l Appendix to Chapter 1
l Mathematics Used in Microeconomics
l Functions of One Variable
l Variables: The basic elements of algebra, usually called X, Y, and so on, that may be given any numerical value in an equation
l Functional notation: A way of denoting the fact that the value taken on by one variable (Y) depends on the value taken on by some other variable (X) or set of variables
l Independent and Dependent Variables
l Independent Variable: In an algebraic equation, a variable that is unaffected by the action of another variable and may be assigned any value
l Dependent Variable: In algebra, a variable whose value is determined by another variable or set of variables
l Two Possible Forms of Functional Relationships
l Y is a linear function of X
– Table 1.A.1 shows some value of the linear function Y = 3 + 2X
l Y is a nonlinear function of X
– This includes X raised to powers other than 1
– Table 1.A.1 shows some values of a quadratic function Y = -X2 + 15X
l Table 1A.1: Values of X and Y for Linear and Quadratic Functions
l Graphing Functions of One Variable
l Graphs are used to show the relationship between two variables
l Usually the dependent variable (Y) is shown on the vertical axis and the independent variable (X) is shown on the horizontal axis
– However, on supply and demand curves, this approach is reversed
l Linear Function
l A linear function is an equation that is represented by a straight-line graph
l Figure 1A.1 represents the linear function Y=3+2X
l As shown in Figure 1A.1, linear functions may take on both positive and negative values
l Figure 1A.1: Graph of the Linear Function Y = 3 + 2X
l Intercept
l The general form of a linear equation is Y = a + bX
l The Y-interceptis the value of Y when when X equals 0
– Using the general form, when X = 0, Y = a, so this is the intercept of the equation
l Slopes
l The slope of any straight line is the ratio of the change in Y (the dependent variable) to the change in X (the independent variable)
l The slope can be defined mathematically as
l where ) means “change in”
l It is the direction of a line on a graph.
l Slopes
l For the equation Y = 3 + 2X the slope equals 2 as can be seen in Figure 1A.1 by the dashed lines representing the changes in X and Y
l As X increases from 0 to 1, Y increases from 3 to 5
l Figure 1A.1: Graph of the Linear Function Y = 3 + 2X
l Slopes
l The slope is the same along a straight line.
l For the general form of the linear equation the slope equals b
l The slope can be positive (as in Figure 1A.1), negative (as in Figure 1A.2) or zero
l If the slope is zero, the straight line is horizontal with Y = intercept
l Slope and Units of Measurement
l The slope of a function depends on the units in which X and Y are measured
l If the independent variable in the equation Y = 3 + 2X is income and is measured in hundreds of dollars, a $100 increase would result in 2 more units of Y
l Slope and Units of Measurement
l If the same relationship was modeled but with X measured in single dollars, the equation would be Y = 3 + .02 X and the slope would equal .02
l Changes in Slope
l In economics we are often interested in changes in the parameters (a and b of the general linear equation)
l In Figure 1A.2 the (negative) slope is doubled while the intercept is held constant
l In general, a change in the slope of a function will cause rotation of the function without changing the intercept
l FIGURE 1A.2: Changes in the Slope of a Linear Function
l FIGURE 1A.2: Changes in the Slope of a Linear Function
l Changes in Intercept
l When the slope is held constant but the intercept is changed in a linear function, this results in parallel shifts in the function
l In Figure 1A.3, the slope of all three functions is -1, but the intercept equals 5 for the line closest to the origin, increases to 10 for the second line and 12 for the third
– These represent “Shifts” in a linear function.
l FIGURE 1A.3: Changes in the Y-Intercept of a Linear Function
l FIGURE 1A.3: Changes in the Y-Intercept of a Linear Function
l FIGURE 1A.3: Changes in the Y-Intercept of a Linear Function
l Nonlinear Functions
l Figure 1A.4 shows the graph of the nonlinear function Y = -X2 + 15X
l As the graph shows, the slope of the line is not constant but, in this case, diminishes as X increases
l This results in a concave graph which could reflect the principle of diminishing returns
l FIGURE 1.A.4: Graph of the Quadratic Function Y = X2 + 15X
l FIGURE 1.A.4: Graph of the Quadratic Function Y = X2 + 15X
l The Slope of a Nonlinear Function
l The graph of a nonlinear function is not a straight line
l Therefore it does not have the same slope at every point
l The slope of a nonlinear function at a particular point is defined as the slope of the straight line that is tangent to the function at that point.
l Marginal Effects
l The marginal effect is the change in Y brought about by one unit change in X at a particular value of X (Also the slope of the function)
l For a linear function this will be constant, but for a nonlinear function it will vary from point to point
l Average Effects
l The average effectis the ratio of Y to X at a particular value of X (the slope of a ray to a point)
l In Figure 1A.4, the ray that goes through A lies about the ray that goes through B indicating a higher average value at A than at B
l APPLICATION 1A.1: Property Tax Assessment
l The bottom line in Figure 1 represents the linear function Y = $10,000 + $50X, where Y is the sales price of a house and X is its square footage
l If, other things equal, the same house but with a view is worth $30,000 more, the top line Y = $40,000 + $50X represents this relationship
l FIGURE 1: Relationship between the Floor Area of a House and Its Market Value
l FIGURE 1: Relationship between the Floor Area of a House and Its Market Value
l Calculus and Marginalism
l In graphical terms, the derivative of a function and its slope are the same concept
l Both provide a measure of the marginal inpact of X on Y
l Derivatives provide a convenient way of studying marginal effects.
l APPLICATION 1A.2: Progressive and Flat Taxes
l Advocates of tax fairness argue that income taxes should progressive so that richer people should pay a higher fraction of their incomes in taxes
– This is illustrated in Figure 1 by the nonlinear line OT that becomes steeper as taxable income increases
– This represents an increasing marginal tax rate
l FIGURE 1: Progressive Rates Compared to a Flat Tax Schedule
l APPLICATION 1A.2: Progressive and Flat Taxes
l Opponents of progressive taxes have argued for a flat tax
l The straight line OT’ represents a proposal where the first $18,000 of taxable income would not be taxed with a flat tax of 17 percent on additional taxable income
– This would also be progressive but not as much as in the current system
l FIGURE 1: Progressive Rates Compared to a Flat Tax Schedule
l Functions of Two or More Variables
l The dependent variable can be a function of more than one independent variable
l The general equation for the case where the dependent variable Y is a function of two independent variables X and Z is
l A Simple Example
l Suppose the relationship between the dependent variable (Y) and the two independent variables (X and Z) is given by
l Some values for this function are shown in Table 1A.2
l TABLE 1A.2: Values of X, Z, and Y that satisfy the Relationship Y = X·Z
l Graphing Functions of Two Variables
l Contour lines are frequently used to graph functions with two independent variables
l Contour lines are lines in two dimensions that show the sets of values of the independent variables that yield the same value for the dependent variable
l Contour lines for the equation Y = X·Z are shown in Figure 1A.5
l FIGURE 1A.5: Contour Lines for Y = X·Z
l Simultaneous Equations
l These are a set of equations with more than one variable that must be solved together for a particular solution
l When two variables, say X and Y, are related by two different equations, it is sometime possible to solve these equations to get a set of values for X and Y that satisfy both equations
l Simultaneous Equations
l Changing Solutions for Simultaneous Equations
l Graphing Simultaneous Equations
l The two simultaneous equations systems, 1A.17 and 1A.19 are graphed in Figure 1A.6
l The intersection of the graphs of the equations show the solutions to the equations systems
l These graphs are very similar to supply and demand graphs
l Figure 1A.6: Solving Simultaneous Equations
l Figure 1A.6: Solving Simultaneous Equations
l APPLICATION 1A.3: Can Iraq Affect Oil Prices?
l APPLICATION 1A.3:Can Iraq Affect Oil Prices?
l FIGURE 1: Effect of OPEC Output Restrictions on World Oil Market
l FIGURE 1: Effect of OPEC Output Restrictions on World Oil Market
l Empirical Microeconomics and Econometrics
l Economists test the validity of their models by looking at data from the real world
l Econometrics is used for this purpose
l Two important aspects of econometrics are
– random influences
– the ceteris paribus assumption
l Random Influences
l No economic model exhibits perfect accuracy so actual price and quantity values will be scattered around the “true” demand curve
l Figure 1A.7 shows the unknown true demand curve and the actual points observed in the data from the real world
l The problem is to infer the true demand curve
l FIGURE 1A.7: Inferring the Demand Curve from Real-World Data
l Random Influences
l Technically, the problem is statistical inference: the use of actual data and statistical techniques to determine quantitative economic relationships
l Since no single straight line will fit all of the data points, the researcher must give careful consideration to the random influences to get the best line possible
l The Ceteris Paribus Assumption
l To control for the “other things equal” assumption two things must be done
– Data should be collected on all of the other factors that affect demand, and
– appropriate procedures must be used to control for these measurable factors in the analysis
l Generally the researcher has to make some compromises which leads to many controversies in testing economic models
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