Source: Microeconomic Theory, Texas A&M University
Tags: functions, slopes, derivatives, linear equations, inverse demand, supply curves, y-intercept, tangent line, calculus for economics, ECO 101
These notes cover the mathematical toolkit you need before tackling any pricing or profit problem in microeconomics. That means functions, slopes, derivatives, and how to read and rearrange demand and supply equations. If you can take a derivative and rearrange a linear equation, you can do most of the maths in this course.
Function
A mathematical object that describes the relationship between input and output variables. In economics, the inputs are things like price or quantity; the output is something like profit or quantity demanded.
Slope (m)
The change in y divided by the change in x. For a straight line, the slope is constant everywhere. For a curve, the slope varies from point to point.
Derivative
Measures the slope of the tangent line at any given point on a curve. It tells you the instantaneous rate of change of one variable with respect to another.
Y-intercept (b)
In the equation y = mx + b, the value b is the y-intercept, the point where the line crosses the vertical axis (where x = 0).
Inverse demand curve
The demand relationship rewritten so that price (P) is on the left-hand side and quantity (Q) is on the right. If demand is Q = 6 − 3P, the inverse demand is P = 2 − 0.33Q.
Choke price
The price at which quantity supplied (or demanded) drops to zero. It is the intercept on the price axis.
Line vs. curve
Lines have constant slopes. Curves do not have constant slopes. This distinction matters when you are finding marginal values, because the derivative of a line is a constant but the derivative of a curve changes with the variable.
A function maps inputs to outputs. The general linear form is y = mx + b.
The slope m tells you how much y changes for a one-unit change in x.
The slope can tell you the direction of a relationship (positive or negative), the steepness, and the rate of change.
Multi-variable functions
Consider grade = b + m1(attendance) + m2(hours of study) + m3(sleep).
The coefficient m2 represents the relationship between time spent studying and the grade.
Each coefficient isolates the marginal effect of that one input, holding the others constant.
Supply equation example: Q = −50 + 5P
The coefficient on P (which is 5) tells you that quantity supplied increases by 5 units for every dollar increase in price.
Supply curve example: Q = 1 + 2P
If the price of an ice cream cone is 4, plug in: Q = 1 + 2(4) = 9.
Note: the quiz answer listed is 3. Double-check the original wording of the question, as Q = 1 + 2(4) = 9 by standard arithmetic. The quiz may have used a different equation such as Q = −5 + 2P or similar.
Supply choke price example: Qs = 4P − 4
Set Qs = 0 and solve: 0 = 4P − 4, so P = 1. Below a price of 1, no producer is willing to supply the good.
You need to be comfortable flipping between Q-form and P-form.
Demand curve Q = 6 − 3P becomes inverse demand P = 2 − 0.33Q (divide both sides by 3 and rearrange).
Inverse demand P = 2 − Q becomes demand curve Q = 2 − P (just rearrange).
With two points (P, Q), you can find the equation of the inverse demand line using the standard slope formula and point-slope form.
The first derivative of y = 20 − 10x with respect to x is dy/dx = −10.
The constant 20 drops out; the derivative of −10x is −10.
The first derivative of y = 100 with respect to x is dy/dx = 0.
A constant has no rate of change.
At the maximum of a function, the slope (first derivative) equals zero. This is the key rule for finding profit-maximising output.
Example: QD = 100 − 4P and QS = −20 + 2P.
Set QD = QS: 100 − 4P = −20 + 2P.
Solve: 120 = 6P, so P = 20.
Plug back in: Q = 100 − 4(20) = 20.
Equilibrium: P = 20, Q = 20.
Formula | What it does |
|---|---|
y = mx + b | General linear function; m = slope, b = y-intercept |
m = Δy / Δx | Slope of a line between two points |
dy/dx of axⁿ = naxⁿ⁻¹ | Power rule for derivatives |
Set QD = QS | Solve for equilibrium price and quantity |
Set Qs = 0, solve for P | Find the supply choke price |
⚠️ Know how to flip between demand (Q as a function of P) and inverse demand (P as a function of Q). This comes up repeatedly.
⚠️ The derivative of a constant is zero. The derivative of a linear term like 8Q is just its coefficient (8). These are the building blocks for the profit-maximisation problems.
⚠️ "At the maximum of a function, the slope is zero" is a direct exam statement. Commit it to memory.
⚠️ When finding equilibrium, set QD = QS and solve for P first, then plug P back into either equation to get Q.
Q: In the equation y = mx + b, what does b represent?
A: The y-intercept.
Q: What does the derivative measure?
A: The slope of the tangent line at a given point.
Q: What is the first derivative of y = 20 − 10x with respect to x?
A: −10.
Q: What is the first derivative of y = 100 with respect to x?
A: 0.
Q: If the supply curve is Qs = 4P − 4, what is the choke price?
A: P = 1 (set Qs = 0 and solve).
Q: Given QD = 100 − 4P and QS = −20 + 2P, what are the equilibrium price and quantity?
A: P = 20, Q = 20.
Q: What is the inverse demand curve for Q = 6 − 3P?
A: P = 2 − 0.33Q.
Q: Do lines have constant slopes?
A: Yes. Curves do not.
function, slope, derivative, tangent line, y-intercept, linear equation, inverse demand, demand curve, supply curve, choke price, equilibrium, power rule, calculus for economics, rate of change, first derivative, ECO 101, microeconomic theory, Texas A&M