Mathematical Modeling of Population and Market Economics

Linear Modeling of Population Projections

• Case Study: U.S. Female Population (Age 18) Projections     - This model utilizes projections from the year $2020$ to the year $2060$.     - The specific demographic tracked is the number of females in the United States who are $18$ years of age.     - The population size is measured in millions.     - Mathematical Model: The population is modeled by the linear function: $n = 0.15x + 37.4$     - In this formula, $n$ represents the number of females (aged $18$) in millions.     - The variable $x$ represents the number of years that have elapsed since the base year of $2020$.

• Dissecting the Linear Function Components ($n(x) = mx + b$)     - Slope ($m$): The slope of the function is $0.15$.         - Interpretation of Magnitude: This value represents the rate of change in the population over time.         - Calculation: In the context of millions, $0.15 \times 1,000,000 = 150,000$.         - Practical Meaning: The slope indicates an increase of $150,000$ females per year.         - General Ratio: This is the change in the number of females ($n$) compared to the change in time ($x$).     - Intercept ($b$): The $y$-intercept (or $n$-intercept) is $37.4$.         - Calculation: In absolute numbers, this is $37,400,000$.         - Practical Meaning: This value represents the starting population count at the beginning of the model in the year $2020$.

• Predictive Modeling for the Year 2040     - To predict the population for the year $2040$, we must determine the value of $x$.     - Calculation: $2040 - 2020 = 20$ years.     - We evaluate the function at $x = 20$: $n(20) = 0.15 \times 20 + 37.4$     - Intermediate step: $0.15 \times 20 = 3$     - Result: $n(20) = 3 + 37.4 = 40.4$     - Conclusion: It is predicted that in the year $2040$, there will be $40,400,000$ females aged $18$ in the United States.

Fundamental Concepts of Supply and Demand in Economics

• The Economic Model Variables     - Price ($p$): Plotted on the vertical axis ($y$-axis).     - Quantity ($q$): Plotted on the horizontal axis ($x$-axis), representing the number of units (e.g., pairs of shoes) being sold.

• The Demand Function     - Definition: Represents the number of items that consumers are willing to purchase or "demand."     - Graph Behavior: The demand function has a negative slope.     - Economic Principle: Higher prices lead to lower demand. For instance, very expensive shoes will have fewer buyers than more affordable ones.

• The Supply Function     - Definition: Represents the number of items that manufacturers are willing to produce and put onto the market.     - Graph Behavior: The supply function has a positive slope.     - Economic Principle: Higher prices act as an incentive for manufacturers. The more they can sell an item for, the more units they are willing to make.

• Market Disequilibrium: Shortages and Surpluses     - Shortage (Shortfall): Occurs in the region of the graph where the demand ($D$) is greater than the supply ($S$). This happens when the current price is below the equilibrium point, leading to too few shoes for the number of people who want to buy them.     - Surplus: Occurs in the region of the graph where the supply ($S$) is greater than the demand ($D$). There are more products on the market than buyers are willing to purchase. This indicates that the current price is likely too high for the general consumer base.

• Market Equilibrium     - Definition: The point where the demand function and the supply function intersect.     - Characteristics: At this point, the supply and demand are equal. This intersection identifies the equilibrium price and the equilibrium quantity.

Analyzing Supply and Demand Functions: A Numerical Example

• Given Equations for Shoe Market     - Demand Equation: $2p + 5q = 200$         - Solved for price: $p = -2.5q + 100$     - Supply Equation: $p - 2q = 10$         - Solved for price: $p = 2q + 10$     - In these equations, $p$ is the price and $q$ is the number of pairs of shoes.

• Scenario Analysis at Price ($p$) = 60     - Step 1: Find Quantity Demanded         - Equation: $60 = -2.5q + 100$         - Subtract $100$ from both sides: $-40 = -2.5q$         - Divide by $-2.5$: $q = 16$         - Result: At a price of $60$, $16$ pairs of shoes are demanded.     - Step 2: Find Quantity Supplied         - Equation: $60 = 2q + 10$         - Subtract $10$ from both sides: $50 = 2q$         - Divide by $2$: $q = 25$         - Result: At a price of $60$, $25$ pairs of shoes are supplied.     - Step 3: Determine Market State         - Since the supply ($25$) is greater than the demand ($16$), there is a surplus.         - Analysis: The price of $60$ is determined to be "too high" because the surplus indicates items are not being purchased at that rate.

Determining Market Equilibrium through Systems of Equations

• The Mathematical Process     - Equilibrium occurs when the demand for price and the supply for price are identical.     - Set the two expressions for $p$ equal to each other: $-2.5q + 100 = 2q + 10$

• Step-by-Step Solution for Quantity ($q$)     - Group quantities on one side: add $2.5q$ to both sides: $100 = 4.5q + 10$     - Group numbers on the other side: subtract $10$ from both sides: $90 = 4.5q$     - Solve for $q$: $q = \frac{90}{4.5}$ $q = 20$     - The equilibrium quantity is $20$ pairs of shoes.

• Solving for Equilibrium Price ($p$)     - To find the price, substitute the quantity ($20$) back into either original function to verify consistency.     - Using the Demand Function: $p = -2.5 \times 20 + 100$ $p = -50 + 100 = 50$     - Using the Supply Function: $p = 2 \times 20 + 10$ $p = 40 + 10 = 50$     - Both equations yield the same value, confirming the result.

• Final Equilibrium Values     - Equilibrium Quantity: $20$ pairs.     - Equilibrium Price: $50$.