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Dynamic effects of consumption tax reforms with durable consumption

  • Qian Li EMAIL logo

Abstract

This paper introduces durables into a dynamic general equilibrium overlapping generation model with idiosyncratic income shocks and endogenous borrowing constraints, which depend on durables. The aim of this paper is to evaluate the welfare effects of consumption tax reforms in a richer model that captures the difference between nondurable and durable consumption. When durables are considered, the standard results that a shift to consumption taxes is welfare improving are overturned. The mechanism of this opposing result is that consumption tax makes durable consumption more expensive without relaxing the borrowing constraint. The inability of borrowing to insure against income risk deviates the economy further away from market completeness and particularly hurts young and poor households. As a result, welfare decreases, coupled with negative redistribution.

JEL Classification: D52; E2; E62; H21

A Definition of welfare gain

A.1 Welfare gain without durables

The welfare gain and the decomposition of the welfare gain are defined in the same way as Domeij and Heathcote (2004), except that we have overlapping generations and that the long-run and short-run welfare gain take different forms.

Specifically, the long run average welfare gain is defined as how much consumption need to be given to newborns in the future in order for them to be indifferent about the reform. Let cjNR and cjR are pre- and post-reform consumption of a newborn household at age j, similarly, ljNR and ljR are the hours worked (note that hours worked becomes 0 after retirement age J0), then the long run average welfare gain ΔLR is the solution of the following equation,

A × E E j = 1 J 1 β j ( Π s = 1 j ϕ s ) ( ( c j R ) 1 σ 1 σ B ( l j R ) 1 + 1 / χ 1 + 1 / χ ) d Ψ ^ 1 R ( a , ϵ ) = A × E E j = 1 J 1 β j ( Π s = 1 j ϕ s ) ( ( ( 1 + Δ L R ) c j N R ) 1 σ 1 σ B ( l j N R 1 + 1 / χ 1 + 1 / χ ) d Ψ ^ 1 N R ( a , ϵ ) ,

where Φ^1NR(a,ϵ) and Φ^1R(a,ϵ) are the conditional cumulative distribution function over assets a and labor efficiency ϵ at age 1 pre- and post-reform. These conditional cumulative distribution functions are different from Ψ~1(a,d,ϵ) where durables d are also an argument.

The short-run welfare gain are in term of the consumption equivalent of the current population. Let ΔSR denote the immediate average welfare gain of the current population in terms of the consumption equivalent, then it is the solution of the following equation

j = 1 J 1 ψ j A × E E t = j J 1 β t j ( Π s = j + 1 t ϕ s ) ( ( c t R ) 1 σ 1 σ B ( l t R ) 1 + 1 / χ 1 + 1 / χ ) d Ψ ^ j R ( a , ϵ ) = j = 1 J 1 ψ j A × E E t = j J 1 β t j ( Π s = j + 1 t ϕ s ) ( ( ( 1 + Δ L R ) c t N R ) 1 σ 1 σ B ( l t N R ) 1 + 1 / χ 1 + 1 / χ ) d Ψ ^ j N R ( a , ϵ )

Define c^jR=c^jNRCRCNR, where CNR and CR are the aggregate consumption pre- and post-reform. Similarly, l^jR=l^jNRHRHNR, where HNR and HR are aggregate hours worked pre- and post-reform. The aggregate component of the short run welfare gain ΔaggSRis the solution of the following equation

j = 1 J 1 ψ j A × E E t = j J 1 β t j ( Π s = j + 1 t ϕ s ) ( ( c ^ t R ) 1 σ 1 σ B ( l ^ t R ) 1 + 1 / χ 1 + 1 / χ ) d Ψ ^ j R ( a , ϵ ) = j = 1 J 1 ψ j A × E E t = j J 1 β t j ( Π s = j + 1 t ϕ s ) ( ( ( 1 + Δ L R ) c t N R ) 1 σ 1 σ B ( l t N R ) 1 + 1 / χ 1 + 1 / χ ) d Ψ ^ j N R ( a , ϵ )

At last, the redistribution component ΔdistSR is

Δ d i s t S R = ( 1 + Δ S R ) / ( 1 + Δ a g g S R ) 1.

A.2 Welfare gain with durables

Here the consumption equivalent is defined as how much more consumption bundle of durables and nondurable (γcjξ+(1γ)dj+1ξ)1/ξ should be given in order for a newborn household to be indifferent about the reform. Specifically, denote the average long run welfare gain as ΔLR. It solves the following equation:

A × D × E E j = 1 J 1 β j ( Π s = 1 j ϕ s ) [ ( ( γ ( c j R ) ξ + ( 1 γ ) ( d j + 1 R ) ξ ) 1 / ξ ) 1 σ 1 σ B ( l j R ) 1 + 1 / χ 1 + 1 / χ ] d Ψ ~ 1 R ( a , d , ϵ ) = A × D × E E j = 1 J 1 β j ( Π s = 1 j ϕ s ) [ ( ( 1 + Δ L R ) ( γ ( c j N R ) ξ + ( 1 γ ) ( d j + 1 N R ) ξ ) 1 / ξ ) 1 σ 1 σ B ( l j N R ) 1 + 1 / χ 1 + 1 / χ ] d Ψ ~ 1 N R ( a , d , ϵ ) ,

where cNR and cR are household nondurable consumption over time if there is no reform and if there is a reform. Other variables are interpreted in the same way. Ψ1~ is the conditional cumulative distribution function at age 1.

Similar to the long run welfare gain, here the consumption equivalent is defined as how much more consumption bundle of durables and nondurable (γcξ+(1γ)dξ)1/ξ should be given in order for the existing households to be indifferent about the reform. Let ΔSR denote the average welfare gain in the short run. It solves the following equation:

j = 1 J 1 ψ j A × D × E E t = j J 1 β t j ( Π s = j + 1 t ϕ s ) [ ( ( γ ( c t R ) ξ + ( 1 γ ) ( d t + 1 R ) ξ ) 1 / ξ ) 1 σ 1 σ B ( l t R ) 1 + 1 / χ 1 + 1 / χ ] d Ψ ~ j N R ( a , d , ϵ ) = j = 1 J 1 ψ j A × D × E E t = j J 1 β t j ( Π s = j + 1 t ϕ s ) [ ( ( 1 + Δ S R ) ( γ ( c t N R ) ξ + ( 1 γ ) ( d t + 1 N R ) ξ ) 1 / ξ ) 1 σ 1 σ B ( l t N R ) 1 + 1 / χ 1 + 1 / χ ] d Ψ ~ j N R ( a , d , ϵ ) ,

because ΔSR captures the immediate welfare gain of the reform, the immediate distribution post-reform remains the same as if there is no reform.

We also define the aggregate component of the welfare gain in a similar way. Define

( γ ( c ^ j R ) ξ + ( 1 γ ) ( d ^ j + 1 R ) ξ ) 1 / ξ = ( γ ( c ^ j N R C R C N R ) ξ + ( 1 γ ) ( d ^ j + 1 N R D R D N R ) ξ ) 1 / ξ l ^ j R = l ^ j N R H R H N R

Denote the short run aggregate component by ΔaggSR, it is the solution of the following equation:

j = 1 J 1 ψ j A × D × E E t = j J 1 β t j ( Π s = j t ϕ s ) [ ( γ ( c ^ t R ) ξ + ( 1 γ ) ( d ^ t + 1 R ) ξ ) 1 / ξ ) 1 σ 1 σ B ( l ^ t R ) 1 + 1 / χ 1 + 1 / χ ] d Ψ ~ j N R ( a , d , ϵ ) = j = 1 J 1 ψ j A × D × E E t = j J 1 β t j ( Π s = j t ϕ s ) [ ( ( 1 + Δ a g g S R ) ( γ ( c t N R ) ξ + ( 1 γ ) ( d t + 1 N R ) ξ ) 1 / ξ ) 1 σ 1 σ B ( l t N R ) 1 + 1 / χ 1 + 1 / χ ] d Ψ ~ j N R ( a , d , ϵ ) .

Lastly, the short run redistribution component ΔdistSR is

Δ d i s t S R = ( 1 + Δ S R ) / ( 1 + Δ a g g S R ) 1

B FOC for model with durables

The maximization problem of the benchmark economy is

max { c j , d j + 1 , a j + 1 , l j } E j = 1 J 1 β j ( Π s = 1 j ) u ( c j , d j + 1 , l j )

subject to

c j + d j + 1 + m j + a j + 1 = ( 1 + r ) a j + y j + ( 1 δ d ) d j T a x j a j + 1 + ( 1 θ ) d j + 1 0 c j > 0 , d j + 1 > 0 , 0 < l j 1.

Let βj(Πs=1j)λj be the multiplier of the budget constraint, βj(Πs=1j)ζj be the multiplier of the borrowing constraint, the rest of the constraints are slack. Then we have the Lagrangian function

L = E j = 1 J 1 β j ( Π s = 1 j ) { u ( c j , d j + 1 , l j ) λ j ( c j + d j + 1 + m j + a j + 1 ( 1 + r ) a j y j ( 1 δ d ) d j + T a x j ) + ζ j ( a j + 1 + ( 1 θ ) d j + 1 ) } .

Taking derivatives with respective the arguments:

c j : u c j = λ j a j + 1 : λ j = β ϕ j + 1 ( 1 + ( 1 τ a ) r ) E λ j + 1 + ζ j d j + 1 : λ j = β ϕ j + 1 ( 1 μ ) ( 1 δ d ) E λ j + 1 + u d j + 1 + ( 1 θ ) ζ j , if  d ( 1 δ d ) d l j : u l j = λ j ( y j T a x j ) l j , if  j J 0 .

Rearranging and we get the following intratemporal and intertemporal conditions:

u / c j u / l j = 1 ( y j T a x j ) / l j , if  j J 0 u d j + 1 = β ϕ j + 1 ( ( 1 τ a ) r + μ + δ d μ δ d ) E u c j + 1 + θ ζ j , if  d ( 1 δ d ) d u c j = β ϕ j + 1 ( 1 + ( 1 τ a ) r ) E u c j + 1 + ζ j , if  d = ( 1 δ d ) d .

Because our conclusion rests on the relative tightness of the post-reform borrowing constraint, we write the post-reform borrowing constraint in a more general way, aB(d), where B() is a function of d’. Denote the tax rates of nondurables and durables as τc and τd, then the first order conditions become

u / c j u / l j = 1 + τ c y j / l j , if  j J 0 u d j + 1 = β ϕ j + 1 1 + τ d 1 + τ c ( r + μ + δ d μ δ d ) E u c j + 1 + ( 1 + τ c + B ( d ) ) ζ j , if  d ( 1 δ d ) d u c j = β ϕ j + 1 ( 1 + r ) E u c j + 1 + ( 1 + τ c ) ζ j , if  d = ( 1 δ d ) d .

In Section 4.2, when equal tax rates on nondurables and durables τc=τd, and function B(d) takes the form of B(d)=(1θ)d, the intertemporal condition of adjusting d’ is

u d j + 1 = β ϕ j + 1 ( r + μ + δ d μ δ d ) E u c j + 1 + ( θ + τ c ) ζ j .

With equal tax rates but a zero borrowing constraint B(d)=0 as in Section 4.3.1, the intertemporal condition of adjusting d’ is

u d j + 1 = β ϕ j + 1 ( r + μ + δ d μ δ d ) E u c j + 1 + ( 1 + τ c ) ζ j .

For the more relaxed borrowing constraint in Section 4.3.1B(d)=(1θ)(1+τc)d, the above first order condition becomes

u d j + 1 = β ϕ j + 1 ( r + μ + δ d μ δ d ) E u c j + 1 + θ ( 1 + τ c ) ζ j .

At last, in the section that explores the optimal tax mix, we have τcτd, but B(d)=(1θ)d, the first order condition corresponding to changing durable stock is

u d j + 1 = β ϕ j + 1 1 + τ d 1 + τ c ( r + μ + δ d μ δ d ) E u c j + 1 + ( θ + τ c ) ζ j .

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Published Online: 2020-02-28

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