Monday, January 13, 2014

The Effect of Sales Tax

The Utility Maximization Problem:

Only one person makes a consumption decision to make a living in two-term. Her utility is Cobb-Dauglas version whose elasticity of substitution is 1, which means her labor supply is vertical to real wage.

max  U(c1,k1) + (1+ρ)^(-1)U(c2,k2)
{ci,ki} i=1,2

st. (1 + t1) c1 + (1 + t2)(1+r)^(-1) c2 = w L1 + (1+r)^(-1) w L2
    ki + Li = Ti, i=1,2

U(ci, ki)=ln(ci)+ln(ki): log function of utility
ci: consumption at i=1,2
ki: leisure time at i=1,2
ρ: discount rate and constant
r: real interest rate and constant
w: real wage rate and constant
ti: sales tax at i=1,2
Li: labor time at i=1,2
Ti: total available time at i=1,2 and constant

The conclusion:

(c2-c1)/c1 = (t1-t2)/(1+t2)  if r=ρ....(1)

Equation (1) means that if sales tax is high in the future (t2 is high), then her consumption growth,  (c2-c1)/c1, is decreased, but her labor supply is not influenced by higher tax due to her utility functional form.

(Li - Ti)/ci = (1+ti)/w ...(2)

Equation (2) means that if sales tax is high (ti is high), then her consumption level, ci, is decreased with her labor supply kept constant.

Not only is consumption level of either term but also consumption growth decreased by high sales tax.

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