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Summarizes 2.1, 2.4 and 2.5 of Fundamental Methods of Mathematical Economics, AC Chiang.
A mathematical model (e.g. Partial Market Equilibrium Model and General Market Equilibrium
model) will have variables (whose magnitude can change) and constants/parameters (magnitude is
fixed). The variables may be categorized into endogenous variables (variables whose solution we
seek from the model / variables explained by functions in the model) and exogenous variables
(variables determined by factors outside the model and hence are accepted as given data only). In a
‘supply and demand model’ the price (P) of a good is endogenous because it is set by a producer in
response to consumer demand. On the other hand, a model which seeks to explain consumer
behaviour, price would be an exogenous variable, since the consumer takes it as given (as
determined by the market forces).
The main objective of the economic (mathematical) models we have discussed in class so far: The
Partial Market Equilibrium Model and The General Market Equilibrium Model is to obtain the
solution of the endogenous variables in terms of parameters/constants and/or exogenous variables.
Generally, economic models will have, a conditional equation which states a particular condition
which needs to be satisfied e.g. the condition: 𝑄𝑑 = 𝑄𝑠 needs to be satisfied for a market to be in
equilibrium, and behavioural equations (which appear usually as functions); Behavioural equations
tell us how a variable changes in response to another variable e.g. in a Partial Market Equilibrium
Model the function: 𝑄𝑑 = 𝑎 − 𝑏𝑃 (𝑏 > 0) tells us that quantity demanded is inversely (negatively)
related with price (P).
Functions may be thought of as a rule / special relation between two or more variables. The function
𝑦 = 𝑓(𝑥) tells us that there is a special relation between 𝑥 𝑎𝑛𝑑 𝑦 (every value of 𝑥 has ONLY ONE
corresponding value of 𝑦).The function𝑦 = 𝑓(𝑥) also tells us that the value of 𝑦 depends on the
value of 𝑥 and hence 𝑦 is called the dependent variable(or the value of the function)and 𝑥 is called
the independent variable (or the argument of the function). The possible values that the
independent variable 𝑥 might assume/take is called the domain of the function. And the
corresponding values of the dependent variable 𝑦, is called the range of the function. Refer to
Example 5, p. 19, Ch. 2, Fundamental Methods of Mathematical Economics 4th Ed. for a better
understanding.
Functions may be represented on the ‘coordinate plane’ as graphs (lines or curves). Let’s take the
simple function; 𝑦 = 𝑥. The graph for this function (covered in class) may be used as a reference
point for sketching ANY linear function. For example, let’s take another linear function 𝑦 = 3 + 𝑥.
This function may be obtained by shifting the 𝑦 = 𝑥 line ‘up’ by 3 units along the 𝑦 𝑎𝑥𝑖𝑠. Similarly,
𝑦 = 3 − 𝑥 may be obtained by shifting the 𝑦 = 𝑥 𝑙𝑖𝑛𝑒 ‘down’ by 3 units along the 𝑦 − 𝑎𝑥𝑖𝑠. 𝑦 =
−𝑥is the reflection of the 𝑦 = 𝑥 curve on the 𝑥 𝑎𝑥𝑖𝑠, giving us a downward sloping curve. And
hence the graph of 𝑦 = 4 − 𝑥 may be obtained by moving the 𝑦 = −𝑥 line ‘up’ by 4 units along the
𝑦 𝑎𝑥𝑖𝑠. On the same note, we may use 𝑦 = 𝑥 2 or 𝑦 = −𝑥 2 (covered in class) as a reference point
when we want to sketch quadratic functions. For example, the graph of 𝑦 = −3 + 𝑥 2 may be
obtained by shifting the 𝑦 = 𝑥 2 graph ‘down’ by 3 units along the y-axis and 𝑦 = 3 − 𝑥 2 can be
obtained by shifting the 𝑦 = −𝑥 2 curve ‘up’ by 3 units. This knowledge of sketching should be
enough for us to do the sketches in chapter 3…