Competitive equilibrium

Solve a competitive market model and distinguish an equilibrium condition from an adjustment story.

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1. What you will learn

You will solve a competitive market model and distinguish an equilibrium condition from an adjustment story, showing the calculation and stating the assumptions that make the conclusion valid.

2. Starting point

Use the supplied definitions and units. Separate an accounting identity, a behavioral assumption, and a normative criterion before drawing conclusions.

3. Terms to use precisely

4. Equilibrium connects two contingent plans

A demand schedule describes quantities buyers would choose at alternative prices under specified conditions. A supply schedule describes quantities sellers would choose at those prices under their own conditions. Neither schedule is simply a record of observed transactions. Both are conditional plans. Competitive market clearing asks whether a common price makes the aggregate planned purchase quantity equal the aggregate planned sale quantity.

Write demand as Qd=a-b P and supply as Qs=c+dP, with b and d positive. Quantity is measured in the same units and period on both sides. Price is the payment per unit. The constants summarize other conditions such as income, input costs, technology and the number of participants. Holding those conditions fixed makes each equation a schedule rather than an unrestricted description of everything that might change together.

At an interior competitive equilibrium, Qd=Qs. Substituting the equations gives a-b P=c+dP, hence a-c=(b+d)P. The candidate price is (a-c)/(b+d). Substitution into either schedule gives the candidate quantity. Substituting into both is a useful independent arithmetic check because a sign error can otherwise survive all the way to the final answer.

The equality concerns desired quantities at that price under the model. It does not say buyers and sellers have identical motives, incomes or bargaining power. Nor does it imply that every person can afford everything they want. Buyers' willingness and ability to pay are built into the demand schedule. Equilibrium is a compatibility condition for these modeled plans, with no automatic conclusion about fairness.

Term	What it means
Market clearing	Equality of planned purchases and sales at a common price.
Excess demand	Desired purchases minus desired sales at the quoted price.
Comparative statics	Comparison of equilibria under different stated parameters.
Endogenous variable	A quantity determined within the model's simultaneous conditions.
Stability	Whether a specified adjustment process returns toward equilibrium after a disturbance.

Another way: Solve the equation and then inspect the domain

Suppose Qd=30-2P and Qs=6+P. Equality gives thirty minus six equal to three times price, so P eight and Q fourteen. Demand is thirty minus sixteen, while supply is six plus eight. Both yield fourteen. The calculation has a clear interpretation only if the price and quantity lie within the domain for which these linear schedules were supplied.

A linear expression can predict a negative quantity at some prices. Physical purchases of the good are not normally negative. A model may restrict attention to a relevant price interval or define actual demand as the maximum of zero and the linear expression. These are different mathematical specifications outside the positive segment. Do not silently extend a fitted line through impossible quantities and call the intersection economically meaningful.

For example, if demand is four minus price and supply is ten plus price, the unrestricted intersection has price negative three and quantity seven. Whether a negative price is permitted depends on the good and institutional setting. Some disposal services can involve negative prices for the material itself. An ordinary goods exercise that explicitly restricts price to be nonnegative cannot simply accept that unrestricted candidate. It must analyze its boundary rules and trading mechanism.

Likewise, an intersection at quantity zero may involve a range of prices compatible with no trade when reservation values do not overlap. A positive-slope supply and negative-slope demand on a positive interior region give a unique intersection there, but this special geometry is not a theorem that every conceivable market has exactly one equilibrium. Nonlinearities, discontinuities, capacity limits and indivisibilities require separate analysis.

Another way: Excess demand describes a discrepancy, not an automatic adjustment law

At a quoted price P, define excess demand as Qd-Qs. A positive value means desired purchases exceed desired sales at that price. A negative value means desired sales exceed desired purchases. In the example Qd thirty minus twice price and Qs six plus price, a price of six produces demand eighteen and supply twelve, so excess demand is six units.

Only the smaller of planned purchases and planned sales can be traded if the price is fixed and there are no inventories or outside sources. Which buyers receive the units depends on a rationing institution that the two schedules alone do not describe. Waiting, lotteries, seller discretion and prior contracts can allocate the same limited quantity differently. A shortage number is therefore not a complete account of the market experience.

An adjustment story might posit that price rises when excess demand is positive. That story adds a dynamic rule to the static model. For example, a continuous adjustment equation could make the rate of price change proportional to excess demand. Under the supplied linear slopes, small disturbances then move back toward the intersection. But the conclusion depends on that added mechanism, its timing and the information available to participants.

With delayed production, inventory feedback or discrete price changes, adjustment can oscillate or behave differently. Calling an intersection an equilibrium establishes a fixed point of the modeled plans; it does not prove that a decentralized process reaches it quickly, uniquely or at all. Separate existence, uniqueness and stability. Each is a different question, and an answer to one cannot be substituted for an argument about another.

Another way: Comparative statics changes one stated condition

Comparative statics compares equilibria before and after a parameter change without describing the intervening path. If demand intercept a increases while b, c and d remain fixed, the equilibrium price rises by the increase in a divided by b+d. Quantity rises by d times that price change. The supply slope governs how much of the additional willingness to buy becomes additional output in this model.

With the original schedules, raising the demand intercept from thirty to thirty-six changes equilibrium price from eight to ten and quantity from fourteen to sixteen. The demand curve has shifted. Supply has not shifted: sellers move along their existing schedule in response to the higher price. Saying that higher price shifted supply confuses a change in quantity supplied with a change in the schedule itself.

If a cost reduction raises supply intercept c instead, the candidate equilibrium price falls by the increase in c divided by b+d. Quantity rises by b times that price reduction in absolute value. The response depends on both sides of the market; looking only at the supply shift cannot tell the complete equilibrium price and quantity effect.

Several parameters can move together. A festival might raise visitor demand while a transport disruption lowers supply. The price effects reinforce each other, but the quantity effects can oppose each other. Without their magnitudes, the quantity direction is ambiguous. A diagram showing one shift cannot justify a confident conclusion about a situation in which important omitted shifts may also occur.

Another way: Elasticities and identification limit the interpretation

A slope is a quantity change per unit price change, while an elasticity is a percentage quantity change relative to a percentage price change. Along linear demand, the derivative is minus b, but the point elasticity is minus b times P divided by Q. A straight line with constant slope generally has changing elasticity because the price-to-quantity ratio changes across its points.

Units matter for slopes. Measuring quantity in individual items rather than hundreds changes the numerical slope, even though the underlying demand behavior is the same. An elasticity avoids that particular units problem by using proportional changes. It still depends on the point, the time horizon, the good definition and the conditions held fixed. Neither a slope nor an elasticity is a universal property detached from its model.

Observed price and quantity pairs usually reflect the joint determination of both variables. Connecting several observed equilibria does not automatically trace a demand curve. If supply shifted while demand stayed fixed, those points might reveal movement along demand. If both schedules shifted, the same observations mix several mechanisms. Estimating a causal demand response requires a credible source of variation and assumptions about what remained fixed.

Even a correctly identified demand curve need not be enough for a policy analysis. A tax or entry rule can change behavior on both sides, introduce evasion, or alter quality. The simple market-clearing model can organize a counterfactual, but its conclusions inherit its assumptions. State which variables are endogenous, which parameters are held fixed, and which institutions the calculation leaves unspecified.

Another way: Partial equilibrium holds a surrounding economy fixed

This lesson solves one market while treating outside conditions as given. That is a partial-equilibrium analysis. If the part is a small share of spending and uses a small share of relevant inputs, holding other markets approximately fixed may be a useful first approximation. If a large energy price change alters transport costs, household purchasing power and production across many industries, those feedbacks can be central. A general-equilibrium model brings some of them into the system rather than leaving them inside constants. The distinction concerns the boundary of the analysis, not whether one kind of model is always superior. A useful report names the excluded feedbacks and explains why they are plausibly small for the question, or why the calculation should be treated as only an initial conditional benchmark.

5. A fictional repair-parts exchange

An analyst studies a weekly exchange for standardized replacement parts. For this exercise, planned buyer demand is Qd=48-2P and planned seller supply is Qs=8+2P, with price in accounting credits and quantity in boxes per week. The quantities describe the same part quality and delivery period. Solving the two plans together gives P ten and Q twenty-eight. Checking demand and supply separately confirms the same twenty-eight boxes.

A community event then increases the demand intercept to fifty-six while costs, seller numbers and both slopes remain fixed in the exercise. The new candidate price is twelve and quantity thirty-two. The analyst describes this as a demand shift and a movement along the unchanged supply curve. That wording records the causal assumption instead of claiming that the higher price itself increased the supply intercept.

The exchange manager asks whether posting a price of nine would ensure wider access. Under the original schedules, desired purchases are thirty and desired sales twenty-six. The four-box excess demand does not specify who would receive the available boxes. A line, lottery, voucher system or seller choice could produce different distributions. The analyst therefore reports the planned discrepancy separately from any judgment about the allocation rule.

Finally, real observations show a price increase during a transport delay. The analyst cannot treat that observation as confirmation of the event-demand calculation without checking supply conditions. The exercise's result follows from its stated parameters; an empirical explanation requires evidence that the relevant shifts and fixed conditions match the actual setting.

6. Check the tempting inference

An observed price rise is not itself a demand shift. Check domains after solving; negative quantities do not become feasible because algebra produced them. Market clearing is not a fairness criterion or a proof of dynamic stability.

7. A positive interior intersection

8. A demand shift with supply held fixed

9. A fixed price needs a rationing account

10. Complete a new intersection

11. Guided practice

Qd=30-2P and Qs=6+P. Prices and quantities are nonnegative. Find the equilibrium and signed excess demand at quoted price 6.

12. Guided practice

For Qd=32-P and Qs=8+P, derive the equilibrium and signed excess demand at quoted price 10.

13. Guided practice

Qd=40-2P and Qs=8+2P. Find equilibrium price and quantity, and signed excess demand at quoted price 10.

	Your result
Equilibrium price
Equilibrium quantity
Excess demand at quoted price