics as a discipline and into the way in which the real world operates.
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