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Questions › GATE › GATE 2021 › Q35
2 marksPublic Economics › Taxation: incidence and efficiency › The Laffer curve, tax buoyancy and tax revenueHardabout 3 min

GATE 2021 Question 35: Taxation: incidence and efficiency

Consider a competitive market where the demand and supply functions are given by qD=12−2Pq^D=12-2PqD=12−2P and qS=4Pq^S=4PqS=4P, respectively. The tax rate per unit of output that maximizes the tax yield (revenue) is ______ (in integer).

Answer and worked solution

Answer: 3

With a per-unit tax ttt imposed on suppliers, suppliers receive P−tP-tP−t per unit sold, so the supply curve becomes qS=4(P−t)q^S = 4(P-t)qS=4(P−t).

Step 1 - Find the equilibrium price as a function of ttt. Setting demand equal to (taxed) supply:

12−2P=4(P−t)⇒12−2P=4P−4t⇒12+4t=6P⇒P=2+2t312-2P = 4(P-t) \Rightarrow 12-2P = 4P-4t \Rightarrow 12+4t = 6P \Rightarrow P = 2+\frac{2t}{3}12−2P=4(P−t)⇒12−2P=4P−4t⇒12+4t=6P⇒P=2+32t​

Step 2 - Find the equilibrium quantity.

Q=12−2P=12−2(2+2t3)=8−4t3Q = 12-2P = 12-2\left(2+\frac{2t}{3}\right) = 8-\frac{4t}{3}Q=12−2P=12−2(2+32t​)=8−34t​

Step 3 - Set up total tax revenue.

Revenue(t)=t×Q=t(8−4t3)=8t−4t23Revenue(t) = t\times Q = t\left(8-\frac{4t}{3}\right) = 8t-\frac{4t^2}{3}Revenue(t)=t×Q=t(8−34t​)=8t−34t2​

Step 4 - Maximize over ttt.

d(Revenue)dt=8−8t3=0⇒t∗=3\frac{d(Revenue)}{dt} = 8-\frac{8t}{3}=0 \Rightarrow t^* = 3dtd(Revenue)​=8−38t​=0⇒t∗=3

Therefore the revenue-maximizing tax rate is 3.

The trap: 2

Students sometimes forget to substitute the tax-adjusted equilibrium price/quantity as a FUNCTION of ttt before setting up the revenue-maximization problem, and instead try to maximize using the original (untaxed) equilibrium quantities, which produces an incorrect optimal tax rate.

Plain-text version

Consider a competitive market where the demand and supply functions are given by q^D=12-2P and q^S=4P, respectively. The tax rate per unit of output that maximizes the tax yield (revenue) is (in integer).

Taxation: incidence and efficiency has come up 79 times in 7 exams since 2009.

What this question tests

This question is on The Laffer curve, tax buoyancy and tax revenue, part of Taxation: incidence and efficiency in Public Economics.

How revenue responds to rates and to growth.

TestStop has 11 past-paper questions on this subtopic.

Take this paper as a timed mockMore on The Laffer curve, tax buoyancy and tax revenue

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