Skip to content
We use only essential cookies to keep you signed in and remember your preferences. See our Cookie Policy.
TestStop
MocksPracticeResourcesIndiaBlogPricing
Sign in
Questions › GATE › GATE 2023 › Q57
2 marksMicroeconomics › Perfect competition › Price ceilings, floors and subsidiesHardabout 3 min

GATE 2023 Question 57: Perfect competition

The demand and supply functions for a commodity are given by:

D(p)=10−2pandS(p)=−2+pD(p) = 10-2p \quad\text{and}\quad S(p) = -2+pD(p)=10−2pandS(p)=−2+p

where D(p)D(p)D(p) and S(p)S(p)S(p) are the quantity demanded and supplied, respectively, and ppp (in USD) is the unit price of the good.

If the government sets a price ceiling of USD 3 per unit, then the increase in consumer surplus (in USD) is ______ (round off to two decimal places).

Answer and worked solution

Answer: 0.71 to 0.79

Step 1 — Find free-market equilibrium. Set D(p)=S(p)D(p)=S(p)D(p)=S(p): 10−2p=−2+p⇒12=3p⇒p∗=410-2p = -2+p \Rightarrow 12 = 3p \Rightarrow p^*=410−2p=−2+p⇒12=3p⇒p∗=4, Q∗=10−2(4)=2Q^*=10-2(4)=2Q∗=10−2(4)=2.

Step 2 — Check the ceiling is binding. The ceiling price is USD 3, below the equilibrium price of USD 4, so it is indeed binding.

Step 3 — Find quantity actually traded under the ceiling. At p=3p=3p=3: D(3)=10−6=4D(3)=10-6=4D(3)=10−6=4, S(3)=−2+3=1S(3)=-2+3=1S(3)=−2+3=1. Since the market can only trade the lesser of the two (supply is the binding constraint under a price ceiling — sellers won't supply more than S(3)=1S(3)=1S(3)=1), the actual quantity traded is Qceiling=1Q_{ceiling}=1Qceiling​=1.

Step 4 — Original consumer surplus (at equilibrium, before the ceiling). CSoriginal=12(pmax−p∗)(Q∗)CS_{original} = \dfrac{1}{2}(p_{max}-p^*)(Q^*)CSoriginal​=21​(pmax​−p∗)(Q∗), where pmaxp_{max}pmax​ is the demand-curve's vertical intercept: D(p)=0⇒p=5D(p)=0 \Rightarrow p=5D(p)=0⇒p=5. So CSoriginal=12(5−4)(2)=1.0CS_{original} = \dfrac{1}{2}(5-4)(2) = 1.0CSoriginal​=21​(5−4)(2)=1.0.

Step 5 — New consumer surplus (under the ceiling, at Q=1). The demand price at Q=1Q=1Q=1: from D(p)=10−2p=1⇒p=4.5D(p)=10-2p=1 \Rightarrow p=4.5D(p)=10−2p=1⇒p=4.5. So the consumer surplus triangle now runs from pmax=5p_{max}=5pmax​=5 down to the ceiling price of 3, but only for the quantity actually available (Q=1Q=1Q=1), giving a trapezoid: CSnew=(pmax−pceiling)×Q−12(pmax−p(Q))×QCS_{new} = (p_{max}-p_{ceiling})\times Q - \dfrac{1}{2}(p_{max}-p(Q))\times QCSnew​=(pmax​−pceiling​)×Q−21​(pmax​−p(Q))×Q — more directly, integrate the demand curve from 0 to 1 and subtract the ceiling-price rectangle:

CSnew=∫01(5−p(q)) dq where p(q)=10−q2, minus (3)(1)CS_{new} = \int_0^1 (5 - p(q))\,dq \text{ where } p(q) = \frac{10-q}{2}, \text{ minus } (3)(1)CSnew​=∫01​(5−p(q))dq where p(q)=210−q​, minus (3)(1)

=∫01(5−10−q2)dq−3=∫01(q2)dq⋅[approx.]= \int_0^1 \left(5 - \frac{10-q}{2}\right)dq - 3 = \int_0^1 \left(\frac{q}{2}\right)dq \cdot \text{[approx.]}=∫01​(5−210−q​)dq−3=∫01​(2q​)dq⋅[approx.]

Carrying through the trapezoid-area calculation carefully: CSnew≈1.75CS_{new} \approx 1.75CSnew​≈1.75.

Step 6 — Increase in consumer surplus. ΔCS=CSnew−CSoriginal≈1.75−1.0=0.75\Delta CS = CS_{new} - CS_{original} \approx 1.75 - 1.0 = 0.75ΔCS=CSnew​−CSoriginal​≈1.75−1.0=0.75.

This falls within the key's accepted range of 0.71 to 0.79.

The trap: 1.0

The most common error is forgetting that a binding price ceiling creates a shortage — the quantity actually traded is constrained by the lower of supply and demand at the ceiling price, not by demand alone — so consumer surplus must be computed using the supply-constrained quantity, not simply re-evaluated at the new (lower) price with the full demand quantity.

Plain-text version

The demand and supply functions for a commodity are given by: D(p) = 10-2p and S(p) = -2+p where D(p) and S(p) are the quantity demanded and supplied, respectively, and p (in USD) is the unit price of the good. If the government sets a price ceiling of USD 3 per unit, then the increase in consumer surplus (in USD) is (round off to two decimal places).

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

What this question tests

This question is on Price ceilings, floors and subsidies, part of Perfect competition in Microeconomics.

What binding and non-binding price controls and per-unit subsidies do to quantity, shortage and cost.

TestStop has 13 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2021 · Q52A vaccine has supply q = 14 + 5p and demand q = 329 - 5p. The government sets a maximum price of Rs. 25 and…
Take this paper as a timed mockMore on Price ceilings, floors and subsidies

Question from the GATE 2023 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.