SUPPLEMENT TO TEMPTATION AND TAXATION (Econometrica, Vol. 78, No. 6, November 2010, )

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1 Econometrica Supplementary Material SUPPLEMENT TO TEMPTATION AND TAXATION (Econometrica, Vol. 78, No. 6, November 00, ) BY PER KRUSELL, BURHANETTINKURUŞÇU, ANDANTHONYA. SMITH, JR. GENERAL FRAMEWORK FOR THE PROOFS OF PROPOSITIONS 3 Letting ū(c c ) u(c c ) + v(c c ), the first-orer conitions for the competitive consumer s maximization problem are given by where ( + τ i )ū (c c ) r ū (c c ) an ( + τ i )v ( c c ) r v ( c c ) c r k + w + s ( + τ i )k c r k + w + s ( + τ i ) k c r k + w c r k + w an Using the first-orer conitions of the consumer, it is easy to show that k > k an ū (c c ) v ( c c ) u (c c ) + v (c c ) v ( c c )>0. We will use these below. The value function of the representative agent is given by U( k Pτ i ) ū(r k + w k r k + w ) v(r k + w + τ i ( k k ) k r k + w ) Differentiating the value function with respect to τ i an using the consumer s first-orer conitions, we obtain U ū (c c )τ k ( r i + ū (c c ) k + w { v ( c c ) k k + τ i k ) } v ( c c ) ( r k + w ) PROOF OF PROPOSITION : In partial equilibrium, Therefore, we obtain r 0an w 0. U (ū (c c ) v ( c c ))τ k i v ( c c ){ τ k k } i 00 The Econometric Society DOI: 0.398/ECTA86

2 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. Since k > k an ū (c c ) v ( c c )>0, then U < 0forallτ i 0. Therefore, the optimal tax rate has to be negative. Q.E.D. PROOF OF PROPOSITION 3: In this case, r k + w w ( k ) + r ( k ) k 0 an r k + w r ( k )( k k ) k. Using these relations, U (ū (c c ) v ( c c ))τ i k v ( c c ){ k k }+v ( c c )( k k ) r( k ) (ū (c c ) v ( c c ))τ k i { + v ( c c ){ k k } MRS r( k } ) where MRS v ( c c ), where MRS enotes marginal rate of substitution. v ( c c ) Taking the erivative of the first-orer conition for the actual choice with respect to τ i, we can show that k < 0. We will show that MRS r( k ) > 0. This implies that U < 0forallτ i 0. Thus, the optimal tax is negative, that is, τ i < 0. To show this, note that in equilibrium, r( k ) MRS r( k ) MRS + τ i where MRS ū(c c ) Therefore, it is enough to show that ū (c c ) MRS r( k ) > 0 Taking the erivative of r( k ) MRS + τ i with respect to τ i,weobtain MRS r( k ) MRS r( k ) Given that MRS ū(c c ) an ū (c c ) k < 0, it is then clear that MRS > 0. Q.E.D. PROOF OF PROPOSITION : In this case, w c w. Given these, we obtain U ū (c c )τ i k v ( c c ) { 0, k 0, k 0, c w,an k + τ i k } v ( c c ) r k (ū (c c ) v ( c c ))τ k i ( + v ( c c ) k MRS r ) The key ifference between the previous case an this one is that the consumer consumes his enowment, that is, MRS ū(w w ) MRS. Therefore, ū (w w ) 0 which

3 implies that MRS r 0. Secon, k TEMPTATION AND TAXATION 3 0. Thus, we obtain that U 0 inepenent of τ i, which implies that the consumer is inifferent to any τ i. Q.E.D. For the proof of Proposition 4, see the proof to Proposition 8, which stuies a T -perio economy with logarithmic utility. s.t. PROOF OF PROPOSITION 5: The problem of the consumer can be written as U(k k τ i ) max( + γ) c σ c c σ [ γ max c c σ c σ + δ( + βγ) σ c σ + δβ σ c + c r( k ) ( + τ i) r( k )k + w( k ) + s + w( k ) r( k ) ( + τ i) Y The first-orer conitions are This implies δ( + βγ) + γ r( k ) + τ }{{} i m( k τ i ) an δβ r( k ) + τ i Y c [ /σ an δ( + βγ) + [m( + γ k τ i ) ( σ)/σ [ /σ δ( + βγ) c m( + γ k τ i ) c c Y +[δβ /σ [m( k τ i ) ( σ)/σ an c [δβm( k τ i ) /σ c From these expressions we obtain [ /σ δ( + βγ) + [m( c + γ k τ i ) ( σ)/σ c +[δβ /σ [m( k x τ i ) ( σ)/σ

4 4 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. an c c [ β( + γ) + βγ /σ c c x [ /σ β( + γ) x + βγ Then we can write the objective function of the government, inserting the expressions above, as where x U( k k τ i ) ( + γ) c σ σ [ c σ γ σ c σ c σ + δ( + βγ) σ c σ + δ( + βγ) σ σ + δ c σ σ [ + γ ( x σ ) c σ + ( x σ σ )δβ c σ σ c ( ) k + f( k ) k an c ( ) k + f( k ) Taking the erivative of the objective function with respect to τ i an inserting [ β(+γ) +βγ /σ x,weobtain U( k k τ i ) [ + δr( k ) k + γ[ ( x σ ) + ( x σ )δβr( k ) k [ γ x σ + δβ [ β( + γ) + βγ /σ x σ x Let τ i be the tax rate that maximizes the commitment utility. Then τ i will generate the conition δr( k ) Using the first-orer conition δ(+βγ) +γ m( k τ i ), this implies ( + βγ) + γ m( k τ i ) r( k )

5 TEMPTATION AND TAXATION 5 It is easy to see that U( k k τ i ) 0atσ. Thus the subsiy that maximizes utility uner logarithmic utility is the same as the subsiy that maximizes the commitment utility. We now characterize the conition uner which U( k τ k τ i )<0 for σ> i hols, so that for σ>, the optimal subsiy is larger than the optimal subsiy that maximizes commitment utility. To o this, we take the erivative of U( k k τ i ) with respect to σ an evaluate at σ. If the erivative is )<0forσ marginally above σ. If )>0forσ marginally negative at σ, then U( k k τ i the erivative is positive at σ, then U( k k τ i above σ. First, for later use, we compute the objects x x σ σ [m( k τ i ) ( σ)/σ [ /σ δ( + βγ) [δβ /σ + γ m( k τ i ) [ +[δβ /σ [m( k τ i ) ( σ)/σ σ σ H m( k τ i ) [ β( + γ) + βγ /σ x an H (σ ) [m( k τ i ) δ( β) ( + γ)[ + δβ Secon, to fin k, take the erivative of the expression δ(+βγ) +γ m( k τ i ) with respect to τ i to obtain k [ σ + σ m( H k τ i ) δ( + βγ) + γ δ( + βγ) m( + γ k τ i )r( k ) m( k τ i )

6 6 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. We know that k < 0 an thus m( k τ i ) < 0 too. Moreover, H (σ ) (+βγ) +γ c. σr( k )[+/δ At σ, we have that x + γ + δ( + βγ) ( + γ)( + δβ) an x β + βγ + γ + δ( + βγ) ( + δβ) Using the expressions above, we can write U( k k τ i ) as U( k k τ i ) γ [ ( x σ ) + ( x σ )β H c }{{} K σ m( k τ i ) } {{ } K γ σ }{{ σ } K [ [ β( + γ) /σ x σ + δβ x σ + βγ [δr( k ) ( σ)/σ H m( k τ i ) τ }{{ i } K Take the erivative of U( k k τ i ) with respect to σ to obtain [ U( k σ τ k τ ) i i γ K K σ + K K σ K K σ K K σ If we evaluate this expression at σ, we obtain [ U( k σ τ k τ ) i i γ K K σ K K σ ( + βγ) + γ δ σr( k )[ + δ m( k τ i ) K σ δ( β) + γ + δ( + βγ) [m( k τ i ) m( k τ i ) K σ

7 TEMPTATION AND TAXATION 7 where K an K σ σ β log(x σ ) log(x ). Evaluating at σ aninserting m( k τ i ) +γ r( k +βγ ),weobtain [ U( k σ τ k τ ) i i γ m( k τ i ) δ( + βγ) Since m( k τ i ) < 0, if ( [ β β log + βγ ( + γ)r( k ) [( [ β + γ + δ( + βγ) β log + βγ ( + δβ) [ + γ + δ( + βγ) log ( + γ)( + δβ) + γ + δ( + βγ) ( + δβ) [ + γ + δ( + βγ) log ( + γ)( + δβ) ) [ + δ + ( β) + γ + δ( + βγ) ) [ + δ + ( β) + γ + δ( + βγ) > 0 then [ σ U( k k τ ) i < 0atσ. Therefore, it is optimal to increase the γ subsiy for σ>ifthis conition above hols. To show that it hols, let ϕ(β γ δ) β log( β(+γ+δ(+βγ)) ) log( +γ+δ(+βγ) )+ (+βγ)(+δβ) (+γ)(+δβ) ( β)(+δ). First, it is easy to show that lim +γ+δ(+βγ) γ ϕ(β γ δ) 0. Secon, we show that ϕ(βγδ) < 0forallβ δ <, which implies that ϕ(β γ δ) > 0 for all finite γ γ>0anβ δ < : ϕ(βγδ) γ β ( + δβ)( + βγ) β( + γ + δ( + βγ)) ( + βγ) + γ + δ( + βγ) + βγ ( + δβ)( + γ) ( + γ + δ( + βγ)) ( + γ) + γ + δ( + βγ) + γ ( β)( + δ)( + δβ) ( + γ + δ( + βγ))

8 8 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. β + γ + δ( + βγ) { β + βγ + δ } ( + δ)( + δβ) + γ + γ + δ( + βγ) β + γ + δ( + βγ) The numerator of the term in curly brackets is { β + βγ + δ + δβγ + γ + βγ + βγ + δ + δβ + δ β + γ + δ + δβγ [β + βγ + δ + δβγ+[βγ + βγ + δγ + δβγ +[δβ + δβγ + δ + δ βγ +[δβ γ + δβ γ + δ βγ + δ β γ [ + δ + δβ + δ β [γ + δγ + δβγ + δ βγ [βγ + δβγ + δβ γ + δ β γ [βγ + δβγ + δβ γ + δ β γ β + βγ + δ + δ βγ δ β γ δ β γ (β ) + δ ( β) + γ(β ) + δ βγ( β) ( β)[δ + δ βγ γ ( β)[δ ( + βγ) ( + γ) Using this expression in ϕ(βγδ),weobtain γ ϕ(βγδ) γ ( β) δ ( + βγ) ( + γ) ( + γ + δ( + βγ)) ( + γ)( + βγ) Note that δ ( + βγ) < + γ for all δ β <. As a result, ϕ(βγδ) < 0forallδ, γ β<. Next, we show that ( c (τ i (σ)) ) σ c (τ i (σ)) σ > 0. For this purpose, let k (τ i (σ)) be the competitive-equilibrium savings associate with the optimal tax policy τ i (σ) an let k c (σ) be the commitment savings for a given σ. We show that (k σ (τ i (σ)) k c(σ)) σ > 0. Thus, the competitive-equilibrium savings uner the optimal policy is higher than commitment savings when σ is marginally higher than. To see this, first consier the consumer s optimality conitions }

9 TEMPTATION AND TAXATION 9 uner commitment an in competitive equilibrium. ( δf (k c (σ)) y k c(σ) ) σ f(k c(σ)) uner commitment, δ( + βγ) ( + γ)( + τ i (σ)) f ( k (τ i (σ)) ) ( ) σ y k (τ i (σ)) in competitive equilibrium. f(k (τ i (σ))) We can rewrite this problem as F(k c (σ) σ) uner commitment, ( + βγ) ( + γ)( + τ i (σ)) F(kc (σ) σ) in competitive equilibrium, where F(k σ) δf (k )( y k f(k ) )σ. We know that k c () k (τ i ()) ( + βγ) ( + γ)( + τ i (σ)) τ ()<0 i Next, take the erivative of the commitment an competitive-equilibrium optimality conition with respect to σ to obtain k c(σ) σ F (k c () ) σ F (k c () ) k (τ i (σ)) F (k (τ i ()) ) σ F (k (τ i ()) ) + τ ()F(k i (τ i ()) ) ( + τ i ())F (k (τ i ()) ) σ Since k c() k (τ i ()), taking the ifference yiels ( k (τ i (σ)) k c σ (σ)) τ i σ ()F(k (τ i ()) ) ( + τ i ())F (k (τ i ()) ) Note that τ () <0 an it is easy to see that F i (k (τ i ()) ) <0. As a result, (k σ (τ i (σ)) k c(σ)) σ > 0. This irectly implies that ( c (τ i (σ)) ) σ c (τ i (σ)) σ > 0. Q.E.D. PROOF OF PROPOSITION 7: To prove this proposition, we solve the consumer s problem backwar, fin her optimal consumption choices, an use those ecision rules to obtain her value function.

10 0 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. Problem at time T : The consumer s problem reas max ( + γ)log(c T ) + δ( + βγ) log(c T ) c T c T γ max log( c T ) + δβ log( c T ) c T c T subject to the buget constraints c T + ( + τ it )k T r( k T )k T + w(k T ) + s T c T Y T r( k T )k T + w( k T ) an The rest-of-lifetime buget constraint is thus c T + c T + τ it r( k T ) r( k T )k T + w( k T ) + s T c T Inserting c T into the rest-of- The first-orer conition is lifetime buget constraint, we obtain This implies c T + w( k T ) + τ it r( k T ) δ(+βγ) r( k T ) c T +γ +τ it + γ + γ + δ( + βγ) Y T an δ( + βγ) r( k T ) c T Y T + γ + δ( + βγ) + τ it c T + δβ Y T an c T δβ + δβ Y T r( k T ) Y T + τ it Notice that the c an the c are constant multiples of each other. As a result, the value function becomes U T (k T k T τ) log(c T ) + δ log(c T ) + a constant Now rewrite the value function in perio T to be use in the problem of the consumer in perio T by inserting the consumption allocations as functions of Y T. This elivers U T (k T k T τ) ( + δ) log(y T ) + δ log(r( k T )/( + τ it )) + a constant

11 TEMPTATION AND TAXATION Problem at time T : Using the T buget constraint an the rest-oflifetime buget constraint at time T for the consumer, we obtain the restof-lifetime buget constraint at time T as c T + + τ it r( k T ) Y T Y T r( k T )k T + w( k T ) + s T The objective of the government is to maximize + w( k T ) + s T ( + τ it ) r( k T ) + w( k T ) r( k T )r( k T ) ( + τ it )( + τ it ) max ( + γ)log(c T ) c T Y T [ ( ) r( kt ) + δ( + βγ) ( + δ) log(y T ) + δ log + a constant + τ it γ max log( c T ) c T Ỹ T [ ( ) r( kt ) + δβ ( + δ) log(ỹ T ) + δ log + a constant + τ it The first-orer conition is δ( + δ)( + βγ) r( k T ) c T + γ + τ it Y T Using the buget constraint, we obtain c T + γ + γ + δ( + δ)( + βγ) Y T an δ( + δ)( + βγ) r( k T ) Y T Y T + γ + δ( + δ)( + βγ) + τ it Inserting Y T in terms of c T into the consumer s problem, we obtain the Euler equation δ( + δ)( + βγ) r( k T ) c T + γ + δ( + βγ) + τ it c T

12 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. The temptation allocations are given by c T + δβ( + δ) Y T an δβ( + δ) r( k T ) Ỹ T Y T + δβ( + δ) + τ it The objective function of the government is U T (k T k T τ i ) log(c T ) + δ( + δ) log(y T ) ( ) + δ r( kt ) log + a constant + τ it Since c T is a multiple of Y T an c T is a multiple of ( r( k T ) +τ it )Y T, by inserting them we obtain U T (k T k T τ i ) log(c T ) + δ log(c T ) + δ log(c T ) + a constant Problem at T 3: The first-orer conition for the consumer is δ( + δ + δ )( + βγ) r( k T ) c T 3 + γ + τ it 3 Y T δ( + δ + δ )( + βγ) r( k T ) + γ + δ( + δ)( + βγ) + τ it 3 c T U T 3 (k T k T 3 τ i ) log(c T 3 ) + δ log(c T ) + δ log(c T ) + log(c T ) + a constant Continuing this proceure backwar completes the proof. Q.E.D. PROOF OF PROPOSITION 9: We solve the problem of the consumer an fin tax rates that implement the commitment allocation. Proposition 6 implies that the problem of a consumer at age t is given by subject to max c t Y t+ t σ + δβu t+(y t+ ) c t + + τ it Y t+ Y t r t+

13 TEMPTATION AND TAXATION 3 where U t (Y t ) c σ t σ + δβu t+(y t+ ) Yt We guess an verify that U t+ (Y t ) b σ where b t σ T. The optimality conition for the consumer is given by t δβb t+ r t+ + τ it Y σ t+ Inserting this into the buget constraint, we obtain Y t c t ( ) ( σ)/σ rt+ + (δβb t+ ) /σ + τ it ( ) /σ r t+ δβb t+ Y t + τ it Y t+ ( ) ( σ)/σ rt+ + (δβb t+ ) /σ + τ it Using these ecision rules, we obtain ) ( σ)/σ + ( β (δβb t+) /(σ) rt+ + τ it b t ( ( ) ( σ)/σ ) σ rt+ + (δβb t+ ) /(σ) + τ it Note that the optimality conition for the consumer can be written as ( ( ) ( σ)/σ ) σ βb t+ t δr t+ + (δβb t+ ) /σ rt+ + τ it + τ t+ it+ Inserting b t+ yiels ) ( σ)/σ + ( β β (δβb t+) /σ rt+ + τ it+ t δr t+ ( ) + τ ( σ)/σ it rt+ + (δβb t+ ) /σ + τ it+ t+

14 4 P. KRUSELL, B. KURUŞÇU, AND A. A. SMITH, JR. To implement the commitment allocation, the government shoul set ) ( σ)/σ + ( β β (δβb t+) /σ rt+ + τ it+ ( ) + τ ( σ)/σ it rt+ + (δβb t+ ) /σ + τ it+ where r t for all t is the equilibrium interest rate that arises uner commitment, that is, r t r( k t ). The recursive formulas for b t an τ it jointly etermine the sequence of optimal tax rates. We solve these formulas backwar noting that b T an b T + 0. Thus, τ it β anb T σ. Continuing backwar, we obtain τ it an τ it 3 β +βδ /σ r ( σ)/σ T + β(δ /σ r ( σ)/σ T +δ/σ r ( σ)/σ T (+βδ /σ r ( σ)/σ T ), b T +δ/σ r ( σ)/σ T β + δ /σ r ( σ)/σ T τ it 4 (β )/ ( + β ( δ /σ r ( σ)/σ T + δ 3/σ r ( σ)/σ T r ( σ)/σ T r ( σ)/σ T (+βδ /σ r ( σ)/σ T r ( σ)/σ T ) + δ /σ r ( σ)/σ T )) (+δ /σ r ( σ)/σ T ) (+δ /σ r ( σ)/σ T r ( σ)/σ T )) σ, One can notice the pattern in the expressions above, which implies the optimal tax for perio t is given by τ it + β β { T (δ /σ ) m (t+) mt+ m nt+ r( k n ) ( σ)/σ } We can also show that as T the optimal tax rate converges to a negative value. To see this, let {c c t } t0 be the consumption sequence associate with the commitment solution. Inserting the commitment Euler equation cc t+ ct c (δr t+ ) /σ into the tax expression, we obtain β τ it + β [ c c t+ + cc t+3 c c T + + c c t+ r t+ r t+ r t+3 r t+ r t+3 r T Note that c c + cc t+ t+ + cc t+3 c c T + + Y c r t+ r t+ r t+3 r t+ r t+3 r t+ T

15 TEMPTATION AND TAXATION 5 where Y c t is the lifetime income at time t associate with the commitment solution. Thus, the optimal tax rate can be written as τ it (β ) cc t+ Y c t+ ( β) cc t+ + β Y c t+ Note that since c c t+ /Yc > 0foranyt an T, we obtain that τ t+ it < 0forallt. Moreover, since the equilibrium allocation uner the optimal tax sequence is the same as the allocation associate with the commitment solution an since self-control cost is zero, the optimal tax policy elivers first-best welfare. Q.E.D. Institute for International Stuies, Stockholm University, 06 9 Stockholm, Sween; tony.smith@yale.eu, Economics Dept., University of Toronto, 50 St. George St., Toronto, Ontario M5S 3G7, Canaa; burhan.kuruscu@utoronto.ca, an Economics Dept., Yale University, 8 Hillhouse Avenue, New Haven, CT 0650, U.S.A.; tony.smith@yale.eu. Manuscript receive June, 009; final revision receive April, 00.