# 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.

1 Watch or download the video & animation | 2 View related question material |

#### 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, rate or comment on YouTube Permalink Download (51 MB)Flash animation only
| Related MathsEG materialProfit, 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, rate or comment on YouTube Permalink Download (28 MB)
| Related MathsEG materialProfit, 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, rate or comment on YouTube Permalink Download (41 MB)Flash animation only
| Related MathsEG materialOptimisation - single variable |

#### 5.04 Finding the optimal number of floors in hotel construction - part two

(5 minutes)

View, rate or comment on YouTube Permalink Download (25 MB)Flash animation only
| Related MathsEG materialOptimisation - single variable |

#### 5.05 Finding the optimal number of floors in hotel construction - Conclusion

Concluding film. (1 minute 17 seconds).

View, rate or comment on YouTube Permalink Download (17 MB)
| Related MathsEG materialOptimisation - 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, rate or comment on YouTube Permalink Download (57 MB)Flash animation only
| Related MathsEG materialDifferentiation 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, rate or comment on YouTube Permalink Download (28 MB)Flash animation only
| Related MathsEG materialElasticity 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, rate or comment on YouTube Permalink Download (24 MB)Flash animation only
| Related MathsEG materialElasticity 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).

View, rate or comment on YouTube Permalink Download (21 MB)
| Related MathsEG materialElasticity of Demand |

#### 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, rate or comment on YouTube Permalink Download (66 MB)Flash animation only
| Related MathsEG materialPartial Differentiation (Polynomial) |

#### 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, rate or comment on YouTube: 1 / 2 Permalink Download (71 MB)Flash animation only
| Related MathsEG materialPartial Differentiation (Polynomial) |

#### 5.12 Partial Differentiation - Multiproduct pricing - part two

Using partial differentiation to understand pricing decisions in diversified organisations. (5 minutes 38 seconds).

View, rate or comment on YouTube Permalink Download (26 MB)Flash animation only
| Related MathsEG materialPartial Differentiation (Polynomial) |

#### 5.13 Partial Differentiation - Conclusion

Concluding clip. (41 seconds).

View, rate or comment on YouTube Permalink Download (8 MB)
| Related MathsEG materialPartial Differentiation (Polynomial) |

#### 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, rate or comment on YouTube Permalink Download (60 MB)Flash animation only
| Related MathsEG materialIntergration of polynomials |

#### 5.15 The Diamond Water Paradox - part two

Integration is used to determine total utility of water versus diamonds. (6 minutes 9 seconds).

View, rate or comment on YouTube Permalink Download (29 MB)
| Related MathsEG materialIntergration of polynomials |

#### 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.

View, rate or comment on YouTube Permalink Download (27 MB)
| Related MathsEG materialIntergration of polynomials |