Optimal Fiscal Policy in the Romer Model

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1 Optimal Fiscal Policy in the Romer Model Abuzer BAKI July 13, 2006 Preliminary version Abstract This paper studies the issues of optimal taxation and optimal subsidies in the Romer (1990) model in a second-best world. To make the problem relevant the labor is assumed endogenous. We have following results: (i) the optimal eective tax on capital income is negative; (ii) the optimal research to subsidy is positive and time varying in transition and constant on the balanced growth path; (iii) the optimal labor tax is positive and depends positively on the initial public debt; and (iv) the growth rate is higher than that of laisser-faire equilibrium, while lower that of rst-best. Keywords: Optimal taxation, eciency, endogenous growth, externalities. JEL Classication Numbers: H21, D62. 1 Introduction Initially, Ramsey approach has been studied intensively for optimal commodity taxation 1. In last years, this insight has been applied to dynamic general equilibrium models 2, hence the question becomes rather optimal Centre d'economie de la Sorbonne (Université Paris1), M. S. E., Bd. de l'hôpital Paris cedex 13, France. Galatasaray University, 36 Çra gan cad Ortaköy-Istanbul, Turkey. abakis@gsu.edu.tr I wish to thank Antoine d'autume and Katheline Schubert for the comments and improvements they have made; the GNU Project for the software that they made freely available. 1 See Auerbach (1985) for a detailed survey. 2 See Auerbach (2002) for a detailed and extended up-to-date review of optimal taxation literature that includes also nonlinear and intertemporal taxation issues. Stiglitz (1987) is the main survey article for optimal income taxation both in static and dynamic setups. 1

2 (factor) income taxation. The main articles are Chamley (1985, 1986), Judd (1985, 1999), Jones et al. (1993, 1997), and Chari and Kehoe (1999). With the emergence of endogenous growth models corrective public policies have become one of the favorite topic of research for most economists. The goal was to nd out how to use scal instruments in order to achieve a Pareto improving allocation in a decentralized economy. In the basic models of endogenous growth (Romer [1986, 1990], Lucas (1988), Grossan and Helpman (1991) among others) one shows that the decentralized equilibrium is suboptimal. The immediate question is how to correct this ineciency. If lump-sum taxes were available it would be a trivial question, but when this is not the case we have to deal with a complicated one. Jones et al. [1993, 1997], Lucas (1990), Chamley 3 (1993), Devereux and Love (1994), Laitner (1995), Mino (1996), Ortigueira (1998), Chari and Kehoe (1999), Judd (1999) all use an Uzawa-Lucas type model (two sector endogenous growth models with human and physical capital) with distortionary taxation. The common conclusion is likely to be that the optimal labor and capital taxes are zero on the balanced growth path. However, the welfare cost associated with each tax is dierent and all depends on the details of the model specication 4. Curiously, there is not any paper (as soon as I know) that studies the question of optimal taxation/subsidies in Romer (1990) model (henceforth Romer model) with distortionary taxation. Arnold (2000a) is the rst paper that show that optimal subsidy to research is not constant, out of Balanced Growth Path (henceforth BGP) in the Romer model. Jones and Williams (2000) and Grimaud and Tournemaine (2004) use an extended Romer model to study the impact of various parameters in a world where lump-sum taxes are available. Schmidt (2003) is a detailed book on Romer model. Particular attention is devoted to the study of transitional dynamics. There are optimal tax/subsidy rates nanced always in a lump-sum manner. His problem being essentially the study of transitional dynamics, the second-best (henceforth 3 The optimal capital tax is zero if capital tax rates are time varying, while it is positive when we have constant capital tax rates over time. 4 Lucas (1990) estimates the welfare gain from abolition of capital tax in a such way that the government budget is balanced by increases in labor tax. If labor supply is inelastic the welfare gain is equal to be 2.7 % of total consumption but when labor is endogenous, the same policy causes a welfare loss of 18 %. In BGP analysis, this policy causes a slight loss in growth rate. In a more general model, Chamley (1993) shows that, for plausible parameter values, a time invariant capital tax may increase the welfare. Devereux and Love (1994) nd that labor tax decreases growth rate more than capital tax, while in revenue equivalent terms capital tax has a greater welfare cost. Using the same model, de Hek shows that taxing only capital, while labor is not taxed, may increase the growth rate if intertemporal elasticity of substitution is not very high. 2

3 SB) analysis is neglected. The problem is that all previous studying optimal policy in Romer model assume lump-sum taxes to study optimal tax/subsidy scheme. The issue becomes one of optimal subsidies more than optimal taxation. Our aim is to give a complete analysis of second-best in the Romer model. To make the problem more relevant the labor is assumed endogenous. The ndings are: in the rst best all taxes are zero while in the second best we have following results. - the optimal eective tax on capital income ((1 + s)(1 τ k )) is negative and equal to its rst-best (henceforth FB) value, which is equal to mark-up rate. There are many ways to implement that policy, the trivial one is not to tax the capital income, while intermediate good production is subsidized at the mark-up rate. - the optimal research to subsidy is positive, time varying in transition and constant on the balanced growth path. The same formula characterizes both the FB and the SB optimal subsidy rate. - the optimal labor tax is positive and constant. It depends implicitly on the initial public debt. The higher the wage elasticity of the labor, the lower is the optimal labor tax. - the growth rate under SB is greater than the one in the pure competitive equilibrium, but lower than the one in the FB. The paper is organized as follows. Section 2 describes the briey the Romer model, sections 3 and 4 analyzes respectively the social optimum and the competitive equilibrium. Section 5 studies the optimal tax/subsidies when lump-sum taxes are available while section 6 is about optimal SB taxation. Section 7 gives some numerical results for variables of interest and nally section 8 concludes. 2 Model We will neglect the row labor in Romer model, since this does not change any qualitative result but simplies the presentation. Our focus will be essentially on the allocation of human capital between dierent sectors and behavior of savings. Let us assume the production function for the consumption good to be At Y t = ΓL 1 α Y t x α itdi (1) 0 L Y is the part of human capital used in the nal good sector. x i is the amount of intermediate input i used by the rm. A t is the stock of the 3

4 (technical) knowledge available in period t. As in the original formulation A is supposed to be non-rival but excludable by patent/copyright laws. There are constant returns to scale for private inputs x i, L Y for given A. For nal good producing rms A forms an externality that they do not pay for. To see it more clearly suppose, as we will see later, that there is a given amount of capital. The rm has two choices: increasing variety by diminishing the amount of each input, or diminishing variety by increasing the quantity of each input. To simplify further, suppose that we have the following alternative: A inputs of quantity x (Y 1 ) or 2A inputs of quantity x/2 (Y 2 ). We have Y 2 > Y 1 while α < 1. In order to have diminishing marginal returns with respect to private input x, we will assume that α < 1. So, the rms producing nal good would like to have more variety. Intermediate goods are produced only using capital good, K. η units of forgone consumption is sucient to produce one unit of intermediate good. Each producer has a patent for the production of a particular intermediate good, i. So, there are A t rms producing intermediate good in period t. Intermediate good production in period t is limited by the available capital stock At K t = η x it di (2) 0 In any period t, investment and consumption are constrained by the nal good production. We assume that there is no government spending as in the original Romer model. K t + C t = Y t (3) Knowledge is produced by human capital. There is an externality due to previous period's knowledge stock. A t = δa t L At (4) One of the novelty in this paper is that total labor supply is endogenous; in any period t the total time endowment L t is allocated between L Y t and L At. L t = L At + L Y t (5) The utility function is separable between consumption and labor. U(C t, L t ) = log C t γ L1+1/ɛ t 1 + 1/ɛ (6) 4

5 3 Social optimum Firstly, we will characterize the rst best of our model. In this section there is no novelty, except the introduction of leisure, we will follow closely Romer (1990). The social optimum is to maximize the sum of discounted utilities (suppose it is given by (6)) subject to physical constraints: (1),(3), (4) and (5). We suppose that x it = x t = K t /(ηa t ) because all intermediate goods have diminishing marginal returns, thus it is optimal to have the same quantity for each one. H c = log C t γ L1+1/ɛ t 1 + 1/ɛ + µ kt(γη α L 1 α Y t K α t A 1 α t C t ) + µ at δa t (L t L Y t ) The rst order conditions (FOCs) with respect to, respectively, C, L, L Y, K, A are U C(t) = µ kt U L(t) = µ at δa t µ at δa t = µ kt (1 α) Y t L Y t µ kt = ρµ kt µ kt α Y t K t µ at = ρµ at µ kt (1 α) Y t A t µ at δ(l t L Y t ) and the transversality conditions are (7a) (7b) (7c) (7d) (7e) lim t e ρt µ kt K t = 0, lim e ρt µ at A t = 0 t On a BGP non-growing variables will be constant, i.e. LAt = L Y t = 0 and growing variables will have a constant (in our model also unique) and growth rate g = Ċt/C t = K t /K t = Ȧt/A t = Ẏt/Y t. Let y t = Y t /K t and q t = C t /K t, then, (7a) and (7d) imply g = αy ρ (8a) Using (7c) we see µ at δa t = µ kt (1 α) Y t L Y t µ at µ at = µ kt µ kt = g 5

6 (7c) and (7e) imply (7a),(7b) and (7c) imply (3) implies (4) and (5) imply L = g = δl ρ (8b) ( (1 α)y ) ɛ (8c) γl Y q g = y q g = δ(l L Y ) (8d) (8e) We can solve for L Y, L, y, q, g as we have 5 equations, (8), and 5 unknowns. 4 Equilibrium Now we can analyze the decentralized equilibrium of our economy. 4.1 Producers Final good producer We suppose that there is a representative rm which uses all available intermediate goods and human capital to produce nal good. The prot maximization follows from Max L Y,x πf = ΓL 1 α Y t At 0 At x α itdi p it x it di w Y t L Y t 0 w Y t = (1 α) Y t L Y t p it = αγl 1 α Y t x α 1 it (9a) (9b) Intermediate good producer Intermediate good sector is characterized by monopolistic competition. In order to make market produce the socially optimal level of intermediate goods let us assume that there is a subsidy s t to the sale of intermediate goods. Max x π i = (1 + s t )p it (x it )x it r t ηx it = (1 + s t )αγl 1 α Y t x α it r t ηx it 6

7 The prot maximization yields And the prots are given by p it = p t = ηr t α(1 + s t ) π it = π t = (1 α)(1 + s t )p t (x t )x t (10a) (10b) RD good producer Research sector is where we invent new patents for new intermediate goods. The rm that buys this patent may produce intermediate goods for an unlimited time. In order to correct for the externality in research sector let us assume that there is a subsidy b t to the sale of patents. Prot maximization follows from Max π r = (1 + b t )P At δa t L At w At L At L A The labor demand is given by the equation w At = (1 + b t )P At δa t (11) There are two no-arbitrage conditions in the economy. Firstly, the representative consumer/worker should be indierent between working in research or nal good sector, and secondly the cost of patent should be equal to the sum of discounted prots of intermediate good. (1 α)y t w At = w Y t P At = (12a) (1 + b t )L Y t δa t P At = e R v t r(s)ds P π(v)dv r t = At + π t (12b) P At t Out of BGP we will have (using (10a) and x t = K t ηa t ) P At = r t P At π t = r t [P At (1 α)(1 + s t)p t x t r t ] = r t [P At 1 α α P At K t A t ] (13) We will use these relations later both for the second best analysis and the comparison between equilibrium and social optimum. This is a condition that needs to be imposed on endogenous variables (A, K, L Y ) and the subsidy to the RD activities to be sure that the considered set of allocation in a SB world will respect the competitive economy. 7

8 4.2 Consumer The representative consumer maximizes her lifetime utility subject to her intertemporal budget constraint. H c = log C t γ L1+1/ɛ t 1 + 1/ɛ + λ t[ˆr t B t + ŵ t L t (1 + τt c )C t T t ] with ŵ t = (1 τ w t )w t and ˆr t = (1 τ k t )r t. The rst order conditions are and the transversality condition is U C = λ t (1 + τ c t ) U L = ŵ t λ t λ t = λ t (ρ ˆr t ) (14a) (14b) (14c) lim t e ρt λ t B t = 0 As it's standard in the literature we will suppose that consumption tax is given and constant for all periods. Technically we need this hypothesis because the system (14) is over determined (4 unknowns: τ k, τ w, τ c, λ for 3 equations). Another reason si that both labor and consumption taxes aect the static labor-consumption arbitrage in the same way. We can x any of two freely. Let us put it to zero, as usual, τt c = 0, t. Since after tax interest rate is given 5 by ˆr t = (1 τt k )(1+s t )α 2 Y t /K t, what matters is the multiplicative product of 1 + s t and 1 τt k for the dynamical system. Hence, we can x one of them freely. Let us put τt k = 0, t. On the BGP, using the denitions of variables that we made in section 3, we may rewrite the system (14) like L = ( (1 τ w (1 α)y ) ɛ ) (15a) γl Y q g = α 2 (1 + s)y ρ (15b) We always have (8d) and (8e), so that in equilibrium as well as the social optimum we have 5 unknowns (L, L Y, y, q, g) contre 5 equations. (8d), (8e), (25), (15a), (15b) 5 Using (9) and (10a). 8

9 4.3 Capital market In any period consumer's asset is divided between dierent alternative uses: physical capital, public debt, new designs. 5 Lump-sum taxation B t = K t + D t + P At A t (16) In the Romer model with lump sum taxation the subsidy to intermediate goods is always constant but the subsidy to RD will not be constant in general. However, in a BGP the latter is constant too. Social optimum g = y q Equilibrium g = y q g = δ(l L Y ) g = δ(l L Y ) ( (1 α)y ) ɛ ( L = L = (1 τ w (1 α)y ) γl Y q γl Y q g = αy ρ g = α 2 (1 + s)y ρ g = δl ρ αy = (1 + b)δl Y ) ɛ Table 1: Comparison of equilibrium and social optimum Proposition 1 If lump-sum taxes are available in the described economy then i-) the FB optimal intermediate good subsidy is constant and equal to mark-up both on and out of BGP: s t = s = 1 α 1 (17) ii-) the FB optimal labor tax is zero which means that labor-consumption relative prices are not distorted: τ w t = τ w = 0 (18) and iii-) the FB optimal research subsidy is constant on the BGP, b = L A L Y (19) 9

10 but variable out of the BGP whose dynamics are given by ḃ t 1 + b t = (1 + b t )δl Y t δl t (20) Proof. See the Appendix (A) The important aspect of optimal subsidy, as showed by Arnold (2000) in the lump-sum taxation version of the model, is that it is not constant out of BGP. We show in the Appendix (A) that this result holds also for the distortionary taxation version of the model. Basically, there are two distortions in the Romer model; the research externality and the monopolistic competition/mark-up pricing. The optimal FB policy corrects for these two distortions. If the growth rate is equal to the optimal one in equilibrium, than L A should be identical via the growth rate of A t in both cases. A comparison of (8a) with (15b) implies that y also should be identical. So, the ressource constraint implies a unique value of q for the equilibrium and social optimum to have the same growth rate. From (8c) and (15a) we see that LL ɛ Y is also identical. Since L A is identical, L Y must be identical too. Hence L also is identical. Both growth rates and levels in equilibrium are identical to the optimum values. 6 Ramsey taxation Ramsey (1927) is the rst one who introduces the method which is called in modern economics second-best taxation. The question is how to choose tax rates to maximize social welfare subject to the constraints that a given amount of revenue should be raised and resulting allocations must be consistent with competitive equilibrium with distortionary taxes. Ramsey, himself, uses what is called, dual approach. The government chooses tax rates being aware of the fact that agents will react to the change in prices due to taxes. Atkinson and Stiglitz (1972) have introduced a novel approach that is absolutely compatible with Ramsey taxation: primal approach. The idea is simple: one may think that social planner, instead of choosing tax rates directly, chooses optimal allocation subject to constraints which guarantee that chosen allocation could be implemented in a decentralized economy set-up with appropriate tax rates. Since we use the primal approach to analyze optimal taxation issue in this paper, we need to take into account the equilibrium constraints. These 10

11 are implementability constraint, resource constraint and the non-arbitrage condition described by (13). In fact, we do not need to impose (13) as an additional condition to the usual constraints, because the optimal choice of the other endogenous variables will determine indirectly b t such that (13) is respected. The resource constraint does not contain prices, so, we are left with implementability constraint. To derive it, I follow a novel method proposed to me by A. d'autume. The advantage of this approach is that the implementability constraint is in dierential form. Let us dene Q t = λ t B t as a new variable that is a product of consumer asset and co-state variable of her maximization problem. Then, using the rst order conditions of the consumer we get (assuming that there is no lump-sum taxes) Q t = Ḃtλ t + B t λt = (ˆr t B t + ŵ t L t (1 + τ c t )C t ) λt + B t λ t (ρ ˆr t ) = ρq t U L(t)L t U C(t)C t with Q 0 = B 0 U c(0) 1 + τ c 0 (21) Let us suppose that there are two alternatives Ramsey problems; P with the constraint (13), and P without it, i.e. φ t = 0. H c = log C t γ L1+1/ɛ t 1 + 1/ɛ + ψ qt[ρq t U L(t)L t U C(t)C t ] + ψ kt (η α ΓL 1 α Y t K α t A 1 α t C t ) + ψ at δa t (L t L Y t ) + φ t α 2 (1 + s t ) Y ( t P At 1 α K ) t K t α A t ψ K, ψ A, ψ q are the co-state variables of, respectively, the resource constraint, knowledge constraint, and implementability constraint. ψ K, ψ A are positive while ψ q is negative and reects the marginal cost of distortionary taxation in terms of utility. (P) 11

12 ψ kt = U C(t) [ 1 ψ qt (1 + E C t ) ] ψ at δa t = U L(t) [ 1 ψ qt (1 + E L t ) ] ψ at δa t = ψ kt (1 α)y t ψ qt = 0 L Y t ψ kt = ρψ kt ψ kt αy t K t ψ at = ρψ at ψ kt (1 α)y t A t P + φ At t (s t, L Y t, K t, A t, P At ) L Y t P φ At t (s t, L Y t, K t, A t, P At ) K t (22a) (22b) (22c) (22d) (22e) ψ at δl At φ t P At A t (s t, L Y t, K t, A t, P At ) φ P t = ρφ t φ At t (s t, L Y t, K t, A t ) P At ( Et i Ct U Ci = (t) U i (t) + L tu Li (t) ) U i (t), i = C, L E C t = 1, E L = 1/ɛ (22f) (22g) I will solve for the rst-order conditions associated with equation P. Under the assumption that P and the proper Ramsey problem P converge to a unique BGP, one can show that along a BGP the constraint (25) (steady state version of (13)) is satised even though it has not been imposed. Hence, if we show that the BGP solution of P implies that the constraint (25) is already satised, we can put safely φ t = 0, thanks to convergence to unique BGP. Proposition 2 The SB optimal research subsidy formula is the same as the FB one both on and out of a BGP, if P and P converge to the same BGP. Proof. I follow the approach developed by Jones et al. (1997, pp ) to prove the above proposition. Firstly, suppose that we have the program P (or equivalently φ t = 0 in (22)). Then (22c) implies ψ at δa t = ψ kt (1 α)y t L Y t (23) In a BGP ψ at and ψ kt will grow with same rate (since this is the case for C t and A t ). (23),(22f) and (22e) imply α Y t K t = δ(l Y + L A ) (24) 12

13 Secondly, the equation (13)which is associated with the program Pis equivalent to the following equation on the BGP 6 αy t K t = (1 + b t )δl Y t (25) (24) is equal to (25) only for certain values of b. This value of b, let us call it b sb, is given by δ(l Y + L A ) = (1 + b sb )δl Y b sb = L A L Y What is important is that for that value of b, the programs P and P converge to the same BGP. From (19) we see that this is the formula of the rst-best subsidy rate to research! For the second part of the proof concerning the transition see the Appendix (B). This result can be seen an extension of Sandmo (1975) and Cremer et al. (1998) who show that the optimal tax formula on the externality generating good is equivalent to the Pigovian tax 7 in a second-best world. So, the externality generating good, here knowledge, is subsidized at a rate that which reects its social benet. As we see later in the the section 7, the fact that the same formula characterizes both the FB and the SB does not mean that the level of the optimal subsidy rate is equal in the two environments. Actually the optimal subsidy to research will be lower in the SB. Once it is shown that we may use the program P for the Ramsey problem, it is relatively simple to derive other optimal policies. The following Proposition makes the point. Proposition 3 If the solution of decentralized equilibrium with distortionary taxation and that of the Ramsey problem converge to the same BGP, then (i) the optimal intermediate good subsidy is the same as the FB one both on the BGP and in the transition: s sb t = s sb = s = 1 α 1 (26) 6 On the BGP P A will be constant because L Y and x are. Remember that the nonarbitrage condition in the labor market implies w t = (1 α)γl α Y t A t x α t = P At δa t Thus, P A will be necessarily constant on the BGP. 7 Actually, one additional condition is required for that result, as showed by Cremer et al. (1998): all agents should have the same marginal rate of substitution between any two goods. Since we have representative agent this condition is already satised. 13

14 (ii) the optimal labor tax is constant but dierent from the rst-best one both on the BGP and in the transition: τ w,sb t = τ w,sb = ψ q (1 + 1/ɛ) (27) What determine the labor tax are two factors, the wage elasticity of the labor supply and the marginal cost of distortionary taxation. Proof. See the Appendix (B). The optimal intermediate good subsidy essentially corrects for the markup pricing as in the FB, hence it is constant. The fact that the optimal labor tax is constant in the SB is specic to the utility function specication. With an utility function with leisure instead of labor the optimal labor tax will be time varying. We see that the FB is the case where Ω(ψ q ) = 1 (comparing the Table (1) and the Table (2)) which is possible only when ψ q = 0. This is equivalent to lump-sum taxation. The Table (2) compares the equilibrium and the second-best on the BGP. Second best g = y q Equilibrium g = y q g = δ(l L Y ) g = δ(l L Y ) ( 1 (1 α)y ) ɛ ( L = L = (1 τ w (1 α)y ) 1 ψ q (1 + 1/ɛ) γl Y q γl Y q g = αy ρ g = δl ρ g = α 2 (1 + s)y ρ αy = (1 + b)δl Y ) ɛ Table 2: Comparison of equilibrium and second-best 7 Numerical analysis I will use GNU Scientic Library 8 (GSL) for my numerical work and Maxima 9 for symbolic calculations. The system to be simulated is (see the Ap

15 pendix (B)) 1 q 0 + ẏ t = 1 α ( ) δlt αy t y t α q t = q t ρ (1 α)y t q t L Y t = δl Y t + 1 α L Y t α δl t q t ( 1 (1 α)y ) t ɛ L t = 1 ψ q (1 + 1/ɛ) γl Y t q t (1 α) y 0 = L Y 0 δ q 0 t=0 e ρt (1 γl 1+1/ɛ t )dt There are two important points for the simulation of the above system: (i) the determination of initial conditions and, then (ii) the solution algorithm. As pointed out by Arnold (2000b) the determination of initial conditions is not trivial given that none of y, q, L Y is predetermined. The only given variables are K 0, A 0 which are not stationary. Let A 0 /K 0 = a 0 and η α Γ = 1, then from the production function one has f(y 0, L Y 0 ) = (a 0 L Y 0 ) 1 α y 0 = 0 which puts a constraint between y 0 and L Y 0. This is our rst boundary condition. We need two others to get the complete solution. Let us use the steady state values of y t, q t, i.e. y(ψ q ) and q(ψ q )where ψ q is just a guessas other boundary values. 7.1 calibration In order to focus on optimal subsidies nanced by distortionary taxes I did not introduce government spending. So, all taxation revenue will be used to subsidy research and intermediate goods production. As there is not reliable estimates about subsidy rates I set the subsidy rates to be zero, i.e. s = b = 0 in calibration step. Therefore, τ k = τ w = τ c = 0 in order to ensure that GBC is respected. Another advantage of this special calibration is to compare the pure laisser-faire (no government) equilibrium with second-best optimal one. Our utility function choice already implies the value of elasticity of intertemporal substitution, σ to be 1. But, in the literature σ is assumed to be inferior to 1, see Jones et al. (1993) and Rebelo and Stokey (1995) among others. The share of capital is about 1/3 but that value can not explain well 15

16 cross-country dierences (see Barro and Sala-i-Martin (1995)), so we will use α =.7. We will choose rstly plausible values for key economic variables. Cahuc and Zylberberg [2001, p.41] advocate the wage elasticity of labor to be inferior to unity 10. Let us set 11 L = 2, r =.07, g =.02, ɛ =.8 and then we will nd model parameters that yield our preferred calibration. Using equilibrium equations on the BGP, we get ρ =.05, δ =.06, L Y = 1.666, γ =.088, q =.122, y =.142. Now we are ready. Let us assume that a 0 =.03 which means that physical capital is relatively scarce. Then the solution algorithm is as follows: 1. Fix ψ q. 2. Guess initial values for q 0 and y Get L Y 0 from f(y 0, L Y 0 ) = 0. Given y 0, L Y 0, q 0 get L Given y i, L Y i, q i, L i get y i+1, L Y i+1, q i+1, L i+1 for i = 0, 1,..., N 1 (N being a large number). 5. If both y N y < ε and q N q < ε, then verify if the implementability constraint is respected. If yes, exit; if not, adjust ψ q and go to step 2. If y N y > ε and/or q N q > ε then adjust y 0 and/or q 0 and go to step Results The optimal eective tax rate on capital income is negative and constant, i.e. this is a constant subsidy which is given by s = In the same manner the optimal labor tax is constant and equal to τ w =.55. The only variable policy is research subsidy. In the steady state it is equal to b = but its time path can be seen in the Figure (1). It is decreasing over initial periods and then constant on the BGP. To derive it I used the equation (20), i.e. ḃ t 1 + b t = (1 + b t )δl Y t δl t 10 Cahuc and Zylberberg cite numerous empirical work who has non-conclusive ndings. For men it goes from.23 to.03 while for women the range is from.1 to g is taken from Jones et al. (1993), r from Jones and Williams (2000); L = 2 is just a normalization. 16

17 SB_subsidy time Figure 1: Time path of optimal research subsidy in the SB where the values b 0 = 0 and b = are imposed as boundary values. The Table (3) compares the steady state values of stationary variables for equilibrium, FB and SB. The dierence between growth rates is enormous. The optimal SB policy implies a growth rate which is 6 times higher then the equilibrium one. This result is considerably higher than the previous works cited in Introduction that essentially study the eect of Ramsey policies in a Lucas type endogenous growth model. Equilibrium SB FB g y q L L Y Table 3: Equilibrium-FB-SB steady state comparisons. 8 Conclusion I have studied the issues of optimal taxes and optimal subsidies in the Romer model in a second-best world. The optimal eective subsidy to capital is 17

18 constant and equal to its rst-best value which corrects for mark-up eect. The optimal subsidy to research is time-varying in general but constant on the balanced growth path. More importantly, we have the same formula both in the second-best and in the rst-best. The optimal labor tax is positive and constant. It decreases with the wage elasticity of labor supply and increases with the marginal cost of distortionary taxationwhich is itself positively correlated with the initial public debt. The growth rate under SB is greater than the one in the pure competitive equilibrium, but lower than the one in the FB. This underlies the cost of distortionary taxation that prevents the planner from replicating the rst-best allocations in a second-best world. My numerical work focused on the characterization of optimal policies both on the BGP and over the transition to the BGP. Hence, I give a complete characterization of the optimal policies in the Romer model. However, this is not a complete paper that can be readily used for policy application. Distributional issues, borrowing constraints, endogenous human capital formation and numerous other aspects of real life have been neglected. A First-best scal policy We will nd optimal research subsidy of the rst-best by comparing the systems of equations that characterize social optimum and equilibrium with lump sum taxes. For that, we need to have the same equation sets, and the same variables. So, one needs to transform both social optimum, and the equilibrium with lump-sum taxes in comparable terms. We begin by the social optimum. Let us dene V t = µ at /µ kt for the social optimum. The denition of V t combined with (7d) and (7e) gives the growth rate of V t in the social optimum. V t V t = µ at µ at = αy t K t µ kt µ kt µ kt (1 α)y t δl At µ at A t = αy t K t δ(l At + L Y t ) (28) We have already y t = Y t K t, q t = C t K t and L = L A + L Y. So V t V t = αy t δl t (29) 18

19 In the other hand (7c) implies V t = (1 α)y t δa t L Y t (30) so, we can derive another expression for V t V t from (30), (1) and (2). Let us rewrite the production function in symmetric case as And so Y t = Γη α L 1 α Y t A 1 α t Kt α Y t = y t = Γη α( A t L ) Y t 1 α K t K t ( yt η α A t L Y t = K t Γ combining this equation with (30) yields Taking the time derivatives makes the point. ) 1 1 α (31) (1 α) V t = Y t (32) δa t L Y t (1 α) = Γη α( L Y t A ) t α (33) δ K t (1 α) = Γη α( y t η α ) α 1 α ( 30') δ Γ V t = α y t (34) V t 1 α y t Finally use this last equation and (29), to obtain our rst equation. ẏ y = 1 α α (δl t αy t ) (35a) Then one uses (3),(7a), (7d) and q = C/K to get the second equation q q = Ċt C K t K q q = αy ρ y + q We can use (3) and (31) for our third equation L Y t L Y t = K t K t A t + 1 A t 1 α y t y t (35b) 19

20 more precisely L Y t = y t q t δ(l t L Y t ) + 1 L Y t 1 α with (8c) which we repeat here for convenience y t y t (35c) L t = ( (1 α)yt γl Y t q t ) ɛ (35d) In equilibrium, we will use (13) and (12a) to get our rst equation P At = (1 α)y t = (1 α)(y tη) (1 + b t )δa t L Y t (1 + b t )δ α 1 α P At = ḃt α y t (36) P At 1 + b t 1 α y t Given interest rate and prots in the intermediate good sector r t = α 2 (1 + s t ) Y t K t = α 2 (1 + s t )y t We may write π = (1 α)(1 + s t )α Y t A t P At P At = r t π t P At = α(1 + s t )[αy t (1 + b t )δl Y t ] and combining this last equation with (36) yields (37a) y t = 1 α ( ) α(1 + s t )[(1 + b t )δl Y t αy t ] ḃt y t α 1 + b t (37a) (37b) follows from the Euler equation coming from q = C/K, (14a),(14c) and (3) q t = (1 + s t )α 2 y t ρ y t + q t (37b) q t And from the production function (as in social optimum) we get L Y t = y t q t δ(l t L Y t ) + 1 L Y t 1 α y t y t (37c) 20

21 with L = ( (1 τ w t ) (1 α)y t γl Y t q t ) ɛ (37d) Now, we have two systems, (35) and (37), which characterize the rst-best and the competitive equilibrium of our economy. If, the two systems are identical then we are able to replicate the rst-best as a competitive equilibrium (by the mean of lump-sum taxes). A comparison of (35b) and (37b) shows that s = 1/α 1, i.e. (17) is satised. This is the constant optimal subsidy rate to intermediate goods. Similarly, comparing (35d) and (37d) yields the constant labor tax which was given by (27). But the optimal subsidy to research will be constant only in the steady state and variable out of the steady state. To see it note that it's given by comparing (35a) and (37a). Putting (1 + s t )α = 1 we get (20), i.e., ḃ t 1 + b t = (1 + b t )δl Y t δl t And nally a comparison of (35d) and (37d) gives (18), i.e. the constant optimal labor tax, which is zero: τ w = 0. B Second-best scal policy Following the same steps as in the section (A) we get the following equations for the second-best from the program P. ẏ t = 1 α ( ) δlt αy t y t α q t = αy t ρ y t + q t q t L Y t = y t q t δ(l t L Y t ) + 1 L Y t 1 α ( 1 L t = 1 ψ q (1 + 1/ɛ) y t y t (38a) (38b) (38c) (1 α)y t γl Y t q t ) ɛ (38d) The two systems to be compared are now (38) and (37). If, both systems converge to the same BGP then we must have the same system of dierential equations. A comparison of (38b) and (37b) shows that s t = 1/α 1, i.e. (26) is satised. As in the rst-best case, this is the constant optimal subsidy rate to intermediate goods. Once again the comparison of (37d) and (38d) yields the constant labor tax which was given by (27). The optimal subsidy 21

22 to research will be once more constant only in the steady state and variable out of the steady state. A comparison of (38a) and (37a) makes the point. Let us put (1 + s t )α = 1, then we get (20) Comparing (37d) and (38d) we get (27): ψ q will be determined by t=0 or equivalently (τ c = 0) where t=0 e ρt( U C(t)C t + U L(t)L t ) dt B0 U c(0) 1 + τ c e ρt (1 γl 1+1/ɛ t )dt B 0 C 0 = B 0/K 0 q 0 B 0 = K 0 + D 0 + P A0 A 0 Assume that there is no initial debt, D 0 = 0. Using (12a) and b 0 = 0 Therefore, t=0 B 0 A 0 (1 α) = 1 + P A0 = 1 + K 0 K 0 L Y 0 δ y 0 e ρt (1 γl 1+1/ɛ t )dt 1 q 0 + (1 α) y 0 L Y 0 δ q 0 If labor is constant, as this will be the case on the BGP, the left hand side is equal to References t=0 e ρt (1 γl 1+1/ɛ t )dt = 1 γl1+1/ɛ ρ [1] Arnold, L. G. (2000a): Endogenous technical change: a note on stability, Economic Theory, 16, [2] Arnold, L. G. (2000b): Stability of the market equilibrium in Romer's model of endogenous technical change: a complete characterization, Journal of Macroeconomics, 22, [3] Atkinson, A. B. and J. E. Stiglitz (1972): The structure of indirect taxation and economic eciency, Journal of Public Economics, 1,

23 [4] Barro, R. J. and X. Sala-i-Martin (1995): Economic Growth. New York: McGraw-Hill. [5] Cahuc, P. and A. Zylberberg (2001): Le marché du travail. Bruxelles: De Boeck Université. [6] Chamley, C. (1985): Ecient taxation in a stylized model of intertemporal general equilibrium, International Economic Review, 26, [7] Chamley, C. (1986): Optimal taxation of capital income in general equilibrium with innite lives, Econometrica, 54, [8] Chamley, C. (1993): Externalities and dynamics in models of `Learning or doing', International Economic Review, 34, [9] Chari, V. V. and P. J. Kehoe (1999): Optimal scal and monetary policy, Chapter 26, Vol. 1 (C), in Handbook of Macroeconomics, ed. by J. B. Taylor and M. Woodford. Amsterdam: Elsevier. [10] Cremer, H., F. Gahvari and N. Ladoux (1998): Externalities and optimal taxation, Journal of Public Economics, 70, [11] Devereux, M. B. and D. R. F. Love (1994): The eects of factor taxation in a two-sector model of endogenous growth, Canadian Journal of Economics, 27, [12] Grimaud, A. and F. Tornemaine (2004): Social values, distortions, and R&D investments: rst best versus second best equilibria in growth models, IDEI Working Paper 279. [13] de Hek, P. A. (2006): On taxation in a two-sector endogenous growth model with endogenous labor supply, Journal of Dynamics and Control, forthcoming. [14] Jones, L., R. E. Manuelli and P. Rossi (1993): Optimal taxation in models of endogenous growth, Journal of Political Economy, 101, [15] Jones, L., R. E. Manuelli and P. Rossi (1997): On the optimal taxation of capital income, Journal of Economic Theory, 73, [16] Jones, C. I. and J. C. Williams (2000): Too much of a good thing? The economics of investment in R&D, Journal of Economic Growth, 5,

24 [17] Judd, K. L. (1985): Redistributive taxation in a simple perfect foresight model, Journal of Public Economics, 28, [18] Judd, K. L. (1999): Optimal taxation and spending in general competitive growth models, Journal of Public Economics, 71, [19] Laitner, J. (1995): Quantitative evaluations of ecient tax policies for Lucas' supply side models, Oxford Economic Papers, 47, [20] Lucas, R. E. (1988): On the mechanics of economic development, Journal of Monetary Economics, 22, [21] Lucas, Robert E. (1990): Supply side economics: an analytical review, Oxford Economic Papers, 42, [22] Mino, K. (1996): Analysis of a two-sector model of endogenous growth with capital income taxation, International Economic Review, 37, [23] Ortigueira, S. (1998): Fiscal policy in an endogenous growth model with human capital accumulation, Journal of Monetary Economics, 42, [24] Ramsey, F. P. (1927): A contribution to the theory of taxation, Economic Journal, 37, [25] Rebelo, S. and N. L. Stokey (1995): Growth eects of at-rate taxes, Journal of Political Economy, 103, [26] Romer, P. M. (1990): Endogenous technological change, Journal of Political Economy, 98, S71-S102. [27] Sandmo, A. (1976): Optimal taxation in the presence of externalities, Swedish Journal of Economics, 77, [28] Schmidt, G. W. (2003): Dynamics of Endogenous Economic Growth : a case study of the Romer model. Amsterdam: Elsevier. 24