1.2.3.2 Mathematical approach to equilibrium analysis

Equilibrium analysis in economics can be approached mathematically using the concepts of demand and supply functions. Demand and supply functions are mathematical representations of the relationship between the price of a good and the quantity demanded and supplied, respectively. Equilibrium occurs when the quantity demanded equals the quantity supplied, which can be determined by solving the equations representing the demand and supply functions simultaneously. Let’s explore the mathematical approach to equilibrium analysis:

1. Demand Function: The demand function represents the relationship between the price of a good (P) and the quantity demanded (Qd) by consumers. It is typically written as:

Qd = f(P)

In this function, Qd is the quantity demanded, and f(P) represents the mathematical relationship between price and quantity demanded. The function f(P) can take different mathematical forms, such as linear, quadratic, or exponential, depending on the characteristics of the demand relationship.

2. Supply Function: The supply function represents the relationship between the price of a good (P) and the quantity supplied (Qs) by producers. It is typically written as:

Qs = g(P)

In this function, Qs is the quantity supplied, and g(P) represents the mathematical relationship between price and quantity supplied. Similar to the demand function, the function g(P) can take different mathematical forms, depending on the characteristics of the supply relationship.

3. Equilibrium Condition: Equilibrium occurs when the quantity demanded (Qd) equals the quantity supplied (Qs), which can be mathematically represented as:

Qd = Qs

To find the equilibrium price (Pe) and equilibrium quantity (Qe), we set the demand and supply functions equal to each other:

f(Pe) = g(Pe)

Solving this equation for Pe will yield the equilibrium price, and substituting the equilibrium price back into either the demand or supply function will give us the equilibrium quantity.

4. Graphical Approach: The equilibrium price and quantity can also be determined graphically by plotting the demand and supply curves on the same graph. The point where the demand and supply curves intersect is the equilibrium point, representing the equilibrium price and quantity.