EF

Intro to Economics and Mathematics Vocabulary (Lecture Notes)

Course logistics and materials

  • At the start of every lecture topic, the instructor will hand out course sheets and post them on Canvas as a backup. These sheets include in-class activities, practice problems, and some extra problems not discussed in class.
  • Practice problems are for study and do not need to be turned in; they are meant to supplement learning and preparation.
  • In-class work will be done in groups; practice problems reinforce understanding of concepts.
  • Emphasis on practice as the best way to retain complex analytical problems.
  • The instructor will provide as many materials as needed to study effectively.

Economics: scope and course structure

  • Economics is fundamentally about allocating scarce resources.
    • Scarcity applies to labor (time) and many other resources.
    • College life itself is a lesson in allocating time and effort.
  • Course focus split:
    • Microeconomics: households and firms in the marketplace (decisions about work, purchases, savings, and investment).
    • Macroeconomics: economy-wide forces (inflation, growth, unemployment, etc.), across countries or globally.

Topics economists study (examples and relevance)

  • Minimum wage and labor market outcomes: effects on average wages, employment, hours, and non-labor outcomes.
  • Agricultural economics: shocks in the environment affecting production and local economy.
  • Market concentration and pricing power: how dominant firms can raise prices when competition is weak (e.g., game consoles, smartphones).
  • Monopoly power and innovation: lack of competition can reduce innovation incentives.
  • Empirical econ findings:
    • Airbnb effects: increased housing supply by hosts can reduce nearby property values.
    • Marijuana legalization in CO/WA: associated with reductions in drunk-driving incidents.
    • COVID-era sports with empty arenas: athletes may choke less in the absence of fans.
    • Primary election reform: can influence political extremism (time/resource is the scarce input).
  • Economics as a social science: includes data-driven analysis of social interactions and policy.

Career paths with an economics degree (overview)

  • Corporate finance and financial analytics (analyst, operations, supply chain).
  • Law school preparation (econ as a strong pre-law major, especially for contract/business law).
  • Investment banking (common for econ majors; often paired with finance or management).
  • Economic consulting (varied applications across industries).
  • Politics and policy analysis (think tanks, campaigns; data-driven policy work).
  • Data science and econometrics (coding, modeling, regression analysis; strong data analytics linkage).

The math toolkit for this course (three core areas)

  • Functions and interpretation: mapping input to output, notation y = f(x).
  • Solving systems of equations: when multiple functions share the same input; find where they intersect (f(x) = g(x)).
  • Area between curves: geometry/graphical intuition for regions formed between curves; often involves triangles and simple shapes.

Functions: definition, notation, and interpretation

  • A function maps one variable to another: input x → output y.
  • Notation: y = f(x) where x is the input, y is the output, and f is the process/function.
  • Linear functions (slope-intercept form): y = m x + b
    • m is the slope (rate of change): as x increases by 1 unit, y changes by m units.
    • b is the y-intercept (the base value when x = 0).
  • Interpreting a simple linear example: y = 2x + 1
    • If x = 0, then y = 1 (the intercept). In words: starting value is 1 unit of output.
    • If x increases by 1, y increases by 2 (slope m = 2).
  • Coffee-hours example (to illustrate interpretation): if x is cups of coffee and y is hours worked, then
    • intercept b is the base hours worked with no coffee.
    • slope m indicates how many additional hours are added per extra cup of coffee.
  • Solving a single linear equation (example):
    • If the model is y = 2x + 1 and we want to know how many cups of coffee (x) are needed to work 6 hours (y = 6):
    • Solve: 6 = 2x + 1 \ 2x = 5 \ x = rac{5}{2} = 2.5.
  • Memorization aids: a 3×5 note card for formulas/identities can be helpful during practice if memory is challenging.

Systems of equations: intersection of two functions

  • When two outputs share the same input x (e.g., y = f(x) and z = g(x)), we look for the intersection where the two outputs are equal: f(x) = g(x).
  • Example setup from the lecture:
    • Let y = 20x and z = 150 - 10x.
    • Set them equal: 20x = 150 - 10x.
    • Solve: add 10x to both sides → 30x = 150 → x = 5.
    • Then y = 20x = 20 imes 5 = 100. (illustrative value for the intersection point)
  • Graphical intersection intuition: the intersection point corresponds to where the two curves cross on the graph.

Area between curves (geometric interpretation)

  • When two curves enclose a region, the area can be computed by geometric shapes (often triangles in simple cases).
  • General triangle area formula: A = rac{1}{2} imes ext{base} imes ext{height}.
  • In a typical intersection scenario, the base is the horizontal distance between the x-values of the intersection (i.e., \text{base} = |x2 - x1|), and the height is the vertical distance between the curves (i.e., the difference in y-values over that horizontal span).
  • Example from the lecture (top triangle):
    • If the left and right intersection x-values are 1 and 8, base = |8 - 1| = 7.
    • If the vertical height (difference in y-values) is 7, then area = A = rac{1}{2} imes 7 imes 7 = rac{49}{2} = 24.5.
  • Example from the board for a more complex shaded region:
    • Top triangle: base = 6 (e.g., 8 − 2 or 14 − 8 depending on the setup described), height = 50, so A_{ ext{top}} = rac{1}{2} imes 6 imes 50 = 150.
    • Bottom triangle: base = 10, height = 50, so A_{ ext{bottom}} = rac{1}{2} imes 10 imes 50 = 250.
  • Note: these illustrate how area calculations arise when you have a region bounded by lines/curves between intersection points.

Worked problems and key takeaways (from the lecture)

  • Problem 4 (linear equation solve):
    • Given y = -225 + 15x and y = 240. Solve for x:
    • 240 = -225 + 15x \ 240 + 225 = 15x \ 465 = 15x \ x = rac{465}{15} = 31.
    • Therefore, x = 31.
  • Problem 5 (system equation / inequality setup):
    • Given two expressions in a system, and we want a relationship between them (an inequality scenario was discussed):
    • Start by rearranging the equations to isolate like terms:
    • Example steps observed:
    • Subtract 10 from both sides: 4x = 2x + 12.
    • Subtract 2x from both sides: 2x = 12.
    • Divide by 2: x = 6.
  • Problem 1 (interpretation of a linear function):
    • Given y = f(x) = 10x + 20, interpret:
    • Slope m = 10 (change in y per unit increase in x).
    • Intercept b = 20 (value of y when x = 0).
  • Problem 2 (linear relation with rainfall):
    • Given y = 300 - 8x, where x is days of rain and y is ice cream cones sold.
    • Slope is negative (-8), indicating an inverse relationship: more rain reduces sales.
    • Intercept y = 300 means that with zero days of rain, sales would be 300 cones.
  • Problem 3 (linear growth with zero intercept):
    • Given y = 35x, intercept is 0, so if x = 0, y = 0; sales are zero without any input (e.g., wings eaten proportional to opening day promotion).

Practice and review strategy (soon-to-be quizzes and assignments)

  • First lecture quiz is scheduled for Monday; some practice problems (Problems 4–7) will be discussed in class, with additional problems (8–9) available for practice.
  • The instructor plans to post a short homework assignment on Canvas; due dates will be announced (likely the following Friday or as indicated).
  • Office hours: if there are no hours before the quiz, students can email to arrange a meeting (before 5 PM on Monday or later in the day).

Quick recap of the core ideas (to anchor study)

  • Economics centers on allocating scarce resources (time, labor, goods, capital).
  • Microeconomics analyzes individual decision-making; macroeconomics analyzes economy-wide aggregates.
  • A function expresses a relationship between inputs and outputs: y = f(x).
  • Linear functions are particularly important: y = m x + b with slope m and intercept b.
  • Intersections of two functions (same input) occur where f(x) = g(x); solving yields the x-coordinate of the intersection and the corresponding y-value(s).
  • When a region is bounded by curves, its area can be computed using geometric formulas like A = rac{1}{2} imes ext{base} imes ext{height} for triangular regions; the base is a horizontal distance between x-values, and the height is the vertical distance between curves.
  • Practice problems involve interpreting slope and intercept, solving for unknowns, and applying algebra to real-world economic questions.

Next steps for students

  • Review Problems 1–3 for interpretation of slope and intercept in context.
  • Practice solving for x in single-equation models and identifying intercepts and slopes.
  • Practice finding intersections of two functions and computing the area between curves where applicable.
  • Attend or review office hours for help with the upcoming quiz and to solidify understanding of these core concepts.