Difficulty: Introductory to Intermediate | Prerequisites: Basic algebra
This material covers the mathematical toolkit and the core supply-and-demand framework that underpin the rest of ECON 207. Units 1A and 1B establish the language, graphs and algebra you will use in every subsequent topic, from elasticity through welfare analysis. If you are comfortable with slopes, simultaneous equations and the idea that markets clear where two curves cross, you are ready for elasticity.
Before you touch supply and demand, you need the maths: slopes, derivatives, simultaneous equations and areas of triangles. Once that toolkit is solid, the core model is straightforward: demand slopes down, supply slopes up, and the market clears where the two cross. Most exam questions test whether you can find that crossing point algebraically and then show what happens when something shifts one of the curves.
Quantity demanded (Qd)
The specific amount of a good buyers are willing and able to purchase at a given price, holding everything else constant. Think of it as one point on the demand curve, not the whole curve.
Law of demand
All else equal, as the price of a good rises, quantity demanded falls (and vice versa). In simple terms, higher prices mean people buy less.
Change in quantity demanded
A movement along a fixed demand curve caused by a change in the good's own price.
Change in demand
A shift of the entire demand curve, caused by a change in one of the determinants of demand (income, tastes, prices of related goods, expectations, number of buyers). In simple terms, something other than price changed.
Determinants of demand
Factors that shift the demand curve: consumer income, tastes and preferences, prices of substitutes and complements, consumer expectations, and the number of buyers in the market.
Quantity supplied (Qs)
The specific amount of a good sellers are willing and able to offer at a given price, holding everything else constant. One point on the supply curve.
Law of supply
All else equal, as the price of a good rises, quantity supplied rises. In simple terms, higher prices give producers more incentive to sell.
Change in quantity supplied
A movement along a fixed supply curve caused by a change in the good's own price.
Change in supply
A shift of the entire supply curve, caused by a change in one of the determinants of supply.
Determinants of supply
Factors that shift the supply curve: input prices, technology, expectations, number of sellers, taxes and subsidies, and prices of related goods in production.
Equilibrium price (P)*
The price at which quantity demanded equals quantity supplied. The market clears, with no surplus or shortage.
Equilibrium quantity (Q)*
The quantity bought and sold at the equilibrium price.
Surplus (excess supply)
When the market price is above equilibrium, Qs > Qd. Unsold goods push the price down.
Shortage (excess demand)
When the market price is below equilibrium, Qd > Qs. Unmet demand pushes the price up.
Exponents
Standard rules apply: x^a * x^b = x^(a+b), (x^a)^b = x^(ab), x^0 = 1.
Negative exponents flip to denominators: x^(-n) = 1/x^n.
You will use these when manipulating demand and supply functions.
Slope of a straight line
Slope = rise/run = (Y2 - Y1) / (X2 - X1).
Between two points: pick any two coordinates on the line and plug in.
At a point on a straight line: the slope is the same everywhere, so use any two points.
In economics, be careful about which axis is which. Price is typically on the vertical axis, quantity on the horizontal, even though P is the independent variable in the equation.
Slope of a curve (derivatives)
The slope at a point on a curve is the derivative of the function at that point.
For y = ax^n, the derivative is dy/dx = n * a * x^(n-1).
This gives you the instantaneous rate of change, which matters for elasticity calculations later.
Calculating area (triangles and rectangles)
Area of a triangle = (1/2) * base * height. You will use this constantly for consumer surplus, producer surplus and deadweight loss.
Area of a rectangle = base * height. Used for revenue rectangles and parts of surplus calculations.
Graphing coordinates with three variables
When a function has three variables (e.g. Qd = f(P, Income)), you plot two on the axes and hold the third constant.
Changing the held-constant variable shifts the entire curve on the graph.
Solving simultaneous equations
Set Qd = Qs (or whatever the two equations are) and solve for the unknown.
This is how you find equilibrium: you have two equations and two unknowns (P and Q), and you solve by substitution or elimination.
The law of demand
Holding all else constant, there is an inverse relationship between price and quantity demanded. On a standard graph (P on the vertical axis, Q on the horizontal), the demand curve slopes downward from left to right.
Quantity demanded vs. demand: a critical distinction
A change in the good's own price causes a movement along the demand curve. This is a change in quantity demanded.
A change in any other factor (income, tastes, related goods' prices, expectations, number of buyers) shifts the entire curve. This is a change in demand.
Rightward shift = increase in demand (more is demanded at every price).
Leftward shift = decrease in demand (less is demanded at every price).
The determinants of demand (demand shifters)
Consumer income: for normal goods, higher income shifts demand right. For inferior goods, higher income shifts demand left.
Tastes and preferences: if a good becomes more popular, demand shifts right.
Price of related goods: if the price of a substitute rises, demand for the good shifts right. If the price of a complement rises, demand for the good shifts left.
Consumer expectations: if consumers expect prices to rise in the future, current demand shifts right.
Number of buyers: more buyers in the market shifts demand right.
The law of supply
Holding all else constant, there is a positive relationship between price and quantity supplied. On a standard graph, the supply curve slopes upward from left to right. Higher prices make production more profitable, so firms offer more.
Quantity supplied vs. supply: the same distinction as demand
A change in the good's own price causes a movement along the supply curve. This is a change in quantity supplied.
A change in any other factor shifts the entire curve. This is a change in supply.
Rightward shift = increase in supply (more is offered at every price).
Leftward shift = decrease in supply (less is offered at every price).
The determinants of supply (supply shifters)
Input prices (costs of production): higher input costs shift supply left (less profitable to produce at any given price).
Technology: improved technology shifts supply right (can produce more at the same cost).
Expectations: if firms expect prices to rise, they may reduce current supply.
Number of sellers: more sellers shifts supply right.
Taxes and subsidies: a tax on producers shifts supply left; a subsidy shifts supply right.
Prices of related goods in production: if a farmer can grow wheat or corn, a rise in the price of corn shifts the supply of wheat left.
Graphically
Equilibrium is the point where the demand and supply curves intersect. Read P* off the vertical axis and Q* off the horizontal axis at that intersection.
Algebraically
Set Qd = Qs and solve for P. Then substitute P back into either equation to find Q. For example, if Qd = 40 - 2P and Qs = 2P - 8, set 40 - 2P = 2P - 8, which gives 48 = 4P, so P* = 12. Then Q* = 40 - 2(12) = 16.
Why competitive markets adjust to equilibrium
If price is above equilibrium, there is a surplus. Sellers compete by lowering prices, which increases Qd and decreases Qs until the surplus is eliminated.
If price is below equilibrium, there is a shortage. Buyers compete by bidding prices up, which decreases Qd and increases Qs until the shortage is eliminated.
The market self-corrects through the price mechanism.
Shifts in demand or supply
When one curve shifts, the old equilibrium no longer clears the market. A surplus or shortage emerges at the old price, and the price adjusts until a new equilibrium is reached.
Increase in demand (rightward shift): P* rises, Q* rises.
Decrease in demand (leftward shift): P* falls, Q* falls.
Increase in supply (rightward shift): P* falls, Q* rises.
Decrease in supply (leftward shift): P* rises, Q* falls.
Simultaneous shifts
When both curves shift at the same time, the direction of change in either P* or Q* becomes ambiguous (depends on the relative size of the shifts).
Both increase: Q* definitely rises, but the effect on P* is ambiguous.
Both decrease: Q* definitely falls, but the effect on P* is ambiguous.
Demand increases and supply decreases: P* definitely rises, but Q* is ambiguous.
Demand decreases and supply increases: P* definitely falls, but Q* is ambiguous.
\text{Slope (between two points)} = \frac{Y_2 - Y_1}{X_2 - X_1}\text{Derivative of } y = ax^n: \quad \frac{dy}{dx} = n \cdot a \cdot x^{n-1}\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}\text{Equilibrium condition:} \quad Q_d = Q_s \quad \Rightarrow \quad \text{solve for } P^*, \text{ then } Q^*Students often say "demand increased" when the price rose and people bought less. That is a decrease in quantity demanded (movement along the curve), not a change in demand (shift of the curve). The vocabulary distinction is worth marks.
Students sometimes confuse the slope of the demand equation with the slope on the graph. If Qd = 40 - 2P, the coefficient on P is -2, but because P is on the vertical axis and Q is on the horizontal, the graphical slope (rise/run) is the inverse: -1/2.
When both supply and demand shift simultaneously, students often try to pin down the direction of both P* and Q*. One of them is always ambiguous unless you know the relative magnitude of the shifts.
A common error when solving for equilibrium is setting the equations equal but forgetting to substitute back to find Q*. Always do both steps.
Expect to solve for equilibrium algebraically given linear demand and supply equations. This is near-guaranteed on the quiz.
You will likely be asked to identify what shifts a curve vs. what causes a movement along a curve. Know the determinants cold.
Simultaneous shifts are a favourite for multiple-choice questions because one variable is always ambiguous.
The maths from Unit 1A (slope, area of triangles, derivatives) reappears in every subsequent unit, especially in elasticity and surplus calculations.
True or false: A rise in the price of a good causes the demand curve to shift left. (False. It causes a movement along the demand curve.)
Fill in the blank: The derivative of y = 5x^3 is dy/dx = ___. (15x^2)
True or false: If both demand and supply increase, the equilibrium quantity definitely rises. (True.)
Fill in the blank: The area of a triangle with base 10 and height 6 is ___. (30)
True or false: A rise in input prices shifts the supply curve to the right. (False. It shifts supply left.)
Q: Given Qd = 60 - 3P and Qs = 2P - 10, find the equilibrium price and quantity.
A: Set 60 - 3P = 2P - 10. That gives 70 = 5P, so P* = 14. Substituting back: Q* = 60 - 3(14) = 18.
Q: If the price of a complement to good X falls, what happens to the demand for good X?
A: Demand for good X increases (shifts right). A cheaper complement means consumers buy more of both goods together.
Q: Suppose both demand and supply for coffee increase. What can you say for certain about the new equilibrium?
A: Equilibrium quantity definitely rises. The effect on equilibrium price is ambiguous: it depends on whether the demand shift or the supply shift is larger.
Q: A new tax on producers raises their costs. Is this a shift of the supply curve or a movement along it?
A: A shift of the supply curve to the left. The tax changes the cost of production at every price level, which is a determinant of supply.
Q: On a graph, the demand curve is P = 20 - 0.5Q. What is the slope of this line as drawn?
A: The slope on the graph (with P on the vertical axis) is -0.5. The demand equation written as Q = 40 - 2P has a coefficient of -2 on P, but that is dQ/dP, not the graphical slope.
The equilibrium-solving technique from this unit is the foundation for every policy analysis in Unit 1D (quotas, price floors, price ceilings, taxes). You will set the same equations equal, then compare the policy outcome to the free-market result.
Slope and derivatives feed directly into elasticity (Unit 1C). The formula for point elasticity uses the derivative of the demand or supply function, and the midpoint formula uses the slope between two points.
Area calculations (triangles, rectangles) are used to compute consumer surplus, producer surplus, deadweight loss and tax revenue in Unit 1D.
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