Verifying Equilibrium Solutions And Units

Why an Equilibrium Calculation Needs a Check

Solving supply and demand equations produces a candidate price and quantity. A separate check is needed to establish that the candidate actually satisfies both equations, that the quantities use compatible units, and that the result is allowed by the model. An algebraic answer can look plausible while failing one of these tests.

This verification is especially important when equations are given in different scales, such as hundreds and thousands of units, or over different time periods, such as a week and a month. It is also necessary when equations are defined only over a stated price range. The calculation is complete only after the candidate has been checked against the original model, not merely against an altered or extrapolated version of its equations.

Definition: A market equilibrium is a price $P^*$ and quantity $Q^*$ at which the quantity demanded equals the quantity supplied, with both quantities expressed in the same units and time period. A candidate is a feasible equilibrium only if it also satisfies the model’s stated domain and other constraints.

A Verification Method

For equations written as $Q_d(P)$ and $Q_s(P)$, a direct check has three parts. First, use the candidate price in the demand equation and calculate $Q_d(P^*)$. Second, use that same price in the supply equation and calculate $Q_s(P^*)$. Third, compare the results only after confirming that they measure the same thing over the same period. If the model restricts the domain of $P$, check that restriction as well.

When demand and supply are linear, a familiar form is $Q_d(P)=a-bP$ and $Q_s(P)=c+dP$, where $b>0$ and $d>0$. Before using this form, assume that both equations describe the same market, their quantity units and time periods have been aligned, and the price is within their stated domain. Equating them gives a candidate price:

a-bP=c+dP \quad\Longrightarrow\quad P^*=\frac{a-c}{b+d}.

The denominator is positive under the stated slope assumptions. The algebra establishes an intersection of the two linear expressions; it does not, by itself, establish that the price is in the allowed domain or that the resulting quantity is nonnegative. Those are separate checks. Also, $a$ and $c$ must be expressed in a common quantity unit, while $b$ and $d$ must have compatible quantity-per-price units. Otherwise, the equation is not a valid comparison of quantities.

Why the substitution check works: At an equilibrium, one price must be consistent with both sides of the market. Substituting $P^*$ into the two model functions tests exactly that requirement: if $Q_d(P^*)=Q_s(P^*)$, the common value is the equilibrium quantity. If the two values differ, the proposed pair does not satisfy the defining equality, even if it arose from some intermediate algebra.

List the units and restrictions.
Record the price unit, quantity unit, time period, price domain, and any nonnegativity conditions stated in the problem.

Put quantities on a common basis.
Convert scales such as thousands to hundreds, and convert time periods if one equation is weekly and the other is monthly.

Substitute the candidate price into both equations.
Calculate demand and supply separately using the original model equations, with any necessary conversions shown.

Check equality and feasibility.
Confirm that the quantities match, then check that the price and quantity obey the stated constraints.

Worked Equilibrium Checks

Worked Example: Verifying a Candidate in Matching Units

A market has demand $Q_d(P)=96-3P$ and supply $Q_s(P)=12+3P$. Quantities are measured in hundreds of units per month, and $P$ is dollars per unit. The equations apply for $0\leq P\leq20$, with quantities required to be nonnegative. Find and verify the equilibrium.

Method. Set the two expressions equal, solve for the candidate price, and substitute that price into each original equation. Both equations already use the same quantity and time units.

Equating demand and supply gives:

96-3P=12+3P \quad\Longrightarrow\quad 84=6P \quad\Longrightarrow\quad P^*=14.

Substitute $P=14$ into demand:

Q_d(14)=96-3(14)=96-42=54 \quad\text{hundreds of units per month}.

Substitute the same price into supply:

Q_s(14)=12+3(14)=12+42=54 \quad\text{hundreds of units per month}.

The quantities agree. In individual units, $54$ hundreds is $54\times100=5{,}400$ units per month. The price $14$ is within the stated range, and the common quantity is nonnegative. Therefore the feasible equilibrium is $P^*=\$14$ per unit and $Q^*=5{,}400$ units per month.

Check. The original equations give the same quantity at the calculated price, and the price and quantity meet the constraints. This avoids the mistake of reporting a price obtained from algebra without checking the resulting quantity in both equations.

Worked Example: Converting Different Quantity Scales

A hypothetical market has demand $Q_d(P)=150-4P$, measured in thousands of units per quarter. Supply is $Q_s(P)=0.6+0.2P$, measured in ten-thousands of units per quarter. Price is in dollars per unit, and the equations apply for $0\leq P\leq30$. Convert the quantities to a common scale, then find and verify the equilibrium.

Method. Convert supply from ten-thousands to thousands before equating the equations. One ten-thousand units is ten thousand units, so the supply expression in thousands is multiplied by $10$:

Q_s(P)=10(0.6+0.2P)=6+2P \quad\text{thousands of units per quarter}.

Now both equations use thousands of units per quarter. Equating them gives:

150-4P=6+2P \quad\Longrightarrow\quad 144=6P \quad\Longrightarrow\quad P^*=24.

At this price, demand is:

Q_d(24)=150-4(24)=150-96=54 \quad\text{thousand units per quarter}.

In the original supply units, supply is:

Q_s(24)=0.6+0.2(24)=0.6+4.8=5.4 \quad\text{ten-thousands of units per quarter}.

Convert that result to units: $5.4\times10{,}000=54{,}000$ units, or $54$ thousand units. Thus both sides give $54{,}000$ units per quarter. The price $24$ is within the permitted interval, and the quantity is positive. The feasible equilibrium is $P^*=\$24$ per unit and $Q^*=54{,}000$ units per quarter.

Common mistake. Equating the unconverted expressions would give $150-4P=0.6+0.2P$, hence $P=149.4/4.2\approx35.5714$. That compares numerical values in different scales. At that price, demand is approximately $7.7143$ thousand units, while supply is approximately $7.7143$ ten-thousands of units, or $77.1429$ thousand units. The numbers in the unconverted expressions look equal, but the physical quantities are not. Convert first; then substitute into the original equations as a check.

Worked Example: Aligning the Time Period

For a hypothetical product, demand is $Q_d(P)=2{,}500-50P$ units per four-week month. Supply is $Q_s(P)=400+25P$ units per week. Price is in dollars per unit. Both equations are stated for $0\leq P\leq20$, with nonnegative quantities. Find and verify a market equilibrium on a four-week basis.

Method. Convert weekly supply to units per four-week month by multiplying by four. This is a four-week period as specified in the problem, not an assumption that every calendar month has exactly four weeks.

Q_s(P)=4(400+25P)=1{,}600+100P \quad\text{units per four-week month}.

Now the quantity equations use the same time period. Solve for the candidate:

2{,}500-50P=1{,}600+100P \quad\Longrightarrow\quad 900=150P \quad\Longrightarrow\quad P^*=6.

Demand at $P=6$ is:

Q_d(6)=2{,}500-50(6)=2{,}500-300=2{,}200 \quad\text{units per four-week month}.

Supply at the same price is $400+25(6)=400+150=550$ units per week. Converting to the stated four-week period gives $4(550)=2{,}200$ units per four-week month. The quantities agree. The price is in the stated domain, and the quantity is nonnegative, so $P^*=\$6$ per unit and $Q^*=2{,}200$ units per four-week month.

Common mistake. Comparing $2{,}200$ units per four-week month directly with $550$ units per week would incorrectly suggest unequal quantities. The quantities describe different periods; multiplying the weekly amount by four makes the comparison valid.

Algebraic Intersections and Feasible Solutions

A solved equation may produce a price outside the range on which the model is defined. In that case, the formula values at the algebraic intersection are extrapolations, not evaluations within the stated model domain. They cannot be reported as a feasible equilibrium unless the problem explicitly allows extending the equations beyond that range.

For example, suppose $Q_d(P)=90-3P$ and $Q_s(P)=30+P$, but both equations are specified only for $0\leq P\leq12$. Equating the formulas gives $90-3P=30+P$, so $P=15$. This is outside the domain. If the formulas are algebraically extended to $P=15$, they each give $45$, but those extrapolated values do not establish an equilibrium in the specified model. Within the allowed price range, there is no price at which the two stated expressions are equal.

More generally, a candidate should be rejected as a feasible equilibrium if it violates a stated price restriction or produces an impermissible quantity. Do not silently discard a constraint because the algebra has already produced a neat intersection. State whether the result is an in-domain solution, an out-of-domain extrapolation, or a case with no feasible equilibrium under the stated model.

Why Units and Constraints Matter

Equilibrium is an equality between economic quantities, not merely between two numerals. “Four hundred units per week” and “four hundred units per month” are different quantities. Likewise, “five thousand units” and “five hundred units” do not become equal because their equations happen to return the same number. Writing the units beside each calculation makes errors visible and helps ensure that a reported answer is interpretable.

The same discipline applies to the price. If $P$ is dollars per unit, a result for $P$ should be reported in those units; it should not be confused with a price per dozen or a total expenditure. Keep each original equation, conversion, and substitution labelled so the path from the model to the answer can be checked.

Substitute into both equations. Checking only the equation used in the final arithmetic does not verify the equality required for equilibrium.
Convert before comparing. Align both the quantity scale and the time period.
Respect the domain. An intersection outside the stated domain is not a feasible solution to that model.
Report the common quantity with its units. If necessary, show both the model’s scale and the equivalent number of individual units.

Key takeaway: A candidate equilibrium is verified only when substituting its price into both model equations gives the same quantity in a common unit and time period, and the price and quantity satisfy every stated constraint.

Check Your Understanding

For each problem, check the units and stated conditions as well as the algebra.

Demand is $Q_d(P)=72-2P$ and supply is $Q_s(P)=12+P$, both in units per day. Price is in dollars per unit and the domain is $0\leq P\leq30$. Find and verify the equilibrium. (Answer: $P^*=\$20$ per unit; $Q^*=32$ units per day.)
Demand is $Q_d(P)=160-4P$ hundreds of units per month. Supply is $Q_s(P)=1+0.2P$ thousands of units per month. Price is in dollars per unit and the domain is $0\leq P\leq30$. Convert, find, and verify the equilibrium. (Answer: $P^*=\$25$ per unit; $Q^*=60$ hundreds, or $6{,}000$ units per month.)
Demand is $Q_d(P)=3{,}000-100P$ units per four-week month. Supply is $Q_s(P)=250+25P$ units per week. Price is in dollars per unit and the domain is $0\leq P\leq20$. Find and verify the equilibrium on a four-week basis. (Answer: $P^*=\$10$ per unit; $Q^*=2{,}000$ units per four-week month.)
Demand is $Q_d(P)=90-3P$ and supply is $Q_s(P)=30+P$, both in units per week. The equations apply only for $0\leq P\leq12$, and quantities must be nonnegative. Does the model have a feasible equilibrium? (Answer: No. The algebraic intersection is $P=15$, outside the stated domain; it is not a feasible equilibrium.)
Demand is $Q_d(P)=600-20P$ units per day. Supply is $Q_s(P)=0.1+0.1P$ thousands of units per day. Price is in dollars per unit and the domain is $0\leq P\leq20$. Convert, solve, and verify, reporting the quantity to one decimal place. (Answer: $P^*=\$25/6$ per unit; $Q^*=1550/3\approx516.7$ units per day.)

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