Linear and Non-linear Equations

Describing and Sketching Economic Functions:
- Supply and demand functions.
- Consumption and savings functions.
- Cost, revenue, and profit functions.
- Understanding linear vs. non-linear functions.
Setting Up and Solving Simple Models:
- Calculate equilibrium price and quantity.
- National income equilibrium.
- Elasticity calculations.
- Identify output levels to maximize profit and total revenue.
Graphing Equations using Excel:
- Model elasticity and national income.
Solving Quadratic Equations:
- Finding roots of equations.
- Using the quadratic formula.
Working with Indices, Exponential, and Logarithmic Functions.

Trade-off Concept:
- Limits on enjoyment can be modeled (e.g., time constraints).
- Example: 10𝑥 + 4𝑦 ≤ 20, where x = crossword clues, y = songs.
Ability to graph these relations for better understanding.

Types of Functions Used in Economics:
1. Linear Function: y = f(x) = 3x + 4
2. Power Function: y = f(x) = 5x
3. Exponential Function: y = f(x) = 2^(0.5x)
4. Logarithmic Function: y = f(x) = log(x)

Linear Dependence: Constant proportional change in y for change in x.
Turning Points: Important for maximizing or minimizing functions.
Exponential Growth: y increases faster than x, relevant for growth models.
Diminishing Returns: Growth rate of y decreases as x increases, applicable in production and utility models.

Roots: Values of x where y = 0.
Linear Equations: The formula y = ax + b can be manipulated to find x when y = 0.

Roots in Quadratic Equations: Quadratics typically have two roots, can be solved using:
- Quadratic formula: x = (-b ± √(b² - 4ac))/(2a).
Example solutions can be calculated to understand the behavior of these equations.

Purpose: Explain behavior and predict outcomes based on changes in the economic environment.
- Market Demand and Supply Dynamics:
  - Demand increases as price decreases, while supply increases as price rises.
- Equilibrium: Market is in equilibrium when demand equals supply.
- Structural vs. Reduced Forms: Structural equations form the base from which reduced forms are derived, explaining endogenous variables as functions of exogenous variables.

Understanding Tax Impact:
- Example of a tax increase on alcohol units affecting pricing in markets.

Approach to Economic Problems:
1. Statement of the problem.
2. Assess available and needed information.
3. Explore possible solutions.
4. Verify the mathematical and economic sense of the solution.

Components of Aggregate Expenditure:
- C (Consumer spending), I (Investment spending), G (Government spending), X (Exports), M (Imports).
- Aggregate Expenditure Formula: E = C + I + G + X - M.

Understanding Multipliers:
- Illustrates the effects of aggregate demand changes on national income.
- Equilibrium condition: Y = E (national income = aggregate demand).

Price Elasticity of Demand:
- Defined as the percentage change in demand relative to the percentage change in price.
- E = % change in demand / % change in price; where E < 1 is inelastic, E > 1 is elastic.
- E = change in Q/ change in P
- E = -(P/Q) x (Change in Q/change in P)
Income Elasticity of Demand:
- Measure how demand changes with consumer income changes, indicating necessity (E < 1) vs. luxury (E > 1).
- D = (a0) x (P^a)
- change in Q/Q = change in Log Q (same for P)
- W = change Log Q/ change Log P
- in constant elasticity demand curve, elasticity is always equal to exponent on the price