Why Solve the Same Market in Different Ways?
A market may be described by a table of quantities at several prices, drawn as demand and supply curves, or given by equations. Each representation contains the same basic information, but each makes a different part of the problem easy to see. A schedule makes it straightforward to compare quantities at listed prices. A diagram makes the intersection visible. Equations allow a precise solution, including when the equilibrium lies between listed values or grid lines.
The method is to identify the price and quantity variables, state the equilibrium condition, and translate the information into that condition. After solving, check that the proposed price produces equal quantities on both sides of the market. This tutorial uses the standard competitive-market equilibrium model, where the market-clearing price is the price at which quantity demanded equals quantity supplied. The approach is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The model treats demand and supply as relationships between price and quantity, holding the other relevant conditions fixed. A price below the equilibrium price typically creates excess demand, or a shortage: \(Q^d>Q^s\). A price above equilibrium typically creates excess supply, or a surplus: \(Q^s>Q^d\). These comparisons are useful checks, but they do not replace solving for the price where the quantities match.
Worked Example: Finding Equilibrium From a Schedule
Worked Example: Finding Equilibrium From a Schedule
A hypothetical market for insulated delivery containers is described by the schedule below. Find the equilibrium price and quantity, then calculate the shortage or surplus at a price of $15.
| Price ($) | Quantity demanded | Quantity supplied |
|---|---|---|
| 10 | 90 | 30 |
| 15 | 80 | 50 |
| 20 | 70 | 70 |
Method. Compare the two quantities row by row. At $10, \(90\neq30\); at $15, \(80\neq50\); at $20, both quantities are 70. Therefore, the schedule gives an equilibrium price of $20 and an equilibrium quantity of 70 containers.
Check at $15. Quantity demanded is 80 and quantity supplied is 50, so excess demand is \(80-50=30\) containers. This is a shortage, not a surplus, because buyers want more than sellers offer at that price.
The schedule also permits a translation into equations if we assume the listed relationships are linear between and beyond these prices. From $10 to $15, price rises by 5 while quantity demanded falls by 10, so the demand slope is \(-10/5=-2\) containers per dollar. Using the point \((P,Q^d)=(10,90)\), the demand equation is \(Q^d=110-2P\). For supply, quantity rises by 20 when price rises by 5, giving slope \(20/5=4\). Using \((10,30)\), the supply equation is \(Q^s=-10+4P\).
Set the equations equal: \(110-2P=-10+4P\). Thus \(120=6P\), so \(P=20\). Substitution gives \(Q^d=110-2(20)=70\) and \(Q^s=-10+4(20)=70\), agreeing with the schedule. The common mistake this avoids is selecting a row with the smallest gap and calling it equilibrium; equilibrium requires an exact equality, or an explicit interpolation assumption when no listed row has one.
Reading Equilibrium From a Diagram
A standard supply-and-demand diagram places quantity \(Q\) on the horizontal axis and price \(P\) on the vertical axis. Demand usually slopes downward from left to right, while supply usually slopes upward. Their intersection represents the price and quantity at which both curves give the same quantity. Read the coordinates in the order dictated by the axes: first the quantity on the horizontal axis, then the price on the vertical axis.
A drawn intersection may be estimated from the graph, but a graph’s grid and line thickness can limit precision. If the graph provides points on straight lines, use those points to derive equations and solve exactly. This makes clear whether an apparent intersection falls between grid lines and prevents an axis-reading error.
Worked Example: Translating a Diagram Into Equations
A hypothetical market diagram has price on the vertical axis and quantity on the horizontal axis. The straight demand line passes through \((Q,P)=(0,48)\) and \((96,0)\). The straight supply line passes through \((0,0)\) and \((96,32)\). Find the intersection.
Demand line. Its slope in the diagram’s coordinates is \((0-48)/(96-0)=-48/96=-0.5\) price units per quantity unit. Because the price intercept is 48, the inverse-demand equation is \(P=48-0.5Q\).
Supply line. Its slope is \((32-0)/(96-0)=1/3\), and it passes through the origin. Its equation is \(P=Q/3\).
At the intersection, both lines have the same price:
Multiply both sides by 6: \(288-3Q=2Q\). Therefore, \(Q=57.6\). Substituting into supply gives \(P=57.6/3=19.2\). The diagram’s equilibrium is quantity 57.6 and price 19.2.
Check. Demand gives \(P=48-0.5(57.6)=48-28.8=19.2\), which equals supply’s price. Equivalently, inverse demand implies \(Q^d=96-2P=96-38.4=57.6\), while supply implies \(Q^s=3P=57.6\). The common mistake here is reversing the axes and reporting the coordinates as price 57.6 and quantity 19.2. The stated axis labels show that quantity is 57.6 and price is 19.2.
Solving Linear Demand and Supply Equations
When equations are provided directly, the equilibrium condition turns the problem into one equation with one unknown price. After finding price, substitute it into either quantity equation. A final substitution into both equations verifies that the same quantity results.
The formula follows directly from the market-clearing condition. Rearranging \(a-bP=c+dP\) gives \(a-c=(b+d)P\). If \(b+d\neq0\), divide both sides by \(b+d\) to obtain the equilibrium price. Substituting that price back into either original quantity equation gives the equilibrium quantity. The slope assumptions \(b>0\) and \(d>0\) mean that demand falls and supply rises as price increases; the result is not a general formula for nonlinear curves.
Worked Example: Solving and Checking Linear Equations
A hypothetical market for reusable cold-storage packs has demand \(Q^d=150-5P\) and supply \(Q^s=30+3P\), where price is in dollars per pack and quantities are in packs. Find equilibrium and determine the market condition at a price of $12.
Equilibrium calculation. Set quantity demanded equal to quantity supplied:
Subtract 30 from both sides and add \(5P\) to both sides: \(120=8P\). Thus \(P=15\) dollars. At this price, demand is \(150-5(15)=150-75=75\) packs, and supply is \(30+3(15)=30+45=75\) packs. The equilibrium is therefore $15 and 75 packs.
Condition at $12. Demand is \(150-5(12)=150-60=90\) packs. Supply is \(30+3(12)=30+36=66\) packs. Excess demand is \(90-66=24\) packs, so the market has a shortage of 24 packs at $12.
Check. The equilibrium price is below $15, and the calculated quantity demanded exceeds quantity supplied, consistent with a shortage. A common mistake is to subtract the quantities in the wrong order and report a negative shortage. State the two quantities first; since \(90>66\), the shortage amount is \(90-66=24\), not a negative quantity.
A Reliable Translation and Checking Routine
Use a consistent routine when a question changes representation. A schedule gives pairs of quantities and prices directly; a diagram gives coordinates or permits an estimate; equations give relationships that must be solved. In every case, the target is the same equality, \(Q^d(P)=Q^s(P)\).
Confirm which variable is price and which is quantity. Check whether price is stated in dollars, dollars per unit, or another unit, and whether plotted points are exact or approximate.
Write that quantity demanded equals quantity supplied at the same price. For a schedule, compare rows; for a graph, locate the intersection; for equations, set the quantity expressions equal.
Find the equilibrium price and then calculate the common quantity. If using a diagram, report the horizontal-axis coordinate as quantity and the vertical-axis coordinate as price.
Substitute the price into demand and supply. Equal quantities confirm equilibrium. Away from equilibrium, compare them in context: demand above supply is a shortage; supply above demand is a surplus.
Common Mistakes to Avoid
- Mixing up price and quantity. On the standard diagram, price is vertical and quantity is horizontal. Report each coordinate with its variable and unit.
- Using the wrong sign for a slope. In a linear demand equation, quantity demanded falls as price rises; in a standard supply equation, quantity supplied rises as price rises.
- Stopping after solving for price. Equilibrium requires both the price and the common quantity. Substitute the price into a quantity equation.
- Calling a shortage a negative quantity. The gap is a nonnegative amount. Identify which side is larger, then subtract the smaller quantity from the larger one.
- Assuming a table is continuous without justification. If no row has equal quantities, interpolation requires an assumption such as straight-line relationships between the listed points. A discrete schedule may not have an equilibrium at a listed price.
Check Your Understanding
For each problem, state the equilibrium or market condition requested, and check that the answer is consistent with the given representation.
- A schedule lists prices of $5, $10, and $15. At those prices, quantities demanded are 70, 60, and 50, while quantities supplied are 20, 40, and 60. Assuming straight-line relationships between the listed points, find equilibrium price and quantity. (Answer: price \(13.333\ldots\), quantity \(53.333\ldots\))
- In a diagram, demand is \(P=36-0.4Q\) and supply is \(P=0.2Q\). Find the intersection, reporting quantity before price. (Answer: quantity 60, price 12)
- A market has \(Q^d=210-7P\) and \(Q^s=30+2P\). Find equilibrium price and quantity, and verify both quantities. (Answer: price 20, quantity 70)
- For \(Q^d=96-2P\) and \(Q^s=12+2P\), find equilibrium. Then determine whether a price of $18 produces a shortage or surplus and calculate its size. (Answer: equilibrium price 21 and quantity 54; shortage of 12)
- A diagram’s straight demand line passes through \((Q,P)=(0,50)\) and \((100,0)\); its supply line passes through \((0,5)\) and \((100,30)\). Derive the two price-as-a-function-of-quantity equations and find their intersection. (Answer: demand \(P=50-0.5Q\), supply \(P=5+0.25Q\); equilibrium quantity 60 and price 20)