Illustrating Complex Relationships

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Illustrating Complex Relationships

• In economics you will often see a complex set of
  relations represented graphically.
• You will use graphs to make interpretations about
  what is happening as variables in a relationship
  change.

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Changes in the supply of corn
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• A change in one or more of the determinants of
  supply will cause a change in supply.
• An increase in supply shifts the supply curve to
  the right as from S1 to S2.
• A decrease in supply is shown graphically as a
  shift of the curve to the left, as from S1 to S3.

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• A change in the quantity supplied is caused by a
  change in the price of the product as is shown by
  a movement from one point to another--as from a
  to b--on a fixed supply curve.

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Market Equilibrium
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• The market equilibrium price and quantity comes
  at the intersection of supply and demand curves.
• At a price of 3 at point C, firms willingly
  supply what consumers willingly demand.

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• When price is too low (say 2), quantity demanded
  exceeds quantity supplied, shortages occur, and
  prices are driven up to equilibrium.
• What occurs at a price of 4?

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The skills you will learn in this book are to

• Describe how changing the y-intercept of a line
  affects the graph of a line.
• Describe how changing the slope of a line affects
  the graph of a line.
• Describe what has happened to an equation after a
  line on a graph has shifted.

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• Identify the intersection of two lines on a
  graph.
• Describe what happens to the x and y coordinate
  values of intersecting lines after a shift in a
  line on the graph.
• Identify the Point of Tangency on a curve.

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• Determine whether a line is a tangent line.
• Calculate the slope at a point on a curve.
• Determine whether the slope at a point on a curve
  is positive, negative, zero, or infinity.

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• Identify maximum and minimum points on a curve.
• Determine whether a curve does or does not have
  maximum and minimum points.

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Analyzing Lines on a Graph

• After reviewing this section you will be able to
  
• Describe how changing the y-intercept of a line
  affects the graph of a line.
• Describe how changing the slope of a line affects
  the graph of a line.
• Describe what has happened to an equation after a
  line on a graph has shifted.

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The Equation of a line
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• The slope is used to tell us how much one
  variable (y) changes in relation to the change in
  another variable (x).

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• The constant labeled "a" in the equation is the
  y-intercept.
• The y-intercept is the point at which the line
  crosses the y-axis.

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Comparing Lines on a Graph

• By looking at this graph, we can see that the
  cost of our plain pizza is 7.00, and the cost
  per topping is our slope, 75 cents.
• This line has the equation of y 7.00 .75x.

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Shift Due to Change in y-intercept

• In the graph at the right, line P shifts from its
  initial position P0 to P1.
• Only the y-intercept has changed.
• The equation for P0 is y 7.00 .75x, and the
  equation for P1 is y 8.00 .75x.

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Shift Due to Change in Slope

• In the graph at the right, line P shifts from its
  initial position P0 to P1.
• Line P1 is steeper than the line P0. This means
  that the slope of the equation has gone up.
• The equation for P0 is y 7.00 .75x, and the
  equation for P1 is y 7.00 .x.

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Identifying the Intersection of Lines

• After reviewing this section you will be able to
  
• Identify the intersection of two lines on a
  graph.
• Describe what happens to the x and y coordinate
  values of intersecting lines after a shift in a
  line on the graph.

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Intersection of Two Lines

• Many times in the study of economics we have the
  situation where there is more than one
  relationship between the x and y variables.
• You'll find this type of occurrence often in your
  study of supply and demand.

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• In this graph, there are two relationships
  between the x and y variables one represented by
  the straight line AC and the other by straight
  line WZ.

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• In one case, the two lines have the same (x, y)
  values simultaneously.
• This is where the two lines RT and JK intersect
  or cross.
• The intersection occurs at point E, which has the
  coordinates (2, 4).

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Examining The Shift of a Line

• In any situation where you are given a shift in a
  line
• identify both the initial and final points of
  intersection, then
• compare the coordinates of the two.

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Before the Shift

• This graph contains the two lines R and S, which
  intersect at point A (2, 3).
• Lines shifts to the right.
• What happens to the intersection of the two lines
  if one of the lines shifts?

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After the Shift

• On the graph below, line S0 is our original line
  S.
• Lines S1 represents our new S after it has
  shifted.
• The new point of intersection between R and S is
  now point B (3, 4).

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Example

• Compare the points A (2, 3) and B (3, 4) on this
  graph.
• The x-coordinate changed from 2 to 3.
• The y-coordinate changed from 3 to 4.

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Nonlinear Relationships

• After reviewing this unit, you will be able to
• Identify the Point of Tangency on a curve.
• Determine whether a line is a tangent line.
• Calculate the slope at a point on a curve.
• Determine whether the slope at a point on a curve
  is positive, negative, zero, or infinity.

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• Identify maximum and minimum points on a curve
• Determine whether a curve does or does not have
  maximum and minimum points.

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Introduction

• Most relationships in economics are,
  unfortunately, not linear.
• Each unit change in the x variable will not
  always bring about the same change in the y
  variable.
• The graph of this relationship will be a curve
  instead of a straight line.

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• This graph shows a linear relationship between x
  and y.

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• This graph below shows a nonlinear relationship
  between x and y.

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Determining the Slope of a Curve

• One of the differences between the slope of a
  straight line and the slope of a curve is that
• the slope of a straight line is constant,
• while the slope of a curve changes from point to
  point.

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• To find the slope of a line you need to
• Identify two points on the line.
• Select one to be (x1, y1) and the other to be
  (x2, y2).
• Use the equation

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• From point A (0, 2) to point B (1, 2.5)

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• From point B (1, 2.5) to point C (2, 4)

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• From point C (2, 4) to point D (3, 8)

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• The slope of the curve changes as you move along
  it.
• For this reason, we measure the slope of a curve
  at just one point.
• For example, instead of measuring the slope as
  the change between any two points, we measure the
  slope of the curve at a single point (at A or C).

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Tangent Line

• A tangent is a straight line that touches a curve
  at a single point and does not cross through it.
• The point where the curve and the tangent meet is
  called the point of tangency.
• Both of the figures below show a tangent line to
  the curve.

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• This curve has a tangent line to the curve with
  point A being the point of tangency.
• In this case, the slope of the tangent line is
  positive.

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• This curve has a tangent line to the curve with
  point A being the point of tangency.
• In this case, the slope of the tangent line is
  negative.

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• The line on this graph crosses the curve in two
  places.
• This line is not tangent to the curve.

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• The slope of a curve at a point is equal to the
  slope of the straight line that is tangent to the
  curve at that point.

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Example

• What is the slope of the curve at point A?

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• The slope of the curve at point A is equal to the
  slope of the straight line BC.
• By finding the slope of the straight line BC, we
  have found the slope of the curve at point A.
• The slope at point A is 1/2, or .5.
• This is the slope of the curve only at point A.

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Slope of a Curve Positive, Negative, or Zero?

• If the line is sloping up to the right, the slope
  is positive ().

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• If the line is sloping down to the right, the
  slope is negative (-).

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• Horizontal lines have a slope of 0.

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Slope of a Curve Positive, Negative, or Zero?
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• Both graphs show curves sloping upward from left
  to right.
• As with upward sloping straight lines, we can say
  that generally the slope of the curve is
  positive.
• While the slope will differ at each point on the
  curve, it will always be positive.

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• In the graphs above, both of the curves are
  downward sloping.
• Curves that are downward sloping also have
  negative slopes.
• We know, of course, that the slope changes from
  point to point on a curve, but all of the slopes
  along these two curves will be negative.

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• In general, to determine if the slope of the
  curve at any point is positive, negative, or zero
  you draw in the line of tangency at that point.

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Example

• A, B, and C are three points on the curve.
• The tangent line at each of these points is
  different.
• Each tangent has a positive slope therefore, the
  curve has a positive slope at points A, B, and C.

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• A, B, and C are three points on the curve.
• The tangent line at each of these points is
  different.
• Each tangent has a negative slope since its
  downward sloping therefore, the curve has a
  negative slope at points A, B, and C.

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• In this example, our curve has a
• positive slope at points A, B, and F,
• a negative slope at D, and
• at points C and E the slope of the curve is zero.
  

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Maximum and Minimum Points of Curves

• In economics, we can draw interesting conclusions
  from points on graphs where the highest or lowest
  values are observed.
• We refer to these points as maximum and minimum
  points.

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• Maximum and minimum points on a graph are found
  at points where the slope of the curve is zero.
• A maximum point is the point on the curve with
  the highest y-coordinate and a slope of zero.
• A minimum point is the point on the curve with
  the lowest y-coordinate and a slope of zero.

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Maximum Point

• Point A is at the maximum point for this curve.
• Point A is at the highest point on this curve.
• It has a greater y-coordinate value than any
  other point on the curve and has a slope of zero.

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Minimum Point

• Point A is at the minimum point for this curve.
• Point A is at the lowest point on this curve.
• It has a lower y-coordinate value than any other
  point on the curve and has a slope of zero.

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Example
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• The curve has a slope of zero at only two points,
  B and C.
• Point B is the maximum. At this point, the curve
  has a slope of zero with the largest
  y-coordinate.
• Point C is the minimum. At this point, the curve
  has a slope of zero with the smallest
  y-coordinate.

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• We can have curves that have no maximum and
  minimum points.
• On this curve, there is no point where the slope
  is equal to zero.
• This means, using the definition given above, the
  curve has no maximum or minimum points on it.

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