Notes on Graphs in Economics: Relationships, Graph Types, and Slopes

Price and Quantity: Basic Relationship

• Initial question from the lecture: how many of you have the textbook? The idea is to connect price and quantity demanded—understand what happens to quantity demanded when price changes.
• Key goal: establish a relationship between two (or more) variables; use graphs as a convenient tool to understand that relationship.
• Important caution raised: just from a graph, you cannot claim a causal relationship. You can identify a positive association (positive correlation) but not causation or proportionality.

Graphs and Types of Relationships (Overview for the course)

• The instructor plans to cover different types of graphs you will encounter in this course, with terminology to be explained later.
• When drawing a graph, ask: does an increase in x cause an increase in y? The answer is not always clear from the graph alone; information about the functional form or causality is needed.
• General observation: a positive relationship (positive association) exists in some graphs, but the rate of change may not be proportional; the slope can change with x.

Examples of Different Curves and Their Characteristics

• Some curves show that as x increases, y increases at a rate faster than proportional (super-linear growth):
  • Example narrative: start with y = 1 at some point, increase x by one unit, and observe y moving to values like 3, 4, 5 in a non-linear way.
  • This implies a rate of change that accelerates as x grows: the slope becomes steeper as x increases.
• Other curves show positive increase but at a decreasing rate (concave up to the right, diminishing marginal gains):
  • Mentioned as a total product-like curve, where total output rises with input but the additional output gained from each extra unit of input falls.
• Negative linear relationship (increasing x leads to proportional decreases in y):
  • As x increases, y decreases in a linear, proportional way.
  • This is described as a negative linear resistance.
• Demand curve (typical economic example):
  • On the axes: quantity demanded (x) and price (y).
  • The demand curve generally shows a downward-sloping (negative) relationship between price and quantity demanded, assuming a typical law of demand.
• Production Possibilities Frontier (PPF):
  • Slope is decreasing (the marginal trade-off worsens as you move along the frontier).
  • As x (e.g., allocative quantity) increases, y moves in a way that the slope becomes smaller in magnitude, indicating increasing opportunity costs.
• Revenue and an optimal point:
  • There exists an optimal point (denoted as x^{*}) where revenue is maximized:
  • Up to x^{}, revenue increases; beyond x^{}, revenue begins to fall.
  • Interpreted as: you cannot simply keep selling more and expect revenue to rise indefinitely.
• Fourth type: no relationship (independence) between variables:
  • Example: demand for pizza and quantity of toothpaste may be unrelated.
  • In a plotted graph, increasing x might not change y at all (y remains constant).
  • A simple discrete example: from point A to point B, if y rises from 5 to 10 while x rises by a fixed amount, the slope could be 1, illustrating a constant rate of change in this particular segment.
• Practical implication: these graphical forms help illustrate that the same pair of variables can have very different relationships depending on the context and underlying mechanisms.

Interpreting Slopes and Rates of Change

• Slope basics (discrete changes):
  • If you observe two points (x1, y1) and (x2, y2), the slope is:
  • \text{slope} = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}
  • If the slope equals 1, it means that for every 1 unit change in x, y changes by 1 unit.
• General takeaway: slope can be constant over a linear segment, but on a curve, slope is not constant.
• Derivatives (continuous case):
  • If you know the equation of the curve, you can take the derivative to find the slope of the tangent line at a point:
  • \frac{dy}{dx}
  • The slope of the tangent line at a given point is the instantaneous rate of change of y with respect to x.
• Tangent line concept:
  • A tangent line touches the curve at a single point and has slope equal to the derivative at that point.
  • To determine the slope at a particular point, you conceptually draw the tangent, since the slope along the curve varies from point to point.
• Important caveat from the lecture:
  • If you do not know the explicit equation of the curve, you cannot compute the exact derivative; you rely on the concept of a tangent line and discrete slopes from data.
• Practical note about temperature and multiple variables:
  • Real-world relationships can be influenced by other factors (e.g., weather temperature affecting ice cream prices), so a simple two-variable plot may miss other drivers.

How to Draw Graphs Between Two or More Variables

• The lecture closes with a query: how to draw graphs between two (or more) variables?
• The takeaway is to practice plotting the axes appropriately (e.g., price on the y-axis, quantity on the x-axis for demand), and to recognize the type of relationship (positive, negative, nonlinear, independent).
• When interpreting graphs, consider whether the relationship is causal or merely correlational, and be mindful of possible confounding factors.

Key Takeaways for Exam Preparation

• Graphs are tools to visualize relationships but do not by themselves prove causation; correlation does not imply causation.
• Relationships can be positive, negative, nonlinear, or independent depending on the variables and context.
• Common economic graphs include:
  • Demand: price vs. quantity demanded (typically downward-sloping)
  • Supply: price vs. quantity supplied (typically upward-sloping)
  • Total product/production curves: output vs. input with diminishing marginal returns
  • Production Possibilities Frontier (PPF): combinations of two goods with a concave curve and a decreasing slope
  • Revenue curves: understanding optima where R(x) = x \cdot p(x) is maximized at x^{*}
• Slope concepts:
  • Discrete: \text{slope} = \frac{\Delta y}{\Delta x}; example: if slope = 1, then a 1-unit change in x yields a 1-unit change in y
  • Continuous: slope becomes the derivative \frac{dy}{dx}; the tangent line at a point has this slope
• Be ready to identify when a graph implies an optimum point and why unlimited increases in one variable do not guarantee higher outcomes.
• Expect questions that require distinguishing between related but separate graphs (e.g., independent vs dependent variables, multiple variables with the potential for confounding factors).