Equations and Identities

Equation

An equation is a statement that shows two quantities are equal. It uses the equal sign === to indicate equality between the left-hand side (LHS) and right-hand side (RHS).

Form:

LHS = RHS  

Example:

5x = 15 

This means 5 × x is exactly equal to 15. 

An equation is true only for certain values of the variable(s). In the example above, it is true only when x = 3.

Identity

An identity is a special type of equation that is true for all possible values of the variable(s). It expresses a universal relationship between quantities.

Form:

LHS ≡ RHS

(The symbol ≡ means “identically equal to.”)

Example:

No matter what a and b are, this equality always holds.

Key Difference between Equation and Identity:

○ Equation: True for some values.

○ Identity: True for all values.

Types of Equations (Common in Economics & Mathematics)

1. Definitional Equation

It states a definition in mathematical form. Both sides have exactly the same meaning — one is just a different way of expressing the other.

Example:

Total Profit (π) is defined as Total Revenue (R) minus Total Cost (C):

π ≡ R−C

Application in Economics:

Definitional equations are used to state fundamental relationships like GDP definitions, elasticity formulas, and accounting identities.

Key Points:

○ Cannot be “tested” with data — it’s true by how the terms are defined.

○ Represented using ≡.

2. Behavioral Equation

• It shows how a variable behaves (changes) in response to changes in another variable.

• It is based on observations or theory about behavior. It can be about human behavior (consumption patterns, spending habits) or non-human systems (cost-output relationships).

Example:

C = 75 + 10Q

Where:

C = total cost

Q = quantity of output

This means:

○ Fixed cost = 75

○ Variable cost per unit = 10

Example (Consumption Function):

C = 100 + 0.8Y

Where C is consumption and Y is income.

Application:

Used in demand functions, production functions, cost functions, and investment equations.

Key Points:

○ Can be estimated from data.

○ Reflects real-world behavior, not just definitions.

○ Form may be linear or nonlinear.

3. Conditional Equation

• It expresses a condition that must be satisfied for a specific state to occur.

• It does not always hold — true only when the stated condition is met. It is often used in equilibrium analysis in economics.

Example (Market Equilibrium):

Qd = Qs 

This holds only at equilibrium price.

Example (Macroeconomics):

S = I  (Savings equal investment in equilibrium)

• Application in Economics:

Used in market equilibrium, balance of payments equality, and monetary equilibrium.

Key Points:

○ States a requirement rather than a universal truth.

○ Often combined with behavioral equations to solve for unknowns.

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Key Takeaways

1. Equations show equality for specific values; identities show equality for all values.

2. Definitional equations are true by meaning and use ≡.

3. Behavioral equations describe cause-effect or reaction patterns in variables.

4. Conditional equations express equilibrium or balance conditions that must be met.

5. In economics, combining behavioral and conditional equations often allows us to solve for unknowns