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Flat subsidies, technological change and transitional dynamics: a welfare analysis
Economics Letters 69 (2000) 393–400 www.elsevier.com / locate / econbase
Flat subsidies, technological change and transitional dynamics: a welfare analysis *, Marcos Sanso ´ Pedro Garcıa-Castrillo University of Zaragoza, Faculty of Economics, Gran Via 2, 50005 Zaragoza, Spain Received 1 March 1999; accepted 9 June 2000
Abstract The best policy measures obtained under the assumption that the economy is always in the steady state will be non-optimal in the transition when at least one of the market distortions is dynamic. Using Romer’s endogenous technological change model, we show that one of these measures could even be harmful for some parameter combinations. 2000 Elsevier Science S.A. All rights reserved. Keywords: Endogenous growth; Technical change; Transitional dynamics; Industrial policy; Thresholds JEL classification: H21; O31; O41
1. Introduction Endogenous growth models have provided a framework in which economic policy can influence individual life time welfare and, in some cases, the short and long-run growth rate. This is due to the fact that two elements are playing a key role in these new growth models: imperfect competition and externalities. Both represent two market distortions that prevent the optimal allocation of resources thereby making the case for the proposal of policies aimed at improving economic efficiency. The dynamic context which characterizes endogenous growth models represents a serious complication when the intention is to formulate optimal policies suitable for any possible situation. The problem would be far more simple in a static context. In fact, up to now, the common practice in these models has been to obtain the policy which is optimal when the economy is always in the steady state, a formally static problem, although clearly not the only possibility. As the earliest works on endogenous growth were devoted to the study of the steady state, it has been an unavoidable result that the earlier mentioned type of policy proposal has only considered that situation. However, now *Corresponding author. Tel.: 134-976-761-000, ext. 4685. ´ E-mail address: [email protected] (P. Garcıa-Castrillo). 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 00 )00310-4
2000 Elsevier Science S.A. All rights reserved.
394
´ , M. Sanso / Economics Letters 69 (2000) 393 – 400 P. Garcıa-Castrillo
that the transitional dynamics of these models are well known, the time has come for us to consider what happens with these policies when the economy is in transition. When the models exhibit transitional dynamics and at least one of the market distortions has a time-varying behavior, the optimal policy must be variable. Thus, not only will a policy designed for the steady state be non-optimal, but in some cases it could even be harmful compared with other fixed subsidy alternatives. In this paper we illustrate some of these questions using Romer’s (1990) model of endogenous technological change. Our objective is to show that the policy designed under the assumption that the economy is always in the steady state may be harmful compared to the policy that only subsidizes the hiring of capital goods. Although this would be better than the alternative of no intervention, we show that, for some combinations of parameters, the application of a subsidy to the R&D activity at the flat rate that would be optimal in the steady state in addition to a subsidy to the hiring of capital goods is worse than a policy which only applies this latter subsidy. The rest of the paper is organized as follows. In Section 2 we briefly present the model and the transitional dynamics with public intervention, with the decentralized economy being a particular case. Section 3 considers the different policies we are interested in, and makes a brief reference to the thresholds appearing in every case. In Section 4 we explain how the simulations are carried out, and how welfare must be evaluated under each alternative policy. Section 5 includes the numerical results and finally Section 6 closes the paper with a review of the main conclusions.
2. Transitional dynamics in Romer’s model with flat subsidies In Romer’s simplified model there are three different types of productive activities, with these being undertaken by different types of firms. The firms of the final goods sector use human capital and a set of differentiated inputs to obtain their output, which is sold in a competitive market. The firms of the R&D sector produce designs of new productive inputs using only human capital, and sell the patent in a competitive market. Finally, the firms that produce the different varieties of durable inputs in a context of monopolistic competition use the same technology as the firms of the final goods sector and hire their output to these firms. Government intervention can be introduced in several forms. From among these, we only consider the role that government can play in order to correct the market failures which appear in the model. In Romer (1990) there are two types of distortions. The first is associated with the monopolistic structure of the capital goods market and can be corrected by means of a subsidy for the hiring of capital goods, intended to make the demanding firms pay only the social marginal cost. As long as the monopolist charges a constant mark-up on the unit cost, the appropriate subsidy, which we call s f , will also be constant. The second distortion appears because the agents do not take into account the positive externality of knowledge, in such a way that an amount of human capital lower than the optimal will be devoted to R&D activities. The way to correct this failure is to subsidize the innovation costs so that new incentives are added, leading to an increase in the allocation of human capital to the sector. We call this subsidy to the R&D costs sA , and only consider the case in which it is constant 1 . The 1 As the externality is essentially dynamic, the optimal subsidy should be adapted to the changing situation of the economy. The mere application, in any situation, of the subsidy that would be appropriate in the steady state could provide incentives that are either more or less than the social optimum. However, in this paper we are not going to consider the optimal subsidy.
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government obtains resources from households through a lump-sum tax, employed to subsidize both the costs of the R&D firms and the hiring price of the capital goods, with the government maintaining a balanced budget at all times. The output of the final goods production sector, Y, is given by a constant returns to scale function in two factors: human capital HY (a fraction of total human capital H ) and a set of differentiated capital goods hx( j), j [ Aj where A is the set [0,A] of existing varieties of capital. The aggregated production function is Y 5 H aY eA x( j)12 a dj. In equilibrium, the demand for each capital good is identical, namely, x( j) 5 x. The firms of the R&D sector produce designs of new inputs according to the following linear technology: A~ 5 d (H 2 HY )A. A firm wishing to undertake the production of a variety of capital goods must buy the patent granting the exclusive right to their production. We will assume that the production of one unit of each variety of capital good requires a unit of final good. The government obtains its resources by means of a lump-sum tax on each household. Public expenditure takes two forms. First, the proportional subsidy s f for the hiring of each unit of capital goods and, secondly, the proportional subsidy sA for each unit of the R&D costs. 12 Using u(c) 5 (c s 2 1) /(1 2 s ) as the instantaneous utility function and denoting by r the intertemporal discount rate of the utility, the following system of three differential equations in z 5 K /A, HY and q 5 cH /K characterizes the dynamics of the decentralized economy with public intervention, given the policy (sA , s f ): g(z) 5 H aY z 2 a 2 q 2 d (H 2 HY )
(1a)
S
D
a 2 sf 1 g(HY ) 5 ]]H aY z 2 a 1 d HY 1 1 ]]]]] 2 q 2 d H 1 2 sf (1 2 sA )(1 2 s f )
S
2
(1b)
D
(1 2 a ) r g(q) 5 ]]] 2 1 H aY z 2 a 1 q 2 ], s s (1 2 s f )
(1c)
where K 5 xA is a measure of the stock of physical capital that exists in the economy. Equating Eqs. (1) to zero we obtain the steady state values:
S
D
1 2 sf (1 2 a )d H *Y r 1 1 (sd H 1 r )(1 2 sA )(1 2 s f ) H *Y 5 ] ]]]]]]]], q* 5 ] 2 ] 2 ]]]2 ]]]]] d s (1 2 sA )(1 2 s f ) 1 1 2 a s s (1 2 a ) (1 2 sA )(1 2 s f ) and z* 5f(1 2 a )(1 2 sA )d 21 H Y* a 21g 1 / a The system of Eqs. (1) presents local saddle path stability for the set of parameters defined by
H
J
(1 2 s )(1 2 a )d H (1 2 a )d H (a,d, r,s,H ) [ R 51 u]]]]] $ r . ]]]]]]]] , (1 2 s f )(1 2 sA ) (1 2 s f )(1 2 sA ) 1 1 2 a
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and there is a stable one-dimensional manifold M containing the steady state, in such a way that the solution tends to the steady state for any value of z if the initial point pertains to M 2 . We can derive that when s . (1 2 a )2 /(1 2 s f ) and s f # a, the policy functions of the interior solution HY 5 HY (z) and q 5 q(z), which provide the optimal values of HY and q in the stable path for each value of the ratio z, are decreasing functions. Given that HY (z) is decreasing, then for low levels of z the optimal level of human capital employed in the final goods sector will be greater than the total available human capital. Thus, a threshold in the variable z appears, which we call z u , characterized by HY (z) . H for z , z u and HY (z) 5 H for z 5 z u , indicating that, for lower levels, the capital accumulation of each variety of differentiated input is too low and that the economy requires a greater accumulation before starting the innovation. In this case, the dynamic behavior of the economy is also represented by system (1) with HY 5 H and g(HY ) 5 0.
3. Industrial policies We are interested in the dynamic behavior of the economy in three different policy situations. The first, which we denote by Md , is the case of the decentralized economy without public intervention. Making s f 5 sA 5 0 in Eqs. (1) we obtain the dynamic system corresponding to the decentralized economy without public intervention, and equating the resulting equations to zero we determine the steady state values for this situation. Industrial policies oriented towards driving the decentralized economy to the long-run social optimum are usually suggested by means of a common practice which consists of assuming that the decentralized economy has reached a situation of balanced growth, comparing it with the steady state social optimum, and establishing the taxes and subsidies that equate the values of the relevant variables, pretending thereby to have corrected the market failures that are present. This common procedure to calculate the required subsidies consists in comparing the steady state values of the variables z and HY in the planned solution and in the market economy with public intervention. Imposing the condition of the same behavior, the appropriate types of subsidy are s *f 5 a and s *A 5 s 21 (1 2 r /d H )3 . The second type of economy, which we denote by Mf , corresponds to a situation in which there is a subsidy only for the hiring of capital goods, that is, s f 5 s *f and sA 5 0. The third, which we denote by MA , adds to the previous case a flat subsidy to innovation at the rate sA 5 s *A . At this point we should indicate that the subsidy s *A is not optimum when it is applied along the transition. The subsidy is intended to internalize the technological externality that is essentially dynamic. When K /A is low, the amount of resources devoted to investment in R&D is also low, the increase in the stock of knowledge will be small and, hence, the productivity gains induced by the externality are reduced. Consequently, given that the effect which it is necessary to internalize is reduced, the subsidy required will also be low. As long as physical capital is accumulated relative to
2 3
The proof is available from the authors upon request. ´ (1995). This result can be found in Barro and Sala-i-Martın
´ , M. Sanso / Economics Letters 69 (2000) 393 – 400 P. Garcıa-Castrillo
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knowledge, the amount of labor devoted to R&D increases, as does the externality and, consequently, the rate of subsidy. Thus if the government applies the rate s *A , it would be subsidizing the innovation in excess, diminishing consumption forcefully and affecting welfare. Although the ratio K /A will end-up being optimum, the economy will arrive at an overaccumulation of physical capital and knowledge. It can be shown 4 that the dynamic optimal behavior of sA is s~A 5 d Ly 2 d L(1 2 sA ).
4. Simulations and welfare assessment procedure The main inconvenience of system (1) is that it cannot be explicitly solved. However, following ´ (1993), an approximation can be obtained by eliminating the time of the Mulligan and Sala-i-Martın system of equations and solving the resulting equations numerically. The solutions of these systems give the policy functions. The numerical resolution requires values for the technological and utility parameters. We have used a wide range of parameter values: thus, we have considered 1.25, 2, 4 and 8 as possible values of s ; as values of r we have used 0.02, 0.03 and 0.04; for d we have chosen the values 0.05, 0.1, 0.25 and 1; the human capital endowments (H ) introduced have been 1, 2, 5 and 10; finally, the values 0.3, 0.4, 0.5, 0.6 and 0.7 have been considered for a. Some of these combinations of parameters could be admissible as reflecting real cases, whilst others may clearly make no sense. In any event, our intention has been to obtain a sufficiently wide sample, capable of providing as much complete information as possible on the dynamic behavior, so that the observed regularities can be stated with reliability. Welfare assessment requires that we obtain the consumption paths numerically. For each combination of parameters, initial value z 0 is the threshold z u corresponding to MA . Given the initial ratios z 0 and q0 , the product z 0 q0 represents the aggregated consumption per variety of capital good. 5 Thus, obtaining c 0 requires a value of A 0 and any value would be suitable . Once we have determined c 0 , the solution of system (1) provides the temporal path of z and q until time t u in which z 5 z u . As no labor is devoted to innovation in that period, A remains constant in A 0 and c t 5 z t qt /HA 0 . In t u the economy changes its dynamic regime, in such a way that from then on c t 5 z t qt /HA t and obtaining the consumption path requires the solution of the system of four differential equations resulting from adding equation for A~ to system (1), beginning at the point (t u ,z u ,qu ,H,A 0 ). The solutions of the systems have been obtained using a Runge–Kutta algorithm with fixed pass size, which provides approximations to c t with fixed time interval. The welfare evaluation has been obtained in two steps. In the first, the welfare has been accumulated numerically from t 0 5 0 to t, using the values c t obtained previously, in the form W(0,t) 5 e0t u(c t )e 2 r t dt. The numerical integration of the function has been calculated using Simpson’s rule. Once we consider that the economy has reached 6 the steady state, for example for t *, the second step is the calculus of the welfare accumulated from t* to infinite: 4 The proof is available from the authors upon request. An analogous treatment of an optimum policy along the transition in a model of human capital can be found in Garcia-Castrillo and Sanso (2000). 5 We have selected A 0 in such a way that u(c) is always positive. 6 The criterion used has been that the numerical solution of the variables were very close to their steady state values. 25 Specifically, we have required that u(z t 2 z*) /z*u , e, u(qt 2 q*) /q*u , e and u(HYt 2 H Y* ) /H Y* u , e, with e 510 .
´ , M. Sanso / Economics Letters 69 (2000) 393 – 400 P. Garcıa-Castrillo
398
E u(c )e `
W(t*,`) 5
t*
2 r (t 2t )
t
S
s c 12 1 1 t* dt 5 ]] ]]]] 2 ] 1 2 s r 2 g(1 2 s ) r
D
where g is the steady growth rate of the consumption. The welfare function evaluated at the initial moment will be: W(0,`) 5 W(0,t*) 1 e 2 r t *W(t*,`). As the evaluation of the alternative policies depends on consumption, it is of interest, first, to determine the evolution of this variable in the three models. Secondly, and with respect to each model, we analyze the relative values of the welfare function W(0,t). Finally, taking into account W(0,`), we study the factors that affect the relative welfare difference.
5. Results
5.1. Consumption behavior The consumption path will be increasing 7 under all three policy rules. The consumption growth rate will be decreasing towards the steady state rate, but differences will exist in the consumption levels and in the growth rate, depending on government policy. We will describe the consumptions corresponding to Md , Mf and MA as c d , c f , and c A , respectively. We can conclude that the level of consumption in the economy with two fixed subsidies ends up being greater than that corresponding to Md and Mf , although in the early stages the consumption is lower. This is so due to the type of subsidy applied in MA , which is superior to the optimum during the transition and, consequently, the accumulation of capital and the number of varieties is also superior. Fig. 1 displays the typical difference between c A and c f . An analogous behavior results for the difference c A 2 c d . The relevant question is whether the initial lower consumption is compensated
Fig. 1. Differences in consumption. 7
This is due to the fact that we are only considering a situation in which z 0 is lower than its steady state value.
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Fig. 2. Differences in welfare.
by the higher future consumption, with this depending on the discount rate and the utility function.
5.2. Welfare analysis We have found that the higher future consumption always compensates for the lower initial one in all the possible combinations of parameters when we compare the total welfare –W(0,`)– of MA with Md . Thus, we can state that the two fixed subsidies applied at the rate corresponding to the steady state always represent a better policy than non-intervention. However, this is not the case when we compare MA with Mf . Fig. 2 shows two graphics representing the temporal evolution of the differences between the functions W(0,t) of MA and Mf for two different combinations of parameters. The conclusion derived from Fig. 2a is surprising: Mf leads to a greater welfare than MA . This means that although the policy of fixed subsidies guarantees that the economy will reach the optimal steady state values of K /A, c /K and HY , its effects on the levels taken by each variable lead to an absolute loss of welfare with respect to its alternative, not only for the early generations, but for all of them. But we can also find other combinations of parameters in which the opposite happens. We have an example in Fig. 2b. Here, the overaccumulation induced by the fixed subsidies produces welfare losses at the beginning, but these are compensated by the greater consumption in later periods. The relevant question is to know what determines whether the results will be one or the other. Once having applied the optimum fixed subsidy to the hiring of capital goods, an additional policy of fixed subsidy to the cost of R&D firms at the rate appropriate to the steady state (MA ) can either increase or decrease the welfare gains as compared to Mf . This situation has been graphically illustrated in two particular cases. The relevant question is to conclude what determines that the application of s *A leads to welfare gains or losses against the alternative of only maintaining the subsidy for the hiring of capital goods. In order to answer this question, Fig. 3 contains the relative welfare difference between MA and Mf for all combinations of parameters. For each a and s, the graphics display the value (W A (0,`) 2 W f (0,`)) /W A (0,`) face to d H /r. Note that when s is reduced (1.25), the fixed subsidy s *A leads, in almost all cases, to welfare gains (except for very low values of both a and d H /r ), and that welfare losses are present when s is high and a and d H /r are low. Specifically, as s grows, the range of values of a and d H /r that produce welfare losses widens out.
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Fig. 3. Relative differences in welfare 100 W A (0,`) 2 W f (0,`) /W f (0,`).
6. Conclusions We know that once the distortion due to monopoly power has been corrected in Romer’s (1990) model, the resulting welfare losses in the economy can be avoided by means of an optimal subsidy to the R&D costs. However, the application of an additional fixed subsidy, although optimal in the steady state, could lead to either welfare gains or losses taking into account the transition, depending on the technological and utility parameters. Specifically, this additional policy can be harmful when the labor endowment, the productivity of the innovative sector, or the monopoly power (a ) are low and both the discount rate and the inverse of the elasticity of substitution are high.
References ´ X., 1995. Economic Growth. McGraw-Hill. Barro, R.J., Sala-i-Martın, Garcia-Castrillo, P., Sanso, M. (2000): Human Capital and Optimal Policy in a Lucas-type Model, Review of Economic Dynamics (Forthcoming). ´ X., 1993. Transitional dynamics in two sector models of endogenous growth. Quarterly Mulligan, C.B., Sala-i-Martın, Journal of Economics 108 (3), 737–773. Romer, P.M., 1990. Endogenous technological change. Journal of Political Economy 98, S71–S102.
Flat subsidies, technological change and transitional dynamics: a welfare analysis *, Marcos Sanso ´ Pedro Garcıa-Castrillo University of Zaragoza, Faculty of Economics, Gran Via 2, 50005 Zaragoza, Spain Received 1 March 1999; accepted 9 June 2000
Abstract The best policy measures obtained under the assumption that the economy is always in the steady state will be non-optimal in the transition when at least one of the market distortions is dynamic. Using Romer’s endogenous technological change model, we show that one of these measures could even be harmful for some parameter combinations. 2000 Elsevier Science S.A. All rights reserved. Keywords: Endogenous growth; Technical change; Transitional dynamics; Industrial policy; Thresholds JEL classification: H21; O31; O41
1. Introduction Endogenous growth models have provided a framework in which economic policy can influence individual life time welfare and, in some cases, the short and long-run growth rate. This is due to the fact that two elements are playing a key role in these new growth models: imperfect competition and externalities. Both represent two market distortions that prevent the optimal allocation of resources thereby making the case for the proposal of policies aimed at improving economic efficiency. The dynamic context which characterizes endogenous growth models represents a serious complication when the intention is to formulate optimal policies suitable for any possible situation. The problem would be far more simple in a static context. In fact, up to now, the common practice in these models has been to obtain the policy which is optimal when the economy is always in the steady state, a formally static problem, although clearly not the only possibility. As the earliest works on endogenous growth were devoted to the study of the steady state, it has been an unavoidable result that the earlier mentioned type of policy proposal has only considered that situation. However, now *Corresponding author. Tel.: 134-976-761-000, ext. 4685. ´ E-mail address: [email protected] (P. Garcıa-Castrillo). 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 00 )00310-4
2000 Elsevier Science S.A. All rights reserved.
394
´ , M. Sanso / Economics Letters 69 (2000) 393 – 400 P. Garcıa-Castrillo
that the transitional dynamics of these models are well known, the time has come for us to consider what happens with these policies when the economy is in transition. When the models exhibit transitional dynamics and at least one of the market distortions has a time-varying behavior, the optimal policy must be variable. Thus, not only will a policy designed for the steady state be non-optimal, but in some cases it could even be harmful compared with other fixed subsidy alternatives. In this paper we illustrate some of these questions using Romer’s (1990) model of endogenous technological change. Our objective is to show that the policy designed under the assumption that the economy is always in the steady state may be harmful compared to the policy that only subsidizes the hiring of capital goods. Although this would be better than the alternative of no intervention, we show that, for some combinations of parameters, the application of a subsidy to the R&D activity at the flat rate that would be optimal in the steady state in addition to a subsidy to the hiring of capital goods is worse than a policy which only applies this latter subsidy. The rest of the paper is organized as follows. In Section 2 we briefly present the model and the transitional dynamics with public intervention, with the decentralized economy being a particular case. Section 3 considers the different policies we are interested in, and makes a brief reference to the thresholds appearing in every case. In Section 4 we explain how the simulations are carried out, and how welfare must be evaluated under each alternative policy. Section 5 includes the numerical results and finally Section 6 closes the paper with a review of the main conclusions.
2. Transitional dynamics in Romer’s model with flat subsidies In Romer’s simplified model there are three different types of productive activities, with these being undertaken by different types of firms. The firms of the final goods sector use human capital and a set of differentiated inputs to obtain their output, which is sold in a competitive market. The firms of the R&D sector produce designs of new productive inputs using only human capital, and sell the patent in a competitive market. Finally, the firms that produce the different varieties of durable inputs in a context of monopolistic competition use the same technology as the firms of the final goods sector and hire their output to these firms. Government intervention can be introduced in several forms. From among these, we only consider the role that government can play in order to correct the market failures which appear in the model. In Romer (1990) there are two types of distortions. The first is associated with the monopolistic structure of the capital goods market and can be corrected by means of a subsidy for the hiring of capital goods, intended to make the demanding firms pay only the social marginal cost. As long as the monopolist charges a constant mark-up on the unit cost, the appropriate subsidy, which we call s f , will also be constant. The second distortion appears because the agents do not take into account the positive externality of knowledge, in such a way that an amount of human capital lower than the optimal will be devoted to R&D activities. The way to correct this failure is to subsidize the innovation costs so that new incentives are added, leading to an increase in the allocation of human capital to the sector. We call this subsidy to the R&D costs sA , and only consider the case in which it is constant 1 . The 1 As the externality is essentially dynamic, the optimal subsidy should be adapted to the changing situation of the economy. The mere application, in any situation, of the subsidy that would be appropriate in the steady state could provide incentives that are either more or less than the social optimum. However, in this paper we are not going to consider the optimal subsidy.
´ , M. Sanso / Economics Letters 69 (2000) 393 – 400 P. Garcıa-Castrillo
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government obtains resources from households through a lump-sum tax, employed to subsidize both the costs of the R&D firms and the hiring price of the capital goods, with the government maintaining a balanced budget at all times. The output of the final goods production sector, Y, is given by a constant returns to scale function in two factors: human capital HY (a fraction of total human capital H ) and a set of differentiated capital goods hx( j), j [ Aj where A is the set [0,A] of existing varieties of capital. The aggregated production function is Y 5 H aY eA x( j)12 a dj. In equilibrium, the demand for each capital good is identical, namely, x( j) 5 x. The firms of the R&D sector produce designs of new inputs according to the following linear technology: A~ 5 d (H 2 HY )A. A firm wishing to undertake the production of a variety of capital goods must buy the patent granting the exclusive right to their production. We will assume that the production of one unit of each variety of capital good requires a unit of final good. The government obtains its resources by means of a lump-sum tax on each household. Public expenditure takes two forms. First, the proportional subsidy s f for the hiring of each unit of capital goods and, secondly, the proportional subsidy sA for each unit of the R&D costs. 12 Using u(c) 5 (c s 2 1) /(1 2 s ) as the instantaneous utility function and denoting by r the intertemporal discount rate of the utility, the following system of three differential equations in z 5 K /A, HY and q 5 cH /K characterizes the dynamics of the decentralized economy with public intervention, given the policy (sA , s f ): g(z) 5 H aY z 2 a 2 q 2 d (H 2 HY )
(1a)
S
D
a 2 sf 1 g(HY ) 5 ]]H aY z 2 a 1 d HY 1 1 ]]]]] 2 q 2 d H 1 2 sf (1 2 sA )(1 2 s f )
S
2
(1b)
D
(1 2 a ) r g(q) 5 ]]] 2 1 H aY z 2 a 1 q 2 ], s s (1 2 s f )
(1c)
where K 5 xA is a measure of the stock of physical capital that exists in the economy. Equating Eqs. (1) to zero we obtain the steady state values:
S
D
1 2 sf (1 2 a )d H *Y r 1 1 (sd H 1 r )(1 2 sA )(1 2 s f ) H *Y 5 ] ]]]]]]]], q* 5 ] 2 ] 2 ]]]2 ]]]]] d s (1 2 sA )(1 2 s f ) 1 1 2 a s s (1 2 a ) (1 2 sA )(1 2 s f ) and z* 5f(1 2 a )(1 2 sA )d 21 H Y* a 21g 1 / a The system of Eqs. (1) presents local saddle path stability for the set of parameters defined by
H
J
(1 2 s )(1 2 a )d H (1 2 a )d H (a,d, r,s,H ) [ R 51 u]]]]] $ r . ]]]]]]]] , (1 2 s f )(1 2 sA ) (1 2 s f )(1 2 sA ) 1 1 2 a
396
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and there is a stable one-dimensional manifold M containing the steady state, in such a way that the solution tends to the steady state for any value of z if the initial point pertains to M 2 . We can derive that when s . (1 2 a )2 /(1 2 s f ) and s f # a, the policy functions of the interior solution HY 5 HY (z) and q 5 q(z), which provide the optimal values of HY and q in the stable path for each value of the ratio z, are decreasing functions. Given that HY (z) is decreasing, then for low levels of z the optimal level of human capital employed in the final goods sector will be greater than the total available human capital. Thus, a threshold in the variable z appears, which we call z u , characterized by HY (z) . H for z , z u and HY (z) 5 H for z 5 z u , indicating that, for lower levels, the capital accumulation of each variety of differentiated input is too low and that the economy requires a greater accumulation before starting the innovation. In this case, the dynamic behavior of the economy is also represented by system (1) with HY 5 H and g(HY ) 5 0.
3. Industrial policies We are interested in the dynamic behavior of the economy in three different policy situations. The first, which we denote by Md , is the case of the decentralized economy without public intervention. Making s f 5 sA 5 0 in Eqs. (1) we obtain the dynamic system corresponding to the decentralized economy without public intervention, and equating the resulting equations to zero we determine the steady state values for this situation. Industrial policies oriented towards driving the decentralized economy to the long-run social optimum are usually suggested by means of a common practice which consists of assuming that the decentralized economy has reached a situation of balanced growth, comparing it with the steady state social optimum, and establishing the taxes and subsidies that equate the values of the relevant variables, pretending thereby to have corrected the market failures that are present. This common procedure to calculate the required subsidies consists in comparing the steady state values of the variables z and HY in the planned solution and in the market economy with public intervention. Imposing the condition of the same behavior, the appropriate types of subsidy are s *f 5 a and s *A 5 s 21 (1 2 r /d H )3 . The second type of economy, which we denote by Mf , corresponds to a situation in which there is a subsidy only for the hiring of capital goods, that is, s f 5 s *f and sA 5 0. The third, which we denote by MA , adds to the previous case a flat subsidy to innovation at the rate sA 5 s *A . At this point we should indicate that the subsidy s *A is not optimum when it is applied along the transition. The subsidy is intended to internalize the technological externality that is essentially dynamic. When K /A is low, the amount of resources devoted to investment in R&D is also low, the increase in the stock of knowledge will be small and, hence, the productivity gains induced by the externality are reduced. Consequently, given that the effect which it is necessary to internalize is reduced, the subsidy required will also be low. As long as physical capital is accumulated relative to
2 3
The proof is available from the authors upon request. ´ (1995). This result can be found in Barro and Sala-i-Martın
´ , M. Sanso / Economics Letters 69 (2000) 393 – 400 P. Garcıa-Castrillo
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knowledge, the amount of labor devoted to R&D increases, as does the externality and, consequently, the rate of subsidy. Thus if the government applies the rate s *A , it would be subsidizing the innovation in excess, diminishing consumption forcefully and affecting welfare. Although the ratio K /A will end-up being optimum, the economy will arrive at an overaccumulation of physical capital and knowledge. It can be shown 4 that the dynamic optimal behavior of sA is s~A 5 d Ly 2 d L(1 2 sA ).
4. Simulations and welfare assessment procedure The main inconvenience of system (1) is that it cannot be explicitly solved. However, following ´ (1993), an approximation can be obtained by eliminating the time of the Mulligan and Sala-i-Martın system of equations and solving the resulting equations numerically. The solutions of these systems give the policy functions. The numerical resolution requires values for the technological and utility parameters. We have used a wide range of parameter values: thus, we have considered 1.25, 2, 4 and 8 as possible values of s ; as values of r we have used 0.02, 0.03 and 0.04; for d we have chosen the values 0.05, 0.1, 0.25 and 1; the human capital endowments (H ) introduced have been 1, 2, 5 and 10; finally, the values 0.3, 0.4, 0.5, 0.6 and 0.7 have been considered for a. Some of these combinations of parameters could be admissible as reflecting real cases, whilst others may clearly make no sense. In any event, our intention has been to obtain a sufficiently wide sample, capable of providing as much complete information as possible on the dynamic behavior, so that the observed regularities can be stated with reliability. Welfare assessment requires that we obtain the consumption paths numerically. For each combination of parameters, initial value z 0 is the threshold z u corresponding to MA . Given the initial ratios z 0 and q0 , the product z 0 q0 represents the aggregated consumption per variety of capital good. 5 Thus, obtaining c 0 requires a value of A 0 and any value would be suitable . Once we have determined c 0 , the solution of system (1) provides the temporal path of z and q until time t u in which z 5 z u . As no labor is devoted to innovation in that period, A remains constant in A 0 and c t 5 z t qt /HA 0 . In t u the economy changes its dynamic regime, in such a way that from then on c t 5 z t qt /HA t and obtaining the consumption path requires the solution of the system of four differential equations resulting from adding equation for A~ to system (1), beginning at the point (t u ,z u ,qu ,H,A 0 ). The solutions of the systems have been obtained using a Runge–Kutta algorithm with fixed pass size, which provides approximations to c t with fixed time interval. The welfare evaluation has been obtained in two steps. In the first, the welfare has been accumulated numerically from t 0 5 0 to t, using the values c t obtained previously, in the form W(0,t) 5 e0t u(c t )e 2 r t dt. The numerical integration of the function has been calculated using Simpson’s rule. Once we consider that the economy has reached 6 the steady state, for example for t *, the second step is the calculus of the welfare accumulated from t* to infinite: 4 The proof is available from the authors upon request. An analogous treatment of an optimum policy along the transition in a model of human capital can be found in Garcia-Castrillo and Sanso (2000). 5 We have selected A 0 in such a way that u(c) is always positive. 6 The criterion used has been that the numerical solution of the variables were very close to their steady state values. 25 Specifically, we have required that u(z t 2 z*) /z*u , e, u(qt 2 q*) /q*u , e and u(HYt 2 H Y* ) /H Y* u , e, with e 510 .
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E u(c )e `
W(t*,`) 5
t*
2 r (t 2t )
t
S
s c 12 1 1 t* dt 5 ]] ]]]] 2 ] 1 2 s r 2 g(1 2 s ) r
D
where g is the steady growth rate of the consumption. The welfare function evaluated at the initial moment will be: W(0,`) 5 W(0,t*) 1 e 2 r t *W(t*,`). As the evaluation of the alternative policies depends on consumption, it is of interest, first, to determine the evolution of this variable in the three models. Secondly, and with respect to each model, we analyze the relative values of the welfare function W(0,t). Finally, taking into account W(0,`), we study the factors that affect the relative welfare difference.
5. Results
5.1. Consumption behavior The consumption path will be increasing 7 under all three policy rules. The consumption growth rate will be decreasing towards the steady state rate, but differences will exist in the consumption levels and in the growth rate, depending on government policy. We will describe the consumptions corresponding to Md , Mf and MA as c d , c f , and c A , respectively. We can conclude that the level of consumption in the economy with two fixed subsidies ends up being greater than that corresponding to Md and Mf , although in the early stages the consumption is lower. This is so due to the type of subsidy applied in MA , which is superior to the optimum during the transition and, consequently, the accumulation of capital and the number of varieties is also superior. Fig. 1 displays the typical difference between c A and c f . An analogous behavior results for the difference c A 2 c d . The relevant question is whether the initial lower consumption is compensated
Fig. 1. Differences in consumption. 7
This is due to the fact that we are only considering a situation in which z 0 is lower than its steady state value.
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Fig. 2. Differences in welfare.
by the higher future consumption, with this depending on the discount rate and the utility function.
5.2. Welfare analysis We have found that the higher future consumption always compensates for the lower initial one in all the possible combinations of parameters when we compare the total welfare –W(0,`)– of MA with Md . Thus, we can state that the two fixed subsidies applied at the rate corresponding to the steady state always represent a better policy than non-intervention. However, this is not the case when we compare MA with Mf . Fig. 2 shows two graphics representing the temporal evolution of the differences between the functions W(0,t) of MA and Mf for two different combinations of parameters. The conclusion derived from Fig. 2a is surprising: Mf leads to a greater welfare than MA . This means that although the policy of fixed subsidies guarantees that the economy will reach the optimal steady state values of K /A, c /K and HY , its effects on the levels taken by each variable lead to an absolute loss of welfare with respect to its alternative, not only for the early generations, but for all of them. But we can also find other combinations of parameters in which the opposite happens. We have an example in Fig. 2b. Here, the overaccumulation induced by the fixed subsidies produces welfare losses at the beginning, but these are compensated by the greater consumption in later periods. The relevant question is to know what determines whether the results will be one or the other. Once having applied the optimum fixed subsidy to the hiring of capital goods, an additional policy of fixed subsidy to the cost of R&D firms at the rate appropriate to the steady state (MA ) can either increase or decrease the welfare gains as compared to Mf . This situation has been graphically illustrated in two particular cases. The relevant question is to conclude what determines that the application of s *A leads to welfare gains or losses against the alternative of only maintaining the subsidy for the hiring of capital goods. In order to answer this question, Fig. 3 contains the relative welfare difference between MA and Mf for all combinations of parameters. For each a and s, the graphics display the value (W A (0,`) 2 W f (0,`)) /W A (0,`) face to d H /r. Note that when s is reduced (1.25), the fixed subsidy s *A leads, in almost all cases, to welfare gains (except for very low values of both a and d H /r ), and that welfare losses are present when s is high and a and d H /r are low. Specifically, as s grows, the range of values of a and d H /r that produce welfare losses widens out.
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Fig. 3. Relative differences in welfare 100 W A (0,`) 2 W f (0,`) /W f (0,`).
6. Conclusions We know that once the distortion due to monopoly power has been corrected in Romer’s (1990) model, the resulting welfare losses in the economy can be avoided by means of an optimal subsidy to the R&D costs. However, the application of an additional fixed subsidy, although optimal in the steady state, could lead to either welfare gains or losses taking into account the transition, depending on the technological and utility parameters. Specifically, this additional policy can be harmful when the labor endowment, the productivity of the innovative sector, or the monopoly power (a ) are low and both the discount rate and the inverse of the elasticity of substitution are high.
References ´ X., 1995. Economic Growth. McGraw-Hill. Barro, R.J., Sala-i-Martın, Garcia-Castrillo, P., Sanso, M. (2000): Human Capital and Optimal Policy in a Lucas-type Model, Review of Economic Dynamics (Forthcoming). ´ X., 1993. Transitional dynamics in two sector models of endogenous growth. Quarterly Mulligan, C.B., Sala-i-Martın, Journal of Economics 108 (3), 737–773. Romer, P.M., 1990. Endogenous technological change. Journal of Political Economy 98, S71–S102.