Module 3 – Functionsembaweb/QSW/Module_3_Functions.pdf© 2002 The Regents of The University of...

© 2002 The Regents of The University of MichiganAll Rights Reserved. No duplication without prior, written consent.

Quantitative Skills WorkshopModule 3 – Functions

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• Function Notation: y = f(x)

–Variables x and y are placeholders for unknowns

»We will cover functions of many variables during one of the in-residence QSW lectures.

• x = independentindependent or exogenousexogenous variable

• y = dependentdependent or endogenousendogenous variable

Functions of One VariableBasics

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EconomicsEconomics:: Quantity Demanded = f(Price)Quantity Demanded = f(Price)

MarketingMarketing: Customer Satisfaction = f(Quality): Customer Satisfaction = f(Quality)

FinanceFinance: Portfolio Return = f(Risk): Portfolio Return = f(Risk)

Functions of One VariableBusiness Applications



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Review of X and Y AxesReview of X and Y AxesCartesian Coordinate SystemCartesian Coordinate System

• Since we will be working with functions of two variables and graphing these functions, we need to represent the two variables in two dimensions.

• In a previous module we used a horizontal number line. This will be the x-axis. On top of this we draw a vertical number line, the y-axis.

• This is called the Cartesian Coordinate System.

– We will refer to a point as having coordinates. Listed first is the x-coordinate, followed by the y-coordinate. For example, if a point has coordinates (2, 10) – called an ordered pair of numbers – then the value of x is 2 and the value of y is 10.

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x

y

Origin

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

2

1

-1

-2

-3

-4

0

Review of X and Y AxesReview of X and Y AxesFinding Points (x, y)Finding Points (x, y)



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Review of X and Y AxesReview of X and Y AxesFinding Points (x, y)Finding Points (x, y)

.(4,2)

To get to the point (4,2), you start at the origin and move 4 units to the right along the x-axis and 2

units up along the y-axis.

.(-4,2)

To get to the point (-4,2), you start at the origin and move 4 units to

the left along the x-axis and 2 units up along the y-axis.

.(4,-2).(-4,-2)To get to the point (4,-2), move 4

units to the right and 2 units down.To get to the point (-4,-2), move 4 units to the left and 2 units down.

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4

3

2

1

-1

-2

-3

-4

0

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y = mx + by = mx + bx

y

LINEAR FUNCTIONSGeneral Case



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y = mx + by = mx + bQuantity

Revenue

LINEAR FUNCTIONSGeneral Case

Application:Revenue is a linear function of quantity sold.

– Note that the line is drawn starting from the origin. In many business contexts the only relevant part of the coordinate system is the upper right-hand quadrant, where x and y can be either zero or positive, but not negative.

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y = by = bx

y

b

LINEAR FUNCTIONSSpecial Case – Constant Function



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y = by = b

Application:Overhead costs, such as monthly lease payments, are constant. They do not change as quantity sold changes (i.e., the same rent payment is due whether it is a good sales month or a bad one).

Quantity

Overhead Cost

b

LINEAR FUNCTIONSSpecial Case – Constant Function

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Functions: Basics & LinesFunctions: Basics & LinesExercisesExercises

1. How much a household spends on water each month depends on a number of factors. The most important factors are the price of water, how much water is consumed, the temperature, and the amount of rainfall (if there is a garden that needs water). Which variables are exogenous and which are endogenous to the household’s consumption decision?

2. Draw the line y = 10. Draw the line x = 10. 3. Plot the following pairs (x,y):

x y–4 –10 12 2



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Functions: Basics & LinesFunctions: Basics & LinesExercises: **Answers**Exercises: **Answers**

1. The exogenous variables are the ones that you cannot control or choose. In this case, they would be the price of water, the temperature, and the amount of rainfall. The endogenous variables are the choice variables. Your only choice in this example is the amount of water you consume. (Note that if this were commercial consumption, rather than residential, the company may be able to negotiate over the price. This would make the price variable not strictly exogenous.)

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2. Draw the line y = 10. Draw the line x = 10.

x

y

10

x

y

10



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These points lie along a straight These points lie along a straight line. In a later CDline. In a later CD--ROM module, ROM module,

we will focus exclusively on we will focus exclusively on working with lines and graphing working with lines and graphing

linear equations.linear equations.

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

3

2

1

-1

-2

-3

0

•• ((--4,4,--1)1)

•• (0,1)(0,1)•• (2,2)(2,2)

3. Plot the following pairs:x y

–4 –10 12 2


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y = axy = ax22 + + bxbx + c+ c

NONLINEAR FUNCTIONSQuadratic (Parabola)

x

y



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y = axy = ax22 + + bxbx + c+ c

Application:In some industries, revenue is not related linearly to quantity sold, but is better described as a quadratic function. Revenue increases initially as sales grow. In order to greatly increasequantity sold, however, you must lower the price significantly. As you move further out to the right along the x-axis (the quantity-axis), revenues fall.

Quantity

Revenue


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Parabolas can point in any Parabolas can point in any direction, but the most direction, but the most

common ones that you will common ones that you will run across are “hillrun across are “hill-- shaped” shaped” and “Uand “U--shaped” parabolas.shaped” parabolas. x

y




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Parabolas can point in any Parabolas can point in any direction, but the most direction, but the most

common ones that you will common ones that you will run across are “hillrun across are “hill-- shaped” shaped” and “Uand “U--shaped” parabolas.shaped” parabolas. Quantity

Cost per unit

Application:Unit or average cost curves are often U-shaped. On the one hand, as production increases the firm’s average overhead expenditure declines. On the other hand, the firm’s average variable expenditure may eventually rise if you have to hire more workers or pay overtime, or if more production errors occur as you push the limits of the system.


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y = axy = ax33 + bx+ bx22 + + cxcx + d+ d

NONLINEAR FUNCTIONSCubic

x

y



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Application:Total cost rises in the short run as output increases, first at a decreasing rate, then at an increasing rate as you begin to hit bottlenecks in production and capacity constraints.

Quantity

Total Cost


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Another Application:The typical pattern of product diffusion, the rate at which new users buy a product, looks like an S-curve. Product sales start slowly, increase at an increasing rate, and then plateau. The steepness of the curve depends on such factors as price, product attributes, advertising, and so on.


Time

Cumulative# Adopters



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a a y =y =

xx x

y

A rational function is a ratio of functions. A reciprocal function, which we cover here, is a special case, where the x in the denominator is raised to the power of one.

NONLINEAR FUNCTIONSRational

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a a y =y =

xxx

y

Special Terminology:• Since you cannot divide by zero, these graphs have a

vertical asymptote and a horizontal asymptote, where the curve gets very close to a line but never touches it.• In this graph the vertical asymptote is the y-axis and the

horizontal asymptote is the x-axis.




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a a y =y =

xx

Application:If a consumer always spends $100/month on lottery tickets, regardless of the ticket price, then

P × Q = 100 or P = 100/Q

Quantity

Price


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•• Select a few values for x Select a few values for x •• Solve for y = f(x) Solve for y = f(x) •• Plot the pairs (x,y)Plot the pairs (x,y)•• Connect the pointsConnect the points

In most cases it is not necessary to plot the In most cases it is not necessary to plot the function precisely. It usually suffices to draw a function precisely. It usually suffices to draw a rough sketch that is accurate enough to show the rough sketch that is accurate enough to show the shape and approximate placement of the curve.shape and approximate placement of the curve.

Graphing Nonlinear Functions



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Graph y = 3x2


x y

-3 27-2 12-1 30 01 32 123 27

x

y

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Graph y = 100/x , only for values of x , only for values of x ≥≥ 00


x

y

1

100x y

0 ∞1 1002 5010 10100 11000 0.1



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1. Graph y = -3x2 + 12

2. Graph y = 100/x + 20x, only for values of x + 20x, only for values of x ≥≥ 00

Graphing Nonlinear FunctionsExercises

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1. Graph y = -3x2 + 12

Graphing Nonlinear FunctionsExercises: **Answers**

x y

-3 -15-2 0-1 90 121 92 03 -15

x

y

12

-2 2



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2. Graph y = 100/x + 20x, only for values of x + 20x, only for values of x ≥≥ 00

Graphing Nonlinear FunctionsExercises: **Answers**

x y

0 ∞1 1202 9010 210100 2001

x

y

2

90

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Application: Learning CurveApplication: Learning Curve• As a firm gains experience in production, it is often the case

that unit costs decline, particularly for complex products: workers become more efficient, technical knowledge increases, and product design improvements are made.

• A learning curve describes the relationship between production costs and cumulative output.

• Learning curves (or experience curves) have been estimated for a number of industries. A common experience curve is an “80 percent” curve. This means that unit production costs drop by 20 percent with each doubling of cumulative output. A “90 percent” curve would mean that costs drop by 10 percent as cumulative output doubles.



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Application: Learning CurveApplication: Learning CurveAn example of an “85 percent” learning curve is shown below, using hypothetical data. The learning curve is clearly nonlinear. You can see how quickly labor hours (and therefore labor costs) drop as cumulative output increases.

LEARNING CURVE

Cumulative Units

Labo

r Hou

rs p

er U

nit

0.0010.0020.0030.0040.0050.0060.0070.00

0 100 200 300

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• An 85 percent experience curve just like this was assumed by Douglas Aircraft when they set their pricing strategy for the DC-9. They fixed prices based on an expectation that there would be an 85% experience curve. In other words, the prices they set were below current cost, but they expected costs to fall quickly with time and experience.

• Unfortunately, the expected cost savings did not occur. Douglas lost money and was later acquired by the McDonnell Company.*

* See * See PankajPankaj GhemawatGhemawat “Building Strategy on the Experience Curve,” “Building Strategy on the Experience Curve,” Harvard Business ReviewHarvard Business Review, March, March--April 1985.April 1985.

Application: Learning CurveApplication: Learning Curve

Module 3 – Functionsembaweb/QSW/Module_3_Functions.pdf© 2002 The Regents of The University of...

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