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.

Changes in the supply of corn

• 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.

• 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.

Market Equilibrium

• 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.

• 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?

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.

• 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.

• 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.

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

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.

The Equation of a line

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

• 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.

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.

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.

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.

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.

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.

• 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.

• 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).

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.

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?

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).

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.

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.

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

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.

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

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

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.

• 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:

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

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

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

• 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).

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.

• 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.

• 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.

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

• 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.

Example

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

• 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.

Slope of a Curve: Positive, Negative, or Zero?

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

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

• Horizontal lines have a slope of 0.

Slope of a Curve: Positive, Negative, or Zero?

• 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.

• 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.

• 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.

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.

• 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 it’s downward sloping; therefore, the curve has a negative slope at points A, B, and C.

• 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.

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.

• 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.

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.

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.

Example

• 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.

• 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.