MANAGERIAL ECONOMICS ECON 5133 Lesson Nine ARC MARGINALS

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MANAGERIAL ECONOMICS ECON 5133, Lesson Nine ARC MARGINALS Computing difference quotients for a linear

MANAGERIAL ECONOMICS ECON 5133, Lesson Nine ARC MARGINALS Computing difference quotients for a linear function Conclusion for linear functions Copyright © 2005 by George A. Collier Jr.

I. Marginals A. Introduce the math of change 1. Marginals express the rate of

I. Marginals A. Introduce the math of change 1. Marginals express the rate of change of one variable w. r. t another variable. 2. Ex: speed (mph) expresses the instantaneous rate of change of distance w. r. t. time. P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

B. Two types: approximate & true 1. approximate marginal type Termed an “arc marginal”

B. Two types: approximate & true 1. approximate marginal type Termed an “arc marginal” Measures the Average rate of change of one variable w. r. t. another variable total(whatever) Arc M(? ) = another variable a “difference quotient” P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

2. The “true” marginal is the Instantaneous rate of change of one variable w.

2. The “true” marginal is the Instantaneous rate of change of one variable w. r. t. another variable: change in one variable change in another var. (as this change nears zero) Mathematically: a derivative of a total (? ) w. r. t. another(? ) P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

3. both marginal concepts give the same results (if linear functions) Simplify analysis by

3. both marginal concepts give the same results (if linear functions) Simplify analysis by assigning values B M E EM wish to eliminate. to variables. Ryou ER Ex: Advertising’s influence Let Px = $400, Py = $500 H = 40, 000 I = $20, 000 then Qx = 29000 +. 02 A P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

4. consider arc marginal demand w. r. t. advertising expenditures: Qx A Let A

4. consider arc marginal demand w. r. t. advertising expenditures: Qx A Let A range: 20000 A 30000 move 30000 toward 20000, computing a new difference quotient after each move P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

Difference quotient computations (when Qx = 29000 +. 02 A) Pt A Qx Between

Difference quotient computations (when Qx = 29000 +. 02 A) Pt A Qx Between A = 20000 & 30000 5 30000 29600. 00 4 25000 29500. 00 3 22500 29450. 00 2 20001 29400. 02 1 20000 29400. 00 P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

A ($000) Qx = 29000 +. 02 A 60 One click animates fully 30

A ($000) Qx = 29000 +. 02 A 60 One click animates fully 30 20 1 5 P S Qx Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

Difference quotient computations Between A = 20000 & 30000 Pt A Qx 5 30000

Difference quotient computations Between A = 20000 & 30000 Pt A Qx 5 30000 29600. 00 4 25000 29500. 00 Between A = 20000 & 25000 3 22500 29450. 00 2 20001 29400. 02 1 20000 29400. 00 P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

A ($000) Qx = 29000 +. 02 A 60 One click animates fully 25

A ($000) Qx = 29000 +. 02 A 60 One click animates fully 25 20 Qx 1 4

Difference quotient computations Between A = 20000 & 30000 Pt A Qx 5 30000

Difference quotient computations Between A = 20000 & 30000 Pt A Qx 5 30000 29600. 00 Between A = 20000 & 25000 4 25000 29500. 00 3 22500 29450. 00 Between A = 20000 & 20001 29400. 02 1 20000 29400. 00 P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.

A ($000) Qx = 29000 +. 02 A 60 One click animates fully 20.

A ($000) Qx = 29000 +. 02 A 60 One click animates fully 20. 001 20 Qx 12

5. conclusion (for linear functions) Qx =. 02 (no matter how small A is)

5. conclusion (for linear functions) Qx =. 02 (no matter how small A is) Formally stated: limit Qx A 0 A =. 02 calculation is called: “arc marginal advertising” P S Click the green loudspeaker to start an audio explanation, the P to pause/restart, or the S to stop the audio.