INDEX NUMBER: 8104619
COURSE CODE: ECONS 363
ASSIGNMENT
The quantity demanded after tax t, can be expressed as Q=Q(p(t))
Differentiating quantity, Q(P(t)) with respect to t, is expressed as;
𝑑𝑄
𝑑𝑡
=[
𝑑𝑄
𝑑𝑝
×
𝑑𝑝
𝑝
]×𝑞
𝑑𝑡
𝑄
𝑝
× ,
𝑃
(Thus, multiplying and dividing this expression by 𝑄 , we will find that change in quantity as tax
change) is
𝑑𝑄
𝑑𝑡
𝑄
𝑑𝑝
= ɛ𝑑 (𝑝 ) ( 𝑑𝑡 )
Thus, the closer 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦 is to zero, the less the quantity falls.
Specifying Revenue:
R = P(t) . Q(P(t)), an increase in tax rate changes the revenue by :
𝑑𝑅
𝑑𝑡
𝑑𝑝
𝑑𝑡
𝑑𝑄
𝑑𝑝
𝑑𝑝
𝑑𝑡
( )=( )×Q + ( )×( )
Using the chain rule,
𝑑𝑅
𝑑𝑡
𝑑𝑝
𝑑𝑡
𝑑𝑄
𝑑𝑝
( )=( ) (𝑄 + 𝑃 ( ))
𝑑𝑝
𝑑𝑄
𝑃
= ( 𝑑𝑡 ) (1 + ( 𝑑𝑝 × 𝑄)
𝑑𝑝
=( 𝑑𝑡 ) × 𝑄(1+∈)
Thus the effect of a change in t on R depends on the elasticity of demand, ∈. Revenue rises with the
tax if the demand is inelastic (-1<∈<0 ) and falls if demand is elastic (∈<-1)