macro econ theory

MATH REVIEW1

• Changes ( ) ℎ = −

ℎ = = − 1

The percentage change is the relative change expressed in percentages: ℎ = × 100 % = − 1 × 100 % 1 We assume that everything is well defined

Example 1: The price of a stock is $50 at the beginning of 2015 and $60 at the end of 2015. Then, the change in the price of the stock in 2015 is $60−$50 = $10. The percentage change is $$ × 100% = 20%. When we talk about economic growth, we will often use the term growth rates. The growth rate is the relative change over a period of time. Example 2: The annual growth rate of world population is 0.016 (or 1.6%). If the world population is 7.5 billion in the beginning of the year, then at the end of the year, it would equal (1+0.016) × 7.5 billion = 7.62 billion. So, during the year, the population increased by 0.12 billion people (this is the absolute change).

We will use the Greek letter to denote absolute changes. For example, if we denote world population by N, then:

is the absolute change in world population, and

is the relative change in world population.

From Example 2, = 0.12 billion people, while = 0.016.

Exercise 1: An asset is currently worth $1,000. What would be the percentage change in the value of the asset if in one year it is worth:

(a) $1,005; (b) $950; (c) $3,500

Exercise 2: In December, a dealership sold 2,000 cars for $50,000 each resulting in revenues of $100 million. In January, the dealership sold 5 percent less cars, but at a 2 percent higher average price. What is the percentage change in the dealership revenues?

Consider two variables and and a constant .

Note that the relative change in a constant is 0.

So, we have = 0 because is a constant The following properties hold approximately, so we use the symbol “≈” which denotes “is approximately equal to”

( ) ≈ + ( / )/ ≈ − ( ) ≈

Exercise 3: A firm produces a single product according to the following production function: = . , where Y is firm’s output, i.e., the number of goods produced, and L is the number of workers hired. Firm’s total revenue equals PY, where P is the price of the good. Consider a situation where the firm faces a 2% increase in the price of the good and increases the number of workers by 1% (for simplicity, assume that both increases happen at the same time).

(a) What is the approximate percentage change in firm’s output? (b) What is the approximate percentage change in firm’s total revenue?

Exercise 4: The revenue of a firm that produces a single good decreased by 1.5%. What was the approximate percentage change in firm’s output if the price of the good decreased by 0.75%?

Percentage Points and Basis Points

Example 3: The interest rate in Brazil was 12% at the beginning of 2015 and 14% at the end of 2015. Then, the relative change in the interest rate equals ≈ 0.1667, or 16.67%. In what units do we measure the absolute change? When we measured the absolute change in world population in Example 2, we used number of people because population is measured in number of people. Here, however, the interest rate is measured in percent. If we say that the interest rate increased by 2 percent, this would become confusing. In fact, this would mean that the interest rate went from 12% at the beginning of 2015 to (1 + 0.02) × 12% = 1.02 × 12% = 12.24% at the end of 2005, which is not the case. For this reason, we use the term percentage points (pp). In our case, the interest rate increased by 14 − 12 = 2 percentage points. More formally, a percentage point is the unit of measurement for the absolute change in a variable that is itself measured in percentages.

A basis point (bp) is a hundredth of a percentage point. In our example, the interest rate increased by 2 × 100 = 200 basis points. Exercise 5: The Bank of England cuts the bank rate from 0.5% to 0.25%. Find the absolute change in the bank rate in both percentage points and basis points. Find the relative change in the bank rate and express it in percent.

• Working with powers =

=

=

( ) =

( ) =

= 1

Exercise 6: Solve the following equations for x:

(a) x3/2 = 8

(b) 2x1/3 — 18x = 0

(c) x—2/5 = 100

(d) (xy)1/5y7/5 = (2y3/5) / (y—1)

• Plotting linear functions

Let = + , where a and b are constants a is the slope and b is the (y-axis) intercept

To plot a linear function, you need two points.

Take = 0, then you have y= . Your first point is (0, ). Take = 0, then 0 = + , so = − . Your second point is − , 0 The plot is then just a straight line passing through these two points.

Exercise 7: Plot the following linear functions:

(a) = 3 − 6 (b) = −2 + 8

• System of two linear equations with two unknowns: An illustration 2 + 3 − 6 = 03 − 2 + 4 = 0 o Take one of the equations (say the first) and use it to express one of the

unknowns (say y) as a function of the other: = 3 − 1.5 (1) o Then, plug it into the other equation and solve for the other unknown: 3(3 − 1.5 ) − 2 + 4 = 0 −6.5 = −13 = 2 o Finally, plug the result into (1) to solve for the first unknown: = 3 − 1.5(2) = 0

Exercise 8: Solve the following system of equations: 3 + 2 − 3.5 = 02 − 4 + 3 = 0

• Derivatives of functions of one variable (only what you need for this course)

Let f be a function of x.

We denote the (first, or first order) derivative of f with respect to x as ( )

or ′( ) Let m and n be some constants.

If ( ) = , then ′( ) = 0 If ( ) = , then ′( ) =

If ( ) = , then ′( ) = If ( ) = , then ′( ) =

Exercise 9: Find ′( ) if: (a) ( ) = / [Hint: Note that / = / ] (b) ( ) = 4√ [Hint: Note that √ = / ]

• Derivatives of functions of two variables (only what you need for this course)

Let f be a function of x and y

We denote the partial derivative of f with respect to x as ( , ) or . It

is simply the derivative of f with respect to x when we treat y as a constant.

Similarly, we denote the partial derivative of f with respect to y as ( , ) or . It is simply the derivative of f with respect to y when we treat x as a constant.

Exercise 10: Find and if:

(a) ( , ) = 2 / / . (b) ( , ) = 3 / / .

• Derivative and the slope of a function

• Example 4: Consider ( , ) = 3 / / .

 (a) Let y=1 and plot f(x,1) as a function of x. [For example, take x=1, 8, 27 , 64].

• Does f(x,1) increase with x?

• Does the slope of f(x,1) increase with x?

 (b) Let x=1 and plot f(1,y) as a function of y. [For example, take y=1, 8, 27 , 64].

• Does f(1,y) increase with y?

• Does the slope of f(1,y) increase with y?

• Derivatives and the maximum of a function

• A function of one variable

 Suppose that f is a function of x. For a point to be a maximizer of f, the following necessary condition should be satisfied:

First order condition (FOC): ′( ) =

 Note that this condition is not sufficient for to be a maximizer of f. We need an additional second order condition.

• A function of two variables

 Suppose that f is a function of x and y. For a point , , to be a maximizer of f, the following necessary condition should be satisfied:

First order condition (FOC): , , = and , , =

 Note that this condition is not sufficient for , , to be a maximizer of f. We need an additional second order condition.

Using Some Math to Illustrate the Properties of an Important Production Function

• A production function shows how much output can be produced from

a given amount of inputs. It is taken to reflect available technology.

• Assume that there are two factors of production: capital and labor.

• The Cobb-Douglas production function is defined as follows: = ( , ) = ,

where Y is the amount of the output produced from K units of capital and L units of labor, α is a constant between 0 and 1, and is a positive constant referred to as total factor productivity.

• For example, consider a firm that uses capital (servers) and

labor to provide web hosting services. The firm is small and takes the price of its inputs and output as given. In particular, assume that each client is paying $1 for the firm's service, while the firm is renting capital for $r per hour and hiring labor for $w per hour. The firm's technology is described by a Cobb-Douglas production function: = ( , ) = ,

where Y is the number of firm's clients, K is the number of capital hours used and L is the number of labor hours hired.

• Constant returns to scale (CRS)

o How much will the output increase if the firm uses twice as

much capital and twice as much labor? [Hint: Find F(2K,2L) and compare with F(K,L)]

o The Cobb Douglas production function exhibits constant returns to scale: a proportional increase in all inputs increases output by the same proportion

• The profit of the firm as a function of prices, inputs, and output is: − −

• Using the production function, we express the output as a function of capital and labor inputs and plug it into the expression for the firm's profit:

− −

• Write the FOC with respect to capital:

o Marginal revenue from using an additional unit of capital equals its marginal cost

o Marginal product of capital (MPK) equals r

• Write the FOC with respect to labor:

o Marginal revenue from using an additional unit of labor equals its marginal cost

o Marginal product of labor (MPL) equals w

• Cobb-Douglas & perfect competition

o The marginal product of an input is proportional to its average

productivity = = (1 − )  Check this is indeed the case.

o The ratio of capital income to labor income is constant:  Capital income is × = × =  Labor income is × = ( ) × = (1 − )  The ratio of capital income to labor income equals

o How much is the firm's profit?

− × − × = ?

• Note that in this course, we will not focus on a single firm, but will consider the economy as a whole. Therefore, Y will be interpreted as the economy's total output, while K and L — as economy's total capital and labor respectively.