Externalities and fiscal policy in a Lucas-type model

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1 University of A Corna From te SelectedWors of Manel A. Gómez April, 2005 Externalities and fiscal policy in a Lcas-type model Manel A. Gómez Available at: ttps://wors.bepress.com/manel_gomez/8/

2 Externalities and fiscal policy in a Lcas-type model* Manel A. Gómez** Department of Applied Economics, University of A Corña Abstract. Tis paper devises a fiscal policy capable of decentralizing te optimal growt pat in a Lcas-type model wen average man capital as an external effect in te goods sector and average learning time as an external effect in man capital accmlation. Keywords: Hman capital; Optimal policy, Externalities JEL Classification: O41. * I tan an anonymos referee for valable comments. Financial spport from Spanis Ministry of Edcation and Science and FEDER is grateflly acnowledged. ** Facltad de Ciencias Económicas, Camps de Elviña, A Corña, Spain. Tel.: Fax: mago@dc.es. 1

3 1. I TRODUCTIO Several ators ave considered externalities in endogenos growt models wit man capital accmlation [e.g., Lcas (1988), Camley (1993), Benabib and Perli (1994)]. In tis case, optimal growt pats and competitive eqilibrim pats may not coincide, altog an adeqate government intervention can correct tis maret failre. Actally, García-Castrillo and Sanso (2000) and Gómez (2003) devise fiscal policies tat are capable to decentralize te optimal eqilibrim in te Lcas (1988) model wen average man capital as an external effect in goods prodction. Tis paper devises a fiscal policy by means of wic te optimal eqilibrim can be decentralized in a Lcas-type model. We extend te analyses of García-Castrillo and Sanso (2000) and Gómez (2003) to allow also for te average learning time to ave an external effect on man capital accmlation. Tis isse is interesting since tese external effects are also plasible [e.g., Camley (1993) and Benabib and Perli (1994)], and one may conjectre tat te optimal policy will depend on te class of externalities considered. We assme diminising point-in-time retrns in man capital accmlation, becase tey ave been sown to be essential for captring te caracteristic life-cycle pases of man capital accmlation [e.g., Rosen (1976)]. Te optimal policy proposed by García-Castrillo and Sanso (2000) mst rely on lmp-sm taxation at least in te transitional pase. More realistically, we assme tat te government cannot resort to lmp-sm taxes. Interestingly enog, we find tat a similar policy to tat devised by Gómez (2003), conveniently modified to correct for an externality of average learning time as well, can decentralize te optimal eqilibrim. Section 2 describes te maret economy, and Section 3, te centralized economy. Section 4 devises te optimal policy. Section 5 concldes. 2. THE DECE TRALIZED ECO OMY Consider an economy inabited by a constant poplation, normalized to one, of identical infinitely-lived agents wo derive tility from consmption, c, according to 2

4 0 e ρt ( c 1 σ 1) σ ) dt ρ > 0, σ > 0. (1) Time endowment is normalized to nity, wic can be allocated to wor,, or learning, 1. Hman capital,, is accmlated according to ν & = B( 1 B> 0, ν > 0, 0, ν + 1, (2) a were ( 1 expresses externalities associated to average learning time. a Te government taxes capital income at a rate τ, labor income at a rate τ, and sbsidizes investment in edcation at a rate s. Te sole cost of edcation is foregone earnings, w(1, a fraction s of wic is terefore financed by te government. Te agent s bdget constraint is & = ( 1 τ ) r+ τ ) w c+ s w, (3) ) were r is te interest rate, and w, te wage rate. We sall express (3) eqivalently as were & = ( 1 τ ) r+ ) w c s w, (4) + τˆ =τ + s. (5) Otpt, y, is prodced wit Cobb-Doglas tecnology: β y= A ( a A> 0, 0< β < 1, 0, were a expresses externalities associated to average man capital. Profit maximization by competitive firms implies tat r= β y and w= ( ) y (. Te government rns a balanced-bdget and cannot resort to lmp-sm taxation: or, sing (5), τ r+τ w= s w( 1, ) τ r+τˆ w s w. (6) = Let J be te crrent-vale Hamiltonian of te agent s problem, J σ ν ( c 1 = 1) σ ) + λ (τ ) r + ) w c+ s w) + µ B a. 3

5 Te first-order conditions are c σ =λ, (7a) ν 1 λ ) w= µνb a, (7b) & λ λ = ρ ( 1 τ )r, (7c) ν & µ µ = ρ B( 1 ( λ µ )( ) w s w), (7d) a + ρt ρt lim λ e = limµ e = 0. (7e) t t Hereafter, let = z& z denote te growt rate of te variable z. In wat follows, te z eqilibrim conditions = a and 1 = a will be imposed. From (7a) and (7c) we get c = (( 1 τ ) β y ρ) σ. (8) Using (6), (4) can be expressed as Log-differentiating (7b) yields = y c. (9) ˆ& ˆ. (10) λ τ τ ) + y = µ + ν + From (7b) and (7d), we get 1 µ = ρ B B ( ν+ ν s )). (11) Te following system caracterizes te dynamics of te economy in terms of te variables, ( + ) ( ) x = and c q=, tat are constant in te steady state: q = (( 1 τ ) β σ ) Ax σ + q ρ σ, (12a) x = Ax β + ) B β ) q, (12b) = ( 1 ( β + ν β ( τ Ax q) + B 1 ν s & β + ) + ν (12c) Eq. (12a) is obtained from (8) and (9); and (12b), from (9) and (2). Sbstitting µ 4

6 from (11), λ from (7c) and y from = β + ( ) + β + ) in (10), and ten y replacing from (9) and = B( 1 in te ensing expression, we get (12c). 3. THE CE TRALLY PLA ED ECO OMY Te central planner taes all te relevant information into accont. He maximizes (1) sbject to & = A β β+ 1 c and & = B. Let J be te crrent-vale Hamiltonian of te planner s problem, J 1 σ β + = ( c 1) σ ) + λ ( A c) + µ B. Te first-order conditions are c σ =λ, (13a) β β + 1 λ β ) A = µ ( ν + B, (13b) & λ λ = ρ βa +, (13c) & β β+ µ µ = ρ B ( λ µ )(β + ) A ), (13d) ρt ρt lim λ e = limµ e = 0. (13e) t t Proceeding as in te case of te maret economy, we arrive at q = ( β σ ) Ax σ + q ρ σ, (14a) x = Ax B β + ) β ) q, (14b) β + ) B (β ) + ( ν + β ) β q =. (14c) β )( β + ν Eqating te growt rates to zero yields te steady state. From (14a) and (14b) we get 1 β ) Aβ ) β * x * = (1 β ) σ (1 *) (1 β ) ρ, (15a) B + + q * = B β + )( σ β ) *) ( β ) β ) + ρ β. (15b) Sbstitting (15b) into (14c) and simplifying yields 5

7 B β + ) *) 1 (σ ) *) + ( v+ *) = β ) ρ. (15c) Now, we analyze wen (15c) as a soltion * (0,1 ). From (15c), define P( = B β + ) 1 (σ ) + ( v+ β ) ρ, wic is defined at least in [0,1). P is twice continosly differentiable, and monotonically increasing becase 2 P ( = B β + )( v+ ( σ + v ) > 0, (0,1). If v+<1, lim P( = +, and so, (15c) as a soltion * (0,1) if and only if 1 P ( 0) = Bβ + )σ ) β ) ρ < 0, or, eqivalently, ρ( ) > B β + )σ ). (16a) If v+=1, lim P( = P(1) = B β + ) β ) ρ, and so, (15c) as a soltion 1 * (0,1) if and only if P ( 0) = Bβ + )σ ) β ) ρ < 0, P ( 1) = Bβ + ) β ) ρ > 0, or, eqivalently, B ( + ) > ρβ ) > Bβ + )σ ). (16b) As (15b) and (15c) imply q*>0 and x*>0 if * (0,1), te steady state is feasible. We can also state te following proposition (see te proof in te Appendix). PROPOSITION 1. Te steady state of te optimal-growt problem is locally saddlepat stable. 4. THE OPTIMAL POLICY Tis section devises a fiscal policy capable of decentralizing te optimal eqilibrim. First, note tat (12b) and (14b) coincide. Comparing (12a) wit (14a), tey coincide if τ =0. Eqating (12c) and (14c), wit τ = 0, yields 6

8 s v β v τ (1 ) 1 ( ) + ( + )) ˆ ) & = +. (17) β ) v Bv We gess a constant optimal τˆ, i.e., ˆ & τ = 0. Hence, (17) redces to s β + ) + v =. (18) β ) v ) Solving (6) and (18), wit τ = 0, we obtain = + β + )( v+, (19) wic is constant as gessed, and 0 < 1. Te optimal sbsidy is ten < s = +, (20a) β + )( v+ wic satisfies 0<s <1. Now, (5), (19) and (20a) yield τ = s = +, (20b) β + )( v+ wic satisfies 0<τ <1. Tese reslts are consistent wit te Pigovian tax intition tat a sbsidy directed at te sorce of a positive externality will correct te maret failre. Te following proposition smmarizes tese findings. PROPOSITION 2. Te optimal eqilibrim can be decentralized if capital income is not taxed and investment in man capital is sbsidized at a rate s = +. β + )( v+ Te sbsidy can be financed by taxing labor income at a rate τ = + β + )( v+. Lmp-sm taxation is not reqired to balance te government bdget. Comparing (7a) and (13a), decentralization of te optimal eqilibrim reqires 7

9 λ= λ. Hence λ= wic, sing (7c) and (13c), yields τ = 0 λ. Log-differentiating (7b) and (13b), recalling tat a constant τˆ was gessed, decentralization also reqires µ = µ ; i.e., sing (7d) and (13d), β β+ β β+ ( λ µ ) + s β ) A = ( λ µ )β + ) A. (21) Hence, te optimal policy mst be set so tat private and social marginal prodcts of man capital in te goods sector, measred in terms of man capital as nmeraire, coincide. Tis can be accomplised by setting s according to (18) wic, ts, corrects for te effects of two divergences: te difference between te private and te social marginal prodct of man capital in goods prodction becase of te externality associated to average man capital, and te difference between te impted real price of pysical capital in te centralized economy, λ µ, and tat in te maret economy, λ µ, becase of te externality associated to average learning time [see (7b) and (13b)]. Te government s bdget constraint (6), wit τ = 0, provides te additional condition reqired to determine τˆ and s. Te optimal policy jst derived is similar to tat devised by Gómez (2003), conveniently adapted to accont for te externality of average learning time as well. Eqs. (20a) and (20b) comprise tree terms. Te first term, ( + ) and ( 1 ) β + ), respectively, depends on te relative size of te externality associated to average man capital, and flly corrects for tis external effect in absence of externalities in te edcational sector ( = 0 ). Tis is te optimal policy derived by Gómez (2003). Te second term, ( v+ and ( 1 ) ( v+, respectively, depends on te relative size of te externality associated to average learning time, and flly corrects for tis external effect in absence of externalities in goods prodction ( = 0 ). Te tird term, wic depends on te prodct of te relative size of bot externalities, is nonzero only if bot external effects coexist. 8

10 As Gómez (2003), let s sppose now tat te government sbsidizes te stoc of man capital at a rate ŝ. Te agent s bdget constraint is ten & = ( 1 τ ) r+ ) w c+ sˆ. (22) Handling te first-order conditions as in Section 2, it can be observed tat te relationsip sˆ s w carries over all te calclations, and so, te dynamics of te = maret economy are also smmarized by (12), after being s sbstitted by sˆ w. Ts, from (19), (20a) and s ˆ = s w, we can derive te following reslt. PROPOSITION 3. Te optimal eqilibrim can be decentralized if capital income is not taxed and te stoc of man capital is sbsidized at a rate s (1 y ˆ = β + ) β + )( v+. Te sbsidy can be financed by taxing labor income at a constant rate = + β + )( v+. Lmp-sm taxes are not reqired to balance te government bdget. Te government s size, measred as te sbsidy (or taxes) sare of otpt, is constant and eqals sˆ φ= = + β ). y β + )( v+ 5. CO CLUSIO S We devise a fiscal policy capable of decentralizing te optimal eqilibrim in a Lcas-type model wit externalities in goods prodction and in man capital accmlation. Te optimal growt pat can be decentralized by eeping capital income ntaxed and institting a sbsidy to investment in edcation or to te stoc of man capital, wic can be financed by a labor income tax. Lmp-sm taxation is not needed to balance te government bdget eiter in te steady state or in te transitory pase. 9

11 References Benabib, J. and R. Perli, 1994, Uniqeness and indeterminacy: on te dynamics of endogenos growt, Jornal of Economic Teory 63, Camley, C.P., 1993, Externalities and dynamics in models of learning or doing, International Economic Review 34, García-Castrillo, P. and M. Sanso, 2000, Hman capital and optimal policy in a Lcastype model, Review of Economic Dynamics 3, Gómez, M.A., 2003, Optimal fiscal policy in te Uzawa-Lcas model wit externalities, Economic Teory 22, Lcas, R.E., Jr., 1988, On te mecanics of economic development, Jornal of Monetary Economics 22, Rosen, S., 1976, A teory of life earnings, Jornal of Political Economy 84, S45-S67. 10