Business Calculus: Linear Functions, Supply & Demand, Marginal Analysis, and Regression

Linear Functions in Business Calculus

Definition and Properties of Functions

A function is a rule that assigns to each element x in a set D (the domain) exactly one element y in R (the set of real numbers). The set of all possible output values is called the range of the function.

• Function notation: $y = f(x)$

• Domain: The set D where the function is defined.

• Range: The set $\{y = f(x) | x \in D\}$

Linear Functions

A linear function is a function of the form:

$y = f(x) = mx + b$

• Slope (m): Measures the rate of change of the function.

• Y-intercept (b): The value of the function when $x = 0$.

• Graph: The graph of a linear function is a straight line.

• Domain: For most business applications, the domain is all real numbers $\mathbb{R}$.

Example: Graph the function $f(x) = -2x + 4$.

• Slope $m = -2$, y-intercept $b = 4$.

• Points: $(0,4)$ and $(1,2)$ are on the graph.

Supply and Demand Functions

Basic Economic Definitions

• Supply: The quantity of a good that producers are willing to provide.

• Demand: The quantity of a good that consumers are willing to buy.

Laws of Supply and Demand

• Law of Demand: If the price of an item increases, the demand decreases.

• Law of Supply: If the price of an item increases, the supply increases.

Supply and Demand Functions

• Let p be the price and q the quantity.

• Supply function: $p = S(q)$

• Demand function: $p = D(q)$

• Note: In practice, quantity often depends on price, i.e., $q = S^{-1}(p)$ and $q = D^{-1}(p)$.

Equilibrium

Equilibrium occurs when supply equals demand. The equilibrium price $p_0$ and equilibrium quantity $q_0$ are found where the supply and demand curves intersect.

• Equilibrium condition: $S(q_0) = D(q_0)$

Example: For sugar, let:

• $p = S(q) = 1.4q - 0.6$

• $p = D(q) = -2q + 3.2$

Set $S(q) = D(q)$:

$1.4q - 0.6 = -2q + 3.2$

$3.4q = 3.8$

$q_0 = \frac{3.8}{3.4} \approx 1.118$ (thousand pounds)

Equilibrium price:

$p_0 = S(q_0) = 1.4 \times 1.118 - 0.6 \approx 0.96$ (per pound)

Graphical Representation

The intersection point $(q_0, p_0)$ is where the supply and demand curves meet. This can be visualized on a graph with price on the vertical axis and quantity on the horizontal axis.

Marginal Cost and Linear Cost Functions

Marginal Cost

The marginal cost is the rate of change of the cost function at a given production level. For a linear cost function $C(x) = mx + b$:

• Slope (m): Marginal cost (cost of producing one additional unit)

• Intercept (b): Fixed cost (cost when production is zero)

Example: Cost to produce 100 cups is $11.02, 400 cups is $40.12.

• Find $m$:

$m = \frac{40.12 - 11.02}{400 - 100} = \frac{29.1}{300} = 0.097$

• Find $b$ using $C(100) = 11.02$:

$11.02 = 0.097 \times 100 + b \implies b = 11.02 - 9.7 = 1.32$

• Cost function: $C(x) = 0.097x + 1.32$

• Marginal cost: $m = 0.097$

• Fixed cost: $b = 1.32$

• Cost to produce 1000 cups: $C(1000) = 0.097 \times 1000 + 1.32 = 97 + 1.32 = 98.32$

Revenue, Profit, and Break-Even Analysis

Revenue and Profit Functions

• Revenue: $R(x) = p x$ (where $p$ is price per unit, $x$ is units sold)

• Profit: $P(x) = R(x) - C(x)$

Break-Even Point

• The break-even quantity is where $R(x) = C(x)$ (profit is zero).

• The break-even point is $(x_0, R(x_0))$ or $(x_0, C(x_0))$.

Example: Fixed cost $= 400$, break-even quantity $= 80$.

• Let $C(x) = mx + 400$

• Let $R(x) = px$

• At break-even: $px = mx + 400 \implies p - m = \frac{400}{80} = 5$

• Marginal profit (slope of profit function): $p - m = 5$

Least-Squares Regression and Correlation

Least-Squares Line (Regression Line)

The least-squares line is the line $y = mx + b$ that minimizes the sum of squared vertical distances from the data points $(x_i, y_i)$ to the line.

• Slope: $m = \frac{n \sum x_i y_i - (\sum x_i)(\sum y_i)}{n \sum x_i^2 - (\sum x_i)^2}$

• Intercept: $b = \frac{\sum y_i - m \sum x_i}{n}$

Correlation Coefficient

The correlation coefficient $r$ measures the strength of the linear relationship between two variables:

$r = \frac{n \sum x_i y_i - (\sum x_i)(\sum y_i)}{\sqrt{[n \sum x_i^2 - (\sum x_i)^2][n \sum y_i^2 - (\sum y_i)^2]}}$

• $-1 \leq r \leq 1$

• If $r$ is close to $1$or$-1$, the fit is strong; if$r = 0$, there is no linear correlation.

Example: Consumer Credit Data

The following table shows U.S. consumer credit (in billions of dollars) for selected years:

Year (2000+x)	Consumer Credit (y)
2013	3078.7
2014	3287.7
2015	3527.0
2016	3623.3
2018	3979.2

• Regression line: $y = 116.84x + 812.28$

• Growth rate per year: $116.84$ billion

• Predicted amount in 2030: $y = 116.84 \times 30 + 812.28 = 6111.5$ billion

• Year when debt exceeds $7000$billion:$x = \frac{7000 - 812.28}{116.84} \approx 53$ (year 2053)

• Correlation coefficient: $r = 0.9952$ (strong positive correlation)

Additional info: MATLAB and other computational tools can be used to calculate regression coefficients and correlation for large datasets.