2 marks, −0.667MicroeconomicsPerfect competitionMarket equilibrium and comparative staticsHardabout 4 min

GATE 2021 Question 23: Perfect competition

Consider the following demand-supply model, where

Demand function: $P=Q^2-12Q+35$

Supply function: $4P-3Q=0$

The stable market equilibrium price-quantity combination will be

Tap an option to check your answer.

Answer and worked solution

Answer: (A)

Supply: $4P=3Q \Rightarrow P=\dfrac34 Q$ .

Step 1 - Find all equilibria by setting demand equal to supply:

$Q^2-12Q+35 = \frac34 Q$

Solving this quadratic gives two solutions: $Q=4$ (with $P=3$ ) and $Q=\dfrac{35}{4}$ (with $P=\dfrac{105}{16}$ ) — matching the quantities implied by options (A) and (B).

Step 2 - Check stability at each equilibrium. Under standard price-adjustment dynamics (price rises when there's excess demand, falls when there's excess supply), an equilibrium is locally stable when the demand curve is 'flatter or more negatively sloped' than the supply curve at that point — more precisely, stability requires the supply curve's slope to exceed the demand curve's slope at the equilibrium.

Demand's slope: $\dfrac{dP}{dQ}=2Q-12$ . Supply's slope: $\dfrac{dP}{dQ}=\dfrac34$ (constant).

At $Q=4$ : demand's slope $=2(4)-12=-4$ (negative, normal downward-sloping demand locally). Since the demand slope ( $-4$ ) is less than the supply slope ( $0.75$ ), this is a stable equilibrium — the standard, well-behaved configuration.

At $Q=35/4$ : demand's slope $=2(35/4)-12=5.5$ (positive — demand is locally upward-sloping at this point, an unusual feature of this quadratic demand curve). Since the demand slope ( $5.5$ ) exceeds the supply slope ( $0.75$ ), this configuration is unstable under standard adjustment dynamics.

Therefore the stable equilibrium is $(P^*,Q^*)=(3,4)$ , matching option (A).

The trap: (B)

(P^*,Q^*)=\left(\dfrac{105}{16},\dfrac{70}{8}\right)

Both candidate equilibria are legitimate algebraic solutions to demand=supply, so a student who solves the quadratic correctly but doesn't perform the SEPARATE stability check (comparing slopes at each point) has a 50% chance of picking the wrong one — the question specifically asks for the stable equilibrium, which requires this extra analytical step beyond simply solving the system.

Plain-text version

Consider the following demand-supply model, where Demand function: P=Q²-12Q+35 Supply function: 4P-3Q=0 The stable market equilibrium price-quantity combination will be (A) (P*,Q*)=(3,4) (B) (P*,Q*)=(105/16,70/8) (C) (P*,Q*)=(6,8) (D) (P*,Q*)=(14,3)

Also written as: Consider the following demand-supply model, where Demand function: P=Q^2-12Q+35 Supply function: 4P-3Q=0 The stable market equilibrium price-quantity combination will be

Perfect competition has come up 115 times in 7 exams since 2006.

What this question tests

This question is on Market equilibrium and comparative statics, part of Perfect competition in Microeconomics.

Finding equilibrium from demand and supply and tracing how it moves when either curve shifts.

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