Figure 21.11 Tangent Lines and the Slopes of Nonlinear Curves. Explain how to estimate the slope at any point on a nonlinear curve. The slopes of these tangent lines are negative, suggesting the negative relationship between smoking and life expectancy. We see here that the slope falls (the tangent lines become flatter) as the number of bakers rises. The slope changes all along the curve. This graph below shows a nonlinear relationship between x and y. Consider the following curve drawn to show the relationship between two variables, A and B (we will be using a curve like this one in the next chapter). For example, if we are modeling the yield of a chemical synthesis in terms of the temperature at which the synthesis takes place, we may find that the yield improves by increasing amounts for each unit increase in temperature. As we add workers (in this case bakers), output (in this case loaves of bread) rises, but by smaller and smaller amounts. In Figure 21.10 “Estimating Slopes for a Nonlinear Curve”, we have computed slopes between pairs of points A and B, C and D, and E and F on our curve for loaves of bread produced. Graphs of Nonlinear Relationships. Again, our life expectancy curve slopes downward. They are the slopes of the dashed-line segments shown. This is a nonlinear relationship; the curve connecting these points in Panel (c) (Loaves of bread produced) has a changing slope. We can deal with this problem in two ways. We turn finally to an examination of graphs and charts that show values of one or more variables, either over a period of time or at a single point in time. Again, our life expectancy curve slopes downward. They also get steeper as the number of cigarettes smoked per day rises. Non Linear (Curvilinear) Correlation. It is also possible that there is no relationship between the variables. Instead, we shall have to draw a nonlinear curve like the one shown in Panel (c). Most relationships in economics are, unfortunately, not linear. Daily fruit and vegetable consumption (measured, say, in grams per day) is the independent variable; life expectancy (measured in years) is the dependent variable. Most relationships which are not linear, can be graphed so that the graph is a straight line. When we add a passenger riding the ski … How can we estimate the slope of a nonlinear curve? • Linearity = assumption that for each IV, the amount of change in the mean value of Y associated with a unit increase in the IV, holding all other variables constant, is the same regardless of the level of X, e.g. A tangent line is a straight line that touches, but does not intersect, a nonlinear curve at only one point. A non-linear equation is such which does not form a straight line. Inspecting the curve for loaves of bread produced, we see that it is upward sloping, suggesting a positive relationship between the number of bakers and the output of bread. Indeed, much of our work with graphs will not require numbers at all. Because the graph isn’t a straight line, the relationship between X and Y is nonlinear. Some relationships are linear and some are nonlinear. The slopes of the curves describing the relationships we have been discussing were constant; the relationships were linear. Hence, we have a downward-sloping curve. The cancellation of one more game in the 1998–1999 basketball season would always reduce Shaquille O’Neal’s earnings by $210,000. A nonlinear relationship between two variables is one for which the slope of the curve showing the relationship changes as the value of one of the variables changes. Chapter 1: Economics: The Study of Choice, Chapter 2: Confronting Scarcity: Choices in Production, 2.3 Applications of the Production Possibilities Model, Chapter 4: Applications of Demand and Supply, 4.2 Government Intervention in Market Prices: Price Floors and Price Ceilings, Chapter 5: Macroeconomics: The Big Picture, 5.1 Growth of Real GDP and Business Cycles, Chapter 6: Measuring Total Output and Income, Chapter 7: Aggregate Demand and Aggregate Supply, 7.2 Aggregate Demand and Aggregate Supply: The Long Run and the Short Run, 7.3 Recessionary and Inflationary Gaps and Long-Run Macroeconomic Equilibrium, 8.2 Growth and the Long-Run Aggregate Supply Curve, Chapter 9: The Nature and Creation of Money, 9.2 The Banking System and Money Creation, Chapter 10: Financial Markets and the Economy, 10.1 The Bond and Foreign Exchange Markets, 10.2 Demand, Supply, and Equilibrium in the Money Market, 11.1 Monetary Policy in the United States, 11.2 Problems and Controversies of Monetary Policy, 11.3 Monetary Policy and the Equation of Exchange, 12.2 The Use of Fiscal Policy to Stabilize the Economy, Chapter 13: Consumptions and the Aggregate Expenditures Model, 13.1 Determining the Level of Consumption, 13.3 Aggregate Expenditures and Aggregate Demand, Chapter 14: Investment and Economic Activity, Chapter 15: Net Exports and International Finance, 15.1 The International Sector: An Introduction, 16.2 Explaining Inflation–Unemployment Relationships, 16.3 Inflation and Unemployment in the Long Run, Chapter 17: A Brief History of Macroeconomic Thought and Policy, 17.1 The Great Depression and Keynesian Economics, 17.2 Keynesian Economics in the 1960s and 1970s, Chapter 18: Inequality, Poverty, and Discrimination, 19.1 The Nature and Challenge of Economic Development, 19.2 Population Growth and Economic Development, Chapter 20: Socialist Economies in Transition, 20.1 The Theory and Practice of Socialism, 20.3 Economies in Transition: China and Russia, Nonlinear Relationships and Graphs without Numbers, Using Graphs and Charts to Show Values of Variables, Appendix B: Extensions of the Aggregate Expenditures Model, The Aggregate Expenditures Model and Fiscal Policy. 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