Solving Equilibrium With Nonlinear Curves

Why Nonlinear Curves Require a Careful Check

Supply and demand equations do not have to be linear. A quantity may depend on the square of price, or both curves may have nonlinear terms. The market-clearing condition is unchanged: at equilibrium, the quantity buyers plan to purchase equals the quantity sellers plan to provide. The additional work is algebraic: solve the resulting equation, decide which roots are feasible, and verify the quantity at the selected price.

A nonlinear equation can produce more than one algebraic root. A negative price, a negative quantity, or a price outside the stated range cannot describe a feasible market outcome under the usual model. Even when a root appears plausible, substituting it into both original curves is an important check against sign errors, arithmetic mistakes, or premature rounding.

Definition: Let $Q_D(P)$ be the quantity demanded at price $P$, and let $Q_S(P)$ be the quantity supplied at price $P$. A market equilibrium is a price $P^*$ and quantity $Q^*$ such that $Q_D(P^*)=Q_S(P^*)=Q^*$. A feasible equilibrium must also satisfy the price and quantity restrictions of the model, commonly $P^*\geq 0$ and $Q^*\geq 0$.

A Reliable Solution Method

Assume a single market for one good over a specified time period, with price measured in dollars per unit and quantities measured in units per period. The equations describe buyers’ and sellers’ planned quantities at each price. For the calculations below, prices and quantities are treated as continuous, and an equilibrium is sought only where both equations describe nonnegative quantities. These assumptions matter: if a problem gives a restricted price range or a different feasible domain, use that stated restriction.

The basic procedure is to set the two quantity equations equal, rearrange into a polynomial equation, solve for candidate prices, and test each candidate against the domain. For each retained price, calculate the quantity using both original equations. The equality of the two results is the final check.

Equate quantities.
Write $Q_D(P)=Q_S(P)$. Keep the price variable explicit so that terms are not confused with quantities.

Solve for candidate prices.
Collect terms into a polynomial equation. Factor it when possible; otherwise use an appropriate algebraic method, such as the quadratic formula for a quadratic.

Apply feasibility conditions.
Check each candidate price against the stated price range and calculate its implied quantity. Discard candidates that violate the model’s restrictions.

Verify the equilibrium.
Substitute the selected price separately into demand and supply. Report the price and the common quantity with units.

Why Equating the Curves Works

The curves describe the quantities buyers and sellers plan to trade at a given price. If $Q_D(P)>Q_S(P)$, planned purchases exceed planned sales; if $Q_S(P)>Q_D(P)$, planned sales exceed planned purchases. A market-clearing price is precisely a price at which neither side’s planned quantity exceeds the other’s. Therefore the defining equation is $Q_D(P)=Q_S(P)$, not an equation that sets either quantity to zero.

For example, if $Q_D(P)=a-bP^2$ and $Q_S(P)=c+dP$, with constants $a,b,c,d$, equating the expressions gives $a-bP^2=c+dP$. Moving every term to one side yields $bP^2+dP+(c-a)=0$. Solving this polynomial generates candidate prices; it does not by itself establish that every root is an equilibrium. The original quantity equations and the feasible domain still have to be checked.

Worked Example: Factoring a Nonlinear Demand Equation

Suppose demand and supply, measured in units per week, are $Q_D(P)=81-P^2$ and $Q_S(P)=9+6P$, where $P$ is dollars per unit. Find the feasible equilibrium. The model requires $P\geq0$ and both quantities to be nonnegative.

Method. Set the two quantity equations equal:

$$ 81-P^2=9+6P. $$

Move all terms to the left and simplify:

P^2+6P-72=0 \qquad\Longrightarrow\qquad (P+12)(P-6)=0.

The candidate prices are $P=-12$ and $P=6$. The price $-12$ is outside the feasible domain $P\geq0$, so discard it. At $P=6$, demand is $81-6^2=81-36=45$ units per week. Supply is $9+6(6)=9+36=45$ units per week. Both quantities are nonnegative and agree.

Answer: The equilibrium price is $\$6$ per unit and the equilibrium quantity is 45 units per week. The method avoids treating every algebraic root as an economically feasible price.

Worked Example: Using the Quadratic Formula and Exact Roots

A hypothetical market has $Q_D(P)=75-P^2$ and $Q_S(P)=11+2P$, with quantities in units per month and price in dollars per unit. Find the feasible equilibrium, first exactly and then to three decimal places. Assume $P\geq0$, and require both quantities to be nonnegative.

Method. Set demand equal to supply and rearrange:

75-P^2=11+2P \qquad\Longrightarrow\qquad P^2+2P-64=0.

For $a=1$, $b=2$, and $c=-64$, the quadratic formula gives

P=\frac{-2\pm\sqrt{2^2-4(1)(-64)}}{2(1)} =\frac{-2\pm\sqrt{260}}{2} =-1\pm\sqrt{65}.

Since $\sqrt{65}\approx8.062257$, the roots are approximately $7.062257$ and $-9.062257$. The negative root is infeasible. The feasible price is exactly $P^*=-1+\sqrt{65}$, approximately $\$7.062$ per unit when rounded to three decimal places.

Calculate the equilibrium quantity from supply using the exact price:

Q^*=11+2(-1+\sqrt{65})=9+2\sqrt{65}.

Now verify demand. Squaring the exact price gives $(-1+\sqrt{65})^2=66-2\sqrt{65}$, so

Q_D(P^*)=75-(66-2\sqrt{65})=9+2\sqrt{65}=Q_S(P^*).

Because $\sqrt{65}\approx8.062257$, the exact quantity $9+2\sqrt{65}$ is approximately $25.124515$, or 25.125 units per month to three decimal places. The exact expressions show equality; any tiny discrepancy after substituting rounded values is only a rounding effect.

Answer: $P^*=-1+\sqrt{65}\approx\$7.062$ per unit and $Q^*=9+2\sqrt{65}\approx25.125$ units per month. The method avoids rounding a root before evaluating the nonlinear demand curve.

Worked Example: Nonlinear Supply and Demand

For a hypothetical product, let $Q_D(P)=100-5P$ and $Q_S(P)=3P^2$, measured in units per day, with price in dollars per unit. Find the feasible equilibrium when $P\geq0$ and both quantities must be nonnegative.

Method. Equate the curves, keeping the squared price term intact:

100-5P=3P^2 \qquad\Longrightarrow\qquad 3P^2+5P-100=0.

Factor the quadratic:

$$ (3P+20)(P-5)=0. $$

The candidate prices are $P=-20/3$ and $P=5$. Reject $-20/3$, since it is negative. At $P=5$, demand is $100-5(5)=75$ units per day, while supply is $3(5^2)=3(25)=75$ units per day. The price and common quantity both satisfy the feasibility conditions.

Answer: The equilibrium is $\$5$ per unit and 75 units per day. The method avoids the common error of evaluating the nonlinear supply equation as $3P$ instead of $3P^2$.

Common Mistakes and Checks

A candidate solution is not confirmed merely because it solves a rearranged equation. Use the original demand and supply equations for the final check. In the second example, trying $P=8$ might seem plausible if one estimates the price or misreads the quadratic. But at that price, $Q_D=75-8^2=75-64=11$, whereas $Q_S=11+2(8)=27$. Since 11 is not 27, $P=8$ is not an equilibrium. Solving the equation correctly gives $P=-1+\sqrt{65}$, not 8.

Keeping only one root without checking it. Solve for all algebraic roots first, then apply the model’s domain restrictions. Do not discard a root just because it is negative until you have identified the price domain; in the usual nonnegative-price model, a negative root is infeasible.
Substituting into only one curve. A computed quantity from demand alone does not establish equilibrium. Evaluate supply at the same price and confirm equality.
Using the wrong exponent or sign. For $81-P^2$, the squared term is $P^2$, not $P$; moving it across the equality changes its sign.
Rounding too early. Retain an exact root, or enough digits, while evaluating quantities. Round the reported result only at the final step and state the precision.
Omitting units. A price is in dollars per unit, while a quantity may be in units per week or another stated period. Keep those units attached to the final answer.

Nonlinear curves do not change the meaning of equilibrium; they change the algebra required to find it. A sound solution separates the tasks: form the equality, solve for candidate prices, check feasibility, and verify a common quantity from both curves. This discipline is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, solve for the feasible price and quantity, and use both curves to check the equilibrium.

Demand is $Q_D(P)=64-P^2$ and supply is $Q_S(P)=4+4P$. Quantities are units per day, $P\geq0$, and both quantities must be nonnegative. Find equilibrium. (Answer: $P=6$, $Q=28$ units per day.)
Demand is $Q_D(P)=150-3P^2$ and supply is $Q_S(P)=6+3P$. Find the feasible equilibrium for $P\geq0$, reporting price and quantity to three decimal places. (Answer: $P\approx6.446$, $Q\approx25.339$.)
Demand is $Q_D(P)=90-2P$, and supply is $Q_S(P)=P^2+6$. Find the feasible equilibrium for nonnegative prices and quantities. (Answer: $P=-1+\sqrt{85}\approx8.220$, $Q=92-2\sqrt{85}\approx73.561$.)
A market has demand $Q_D(P)=40-P^2$ and supply $Q_S(P)=4+2P$. Quantities are units per week. Find the nonnegative equilibrium, reporting both values to three decimal places. (Answer: $P=-1+\sqrt{37}\approx5.083$, $Q=2+2\sqrt{37}\approx14.166$ units per week.)
Demand is $Q_D(P)=80-2P^2$, and supply is $Q_S(P)=10+4P$. Find the feasible equilibrium for $P\geq0$, showing why any other algebraic root is not retained. (Answer: $P=5$, $Q=30$; the other root is $P=-7$, which is infeasible.)

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