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  • Marginal Analysis-simple exampleMath165: Business Calculus

    Roy M. Lowman

    Spring 2010, Week4 Lec3

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample

    Given:

    cost per unit: c = $6 per unit, cost to producer

    Demand Relation: q = 100 2p,sometimes written D(p) = 100 2p. Note, as the price perunit increases, the demand decreases.

    production level: q,assume that the number of units sold is the same as thenumber of units produced.

    price per unit: p, selling price

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample

    Given:

    cost per unit: c = $6 per unit, cost to producer

    Demand Relation: q = 100 2p,sometimes written D(p) = 100 2p. Note, as the price perunit increases, the demand decreases.

    production level: q,assume that the number of units sold is the same as thenumber of units produced.

    price per unit: p, selling price

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample

    Given:

    cost per unit: c = $6 per unit, cost to producer

    Demand Relation: q = 100 2p,sometimes written D(p) = 100 2p. Note, as the price perunit increases, the demand decreases.

    production level: q,assume that the number of units sold is the same as thenumber of units produced.

    price per unit: p, selling price

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample

    Given:

    cost per unit: c = $6 per unit, cost to producer

    Demand Relation: q = 100 2p,sometimes written D(p) = 100 2p. Note, as the price perunit increases, the demand decreases.

    production level: q,assume that the number of units sold is the same as thenumber of units produced.

    price per unit: p, selling price

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample

    Given:

    cost per unit: c = $6 per unit, cost to producer

    Demand Relation: q = 100 2p,sometimes written D(p) = 100 2p. Note, as the price perunit increases, the demand decreases.

    production level: q,assume that the number of units sold is the same as thenumber of units produced.

    price per unit: p, selling price

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample

    Given:

    cost per unit: c = $6 per unit, cost to producer

    Demand Relation: q = 100 2p,sometimes written D(p) = 100 2p. Note, as the price perunit increases, the demand decreases.

    production level: q,assume that the number of units sold is the same as thenumber of units produced.

    price per unit: p, selling price

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    Find:

    C(q), Cost function

    R(q), Revenue function

    P(q), Profit function

    qmax production level to maximize profit

    pmax the price to charge for each unit to maximize profit

    maximum profit Pmax

    Cavg =C(q)

    q Average Cost function

    break even point(s), set P(q) = 0 and solve for q

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    There are two standard ways to approach the problem offinding qmax1st solve MR = MC i.e. set R(q) = C(q) and solve for qmax.

    Using this method you never need to actually find the profitfunction. Sometimes this is useful.

    2nd solve MP = 0, i.e. set P(q) = 0 and solve for qmax. Hereyou must first find the profit function and its derivative.

    This should be obvious from the graph:

    -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195

    -40

    -35

    -30

    -25

    -20

    -15

    -10

    -5

    5

    10

    15

    20

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    35

    40

    45

    50

    55

    60

    65

    70

    75

    80

    85

    90

    95

    100

    105

    110

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    There are two standard ways to approach the problem offinding qmax1st solve MR = MC i.e. set R(q) = C(q) and solve for qmax.

    Using this method you never need to actually find the profitfunction. Sometimes this is useful.

    2nd solve MP = 0, i.e. set P(q) = 0 and solve for qmax. Hereyou must first find the profit function and its derivative.

    This should be obvious from the graph:

    -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195

    -40

    -35

    -30

    -25

    -20

    -15

    -10

    -5

    5

    10

    15

    20

    25

    30

    35

    40

    45

    50

    55

    60

    65

    70

    75

    80

    85

    90

    95

    100

    105

    110

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    There are two standard ways to approach the problem offinding qmax1st solve MR = MC i.e. set R(q) = C(q) and solve for qmax.

    Using this method you never need to actually find the profitfunction. Sometimes this is useful.

    2nd solve MP = 0, i.e. set P(q) = 0 and solve for qmax. Hereyou must first find the profit function and its derivative.

    This should be obvious from the graph:

    -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195

    -40

    -35

    -30

    -25

    -20

    -15

    -10

    -5

    5

    10

    15

    20

    25

    30

    35

    40

    45

    50

    55

    60

    65

    70

    75

    80

    85

    90

    95

    100

    105

    110

    Roy M. Lowman Marginal Analysis-simple example

  • Marginal Analysisexample part 1

    There are two standard ways to approach the problem offinding qmax1st solve MR = MC i.e. set R(q) = C(q) and solve for qmax.

    Using this method you never need to actually find the profitfunction. Sometimes this is useful.

    2nd solve MP = 0, i.e. set P(q) = 0 and solve for qmax. Hereyou must first find the profit function and its derivative.

    This should be obvious from the graph:

    -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195

    -40

    -35

    -30

    -25

    -20

    -15

    -10

    -5

    5

    10

    15

    20

    25

    30

    35

    40

    45

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