Page 355 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Maximisation of Tax Revenue

Let p = f(x) and p = g(x) be the market demand and supply of a commodity and a specific tax of
t per unit be imposed. Then under equilibrium, we can write f(x) = g(x) + t.
Let x be the equilibrium quantity obtained by solving the above equation for x. We can write
t
the expression for tax revenue T as T = t.x (note that x is a function of t).
t t
From this we can find t such that T is maximum.

Example: The inverse demand and supply functions of a commodity, in a perfectly
competitive market, are given by p x and p b ax respectively, where a, b, a, b > 0 and b > b.

Find the equilibrium values if p and x. If the government imposes a specific tax @ t per unit,
find post-tax equilibrium values. Also find the value of t for maximum tax revenue.

Solution:
We have demand price = supply price, (in equilibrium)
b
x = b + ax or x , is the equilibrium quantity. We substitute this value in demand
a
function to get the equilibrium price.
b a b
Thus, p
a a
After a specific tax of t per unit, the equilibrium condition becomes:
demand price = supply price + t
b t
or x b ax t x t
a
b t a b t
The post-tax equilibrium price p
a a
t b t
The tax revenue T . t x
t
a
dT b 2t 1
Thus 0, for maximum T, t b .
dt a 2

Example: The inverse demand and supply of a commodity in a perfectly competitive
market are given by p = f(x) and p = g(x), where f x 0 and g x 0 . If a specific tax of t per
unit is imposed, show that equilibrium output decreases as tax rate t increases.
Solution:

Let a specific tax of t per unit be imposed on the commodity with demand and supply function
as p f x and p s g x . Where p denotes price paid by the consumer and p is the price received
s
by the seller. Thus under equilibrium we have
s
p = p + t or f(x) = g(x) + t
Let x (the equilibrium quantity) be the solution of this equation.
t
Therefore, we can write f x g x t
t t
Since x is a function of t, we can differentiate the above equation with respect to t, to get
t

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