Topic 3: Application of Differential Calculus in

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1 Mathematics 2 for Business Schools Topic 3: Application of Differential Calculus in Economics Building Competence. Crossing Borders. Spring Semester 2017

2 Learning objectives After finishing this section you should be able to interpret a derivative as a marginal function and apply this concept to common economic functions. discuss the possible shapes of cost curves and characterize the conditions for a S-shaped cost curve. find the minimum average cost and the minimum average variable cost and discuss their meanings. find revenue or profit maxima. analyse common economic functions and determine their characteristics. apply the concept of the differential to economic functions. calculate and interpret elasticities. 2

3 Essential economic functions and their first derivatives Function First derivative (Total) cost K(x) Marginal cost K (x) Average (total) cost [unit cost] k(x) Marginal average cost k (x) (Total) revenue E(x) Marginal revenue E (x) Profit G(x) Marginal profit G (x) Production x(r) Marginal productivity x (r) Consumption C(Y) Marginal propensity to consume C (Y) Savings S(Y) Marginal propensity to save S (x) 3

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6 The first derivative as the marginal function y f f Δx f x 0 or f f x 0 Δx, if Δx is small! x 4

7 Interpretation of the first derivative The first derivative f (x) indicates by how much the functional value f(x) approximately changes if the independent variable x increases by 1. Examples in an economic context: Marginal cost indicates the incremental cost of producing one additional unit of the output. Marginal revenue is the additional revenue generated by the next unit sold. etc. Specific example: A producer operates at marginal costs of CHF 5. This means: The next unit produced causes additional costs of about CHF 5. 5

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9 Differential If the change in the variable x is very small, infinitesimal in correct terms, then the following approximation is acceptable: Δf f x 0 Δx Then we often restate this as df = f x 0 dx and by doing so assume df and dx as being infinitesimal, i.e., very small changes. We call df and dx «differentials». 6

10 Application of the differential The differential enables to approximately quantify the absolute change f of the functional value f(x) provided that the change Δx of the independent variable x is small. Example: Given the marginal cost function K x = 0.15 x 2 4x + 65 What is the change of total cost if the quantity of x 0 = 12 units is reduced by 0.5 units? 7

11 Typical cost curves (1) Linear cost curve: Monotonicity: K x > 0 Curvature: K (x) = 0 Declining cost curve: Monotonicity: K x > 0 Curvature: K (x) < 0 8

12 Typical cost curves (2) Progressive cost curve: Monotonicity: K x > 0 Curvature: K x > 0 S-shaped cost curve: Monotonicity: K x > 0 Curvature: K x < 0 for x < x s K x > 0 for x > x s 9

13 S-shaped cost curves The cost curve is S-shaped if the following three conditions are satisfied: 1. K does not have any interior extrema, but is strictly monotonically increasing. 2. K has a concave-convex inflection point x s. 3. K is non-negative, thus K(x) 0 for all x 0, i.e., fixed costs are non-negative. S-shaped curve can be modelled by a polynomial of degree 3: The polynomial K x = ax 3 + bx 2 + cx + d describes a S-shaped total cost function if simultaneously a > 0, b < 0, c > 0, d 0, and b 2 < 3ac. 10

14 Minimum of average total cost K x = 0.4x 3 3.6x x Find the quantity x O that minimizes average (total) cost: k x O = K(x O) = 0 x O 11

15 Minimum of average total cost Alternative approach K x K x = 0.4x 3 3.6x x K K (x) = k x = x x O At the tangent point x O, average total cost is at its minimum and equals marginal cost: k x O = K x O x O = K x O 12

16 Minimum of average variable cost Find the quantity x m that minimizes average variable cost: k var x m = K var(x m ) = 0 x m K x = 0.4x 3 3.6x x

17 Minimum of average variable cost Alternative approach K x K K x = 0.4x 3 3.6x x x m At the tangent point x m, average variable cost is at its minimum and equals the marginal cost: k var x m = K var x m x m = K x m 14

18 Minima of average total cost, average variable cost, and marginal costs Marginal cost Average total cost Average variable cost x 0 : x m : x s : Minimum of average total cost Minimum of average variable cost Minimum of marginal cost x s x m x 0 15

19 Comparison K x x s x m x 0 x s x m x 0 x s x m x 0 16

20 Maximum revenue for constant price In a market with a large number of buyers and sellers the price of a homogenous product cannot be influenced by single players: p x = p o E(x) E x = p 0 x Capacity Limit x x max Since E x = p o 0 the extrema of the revenue function must be at the boundaries. 17

21 Maximum revenue for a linear demand function x p = 64 4p (demand function) Maximum E p = E p = E max = 18

22 Profit maximum if the profit function is linear An ice cream shop operates with the cost function K x = x: K Total daily costs x Daily quantity of ice cream produced and sold [in kg] x lim Daily capacity limit 20 kg It can sell its ice cream at an average selling price per kg of p = 18 CHF. Find the breakeven quantity x BE and the profit maximizing quantity x max as well as the maximum daily profit! 19

23 Profit maximum in a monopoly (I) E x c = K x c E x E = 0 E x = 10x x K x = x 3 12x x + 98 Find x E : revenue maximizing quantity x c : profit maximizing quantity (Cournot quantity) x c x E x 20

24 Profit maximum in a monopoly (II) G x = E x K x = 0 E x = K (x) E x = K (x)= x x c x E 21

25 Introductory example Elasticity (I) f x = 2x + 5 f 2 = 9 f 2.2 = 9.4 x = 0.2 f 20 = 45 f 20.2 = 45.4 f =

26 Introductory example Elasticity (II) x 1 = 2 x 2 = 20 x = 0.2 x 1 2 = 10% x = 0.2 x 2 20 = 1% f f % f f % f f x x % 10% = f f x x % 1% =

27 The arc elasticity of a function The ratio ε f,x of the relative (percentage) change of the function value f with respect to the relative (percentage) change of its input x x f ε f,x = f f(x) x x = f x x f(x) is called arc elasticity of f with respect to x. 24

28 The point elasticity of a function The limit of the arc elasticity of f with respect to x for Δx 0 is called point elasticity of f with respect to x. Calculation: ε f,x = lim Δx 0 f f(x) x x f = lim Δx 0 x x f(x) = f x x f(x) In practice, the point elasticity ε f,x of f with respect to x is calculated using: ε f,x = f x f x x 25

29 Interpretation of the elasticity The value of the elasticity ε f,x corresponds (approximately) to the percentage change of f due to a 1% increase in x. Example: ε f,x = means that a 1% increase in x leads to 0.444% increase of f. ε f,x = 0.8 means that a 1% increase in x leads to 0. % decrease of f. 26

30 Economic application Price elasticity of demand The elasticity of the demand function with the price as the independent variable is called price elasticity of demand. In practical use, often no distinction between point elasticity and arc elasticity is made. Both terms are used synonymously. That is why there is only one notation for elasticity ε f,x. 27

31 Example 1: Price elasticity of demand At which price does a 2% price increase cause a 4% decrease in demand? x p = 23p

32 Example 2: Price elasticity of demand a) Find the point elasticity of x p = 180 p 1 at p 0 = 2. b) Find the arc elasticity of x p = 180 p 1 at p 0 = 2 and p 1 = 1.9. x p = 180 p 1 (p > 0) 29

33 Example 3: Price elasticity of demand a) Find the point elasticity of x p = 180 e 0.45 p at p 0 = 2. b) Find the arc elasticity of x p = 180 e 0.45 p at p 0 = 2 and p 1 =

34 Interpretation of the price elasticity of demand Value Descriptive Terms Quantity Effect ε x,p Perfectly elastic Any increase of price will cause demand to drop to zero. 1) ε x,p < 1 Elastic Percentage change in quantity demanded is greater than that in price. ε x,p = 1 Unit elastic Percentage change in quantity demanded is equal to that in price. 1 < ε x,p < 0 Inelastic demand Percentage change in quantity demanded is smaller than that in price. ε x,p = 0 Perfectly inelastic demand Changes in the price do not affect the quantity demanded. 1) Try to sell an ordinary CHF 10 bill for CHF

35 Elasticities for some consumer goods The value of the price elasticity of demand ε x,p corresponds (approximately) to the percentage change of the quantity x demanded due to a 1% price increase. Consumer Good Fresh butter Fresh milk Yoghurt Veal Flour / Bred Approx. 0 Petrol short-tem bis Petrol long-term Price Elasticity of Demand Source: Institut für Agrarwirtschaft, ETH Zürich, 2006; G. Foos: Die Determinanten der Verkehrsnachfrage, Karlsruhe: Loeper, citation: Beck, Bernhard: Volkswirtschaft verstehen. Zürich: vdf, 2008, S