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)