PhD Macroeconomics question

Basic Math Revision[footnoteRef:1] [1: ]

We must revisit First Order and Second Order Derivative concept in this course. Of the several formulas in calculus, we need at least few formulas and concepts in this course.

The derivative of power functions

In practice, we do not want to calculate derivatives repeatedly using the  definition, but we use calculation rules to calculate the derivative directly. We first look at the derivative of a power function.

………………………………………….……………………………….1

Example: y= x5

= 5x4

If we have a power function with a constant in front of it, we can easily factor out the constant.

……………………………………….…………………………………2

Example: y = 2x-3

= -6x-4

The derivative of any constant (which is just a way of saying any number), is zero.

In mathematical notation, we express the derivative of a constant function as:

f'(x) = 0

Example: y = 3x+20 ……………………………………………….………3

(here 20 is constant in the given equation)

x1-1 + 0

= 3x0 ( we know, X0 = 1 )

= 3

Equation 1 through 3, what we have done is First Order Derivative. Sometimes also called as first order condition (f.o.c.) and denoted as or (x)

Let’s do one more complex example including all rules.

Y= 8X3-9X2-7X+100………………………………………………………….4

F.O.C:

8.3.X3-1 – 9.2. X2-1-7X1-1 + 0

= 24 X2-18X-7………………………………………………………………….5

Second Order derivative is denoted as or (x) or s.o.c. (second order condition).

Example: if you are asked to do s.o.c. of equation 4, then you just need to complete the differentiation twice. Once you have done and got equation 5 i.e 24 X2-18X-7

Now, applying s.o.c.

or (x) = 24. 2 X2-1-18 X1-1 -0

= 48X-18

This is the second order derivative of equation 4. This much math is required to understand the profit maximization, cost minimization, derivation of marginal functions such as Marginal revenue, Marginal Cost. These concepts are crucial in weeks 3 and 4.

Application tips:

1. f.o.c. of Total Production function is Marginal Product (MP) function

2. f.o.c. of total revenue (TR) function is Marginal Revenue (MR) function

3. f.o.c. of Total Cost function is Marginal Cost (MC) function

4. Profit maximization rule for any economic agent is:

MR = MC

5. f.o.c. of Demand function is used to estimate Price Elasticity of Demand

6. f.o.c. of Supply function is used to estimate Price Elasticity of Supply

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