ECO607 Handouts by VUBuddy

Lesson 01 INTRODUCTION TO DYNAMIC ANALYSIS

TOPIC 001: NEED AND IMPORTANCE OF DYNAMIC ANALYSIS IN ECONOMICS

Pre-requisites: Basic Mathematical Economics Types of Economic Analyses: Static, Comparative-Static & Dynamic Static: ساکن, Not moving (over time) Comparative-Static: مقابل ًتا ساکن, Moving (due to exogenous variables) Dynamic: متحرک, Moving (over time)

COMPARISON · Grounds · Static/Comparative-Static · Analysis · Dynamic Analysis · 1 Time · dimension

Time dimension is absent as all economic variables refer to the same point in time Static economy/Timeless economy. Time element has pivotal role. Variables can occur in different points of time.

2 Equilibrium

Studies only a particular point of equilibrium. Studies the process by which equilibrium is achieved. Therefore, it deals with both, equilibrium and disequilibrium.

3 Transition

Doesn’t show the path of change – only the condition(s) of equilibrium, like a ‘still picture’ of the market. Shows the path of changes – like a ‘movie’ of the market.

4 Nature · of · Model

‘Static model’ evaluates the endogenous variables that satisfy some specified equilibrium condition(s) ‘Dynamic model’ delineates the time path of some variable, on the basis of the rate of change. and

5 Real-life

Situation

Doesn’t fully represent real-life situations. Based on the unrealistic assumptions of perfect competition, perfect knowledge, etc. Important economic variables like fashions, population, models of production, etc. are assumed to be constant. Closer to reality. Considers that all economic variables as changeable.

6 Complexity

Easy but oversimplified Complex but realistic

Rationale for Dynamic Analysis: Time is ‘unstoppable’ and affect variables of all disciplines including those of economics. Better to conduct analysis over time rather than one point in time only. Examples of Dynamic Analysis in Economics:

1. Decline in global GDP due to Covid-19 during first half of 2020.
2. Rise in unemployment in Pakistan due to Covid-19 during first half of 2020.
3. Decline of fuel (Petrol) prices due to Covid-19 during first half of 2020.
4. Change of share value over time.
5. Daily activity in Pakistan Stock Exchange (PSX).

TOPIC 002: DYNAMIC ANALYSIS: EXAMPLE FROM POPULATION

Say, population size 𝐻 changes over time at the rate:

𝟐 ?

Here 𝑯(𝒕) can be called time path as ‘population’ depends on ‘time’.

𝟐, then its derivative is = 𝟐( 𝟐) 𝒕−𝟏 𝟐 ⇒ 𝟐.

Interpretation: If rate of change of population over time is 𝟐, then the relationship (time

𝟐. However, same

𝟐 ± 𝟏𝟓

then

𝟐 ± 𝟗𝟗

again

𝟐 + 𝒄∗ again

𝒄 is used as general notation for all possible constant values. It’s better to call 𝒄 as the ‘arbitrary constant’. As arbitrary is something which is based on ‘random choice’, rather than any reason. 𝒄 is arbitrary as it can assume any value at random.

𝟐 ± 𝒄 is time path of population, however, without determination of 𝑐 the time path will remain ‘general’ in nature and not ‘definite’. Therefore, we try to determine value of 𝒄, which is possible. We need the starting point of the function for it – Initial condition. Initial condition: Value of the dependent variable when independent variable is zero. Here dependent variable is 𝑯 and independent variable 𝒕.

Now for the initial condition, general time path 𝑯(𝒕) becomes:

𝟐 + 𝒄

𝟐 + 𝑯(𝟎)

𝟐𝒕 𝑯(𝟎)

Arbitrary constant determined is not arbitrary any more 𝒄= 𝑯(𝟎).

Lesson 02 INTEGRATION AND RULES

TOPIC 003: INTRODUCTION TO INTEGRAL CALCULUS: INDEFINITE INTEGRALS

RATIONALE FOR INTEGRATION:

History Perspective of Integration

Calculus is divided into differential and integral calculus. In this discussion, we are concerned with integral calculus. Economists often face the mathematical problem of finding a function from its derivative. This process of reconstructing a function from its derivative can be regarded as the ‘inverse’ of differentiation. Mathematicians call this process integration. There are simple formulas that have been known since ancient times for calculating the area of any triangle and polygon. Over 4000 years ago, however, the Babylonians were concerned with accurately measuring the area of plane surfaces, like circles, that are not bounded by straight lines. Solving this kind of area problem is related to integration. This topic will also include some important applications of integrals that economists are expected to know.

Mathematical Formulation

However, in Integration the shoe is on the other foot – From derived function to back to primitive

Function to be integrated is known as ‘integrand’. Differentiation and Integration are opposite of each other: 𝒅

𝒄 is arbitrary constant of integration. Multiple parentage of the integrand – multiple possibilities of 𝒄.

TOPIC 004: BASIC RULES OF INTEGRATION: CONSTANT FUNCTION RULE & CONSTANT MULTIPLE RULE

1. Constant Function Rule: It is known that a constant function can be written as:

Integrating w.r.t. 𝒚 and 𝒙 on L.H.S and R.H.S, respectively:

𝒄 is the constant of integration.