EC1008 Handout 1: FUNCTIONS AND GRAPHS • Linear Relationships • Nonlinear Relationships Linear Equations Y = a + bX a, b positive parameters Key feature: slope is not a function of position on the line (i.e., not a function of the value of X). Y X a b
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EC1008 Handout 1: FUNCTIONS AND GRAPHS
• Linear Relationships • Nonlinear Relationships
Linear Equations
Y = a + bX a, b positive parameters
Key feature: slope is not a function of position on the line (i.e., not a function of the value of X).
Y
X
a b
Nonlinear Relationships
Key difference between linear and nonlinear relationships: slope becomes a function of X. Four important nonlinear relationships: • Quadratic functions • Power functions • Logarithmic functions • Exponential functions
Quadratic Function
2cXbXaY +++= a, b, c positive parameters
Y
Power Functions
Y = a + bXc, a, b, c parameters. Suppose a, b, c positive, c > 1.
Exponential Function
Y = exp(X) = eX, e ≈ 2.7183
Y
X
a
X
eX
X
Logarithmic Function
Y = ln(X)
Summary
Generic representation of a function: Y = f(X) Five important examples: linear functions: bXaY += quadratic functions 2cXbXaY +++=
Putting it another way:the equation of a line may be described as the formula that allows you to calculate the y co-ordinate for any point on the line, when given the value of the x co-ordinate.
Example: y = x is a line which has a slope = 1, intercept = 0
Example:y = x + 2 is the line which has a slope = 1 , intercept = 2
The equation of a line may be written in terms of the two characteristics, m (slope) and c (intercept) . y = mx + c
Rearrange the equation in the form y = mx + c (see above)Plot y on the vertical axis, against x on the horizontal axisCalculate and plot the vertical and horizontal interceptsJoin the points and label the graph
Adjust the graph of the Budget Constraint: y = 30 - 0.5xwhen the per unit price of good Y decreases from 6 to 3
The adjusted constraint pivots upwards from the unchanged horizontal intercept (see Figure 2.41)Comment: When Y decreases in price, more units of Y are affordable
Figure 2.41 ∆Py and its effect on the budget constraint
Summary: Change in the graph of the Budget Constraint:y = 30 - 0.5x when the budget limit increases
Slope is the same: intercept has changed from 30 to 40When the budget limit increases, the constraint moves upwards, parallel to the original constraintComment: When the budget limit increases, more units of both X and Y are affordable
Figure 2.42 ∆Y and its effect on the Budget constraint
Properties of quadratic functions, illustrated graphically
If c<0 intercept is below x-axis, if c>0 aboveThe quadratic is a minimum type if a > 0, a maximum type if a < 0The graph is symmetrical about the vertical line drawn through the maximum or minimum pointThe roots are at the points of intersection with the x-axisThe roots are equidistant, (one greater, one smaller), from the x-coordinate of the maximum or minimum point
These point are illustrated in Worked Example 4.5, for the quadratic: y = 2x2 - 7x -9
Start with the population of 1750, on the vertical axisDraw a horizontal line across to the graphFrom this point on the graph, drop a vertical line down to the horizontal axisRead off the year. The population is 1750 in the year 2008 approximately
250
2250
1980 1985 1990 1995 2000 2005 2010
1750
753
population =
Year
FUNCTIONS AND GRAPHS: SUMMARY
Let pxy =
If 1−=p 1−= xy exponential function
If 0=p 1=y linear function (b=0)
If 1=p xy = linear function (a=0)
If 2=p 2xy = quadratic function
• Functions of One Variable bxay +=
2cxbxay ++= xey =
• Functions of Two (or more) Variables
czbxay ++=
Functions of More Than 1 Variable
• Multivariate function: the dependent variable, y, is a function of more than one independent variable
• If y = f(x,z) y is a function of the two variables x and z
• We substitute values for x and z to find the value of the function
• If we hold one variable constant and investigate the effect on y of changing the other, this is a form of comparative staticsanalysis
Total and Average Revenue
• TR = P.Q• AR = TR/Q = P
• A downward sloping linear demand curve implies a total revenue curve which has an inverted U shape
• Symmetric: the shape of one half of the curve is the mirror image of the other half
Total and Average Cost
• Total Cost is denoted TC • Fixed Cost: FC is the constant term in TC• Variable Cost: VC = TC – FC• Average Cost per unit output: