Essential Math For Economics

Transcription

1 Essential Math For Economics D I A N N A D A S I L V A - G L A S G O W D E P A R T M E N T O F E C O N O M I C S U N I V E R S I T Y O F G U Y A N A S E P T E M B E R 7,

2 Wk 2 Lecture 1... INTRODUCTION TO GRAPHS

3 GRAPHS AND THEIR MEANING Graphs are extensively used in economics for modeling A graph is a visual representation of the relationship between two variables.

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5 Types of variables We can distinguish between two types of variables 1. Dependent and 2. Independent This is a Means Of Differentiating Which Variable is the Cause And Which Is The Effect.

6 The DEPENDENT VARIABLE THE DEPENDENT VARIABLE IS THE EFFECT OR OUTCOME; IT IS THE VARIABLE THAT CHANGES BECAUSE OF CHANGES IN THE INDEPENDENT VARIABLE.

7 The independent variable THE INDEPENDENT VARIABLE IS THE CAUSE OR SOURCE; IT IS THE VARIABLE THAT CHANGES FIRST.

8 Quick Quiz Identify the dependent and the independent variables below: A painter must measure a room before deciding how much paint to buy. The height of a candle decrease d centimeters for every hour it burns. A veterinarian must weight an animal before determining the amount of medication. A company charges $10 per hour to rent a jackhammer. Camryn buys p pounds of apples at $0.99 per pound. Grade Point Average and Time Spent Studying

9 Quick Quiz A painter must measure a room before deciding how much paint to buy. Dependent: amount of paint Independent: measurement of the room The height of a candle decrease d centimeters for every hour it burns. A veterinarian must weight an animal before determining the amount of medication. Dependent: height of candle Independent: time Dependent: amount of medication Independent: weight of animal

10 Quick Quiz A company charges $10 per hour to rent a jackhammer. Camryn buys p pounds of apples at $0.99 per pound. Dependent variable: cost Independent variable: time Grade Point Average and Time Spent Studying Dependent variable: cost Independent variable: pounds Dependent variable: GPA Independent variable: Time

11 Graphing variables On a Graph how do we reflect independent and dependent variables Cause is plotted on the X-axis Effect is plotted on the Y-axis All Rights Reserved DR. DAVID P ECHEVARRIA 11

12 Example 1

13 Types of Relationships Among variables CORRELATION A POSITIVE RELATIONSHIP A NEGATIVE RELATIONSHIP

14 Positive relationship A POSITIVE RELATIONSHIP IS ONE WHERE THE TWO VARIABLES CHANGE IN THE SAME DIRECTION. THAT IS WHEN ONE INCREASE THE OTHER ALSO INCREASES OR WHEN ONE DECREASES THE OTHER ALSO DECREASES. THE CURVE FOR SUCH A GRAPH IS UPWARD SLOPING, E.G. CONSUMPTION AND INCOME.

15 Positive Correlation As one variable increases, so does the other. y x

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17 Example of positive correlation The longer you exercise, the more calories you burn. *As exercise increases, calories burned increases Can you think of one?

18 Negative relationship A NEGATIVE RELATIONSHIP IS ONE WHERE THE TWO VARIABLES CHANGE IN THE OPPOSITE DIRECTION. THAT IS WHEN ONE INCREASE THE OTHER DECREASES OR WHEN ONE DECREASES THE OTHER INCREASES. IN THIS CASE WE SAY THAT THE RELATIONSHIP IS INVERSE E.G. PRICE AND QUANTITY DEMANDED. THE CURVE FOR SUCH A GRAPH IS DOWNWARD SLOPING, E.G. CONSUMPTION AND INCOME.

19 Negative Correlation: As one variable increases, the other decreases. y x

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21 Example of negative correlation The longer the air conditioner is turned on, the colder the temperature. *As time increases, the temperature decreases Can you think of one?

22 No Correlation: As one variable increases, you cannot tell what the other is doing. y x

23 Example of no correlation The number of students in the classroom and the average height. *As the number of students increases, you cannot tell what the average height will do. Can you think of one?

24 Correlation Coefficient A statistic that quantifies a relation between two variables Can be either positive or negative Falls between and 1.00 The value of the number (not the sign) indicates the strength of the relation The sign indicates the direction of the impact, whether positive or negative

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26 Check Your Learning Which is stronger? A correlation of 0.25 or -0.74?

27 Misleading Correlations Something to think about There is a 0.91 correlation between ice cream consumption and drowning deaths. Does eating ice cream cause drowning? Does grief cause us to eat more ice cream?

28 The Limitations of Correlation Correlation is not causation. Invisible third variables Three Possible Causal Explanations for a Correlation

29 The Pearson Correlation Coefficient A statistic that quantifies a linear relation between two scale variables. Symbolized by the italic letter r when it is a statistic based on sample data. Symbolized by the italic letter p rho when it is a population parameter.

30 Pearson correlation coefficient r Linear relationship [( X M )( Y M r X ( SS )( SS ) X Y Y )]

31 Linear vs non-linear effects

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33 Slopes Lines can be described in terms of their slopes (gradient). The slope of a straight line is the ratio of change in the dependent variable to changes in the independent variable. This tells us how steep a straight line is. Slopes can be positive or negative and depict the underpinning relationship between the two variables.

34 The gradient or gradient of a line is a number that tells us how steep the line is and which direction it goes. If you move along the line from left to right and are climbing, it is a positive gradient. This one has the greatest gradient These are all positive gradients. The steeper the line, the larger the gradient value.

35 rise We could compute the run by looking at the difference between the x values. We compute the gradient by taking the ratio of how much the line rises (goes up) and how much the line runs (goes over) run (x 2, y 2 ) (x 1, y 1 ) y 2 - y 1 If we took two points on the line, we could compute the rise by looking at the difference between the y values. x 2 - x 1 So the gradient or gradient is the rise over the run m risey ( 2y 2 m runx ( x 2 2 y 1y x x ) ) This is the gradient formula gradient is designated with an m

36 Let s figure out the gradient of this line. We know it should be a positive number. 2 y x 1 (1, 2) (0, 0) (2, 4) Choose two points on the line. The rise over the run is 2 over 1 which is 2. Let s compute it with the gradient formula. (-2, -4) What if we'd chosen two different points on the line? m It doesn't matter which two points we pick, we'll always get a constant ratio of 2 for this line.

37 If we look at any points on this line we see that they all have a y coordinate of 3 and the x coordinate varies. y (-4, 3) (-1, 3) (2, 3) x Let's choose the points (-4, 3) and (2, 3) and compute the gradient. m This makes sense because as you go from left to right on the line, you are not rising or falling (so zero gradient) The equation of this line is y = 3 since y is 3 everywhere along the line. In general, the equation of a horizontal line is y = b, where b is the y coordinate of any point on the line.

38 Analysis Infinite and zero slopes 0 slopes show no relationship among variables. For instance, consumption is completely unrelated to the divorce rate. The line parallel to the horizontal axis represents this lack of relationship. C o n s u m p t i o n Divorce Rate Slope = 0

39 If we look at any points on this line we see that they all have a x coordinate of - 2 and the y coordinate varies. (-2, 3) y Let's choose the points (-2, 3) and (-2, - 2) and compute the gradient. (-2, 0) x (-2, -2) m undefined(or infinite) Dividing by 0 is undefined so we say the gradient is undefined. You can't go from left to right on the line since there isn't a left and right. The equation of this line is x = - 2 since x is - 2 everywhere along the line. In general, the equation of a vertical line is x = a, where a is the x coordinate of any point on the line.

40 Analysis Price Of infinite slope Bananas Quantity of computers THE GRAPH OF THIS RELATIONSHIP IS A LINE PARALLEL TO THE VERTICAL AXIS INDICATING THAT THE SAME QUANTITY OF COMPUTER IS PURCHASED NO MATTER WHAT THE PRICE OF BANANAS IS. THE SLOPE OF SUCH LINE IS INFINITE. SIMILARLY, CONSUMPTION IS COMPLETELY UNRELATED TO THE DIVORCE RATE.

41 zero gradient (or no gradient) It is easy to remember 0 gradient because the line does not slope at all (it is horizontal) undefined gradient It is easy to remember undefined gradient because you can t move along from left to right (it is vertical)

42 Slopes and Marginal Analysis ECONOMICS IS LARGELY CONCERNED WITH CHANGES FROM THE STATUS QUO. THE CONCEPT OF SLOPE IS IMPORTANT IN ECONOMICS BECAUSE IT REFLECTS MARGINAL CHANGES- THOSE INVOLVE 1 MORE (OR LESS) UNIT- BETWEEN TWO VARIABLES.

43 Quick Quiz If THE VARIABLES ARE PRICE AND QUANTITY DEMANDED AND THE SLOPE IS -5, WHAT IS THE MEANING OF THIS SLOPE?

44 The vertical Intercept Intercept: The value of Y when X = zero. The intercept of a line is the point where the line meets the vertical axis.