Econometrica Supplementary Material SUPPLEMENT TO TEMPTATION AND TAXATION (Econometrica, Vol. 78, No. 6, November 00, 063 084) 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
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
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 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 )
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 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
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 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 }
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.
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
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
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+
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+
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
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.