Nonlinear Relationships and Graphs without Numbers
It gives the full non-linear relationship between variables. They can very well be approximated to be linear in Newtonian mechanics as a first approximation at. For example, a linear relationship between the height and weight of a person is Nonlinear relationships, in general, are any relationship which is not linear. Each unit change in the x variable will not always bring about the same change in the y This graph below shows a nonlinear relationship between x and y.
Monotonic changes may be smooth or they may be abrupt. For example, a drug may be ineffective up until a certain threshold and then become effective. However, nonlinear relationships can also be non-monotonic. For example, a drug may become progressively more helpful over a certain range, but then may become harmful. Thus the degree of help increases and decreases and this is a non-monotonic, as well as a nonlinear, relationship.
Even when a relationship is monotonic, and the changes in one quantity are smoothly related to the changes in the other quantity a linear relationship is not always the best approximation. It is often useful to generalize to a power law relationship.
The slope of our bread production curve at point D equals the slope of the line tangent to the curve at this point.
In Panel bwe have sketched lines tangent to the curve for loaves of bread produced at points B, D, and F.
Notice that these tangent lines get successively flatter, suggesting again that the slope of the curve is falling as we travel up and to the right along it. In Panel athe slope of the tangent line is computed for us: Generally, we will not have the information to compute slopes of tangent lines.
We will use them as in Panel bto observe what happens to the slope of a nonlinear curve as we travel along it. We see here that the slope falls the tangent lines become flatter as the number of bakers rises. Notice that we have not been given the information we need to compute the slopes of the tangent lines that touch the curve for loaves of bread produced at points B and F.
In this text, we will not have occasion to compute the slopes of tangent lines. Either they will be given or we will use them as we did here—to see what is happening to the slopes of nonlinear curves.
In the case of our curve for loaves of bread produced, the fact that the slope of the curve falls as we increase the number of bakers suggests a phenomenon that plays a central role in both microeconomic and macroeconomic analysis.
As we add workers in this case bakersoutput in this case loaves of bread rises, but by smaller and smaller amounts. Another way to describe the relationship between the number of workers and the quantity of bread produced is to say that as the number of workers increases, the output increases at a decreasing rate.
In Panel b of Figure Indeed, much of our work with graphs will not require numbers at all. We turn next to look at how we can use graphs to express ideas even when we do not have specific numbers.
Graphs Without Numbers We know that a positive relationship between two variables can be shown with an upward-sloping curve in a graph. A negative or inverse relationship can be shown with a downward-sloping curve. Some relationships are linear and some are nonlinear. We illustrate a linear relationship with a curve whose slope is constant; a nonlinear relationship is illustrated with a curve whose slope changes. Using these basic ideas, we can illustrate hypotheses graphically even in cases in which we do not have numbers with which to locate specific points.
Consider first a hypothesis suggested by recent medical research: We can show this idea graphically. Daily fruit and vegetable consumption measured, say, in grams per day is the independent variable; life expectancy measured in years is the dependent variable.
Panel a of Figure Notice the vertical intercept on the curve we have drawn; it implies that even people who eat no fruit or vegetables can expect to live at least a while! The graphs in the four panels correspond to the relationships described in the text. Panel b illustrates another hypothesis we hear often: Here the number of cigarettes smoked per day is the independent variable; life expectancy is the dependent variable.
The hypothesis suggests a negative relationship. Hence, we have a downward-sloping curve.
Correlation and dependence - Wikipedia
As we saw in Figure Strong positive linear relationship Plot 2: Strong negative linear relationship When both variables increase or decrease concurrently and at a constant rate, a positive linear relationship exists. The points in Plot 1 follow the line closely, suggesting that the relationship between the variables is strong. When one variable increases while the other variable decreases, a negative linear relationship exists.
The points in Plot 2 follow the line closely, suggesting that the relationship between the variables is strong. Weak linear relationship Plot 4: Nonlinear relationship The data points in Plot 3 appear to be randomly distributed. They do not fall close to the line indicating a very weak relationship if one exists.