Chapter 2: Basic Mathematical Tools

Question 1

Express the following in summation notation: x1+x2+x3+x4+x5+x6+x7+x8

Answer 1

∑(i=1)^8 xi

Question 2

Let Q^s= -3+1.5P , where Q^s is the quantity supplied of a good and P is the market price:
a) State the interpretation of the slope in economic terms

Answer 2

The slope is the change in the quantity supplied per unit change in market price. The slope here is 1.5, which represents a 1.5 unit increase in the quantity supplied of good due to a one unit increase in market price.

Question 3

Let Q^s= -3+1.5P , where Q^s is the quantity supplied of a good and P is the market price:
b) Calculate the elasticity at P= 10 and at P=50, and state their interpretations.

Answer 3

Elasticity = % change Q^s / % change P = dQ^s / dP * P / Q^s = slope x P / Q^s

When P=10 then Q^s = 12
Elasticity = slope * P/Q^s = 1.5 * 10/12 = 1.25

The elasticity shows the percentage change in Qs associated with a 1 percent change in P. At the point P=10 and Qs = 12, a 1 percent increase in P is associated with a 1.25 percent increase in Qs.

When P =50 then Q^s = 72
Elasticity = slope * P/Q^s = 1.5 * 50/72 = 1.04

At the point P = 50 and Qs = 72, a 1 percent increase in P is associated with a 1.04 percent increase in Qs.

Question 4

Suppose the following equation describes the relationship between the average number of classes missed during a semester and the distance from school measured in miles: number of classes missed = 3 + 0.2 distance

a) Sketch this line, being sure to label the axes. How do you interpret the intercept in this equation?

Answer 4

This is just a standard linear equation with intercept equal to 3 and slope equal to 0.2.

The intercept is the number of classes missed by a student who lives on campus.

Question 5

Suppose the following equation describes the relationship between the average number of classes missed during a semester and the distance from school measured in miles: number of classes missed = 3 + 0.2 distance

b) What is the average number of classes missed for someone who lives five miles away?

Answer 5

The average number of classes missed for someone who lives five miles away is 3 + 0.2(5) = 4 classes. This can also easily be seen in the graph above.

Question 6

Suppose the following equation describes the relationship between the average number of classes missed during a semester and the distance from school measured in miles: number of classes missed = 3 + 0.2 distance

What is the difference in the average number of classes missed for someone who lives 10 miles away and someone who lives 20 miles away?

Answer 6

The difference in the average number of classes missed for someone who lives 10 miles away and someone who lives 20 miles away is (20-10)0.2 = 10(0.2) = 2 classes.

Again, this can also easily be seen in a graph, the number of classes missed at 10 miles is 5 and number of classes missed at 20 miles is 7, thus the difference is 7-5=2.

Alternatively, calculate number of classes missed separately for two distances and find the difference as follows:
For distance = 10 miles, classes missed = 3 + 0.2 (10) = 5 classes

For distance = 20 miles, classes missed = 3 + 0.2 (20)= 7 classes.

The difference is = 7 – 5 = 2 classes.

Question 7

What is Property Sum 1:

∑(i=1)^n c =

Answer 7

∑(i=1)^n c = (c + c +c … +c) = nc

e.g.
∑(i=1)^4 6 = (6 + 6 + 6 +6) = 4 * 6

Question 8

What is Property Sum 2:

∑(i=1)^n cxi =

Answer 8

∑(i=1)^n cxi = (cx1 + cx2 + … + cxn) = c∑(i=1)^n xi

e.g.
∑(i=1)^5 4xi = (4x1 + 4x2 + 4x3 + 4x4 +4x5) = 4(x1 + x2 + x3 + x4 + x5) = 4∑(i=1)^5 xi

Question 9

What is Property Sum 3:

∑(i=1)^n (ax1 + by1) =

Answer 9

∑(i=1)^n (ax1 + by1) = a∑(i=1)^n x1 + b∑(i=1)^n y1

e.g.
∑(i=1)^n (3x1 + 4y1) = 3∑(i=1)^n x1 + 4∑(i=1)^n y1

Question 10

What is Property Sum 4:

∑(i=1)^n x1 ∑(i=1)^n y1=

Answer 10

∑(i=1)^n x1 ∑(i=1)^n y1= (x1 + x2 + … + xn)(y1 + y2 + … + yn)

Question 11

What are the properties of linear functions?

y = B0 + B1x

Answer 11

y is a linear function of x

B0 and B1 are two parameters describing this relationship

The intercept is B0

The slope is B1

The defining feature of a linear function is that the change in y is always B1 times the change in x: Δy = B1Δx
- In oher words, the marginal effect of x on y is constant and equal to B1

Question 12

What is the relative change/proportionate change?

Answer 12

The proportionate change in x moving from x0 to x1, sometimes called the relative change, is simply:

(x1 - x0)/x0 = Δx/x0

Question 13

What is the percentage change?

Answer 13

The percentage change in x in going from x0 to x1 is simply 100 times the proportionate change:

%Δx = 100(Δx/x0)

Question 14

What are derivatives used for?

Answer 14

It helps to recall the derivatives of a handful of functions because we use the derivatives to define the scope of a function at a given point.

We can then use the derive function to find the approximate change in y for small changes in x.

Δy = df/dx * Δx

Question 15

What is the derivative of a constant?

Answer 15

The derivative of a constant is 0.

e.g. if y = 5 then dy/dx = 0

Question 16

What is the derivative of a linear function?

Answer 16

In the linear case, the derivative is simply the slope of the line:

y = β0
= β1x, then dy/dx = β1

Question 17

What is the derivative of a quadratic function?

Answer 17

If y = x^c then dy/dx = cx^c-1

Question 18

What is the derivative of a sum of two functions?

Answer 18

What is the derivative of a sum of two functions is the sum of the derivatives:

d[f(x) + g(x)]/dx = df(x)/dx +dg(x)/dx

Question 19

What are partial derivatives used for?

Answer 19

When y is a function of multiple variables (AKA non linear models), the notion of a partial derivative becomes important.

Suppose that y = f(x1, x2)

Then there are two partial derivatives, one with respect to x1 and one with respect to x2. The partial derivative of y with respect to x1, denoted here by ∂y/∂x1 is just the usual derivative of the above function with respect to x1, where x2 is treated as a constant.

Similarly, ∂y/∂x2 is just the derivative of the function with respect to x2, holding x1 fixed.

Question 20

What is a way to capture diminishing marginal returns

Answer 20

One simple way to capture diminishing marginal returns is to add a quadratic term to a linear relationship.

Question 21

Correlation vs Causation

Answer 21

Most of the time economist are interested in the causal effect of one variable on another
▶ Correlation = ̸ = Causation
▶ Correlation: X and Y are related to each other.
▶ Causation: X caused Y.

Question 22

What are the different types of data?

Answer 22

Experimental:
▶ Data created by a researcher with a treatment and control group.
▶ e.g. What is the effect of drug on diseases?
▶ Non-Experimental, Observational,

Retrospective data:
▶ Focus of this class and econometric methods in general.
▶ Usually what we work on in economics.
▶ e.g. Data on salaries, occupations, level of educations to study gender pay gap

Question 23

Stucture of data

Answer 23

Cross Sectional
▶ N individuals observed 1 time

Time series
▶ 1 individual observed T times across time

Panel data
▶ N individuals observed T times across time

Question 24

How do you calculate the mean?

Answer 24

x̄ = 1/n ∑(i=1)^n xi

▶ Group A: 5 students with grades:
▶ 50, 50, 50, 50, 50
▶ x̄ = (50+50+50+50+50)/5 = 50

Group B: 5 students with grades:
▶ 50, 75, 25, 60, 40
▶ x̄ = (50+75+25+60+40)/5 = 50

Question 25

How do you calculate variance?

Answer 25

Sample variance: V(x) = 1/n ∑(i=1)^n (x1 - x̄)^2

Group A: 5 students with grades:
▶ 50, 50, 50, 50, 50
▶ Var (X) = [(50−50)^2+(50−50)^2 +(50−50)^2 +(50−50)^2+(50−50)^2]/5 = 0

Group B: 5 students with grades:
▶ 50, 75, 25, 60, 40
▶ Var (X) = [(50−50)^2+(75−50)^2 +(25−50)^2 +(60−50)^2+(40−50)^2]/5 = 290

Question 26

How do you calculate standard deviation?

Answer 26

Standard deviation is the square root of the variance.

▶ Std(X) = √Var (X)

Question 27

How do you calculate standard deviation?

Answer 27

Standard deviation is the square root of the variance.

▶ Std(X) = √Var (X)

Question 28

What plays the most important role in econometric analysis?

Answer 28

The natural logarithm.

y = log(x)

Question 29

What are the properties of the log function?

Answer 29

The log function is defined only for x > 0.

When y = log(x) the relationship between y and x displays diminishing marginal returns.

The effect of x on y never becomes negative (unlike a quadratic function), instead the slope of the function gets closer and closer to zero as x gets larger, but the slope never quite reaches zero.

Question 30

What displays diminishing marginal returns?

Answer 30

The log function

Question 31

What is the exponential function?

Answer 31

It is the inverse of the log function.

y = exp(x)