AGEC 317AGEC 317

Introductory Calculus: Introductory Calculus: Marginal AnalysisMarginal Analysis

ReadingsReadings

Review of fundamental algebra concepts Review of fundamental algebra concepts (Consult any math textbook)(Consult any math textbook)

Chapter 2, pp. 23-44, Chapter 2, pp. 23-44, Managerial Managerial EconomicsEconomics

TopicsTopics DerivativesDerivatives

Linear functionsLinear functions Graphical analysisGraphical analysis SlopeSlope InterceptIntercept

Nonlinear functionsNonlinear functions Graphical analysisGraphical analysis Rate of change (marginal effects)Rate of change (marginal effects) Optimal points (minima, maxima)Optimal points (minima, maxima)

Applications to revenue and profit functionsApplications to revenue and profit functions

Marginal profit is the derivative of the profit function (the same is true for cost and revenue). We use this marginal profit function to estimate the amount of profit from the “next” item. 

DerivativesDerivativesDerivatives are all about change ... They show how fast something is changing (called the rate of change) at any point.Y=f(x)First DerivativeFirst Derivative

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Marginal profit is the first derivative of the profit function (the same is true for cost, utility and revenue, etc.). We use this marginal profit function to estimate the amount of profit from the “next” item. 

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DerivativesDerivatives Y=f(x) First DerivativeFirst Derivative

Derivative of Y with respect to X at point A is the slope of a line that is tangent to the curve at the point A.

DerivativesDerivatives

Second Derivative Second Derivative of a function ƒ is the derivative of the derivative of. ƒ. It measures how the rate of change of a quantity is itself changing

On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph.

The graph of a function with positive second derivative curves upwards, while the graph of a function with negative second derivative curves downwards.

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Linear Function: Y=aX+bLinear Function: Y=aX+b

Slope = dY/dX = a Interpretation: a one unit increase in X leads to an

increase in Y of a units.

Intercepts On x-axis; the value of X if y = 0:

aX + b = 0, the x intercept is -b/a; put another way (-b/a, 0)

On y-axis; the value of Y if x = 0: The y intercept is b; put another way (0,b)

Graph Y = -2X + 2, x intercept is 1 or (1,0); y intercept is 2 or

(0,2) Y = 2X + 4, x intercept is -2 or (-2,0); y intercept if 4 or

(0,4)

Application of Linear Function: Application of Linear Function: Revenue & Output Revenue & Output

Total Revenue Output$1.50 1

3.00 24.50 36.00 47.50 59.00 6

Questions:

(a) Slope

(b) Intercepts

Nonlinear function

Locating Maximum and Minimum Values Locating Maximum and Minimum Values of a Functionof a Function

Step 1: Find the derivative of the function with respect to the “independent” variable.

For example, suppose that profit ( π ) = a – bQ + cQ2

The “independent” variable is Q and the “dependent” variable is π

Then the derivative (marginal profit) = -b + 2cQ

Step 2: Set the derivative expression from step 1 to 0 (first-order condition)

so, -b + 2cQ = 0

Step 3: Find the value of the “independent” variable that solves the derivative expression

-b + 2cQ = 0

2cQ = b

Q = b/2c

Locating Maximum and Minimum Values Locating Maximum and Minimum Values of a Function (Con’t)of a Function (Con’t)

Step 4: How to discern whether the value(s) from step 3 correspond to minimum values of the function or maximum values of the function:

Calculate the second derivative with respect to the “independent” variable

first derivative: -b + 2cQ

second derivative: 2c

If the second derivative at the value of the “independent” variable that solves the first derivative expression (step 3) is positive, then that value of the “independent” variable corresponds to a minimum.

If the second derivative is negative at this point, then that value of the “independent” variable corresponds to a maximum.

Locating Maximum and Minimum Values of Locating Maximum and Minimum Values of a Function (con’t)a Function (con’t)

Step 5: Finding the Maximum of Minimum Value of the Function

Simply replace the “optimum” value of the “independent” variable into the function

= a – bQ + b/2c

from step 3, Q = b/2c

from step 4, if c > 0, then Q = b/2c corresponds to a minimum value

if c < 0, then Q = b/2c corresponds to a maximum value.

The minimum (maximum) value of then is

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Locating Maximum Locating Maximum and Minimum and Minimum

Values of a Values of a FunctionFunction

Locating Maximum Locating Maximum and Minimum and Minimum

Values of a Values of a FunctionFunction

Profit

Summary of Algebraic ReviewSummary of Algebraic Review

Mathematical operations with algebraic expressions

Solving equations

Linear functions

Nonlinear functions

Applications to revenue and profit functions