ECON 387
Introduction to Mathematical Economics I
Lecture 1
Economic Models and
Review of Fundamentals
Lucia Vojtassak
Department of Economics
The Concept of Economic Model
• An abstraction from a real world
• Rough representation of the actual economy
- Focus attention to the primary factors and relevant relations
5 elements
1. Economics Agents
- Consumers, workers, firms, governments.
- Agents are characterized by their ability to make decisions and to pursue
goals
2. An Economic Environment
- Agents do not pursue goals in vacuum
- E.g. consumers: no control over the prices
ß exogenous vs endogenous variables
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3. Choices
ß how to best pursue the goal in the face of economic
environment
- Utility maximization model: choices are explicit
- Demand and supply model: the result of choices
4. Equilibrium Analysis
- Model is in equilibrium when there is no tendency for
economic variables to change unless there are the changes
in economic environment.
àequilibrium values: endogenous
5. Comparative Statics
- Analysis of how changes in the environment affect equilibrium
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Ingredients of a Mathematical Model
Variables
- Something whose magnitude can change
o Price, profit, revenue, costs, national income, consumption,
investment, …
à must be represented by symbol instead of specific number
Endogenous vs Exogenous variables
- Endogenous: solution values come from the model
o E.g. market clearing price of wheat, …
- Exogenous: determined by forces external to the model
o E.g. market prices consumer faces in consumer theory
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Constants
- A magnitude that does not change: e.g. 7 or 0.5
- Frequently joined to a variable:
o 7P or 0.5R
o called coefficient of that variable
• numerical: 7P or 0.5R
• symbolic: aP or bR
- in order to attain higher level of generality
à it is a CONSTANT that is VARIABLE
- parametric constant or parameter
- treated as given, as exogenous variable
Notation: symbols a, b, c, … or Greek alphabet α, β, ξ,…
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Equations and Identities
- variables are interesting when they are related to one
another
3 types of equations
i. a definitional equation
- sets up identity between two alternate expressions that have
the same meanings
ii. a behavioural equation
- specifies the manner in which a variable behaves in response
to changes in other variables
iii. a conditional equation
- states a requirement that has to be satisfied
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Concept of Set
• A collection of distinct objects = elements of set
S = {2, 4, 6, 8, 10}
S = {x| x is an even number between 1 and 11}
Consider now three further sets of numbers
Z+ = {x | x is a positive integer}
A = {x ∈ Z+ | x ≤ 11} = {1,2,3,4,5,6,7,8,9,10,11}
B = {x ∈ A | x/2 ∈ Z+} = {2, 4, 6, 8, 10}
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Subsets
XÍY
- If all elements of the set X are also elements of set Y, then X
is subset of Y.
Proper Subsets
XÌY
- If all the elements in a set X are in a set Y, but not all the
elements of Y are in X, then X is a proper subset of Y
Equality of sets
X=Y
- Two sets are equal if they contain exactly the same elements
- Note that the equality of two sets means that they are
identical,
X ⊆ Y and Y ⊆ X
⇔
X=Y
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Operations on Sets
Consider
• universal set, U
• subsets like A and B or X and Y
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i.
The INTERSECTION W of two sets X and Y is the set of
elements that are in both X and Y :
W = X ∩ Y = {x | x ∈ X and x ∈ Y}
ii.
The EMPTY set or the NULL set is the set with no elements.
The empty set is always written ∅.
iii. The UNION of two sets A and B is the set of elements in one
or other of the sets
C = A ∪ B = {x | x ∈ A or B}
iv. The COMPLEMENT of a set X is the set of elements of the
universal set U that are not elements of X, and it is written
X(bar).
X = {x ∈U | x ∉ X}
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