Slope Calculation Rule

8/13/2019 Slope Calculation Rule

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Unit 1.Fundamentals of Managerial

Economics (Chapter 1)


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While there is no doubt that luck, both goodand bad, plays a role in determining thesuccess of firms, we believe that success isoften no accident. We believe that we can

better understand why firms succeed or failwhen we analyze decision making in terms ofconsistent principles of market economics andstrategic action.

Besanko, et. AlEconomics of Strategy (2 nd)


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Microeconomics is the study of how individualfirms or consumers do and/or should make

economic decisions taking into account suchthings as:

1. Their goals, incentives, objectives.

2. Their choices, alternatives, problems.3. Constraints such as inputs, resources, money, time,

technology, competition.4. All (cash & noncash) incremental or marginal

benefits and costs.5. The time value of money.


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Goals, Incentives, Objectives A fundamental economic truth is thatindividual firms or decision makersrespond to economic incentives. Whatthese incentives are (i.e. money,profits, utility, etc.) and how they

influence economic decision making arekey topics for study and analysis inbusiness (or managerial) economics.


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Managerial Choices(examples)

Output quantityOutput qualityOutput mixOutput priceMarketing andadvertising

Production processes(input mix)Input quantityProduction locationProduction incentivesInput procurement


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Michael Porter s Five

Competitive Forces = Decision-making constraints= Factors that influence the sustainability of

firm profits1. market entry conditions for new firms2. Market power of input suppliers3. Market power of product buyers4. Market rivalry amongst current firms5. Price and availability of related productsincluding both substitutes and complements


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Marginal Analysis

Analysis of marginal costs and marginal benefits due to a change Marginal = additional or incrementalCosts and benefits that are constant(i.e. fixed, don t change) are excludedfrom the analysis

Changes occurring at the margin areall that matter


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Marginal Analysis(Examples)

Y XIncremental Y/Incremental X

TR Units of output MRTC Units of output MCTP Units of input MPTRP Units of input MRP

TC Units of input MFCTU Units of good MUProfit Units of output MP


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New Beer Sales Resulting from AmountsSpent on TV and Radio Advertising

Total Spent

New Beer Sales Generated

(in barrels per year)TV Radio

$0 0 0

$100,000 4,750 950

$200,000 9,000 1,800

$300,000 12,750 2,550

$400,000 16,000 3,200

$500,000 18,750 3,750

$600,000 21,000 4,200

$700,000 22,750 4,550$800,000 24,000 4,800

$900,000 24,750 4,950

$1,000,000 25,000 5,000

Max B(T,R)Subject to: T + R = 1,000,000


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Example of a Business (Economic)

Decision Resolved with Marginal Analysis Goal: max dollar sales of a productConstraint: advertising budget

Marginal decision rule: incrementaldollar sales generated per incrementaldollar spent should be the same foreach alternative type of advertising


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Assume you are a member of your company s

Marketing Dept. You believe, and correctly so,1) the market demand for your firm s product islinear,2) if your company charges $5.00 for its product,

quantity sold would be 200 units and3) if your company set price = $3.00, the number ofunits sold would be 400.

Develop alternative ways of explaining to upper-levelmanagement more fully the relationship between thecompany s price and the resulting number of units of product sold.


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Variable Relationships

Example of Alternative Ways of DepictingTabular P Q

$7 0

6 1005 200

4 300

3 400

2 500

1 600

0 700


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Variable Relationships

Example of Alternative Ways of DepictingGraphical


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Variable RelationshipsExample of Alternative Ways of Depicting

Mathematical

Q = 700 100P

P = 7 0.01Q


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Common Math Terms Used in

Economic AnalysisTerm Definition

Variable Something whose value or magnitude (often Q or $in Econ) may change (or vary); usually denoted byletter labels such as Y, X, TR, TC

Parameter orConstant

Something whose value does NOT change

General

equation orfunction

A mathematical expression that suggests the value

of one variable relates to or depends on the value ofanother variable (or set of variables) withoutshowing the precise nature of that relationship [e.g.y = f(x)].


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Common Math Terms Used in

Economic AnalysisTerm Definition

Specificequation orfunction

A mathematical expression that shows precisely howthe value of one variable is related to the value ofanother variable (or set of variables) [e.g. y = 10 +2x].

Inverseequation orfunction

A mathematical expression rewritten so that thevariable previously on the right-hand side of theequal sign now becomes the variable solved for onthe left-hand side of the equal sign [e.g. y = 2x andx = 1/2 y are each an inverse equation of theother].


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Common Math Functions Used in

EconomicsFunction Form

Name ofFunction Graph of Function

Y = a 0 Constant Horizontal straight line withslope = 0

Y = a 0 + a 1 x(or y = mx + b)

Linear Straight line with slope = a 1 (or= m)

Y=a 0+a 1 x+a 2 x 2 Quadratic Parabola (u-shaped curve) witheither minimum or maximumvalue

Y=a 0+a 1 x+a 2 x 2+a 3 x 3 Cubic Curved line (e.g. slope changesfrom getting flatter to steeper

Y=a 0 x -n Hyperbola Curved line (u-shaped) bowedtowards origin


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EXPONENT RULES EXAMPLES

1. xn = x multiplied by itself n times x 3 = x x xx1 = x

2. x0 = 1

3. x-a = 1 x a

x-2 = 1/x2

4. xa x b = xa+b x2 x3 = x5x2 x-1 = x2-1 = x

5. x a

x b = xa-b x2/x3 = x2 x-3 = x -1 = 1/x

6. x1/a = the a th root of x x 1/2 x

= what number multiplied byitself "a" times = x 8 1/3 = 2 (because 2 2 2 = 8)

7. xa ya = (xy) a x2y2 = (xy) 2

8. (xa) b = xab (x2)3 = x6

9. (xy)1/a = x1/a y1/a (xy)1/2 = x . y

3. x-a = 1 x a

x-2 = 1/x2

4. xa x b = xa+b x2 x3 = x5

5. x a

x b = xa-b x2/x3 = x2 x-3 = x -1 = 1/x

6. x1/a = the a th root of x x 1/2 x

9. (xy)1/a = x1/a y1/a (xy)1/2 = x . y


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Y = a + b 1X1 + b nXn=> the value of Ydepends on the values of n differentother variables; a ceteris paribusassumption => we assume that all Xvariable values except one are heldconstant so we can look at how thevalue of Y depends on the value of the

one X variable that is allowed to change


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Given 2 pts on a straight line, how to solve for the

specific equation of that line?

Recall, in general, the equation of a straight line is Y= a + bX, where b = the slope, and a = the verticalaxis intercept. The specific equation has the values of

a and b specified.

Solution procedure:1. Solve for b = Y/ X = (Y 2-Y 1)/(X 2-X1)

2. Given values at one pt for Y, X, and b, solve for a(e.g. a = Y 1 bX1)


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Graphical Concepts (Variable Relationships) Y axis: a vertical line in a graph along which the

units of measurement represent differentvalues of, normally, the Y or dependentvariable.

Y axis intercept:the value of Y when the value of X = 0, or

the value of Y where a line or curveintersects the Y axis; = a in Y = a + bX


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Graphical Concepts (Variable Relationships)X axis: a horizontal line in a graph along which

the units of measurement representdifferent values of, normally, the X orindependent variable

X axis intercept:the value of X when the value of Y = 0, orthe value of X where a line or curveintersects the X axis


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Graphical Concepts (Variable Relationships)

Slope:= the steepness of a line or curve; a +(-) slope =>the line or curve slopes upward (downward) to theright

= the change in the value of Y divided by thechange in the value of X (between 2 pts on a lineor a curve)= Y/ X = 1 st derivative (in calculus)= Y/ X using algebraic notation= the marginal effect, or the change in Y broughtabout by a 1 unit change in X= b if Y = a + bX


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Slope Graphically

y

xriserun

y y x x

2 1

2 1


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Slope Calculation Rules(slope = Y/ X = dy / dx)

Rule Example

1. Slope of a constant = 0 If y=6, slope = 0

2. power rule => slope of afunction y = ax n is (n)(a)x n-1

If y=3x 2, slope = (2)(3)x 2-1 =6xIf y=x, slope = (1)x 1-1 =1

3. Sum of functions rule = slopeof the sum of two functions is thesum of the two functions slopes

If y = x + 3x 2, slope = 1 + 6x


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Mathematics of Optimization

Optimization a decision maker wishesto either MAXimize or MINimize a goal

(i.e. objective function)

For a function to have a maximum orminimum value, the correspondinggraph will reveal a nonlinear curve thathas either a peak or a valley


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Mathematics of Optimization The mathematical equation of the function to be optimized willhave THE VERTICAL AXIS VARIABLE ON THE LEFT-HAND SIDEOF THE EQUATION (e.g. Y = f(x) Y is the vertical axisvariable)

the slope of a curve at either a peak or a valley will = 0; in mathterms, the slope is the first derivative (I.e. dY/dX = 0)

Constrained optimization do the best job of maximizing (orminimizing) a function given constraints; the LagrangianMultiplier Method is a mathematical procedure for solving thesekinds of problems


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Typical Time Value of Money

Problems in BusinessHow to compare or evaluate twodifferent dollar amounts at two different

time periods?

0 t 1 t 2 t 3

$Y$X


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Time Value of Money ConceptsPV = present value

= the number of $ you will be able toborrow [or have to save] presently in

order to payback [or collect] a givennumber of $ in the futureFV = future value

= the number of $ you will have to pay back

[or be able to collect] in the future as aresult of having borrowed [or saved] agiven number of $ presently


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FV1 = PV + PV(r)

= PV(1+r)FV2 = FV 1+FV1(r)

= FV 1(1+r)= PV(1+r)(1+r)

= PV(1+r) 2

FVn = PV(1+r) n


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Time Value ProblemsFVn = PV(1+r) n

Given Solve For

PV,r,n FVn = PV(1+r) n = compounding

FVn,r,n PV=FV n[1/(1+r) n] = discounting

FVn,PV,n r (1+r) n=FV

n /PV ( find in n row)

FVn,PV,r n (1+r) n=FVn /PV ( find in r column)


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r(%)

Borrow or savetoday (= PV)

Pay back or collectin 1 yr (= FV)

$ interest8 1.00 _____ _____

8 _____ 1.00 _____

9 1.00 _____ _____

9 _____ 1.00 _____

r PV _____ _____

r _____ FV _____


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Net Present Value (NPV)

= an investment analysis concept= PV of future net cash flows initial

cost

= PV of MR s PV of MC s = invest if NPV > 0= invest if PV of MR s > PV of MC s


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Internal Rate of Return

= an investment analysisalternative

= value of r that results in a NPV = 0


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Payback Period= an investment analysis alternative= period of time required for the sum

of net cash flows to equal the initialcost= value of n such that

i

n

i NCF C 1


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Firm ValuationThe value of a firm equals the present value of all itsfuture profits

PV it t / ( )1

If profits grow at a constant rate, g


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Time Value of Money(Applied to Inflation)

Can be used to estimate or forecastfuture prices, revenues, costs, etc.

FV n = PV (1+r) n wherePV = present value of price, cost, etc.

r = estimated annual rate of increasen = number of yearsFV = future value of price, cost, etc.

Slope Calculation Rule

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