Optimal taxation with heterogeneous firms and informal sector

Journal of Macroeconomics 35 (2013) 39–61

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Optimal taxation with heterogeneous firms and informal sector Rodrigo A. Cerda a, Diego Saravia b,⇑ a b

Ministry of Finance, Teatinos 120, 8340454 Santiago, Chile Central Bank of Chile, Agustinas 1180, 8340454 Santiago, Chile

a r t i c l e

i n f o

Article history: Received 7 July 2011 Accepted 7 August 2012 Available online 26 October 2012 JEL classification: H21 Keywords: Optimal taxation Heterogeneous firms Capital taxation Informal sector

a b s t r a c t We study steady state optimal taxation in a context where firms differ in productivity and they decide whether to produce in a formal sector where the government raises taxes or an informal sector where it cannot do so. The government taxes capital income, firms’ profits and labor income. In this context, taxation might distort the firms’ decisions to participate in the formal sector (extensive margin) as well as their factor allocations once they decide to produce (intensive margin). We find that the government has incentives to subsidize costs to induce firms into the formal sector, making them taxable and completing the tax system. The optimal capital income tax is negative while the corporate tax rate is positive and the sign of labor income tax is ambiguous. We show numerically that following a Ramsey policy rather than actually observed taxes have significant effects on welfare. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction In this paper we study steady state optimal (Ramsey) taxation in a context where firms are heterogeneous in the sense that they differ in their productivity and decide whether to produce in a formal or an informal sector. The presence of an informal sector makes the analysis especially relevant to developing economies where the coexistence of formal and informal sectors is more important than in developed ones. In fact, in developing countries the average size of the informal sectors is around 35% points of the ‘‘official’’ GDP, in contrast to a 15% points in OECD countries (see, for example, Buehn and Schneider (2007)). The relative importance of informal sectors in developing economies could possibly be explained by come characteristics present in those countries. For example Loayza (1996) argues that informal sectors arise when excessive taxes and regulations are imposed by governments that lack the capability of enforcing compliance or face technological restrictions that impede tax collection in some activities, which is a more common situation in developing economies. Our work does not explain why an informal sector exists but takes its existence as given and analyze the optimal policies of interest, which is a strategy followed by previous works in the literature studying Ramsey problems in models with an informal sector (see for example Nicolini (1998)). In the paper, the decision of producing in the formal or informal sector is taken after comparing the after-tax profits ob`-vis profits that would be obtained in the informal (outside option) sector. tained from production in the formal sector vis-a The government finances an exogenous expenditure path using three tax instruments: capital income tax, labor income tax and a tax on firms’ profits. Also, the government can issue debt to finance its expenditure but, crucially, we assume that the government cannot set taxes in the informal sector. In this context, taxation potentially distorts the firms’ decision to participate in the formal sector, i.e. the extensive margin distortion, as well as the factor allocations of the firms already involved in production, i.e. the intensive margin distortion.

⇑ Corresponding author. E-mail address: [email protected] (D. Saravia). 0164-0704/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2012.08.002

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R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

The distortions created in the extensive margin are key to our results. When there are firms that choose not to produce in the formal sector, it is optimal to set a negative tax on capital income and a positive tax on firms’ profits. There is also a tendency to subsidize labor, although the sign of this tax depends on labor supply considerations. The intuition is that by subsidizing capital income, and possibly labor, the planner induces more firms into production in the formal sector and makes them taxable through the tax on profits. However, when there is no distortion in the extensive margin, i.e. when all firms decide to produce in the formal sector, the optimal capital income tax is equal to zero, as in the celebrated results of Chamley (1986) and Judd (1985). The tax on labor is also zero in this case and all tax revenues are raised using the tax on profits. The intuition is that, in this case, additional capital and labor is not socially desirable since they do not enlarge the taxable set of firms and thus it is optimal not to distort the intensive margins. It is optimal to tax only profits in this case since this tax does not distort any margin acting as a lump-sum tax. The paper is related to several other studies in the literature. Following Chamley (1986) and Judd (1985) several papers showed that optimal capital income taxation is different from zero if the context is modified in some ways. Correia (1996) extends Chamley’s (1986) result to an environment of incomplete taxation, where not all factors can be taxed. She finds that taxing capital would be an indirect way of taxing the untaxed factor and the sign of this tax would depend on the complementarity between capital and the untaxed factor.1 Like Correia (1996), we consider a context of incomplete taxation where the government does not tax the informal sector, but differ in that we analyze an environment where firms are heterogeneous. Firm heterogeneity and the distortions created in the extensive margin are key to our results. In this setting, the planner has incentives to subsidize costs in order to tax firms that would not otherwise be taxed. Thus, the optimal capital income taxation is negative. In the same vein as Correia (1996), other studies find that capital taxes should not be zero as a consequence of incomplete taxation. Jones et al. (1993) study the issue with endogenous growth models, showing that including government expenditures as a productive input leads to an optimal tax rate different from zero. The reason is similar to Correia’s (1996) explanation since government expenditure is not taxed. Jones et al. (1997) also show that the zero income capital tax is no longer optimal when pure profits are generated. Their interpretation of this result is that taxing capital is a way of taxing pure profits in a setting where they cannot be taxed directly. A second line of research related to our study includes Judd (1997), Judd (2002) and Coto-Martinez et al. (2007). Judd (1997) and Judd (2002) argue that the optimal capital income tax rate is negative and the tax on profits is positive in a context of monopolistic competition. Coto-Martinez et al. (2007) add entry and exit of firms to Judd’s (2002) framework, where the entrance of new firms augment the general productivity of the economy but implies a waste of resources in the form of a fixed cost. In Coto-Martinez et al. (2007) optimal taxes depend on the tax code available. When the available taxes are such that the government can control the number of firms through a tax on profits, it is optimal to subsidize capital to correct the markup distortion as in Judd and set a subsidy or a tax on profits depending on the aggregate returns to specialization. When the tax system does not allow to control the number of firms through profits taxation, they find that the optimal capital income taxation is zero if the returns to specialization are zero. The reason is that, in this case, it is not desirable to subsidize the entrance of new firms since there are only losses (fixed cost) associated with them. We also find that it is optimal to subsidize capital but in a different context and for different reasons. Ours is a context of perfect competition (no markup distortions) and without aggregate returns to specialization. In the case, where the yields of the alternative option cannot be taxed, it is optimal to subsidize capital, and possibly labor, to induce firms that are on the margin into production, making their taxation possible. Thus, in our case, the planner provides a subsidy to capital in order to complete the tax system not due to distortions created by imperfect competition or the presence of returns to specialization. While, to the best of our knowledge, the papers most related to ours are those mentioned above, there are other papers that find an optimal capital income tax different from zero. Aiyagari (1995) makes the point in an economy with borrowing constraints, where precautionary savings leads to too much capital in steady state. The optimality of taxing capital income was also obtained in OLG models. Recent work using this approach includes Abel (2005) and Erosa and Gervais (2002). Abel (2005) focuses on a context of consumption externalities between generations and shows that taxing capital is a way to correct the no ‘‘internalization’’ of cohorts’ consumption. Erosa and Gervais (2002) derives the optimality of capital taxation as a way of making taxes age-dependant. In a context of private information about agents’ skills, Golosov et al. (2003) show that it is optimal to have a wedge between the marginal benefit and marginal cost of investing, which is consistent with a positive tax on capital income. Since our paper analyzes a Ramsey problem in a context where there is a sector where taxes cannot be raised the relation to other papers analyzing other types of optimal policies in models with informal sectors is evident. For example Nicolini (1998) and Koreshkova (2006) study a Ramsey problem to determine the optimal inflation tax in a context where it is impossible for the government to raise other taxes in the informal sector. Our interest is in optimal taxation with no inflation tax, with entry and exit of firms and capital accumulation. However, there are same similarities in the essence of the analysis. Similar to these papers, we assume the existence of the informal sector, possibly as a consequence of institutional restrictions, and solve for the relevant optimal policies. While in Nicolini (1998) the technology used in both sectors is the same, in our case as in Koreshkova (2006) the technologies used are different. Another similarity with those papers is that the consumption good is produced in a competitive environment.

1

Correia (1996) suggests that similar results would be obtained if firms present decreasing returns to scale and profits cannot be taxed.

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

41

We perform numerical simulations of our model and study quantitatively the welfare implications of following the optimal policy suggested by this model. In particular, we compare consumption, output and utility in the case of optimal policies with their values in the case where actual tax rates are used. We find that the steady state consumption must be increased approximately by 16% in the model evaluated using actual taxes to reach the same level of utility that would be observed under optimal tax policies. This number is consistent with Jones et al. (1993), who find that consumption must be incremented by around 15% to reach the utility observed using optimal taxes for a given parametrization of the model. The rest of the paper is organized as follows. Section 2 describes the model. Section 3 sets up and solves the Ramsey problem. Section 4 presents numerical examples that confirm our theoretical findings and provides quantitative impacts in the case of standard utility and production functions. Section 5 concludes the paper. 2. The model 2.1. Firms There is a set I of mass one of heterogeneous firms indexed by i that can operate and produce a single good. Firm heterogeneity comes from a productivity parameter, Ait, which is iid across time t and is distributed across firms with cumulative distribution G(Ait) with support [Al, Au], where 0 < Al < Au < 1.2 We assume that there are two sectors of production: one sector uses capital and labor to produce the single good while the other sector uses only labor. In each period t firm i observes its technology shock Ait and decides in which sector to produce. Assumption 1. The government cannot tax either profits or labor from the sector that uses only labor to produce the single good. This assumption implies that there is a formal sector, the one using capital and labor, and an informal one, the one using only labor. The assumption also introduces incompleteness in the tax system which is crucial to our results, as will become clear below. It is common in the literature to model the sector where taxes cannot be raised (the informal sector) as using labor more intensively than capital. See, for example, Turnovsky and Basher (2009), Ihrig and Moe (2004), Larraín and Poblete (2007) and Yuki (2007), where the informal sector is modelled, as we do here, using only labor while the formal sector uses both capital and labor.3   inf Let Aitf(kit, lit) be the production function of a firm with productivity Ait in the formal sector and B lit the production function in the informal sector which is common to all firms producing in it and does not depend on the productivity level that the firm would have if it operated in the formal sector. Production functions in both sectors present decreasing returns to scale and are strictly increasing, strictly concave and satisfy Inada conditions on the factors of production. We further as  inf sume that f(kit, lit) is homogeneous of degree h < 1 in (kit, lit) and that B lit is homogeneous of degree 1  a < 1. The homogeneity assumption does not affect the main results of the paper but simplifies the exposition. Capital is rented from the representative household in each period at the rental rate rt, which, in equilibrium, is the same for all firms. Firms in the formal sector pay an amount wt as compensation for the use of labor while firms in the informal sector pay an amount winf t . In each period, after observing its shock Ai, firm i must decide between entering to produce in the formal or in the informal sector. To make this decision, firms compare the after-tax profits obtained in the formal sector with the profits that would be obtained in the informal sector. The higher the tax on profits in the formal sector, the lower the number of firms that would choose to produce in it. Thus, the tax on profits distorts the extensive margin and this is the only margin that this tax distorts in this model as will be discussed below. Firms’ profits are given by the product obtained minus payments to factors used. The rental rate of capital and the wage rate are determined in competitive markets and are the same for all firms in each sector. Hence, a firm solves the following static problem in period t:

n o max V it ; V inf it where V it ¼ max kit ;lit

and V inf it ¼ max inf

lit



ð1Þ  1  sut ½Ait f ðkit ; lit Þ  r t kit  wt lit 

h   i inf inf B lit  winf t lit :

ð2Þ

ð3Þ

2 Previous work that used iid shocks in models with informal sectors include Fortin et al. (1997) and Russo (2008). The main theoretical results of the paper do not depend on this assumption. 3 This modelling strategy followed by the literature is supported by the observation that, in the real world, informal sectors are much less capital intensive. Modelling informal sectors as using only labor is a benchmark that simplifies the exposition.

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Let Vlt and Vut be the function Vit evaluated at the lowest and highest productivity shocks, Ait = Al and Ait = Au, respectively. The solution to the firm’s problem, given by Eq. (1), is stated in the following lemma. Lemma 1. There exists a threshold technology level At such that firms endowed with technology Ait P At enter into the formal sector, while firms endowed with Ait < At do not enter into the formal sector. When V lt < V inf < V ut , the threshold At is interior it and uniquely determined by:



max kit ;lit

  1  sut At f ðkit ; lit Þ  r t kit  wt lit ¼ V inf it :

ð4Þ

  inf If V lt P V inf it , the threshold At is equal to Al and if V ut 6 V it the threshold At is equal to Au.

Proof. Using envelope arguments it can be seen that the function Vit in (1) is strictly increasing and continuous in Ait while inf inf 4 the function V inf it is independent of Ait. Then, when V it is between Vlt and Vut, there is a unique Ait that makes Vit equal to V it . If  Vlt were larger than V inf , all firms would prefer to participate in the formal sector for any value of A and the threshold A would it t it be given by Al. If Vut were smaller than V inf it then all firms would prefer to participate in the informal sector for any value of Ait h If it were the case that firms with the highest productivity level prefer to participate in the informal sector (i.e. V ut 6 V inf it ) then from Lemma 1 we know that all firms would prefer the informal sector which is not an interesting scenario in our case.5 To rule out this corner situation we assume that the most productive firm in the formal sector (i.e. the one with higher profits in that sector) would always prefer the formal to the informal sector. That is, we assume: Assumption 2. V ut > V inf it . Thus, in our context firms are able to move from one sector to the other after observing their idiosyncratic shock and evaluating in which sector they will earn higher after-tax profits. Although data on informal sectors is problematic by the nature of the issue, there is empirical evidence showing that movements between sectors occur and the magnitude depends on the countries considered. For example, in El Salvador, 77% of the firms that begin activities in the informal sector continues in that status while in Argentina this magnitude is around 90% (see La Porta and Schleifer (2008)). Firms in the formal sector demand capital and labor while those in the informal sector demand only labor. Factor demands are functions of factor prices and the idiosyncratic shocks and are generically given by:

  if kit ¼ kit Ait ; r t ; wt ; winf t kit ¼ 0 if

and

At

At 6 Ait ;

ð5Þ

> Ait

  lit ¼ lit Ait ; rt ; wt ; winf if At 6 Ait ; t   inf inf if At > Ait : lit ¼ lit Ait ; r t ; wt ; winf t

ð6Þ

Markets are competitive, capital and labor are paid their marginal productivity, and the rental and wage rates are the same for all firms in each sector. Hence, the rental and wage rates in the formal sector are given by:

rt ¼ Ait fk ðkit ; lit Þ;

ð7Þ

wt ¼ Ait fl ðkit ; lit Þ

ð8Þ

and the wage rate in the informal sector is given by: inf

winf t ¼ Bl ðl Þ:

ð9Þ

The factor demands in each sector follow from the aggregation of individual factor demands by all the firms that produce in them. That is, the group of firms that get a productivity shock higher than At demand:

K Dt ¼

Z

Au

At

kit dGðAit Þ

ð10Þ

lit dGðAit Þ;

ð11Þ

and

LDt ¼

4 5

Z

Au

At

  @V inf u t it By the envelope theorem @V @Ait ¼ 1  st f ðkit ; lit Þ; while @Ait ¼ 0. In this case the formal sector would be empty and the government would not be able to raise any taxes.

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while the group of firms in the informal sector (i.e. shock smaller than At ) demand:

 inf  LD;inf ¼ lit G At : t

ð12Þ

2.2. The household 1 There is an infinitely lived representative household choosing a consumption path fct g1 t¼0 and a leisure path fht gt¼0 that maximizes:

1 X bt uðct ; ht Þ;

ð13Þ

t¼0

where u() is strictly increasing, strictly concave and three times continuously differentiable in both arguments. We assume also that uch P 0. The household is endowed at time zero with an initial amount of capital K0 and holds the initial stock of government bonds b0. Each period, she decides how much to consume, how much to work and how much to invest in capital and government bonds to be held into next period. It is assumed that capital depreciates at rate d, while total time available for work and rest is H. Capital, Kt, is rented to firms in order to be used in the production process at the rental rate rt. Labor can be supplied in the formal and in the informal sector. It is assumed that labor is homogeneous. Labor offered in the formal sector is denoted by Lt and is rented at rate wt while Linf is the labor rented to firms in the informal sector at a rate winf t . t RA The representative individual receives the after-tax profits that firms obtain in the formal sector, Au V it dGðAit Þ, or alter  t natively, the returns of the firms participating in the informal sector, V inf it G At . In the formal sector, the government taxes k l u the rental rate at rate st , the wage rate at the rate st , firms’ profits at rate st and issues one-period bonds, which pay a gross d interest rate of Rt. Let bt be the stock of bonds held by the representative household. Hence, each period the household faces the following constraints: d

btþ1 inf ~ t Lt þ winf ¼ ~r t K t þ w t Lt þ Rt   d þ V inf it G At þ bt ;

ct þ it þ

Z

Au

At

V it dGðAit Þ ð14Þ

K tþ1 ¼ ð1  dÞK t þ it

ð15Þ

and ht þ Lt þ Linf t ¼ H;

ð16Þ 

 k



 l

~ t  wt 1  st . ~ t as ~r t  rt 1  st and w where following Chamley (1986) we define the variables ~rt and w The solution to the consumer’s problem yields the standard optimality conditions which include the marginal rate of substitution between present and future consumption (17), the intratemporal rates of substitution between leisure and consumption (18) and (19), the non-arbitrage condition (20) and the transversality conditions for capital and government bonds, (21) and (22).

U c ðtÞ ¼ bU c ðt þ 1Þð1 þ ~rtþ1  dÞ; ~ t; U h ðtÞ ¼ U c ðtÞw U h ðtÞ ¼ U c ðtÞwinf t ; Rt ¼ 1 þ ~rtþ1  d; ! T1 Y kTþ1 ¼ 0 R1 lim i

T!1

lim

ð19Þ ð20Þ ð21Þ

i¼0

and T!1

ð17Þ ð18Þ

T1 Y i¼0

! R1 i

bTþ1 ¼ 0: RT

ð22Þ

The distortions created by the taxes we analyze in this framework can be seen from the individual and firms’ optimality conditions. As usual, the tax on the rental rent of capital introduces a wedge between the intertemporal rate of substitution in consumption and the return on investment, as shown in Eq. (17). The labor tax introduces a wedge between the intratemporal marginal rate of substitution between labor and consumption and the marginal productivity of labor in the formal sector, as can be inferred from (18) and (8). In addition to this usual distortion, the labor tax introduces a wedge between the marginal productivity of labor in the formal sector and in the informal sector. In effect, since labor is homogeneous and can be supplied in both sectors without any restriction, it follows from (18) and (19) that the wage rate in the informal sector, winf t , must be equal to the after-tax ~ t in equilibrium. If this were not the case, labor would be supplied only in the sector with wage rate in the formal sector, w higher wages.

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R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

Because wages in both sectors are equal to their respective marginal productivities of labor, the equilibrium condition mentioned in the previous paragraph implies that the productivity of labor employed by firms in the formal sector is different from the marginal productivity of labor in the informal sector, which is a suboptimal situation since it would be desirable to employ additional labor where it is more productive. Finally, the tax on profits in the formal sector only distorts the firms’ decision of whether to produce in the formal or informal sector (i.e. the extensive margin) and does not distort any marginal rates of substitution for the individual. 2.3. The government As is usual in the optimal taxation literature the government collects taxes to finance an exogenous expenditure path fet g1 t¼0 . We assume that government expenditure is wasteful. As noted above, the government finances its expenditure by issuing bonds and levying flat-rate, time-varying taxes on capital income, on labor income and on firms’ profits. To avoid the possibility that the government raises all revenues by taxing initial capital heavily not distorting the economy allocations, we make the standard assumption that the government takes the tax rate on capital income in the first period, sk0 , as given. We also assume that the government can commit itself to a given policy so we do not analyze commitment issues. Hence the government’s budget constraint in period t is:

slt wt Lt þ skt rt K t þ sut

Z

s

Au

At

½Ait f ðkit ; lit Þ  r t kit  wt lit dGðAit Þ þ

btþ1 s  bt P e t Rt

8t:

ð23Þ

The first term on the left hand side is the amount of taxes on labor income, the second is the amount raised from capital income while the third term corresponds to the taxes raised from firms’ profits. The difference between the fourth and fifth terms is the increase in government indebtedness in period t.6 2.4. Equilibrium Given the description of our economy, we may state the following definition of equilibrium. n o1 inf Definition 1. A competitive equilibrium is a sequence of allocations ct ; kit ; lit ; lit ; bt , a sequence of prices n o1 t¼0;i2I  k l u 1   1 ; R , a government policy s ; s ; s ; e and a sequence of threshold technology levels At t¼0 such that: r t ; wt ; winf t t t¼0 t t t t t¼0

1. the household maximizes (13) subject to (14)–(16) taking K0 and b0 as given, 2. each firm solves (1) conditional on Ait, 3. the sequence of threshold technology levels is determined by:

       inf 1  sut ð1  hÞAt f kt ; lt P ð1  aÞB lt 8t; 



where kt ; lt are the optimal capital stock and labor demanded by the firm endowed with the threshold technology level and inf lt

is labor demanded by firms in the informal market,7 4. the government satisfies (23), 5. the capital market clears, i.e.

Kt ¼

Z

Au

At

kit dGðAit Þ 8t;

ð24Þ

6. the labor market clears, i.e.

Lt ¼

Z

Au

At

lit dGðAit Þ 8t;

inf    Linf 8t; t ¼ l t G At

H  ht ¼ Lt þ Linf t

ð25Þ

8t;

7. the bonds market clears s

d

bt ¼ bt

8t;

ð26Þ 8

8. the goods market clears, i.e.

6 Note that this budget constraint is derived considering that marginal productivity of capital and labor is equalized across firms and thus the wage rate and the rental rate of capital are outside the integral. 7 These expressions already consider the homogeneity assumption. 8 The first term in the right hand side is total output produced by firms in the formal sector while the second term is total output produced by firms in the informal sector.

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

ct þ et þ K tþ1 ¼

Z

Au

At

  inf Ait f ðkit ; lit ÞdGðAit Þ þ Bðlt ÞG At þ ð1  dÞK t

8t:

45

ð27Þ

3. The Ramsey problem and the optimal taxes Our goal is to characterize the tax rates that are consistent with the allocations in a second best steady state, assuming that the economy converges to this steady state in the long run. As is standard in the literature, the social planner will choose among the set of competitive equilibria available the one that maximizes the representative individual utility. The planner chooses the allocations, tax rates and threshold technology subject to goods market clearing, consumer budget constraints, government budget constraints and the individual’s and firms’ optimality conditions. Implicit in our setup is the assumption that, although taxes can be raised only in the formal sector, the planner knows both sectors’ production function. In this way the planner can follow the tax policies derived from the constrained maximization that make the competitive equilibrium reach the second best allocation. Therefore, the planner solves the following problem: L¼

max

u ;k ;l ;linf ;b ;w r t ;At t ~ t ;~ t it it it

fct ;s

þ 1  dÞ

Z

Au

At

1

( Z 1 X bt u ct ;H 

gt¼0 t¼0

Z kit dGðAit Þ  ct  et 

Au At

! inf

lit dGðAit Þ  lt GðAt Þ þk1t #

Au

Atþ1

kitþ1 dGðAitþ1 Þ

" þ k2t

sut

Z

"Z

Au At

    inf Ait f ðkit ;lit ÞdGðAit Þ þ B lt G At

Au

At

ð1  hÞAit f ðkit ;lit ÞdGðAit Þ

# btþ1  bt  et 1 þ ~r tþ1  d At At At h  i h  i     inf 3 4 5 ~ t  Bl linf þ kt ½uc ðtÞ  buc ðt þ 1Þð1 þ ~r tþ1  dÞ þ kt ½uh ðtÞ  uc ðtÞw~t  þ kt 1  sut ð1  hÞAt f kt ;lt  ð1  aÞB lt þ k6t w t ) Z Au Z Au           þ k7it Ait fk ðkit ;lit Þ  At fk kt ;lt dGðAit Þ þ k8it ðAit fl ðkit ;lit Þ  At fl kt ;lt ÞdGðAit Þ : þ

Z

Au

hAit f ðkit ;lit ÞdGðAit Þ  ~r t

Z

At

Au

~t kit dGðAit Þ  w

Z

Au

lit dGðAit Þ þ

At

ð28Þ

Note that the above Ramsey problem is written as in the ‘‘dual approach’’, similar to many papers in the literature. In stat~ t . Note that these expressions do not represent prices but ing the problem, we follow Chamley (1986) by including ~rt and w replace the capital income tax and the labor income tax, respectively. The first constraint, associated with k1, is the goods market clearing, given by Eq. (27). The second, related to k2, is the ~ t , ~r t Þ and the fact that the production government budget constraint (23); which already incorporates the definitions of ðw functions of firms in the formal sector are homogeneous of degree h (see Appendix A for the details). The third constraint, accompanying k3, is the intertemporal consumption Euler Eq. (17). The restriction related to k4 is the intratemporal marginal rate of substitution between consumption and leisure (18). This restriction already incorporates the ~ t must be equal to winf fact that w in equilibrium as was explained above.9 t The restriction related to k5 indicates that the marginal firm (i.e. the least productive firm that decides to operate in the formal sector) must earn after-tax profits at least as large as the outside option in the alternative activity. The restriction ~ t is equal to winf associated with k6 is the informal sector firms’ first order condition (9) taking into account that w in equit librium. Finally the last two restrictions, related to k7 and k8, are in place to indicate that the marginal products of labor and capital must be equal among firms.10 As is common in Ramsey problems (see, for example, Lucas and Stockey (1983)), second-order conditions for this optimization problem involve third derivatives which prevent us from verifying that the allocation found as a solution is actually a maximum without imposing further restrictions regarding functional forms. However, an optimal interior policy has to fulfill the optimality conditions derived in the next sections. We assume, as previous works have done (e.g. Lucas and Stockey, 1983), that this optimal policy exists. 3.1. Optimal capital income taxation The optimality condition with respect to kit is:

    k1t b½Ait fk ðkit ; lit Þ þ ð1  dÞgðAit Þ þ k2t b sut ð1  hÞAit fk ðkit ; lit Þ þ hAit fk ðkit ; lit Þ  ret gðAit Þ þ k5t b 1  sut         ð1  hÞAt fk kt ; lt 1 Ait ¼ At þ b k7it Ait fkk þ k8it Ait flk gðAit Þ ¼ k1t1 gðAit Þ;

ð29Þ

  where 1 Ait ¼ At is an indicator variable that takes the value one when the firm is the marginal one and it is equal to zero otherwise. The first term on the left-hand side of this expression is the social valuation of an increase in the quantity of available goods derived from a marginal increase in capital in period t by firm i. The second term is the marginal reduction in the excess burden caused by the marginal increase in capital. The increase in capital allows for an increase in tax revenues that 9 An alternative would have been to write an additional constraint indicating the competitive equilibrium in the informal sector (19) and another condition 4 ~ t ¼ winf reflecting that w t . The multiplier k in the Ramsey problem would be equal to the sum of two multipliers had we used this alternative specification. 10 We do not post the consumer budget constraint because it is redundant by Walras law.

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R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

enables the government to reduce its debt and other taxes by the same amount. The third term on the left-hand side indicates the social value of an additional unit of capital invested by the marginal firm, which relaxes the participation constraint, enlarging the set of firms that are involved in production and, thus, are taxable. The last term indicates the change in social welfare derived from the effect that firm i’s additional unit of capital has on factors’ marginal productivity. Of course, the term on the right-hand side is the social cost of the marginal investment. Integrating (29) over all firms involved in the formal sector, evaluating it in steady state (dropping time subscripts) and taking into account that each firm equates its marginal product of capital to the interest rate yields:

1 k5 bð1  su Þð1  hÞr 1  GðA Þ

Z Au  Z Au k7i ðAi fkk ÞdGðAi Þ þ k8i ðAi flk ÞdGðAi Þ b

k1 b½r þ ð1  dÞ þ k2 b½su ð1  hÞr þ hr  ~r  þ þ

1 1  GðA Þ

A

A

1

ð30Þ

¼k :

In a competitive equilibrium the capital income tax distorts the intertemporal margin implied by Euler Eq. (17). When evaluated in steady state, this equation yields:

1 ¼ bð1 þ rð1  sk Þ  dÞ:

ð31Þ

From (31) and (30) we obtain the following expression:

k5   ð1  su Þð1  hÞr 1  G At

Z Au  Z Au k7i ðAi fkk ÞdGðAi Þ þ k8i ðAi flk ÞdGðAi Þ

½k1 þ k2 ½r  ~r  þ k2 ½ðsu  1Þð1  hÞr þ þ

1 1  GðA Þ

A

A

¼ 0:

ð32Þ

The first term of this expression is the only one that appears in Chamley’s (1986) context and is the basis for his conclusion that the capital tax rate should be zero in steady state. In fact, given that k1 and k2 are positive, the only way to have this expression equal to zero in Chamley’s case is when r equals ~r, which is the case only when sk equals 0. In our setup we also have the second term which is related to the social value of increasing government revenues through the increase in capital, the third term related to the social value of relaxing the extensive margin and the last term indicating the social valuation of the change in firms’ marginal productivity derived from the increase of capital. In Chamley’s setup the second term would be zero since the production function he used is homogeneous of degree one and the third term would not exist since it is related to the profits of the marginal firm, which is implied by our context of heterogeneous firms. Also, the last term would be zero since, as will be shown below, it depends on the existence of the extensive margin. To advance in the exposition of our findings we take the first-order condition with respect to su which, when evaluated in steady state, yields:

k2

Z

Au

A





Ai ð1  hÞf ðki ; li ÞdGðAi Þ ¼ k5 A ð1  hÞf ðk ; l Þ:

ð33Þ

This expression balances the marginal social cost of raising su, given by the social value of displacing the marginal firm out of production in the formal sector, the term on the right hand side containing k5, with the social value of raising government revenues (reducing the tax burden) through an increase in the tax rate on profits. Dividing (32) by r, using (33) and after several algebraical steps (see Appendix A for details) we obtain the following expression for sk: k

s

#

"  ð1  hÞð1  su Þ k5 1 ¼ SY  1    : 1 2 1  GðA Þ gðA Þ k þk

where SY 

ð1hÞA f ðk ;l Þ ð1hÞ

R Au A

dGðA Þ

Ai f ðki ;li Þ1GðAi Þ

ð34Þ

is the share of the marginal firm’s profits in the profits of all firms producing in the formal sec-

tor. Note that SY is positive but less than one since the marginal firm has lower production (profits) than the rest of the firms involved in production. 5 Eq. (34) is not a reduced-form expression for sk. In fact, it depends on other endogenous variables such as A ; k1kþk2 ; su and k k SY. However, it allows us to obtain some intuition about the sign of s . Firstly, note that s 6 0. This result holds because (1) su < 1; if not, no firm would be involved in production, (2) k1, k2 and k5 are nonnegative, (3) SY < 1. Note the relevance of the extensive margin distortion in our results. If k5 were equal to zero, the optimal capital income tax would be zero as in the classical results of Chamley (1986) and Judd (1985). In this case, all firms would be involved in

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

47

production in the formal sector and there would be no distortion in the extensive margin derived from taxation.11 Thus, in this case, the tax on profits would not be distortionary and it would be optimal to set capital income taxes equal to zero as is the case for labor income taxes as shown below. However, if there are firms not involved in production in the formal sector, such that k5 is positive, then it is optimal to subsidize capital income. This incentive to subsidize capital income comes from the extra social benefit derived from the additional unit of capital employed by the marginal firm (the term related to k5 in (32) discussed above). By setting sk less than zero the steady state rental rate faced by firms, r, is depressed as shown in Eq. (31). This provides incentives to firms that are not in the formal sector, but that are in the neighborhood, to enter into it allowing the government to obtain revenue from them through other taxes. In our case, the government cannot obtain fiscal revenues from firms in the informal sector. By inducing some firms into the formal sector through a decrease in the interest rate, the planner enlarges the set of firms that can be taxed using a tax on profits. The planner completes the tax system, at least partially. As can be deduced from Eq. (34) the sign of sk does not depend on whether the public accumulates assets or debt in steady state. As in the case of Chamley (1986), our result on capital income taxation would be obtained even in the case where bond issuance is not allowed. We can summarize the findings of this section in the following proposition: Proposition 1. It is optimal to set sk less than zero if and only if not all firms are producing in the formal sector. In the case that all firms are involved in the formal sector, the optimal sk is zero. 3.2. Optimal labor tax and optimal tax on profits In this section, we focus on the planner’s choice of the tax rates on labor and profits. As in the case of the expression concerning sk, the expressions we obtain next are not reduced-form solutions as they will depend on other endogenous variables. However, as above, we will be able to obtain the signs and intuition about the economic determinants involved. The optimal labor tax is obtained as follows. Similarly to the optimality condition of capital stock given in Eq. (30), we may obtain the optimality condition with respect to the allocation of labor in the ith firm; which yields the following when evaluated in steady state:

  ~ gðAi Þ þ k1 Ai fl ðki ; li ÞgðAi Þ þ k2 ½su ð1  hÞAi fl ðki ; li Þ þ hAi fl ðki ; li Þ  wgðA ~ uh þ k3 uch ð~r  dÞ  k4 uhh þ k4 uch w iÞ     5 u þ k ð1  s Þð1  hÞA fl ki ; li 1½Ai   ¼ A  þ k7i ðAi fkl ÞgðAi Þ þ k8i ðAi fll ÞgðAi Þ ¼ 0;

ð35Þ

⁄

where, as before, 1[Ai = A ] takes the value of one if firm i is the marginal firm and zero otherwise. The social planner balances the social benefit of increasing output through a marginal increase in labor (the term involving k1), the social value of increasing tax revenues (the term involving k2), the social value of the change in the marginal firm’s profits (the term involving k5) and the social value of the change in marginal productivity (terms related to k7 and k8) with the marginal social costs of increasing labor that is given by the direct effect on the consumer’s utility and the effects on the intertemporal and intratemporal wedges in the competitive equilibrium (first term of (35)). Integrating this expression over all the firms involved in production and using the first order conditions with respect to ~ t and ~r t we obtain (see Appendix A): ct ; w



  SY  1 uh  uch  1 ð1  su Þð1  hÞ þ k6 2 ucc  þ uc w SY w

Z Au  Z Au k7i ðAi fkl ÞdGðAi Þ þ k8i ðAi fll ÞdGðAi Þ ;

sl ½k2 ð1 þ rch þ rhh Þ þ uc  ¼ k2 ðrcc þ rch þ rhh Þ þ  ~

1 1  GðA Þ

A



A

ð36Þ

h R i R A ~ AAu li dGðAi Þ is total individual’s income (excludand ~c ¼ ~rð Au ki dGðAi ÞÞ þ bss b þ w

where rcc ¼ uuccc ~c, rch ¼ uuch c, rhh ¼ uhhuð1hÞ h h ing firm transfers) in steady state.12 Since expression (36) contains terms with opposite signs, it is not possible to determine the sign of sl. However it is possible to analyze the forces that influence it. The second term on the right hand side (the one containing SY) has similar components to the expression obtained for sk in (34), and it is negative. The third term on the right hand side is positive.13 A higher tax rate increases the use of labor in the informal sector and this term is related to the social value of the additional 11 12

If all firms are involved in production k5 is equal to zero by complementary slackness. Also, note that

bbss ¼ ð1 þ ~r  dÞbss ; is the gross return on bonds in steady state. inf 13 ~ since the marginal productivity for This is the case because k6 is negative given the way the last restriction was placed in the Ramsey problem: Bl ðl Þ P w labor tends to infinity when the demand of labor in the informal sector tends to zero. Thus, by Khun-Tucker k6 6 0. The term in parentheses accompanying k6 is also negative.

48

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

labor employed in that sector. The sum of the last two terms (the sum of the integrals) is the social value of the change in marginal factor productivity derived from the new labor employed which is positive using similar reasoning as used in the derivation of sk in (34). Note that the terms rcc, rch and rhh are related to the concavity of the utility function. However, it is not possible to know the signs of rcc and rch since they depend on the sign of ~c that includes bond holdings. A sufficient condition for the term multiplying sl and the first term on the right hand side (the sum of rcc, rch and rhh) to be positive is that ~c be positive. This would be the case if the return on capital and labor employed in the formal sector are higher than the interest payment on asset holdings. Further, note that if the whole set of firms is involved in production in the formal sector k5 = 0 and, from Eq. (33), k2 = 0. Note also that k6 would be zero in this case as its related restriction would be non-binding.14 Using a reasoning similar to that used above it can be deduced that when k5 = 0 the last term in the right hand side of (36) is zero (see the derivation of (34) in Appendix A). Consequently, it follows from Eq. (36) that the optimal sl is zero in the case that all firms are in the formal sector, as in the case of capital taxation. The intuition is that as the whole set of firms is already involved in production, the planner’s incentives to subsidize firms’ costs disappear. Fiscal revenue will be obtained from firms’ profits and possibly returns on government bonds, as will be shown below. By setting sl = 0 the government avoids the distortion in the marginal rate of substitution between consumption and leisure. We next focus on obtaining the optimal tax on profits, su. First note that su < 1 because if su were equal to one there would be no firms producing in the formal sector, making it impossible to collect revenues needed to finance the exogenous fiscal expenditure. We will initially analyze the case in which there are firms involved in the formal sector while others operate in the informal one (i.e. interior solution). In this case A⁄ is interior, i.e. Al < A⁄ < Au and k5 > 0. To obtain su, note that Eq. (35) and the first order condition with respect to A⁄ yield (see Appendix A):









su k5 hð1  Sy Þ 1 1 P  6 00 inf ¼ 1 þaþ  þ  l þ k B ðl Þ ;   A gðA Þ 1  su k þ k2 1  GðA Þ ð1  su Þðk1 þ k2 Þð1  hÞA f ðk ; l Þ 1  GðA Þ

ð37Þ

RA RA where P ¼ Au k7i Ai fkl dGðAi Þ þ Au k8i Ai fll dGðAi Þ. Again, this expression is not a closed-form solution since it depends on other endogenous variables. However, it is enough to determine the sign of su, which is positive since the right hand side of (37) is positive. The first term is positive for the reasons given above. The second term is positive since k6 and B00 (linf) are negative. It can be shown that P is positive by the same reasoning used in the derivation of (34). 15 Next we analyze the case in which A⁄ is not interior, i.e. Al = A⁄. In this case Eq. (37) cannot be applied in the analysis since it was obtained using the first order condition with respect to A⁄ that is no longer valid in the case that A⁄ = Al. To obtain su in this corner case note that we must satisfy the government budget constraint, which in steady state is:

su

Z

Au

ð1  hÞAi f ðki ; li ÞdGðAi Þ þ Al

bss  bss ¼ e; 1 þ ~r  d

k

when s = sl = 0. Here bss is the level of government bonds in steady state. It follows that:

su ¼

e þ ð1  bÞbss : R Au Ai f ðki ; li ÞdGðAi Þ A

ð1  hÞ

ð38Þ

l

Note that in the case where all firms are involved in production, the sign of su depends on fiscal expenditure, e, plus steady state interest payments, (1  b)bss.16 We can summarize the preceding discussion in the following proposition. Proposition 2. In the case that some firms are not involved in production in the formal sector, the optimal tax on profits is positive while the sign of the optimal labor tax remains ambiguous. However, in the case that all firms are involved in the formal sector, the optimal tax on profits is different from zero while the labor tax is zero. Since it is not possible to advance more analytically in our findings, in the next section we present numerical exercises to confirm the validity of our analytical findings and shed additional light on our results. 4. Numerical examples In our simulations, we consider the long-run steady state of the Ramsey equilibrium. To obtain our results, we follow the procedure developed in Schmitt-Grohé and Uribe (2006). See Appendix A on the numerical method for a detailed explanation of the procedure. 14 15 16

Demand for labor in the informal sector is zero in this case and, consequently, B0 (linf) tends to infinity. As in the case of sk the sign of su does not depend on the government asset position. It could actually be less than zero if the government holds a stock of assets such that its yields are bigger than fiscal expenditures.

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

49

First, we present the optimal policy’s results found when using our baseline set of parameters and run some robustness checks. Then, we compare welfare when the optimal policies found in the paper are followed with situations where debt levels and tax rates are as in reality. 4.1. Functional forms We use the following standard functional form for the utility function:

U¼

 /1 c/t 2 ðH  lt Þ1/2 /1

:

ð39Þ

The production function for a firm operating in the formal sector is:

  a 1a h Y it ¼ Ait kit lit ;

ð40Þ

while the firm’s production function if it operates in the informal sector is:

 1a  Y inf ;t ¼ W linf ;tinf :

ð41Þ

4.2. Parameters Our solution method requires the specification of several parameters governing preferences, technology and government expenditure. The values of the parameters are chosen according to previous work and empirical evidence. The discount factor b is set to yield an average interest rate of 6.5% (King and Rebelo, 1999) and the rate of capital depreciation is chosen so that the average investment-to-capital ratio is 7.6% (Cooley and Prescott, 1995). The parameters in the utility function are similar to those used in Schmitt-Grohé and Uribe (2006), /1 = 1 and /2 = .75, while the total number of hours available for leisure and work, H ¼ 1. In addition, we consider a level of a = ainf = .36 for both sectors in our baseline calibration.17 The parameter h is equal to .90 in the baseline estimation which is consistent with post-war US labor to capital share (see for example Khan and Thomas (2003)).18 We set the number of firms in the economy, N, equal to 50. Productivity values, Ai, are equally spaced between Al = 1 and Au = 5 with each productivity level having the same probability of occurrence. Productivity in the informal sector is chosen to be nearly 60% of the maximum productivity in the formal sector (Au), which is the parametrization used in Turnovsky and Basher (2009) in their study of fiscal policy with an informal sector. Finally, government expenditure is chosen to give a government-to-output ratio in a range observed in reality; for example this ratio is 15% in Chile while it is around 20% in the United States (see, for example, Galf (1994)). Table 1 summarizes the parameters used in our baseline estimations. 4.3. Results In this section we present the results of the numerical exercises for our baseline parameterization as well as some additional results derived from changing certain parameter values to check for robustness. We also present exercises in which we compare the loss in important dimensions such as consumption and welfare from using actual levels of taxation and debt rather than following the optimal tax rates obtained in the model. For comparison, we first present in Table 2 the results when the level of productivity in the informal sector, W, is equal to zero. Clearly in this case all firms will choose to produce in the formal sector (column 7) and the informal sector’s output would be nonexistent (column 6). Also, as the model predicts for this case, taxes on capital and labor would be zero while taxes on firms’ profits would be positive. Since sk and sl are equal to zero in this context, the remuneration to capital and labor are the same before and after taxes (columns 3 through 6 in the bottom block of the table). The bottom part of column 2 includes the information on welfare levels while the bottom part of column 7 and column 8 refer to the share of profits in total output of the formal and informal sector respectively. In this table, we include the results for three values of government expenditures which are chosen such that the ratio of these expenditures to output is in accordance with values observed in the real world (column 5). In Table 3 we show the same exercise as in Table 2 but considering the value for the productivity in the informal sector as in Table 1. As can be seen in Table 3, when introducing the informal sector the mass of firms producing in the formal sector reduces to a half of total firms. The optimal policy implies a tax on capital and labor income that is always negative and a tax on 17 We are not aware of estimations of this parameters for informal sectors. We use the same value for both sectors following previous work like Turnovsky and Basher (2009). 18 Although different papers use different values for example Bloom et al. (2009) use 0.75 and Bachmann et al. (2006) use 0.82.

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R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

Table 1 Parameter values used in the baseline results. b

/2

/1

H

h

a

d

e

ainf

W

.992

.75

1

1

.90

.36

.076

3

.36

3

Table 2 Optimal taxes without informal sector. (1) e

(2)

(3)

(4)

(5)

(6)

sk

sl

su

e Y

Y inf Yf

2.00 2.50 3.00

0 0 0

0 0 0

0.066 0.067 0.068

e

U

r%

~r%

w

~ w

Proff YT

2.00 2.50 3.00

3.409 3.256 3.102

5.99 5.99 5.99

5.99 5.99 5.99

10.99 10.82 10.65

10.99 10.82 10.65

10.00 10.00 10.00

(7) 1  G(A⁄)

(8) bss/Y

0.500 0.500 0.500

13.575 16.933 20.291 Profinf YT

0.14 0.17 0.21

0 0 0

(7) 1  G(A⁄) 1.00 1.00 1.00 %

(8) bss/Y 9.545 15.24 20.92 Profinf YT

%

– – –

Table 3 Optimal taxes. (1) e

(2)

(3)

(4)

(5)

(6)

sk

sl

su

e Y

Y inf Yf

2.00 2.50 3.00

0.009 0.010 0.011

0.007 0.008 0.009

0.176 0.192 0.208

e

U

r%

~r%

w

~ w

Proff YT

5.99 5.99 5.99

10.742 10.723 10.704

10.815 10.806 10.798

9.95 9.95 9.95

2.00 2.50 3.00

3.420 3.265 3.111

5.93 5.93 5.92

0.143 0.176 0.209

0.005 0.005 0.005

%

%

0.18 0.18 0.17

profits that is always positive which are in accordance with the findings in the theoretical model for an interior solution. Higher level of government expenditures are associated with higher levels of taxes on profits and higher levels of capital and labor income subsidies.19 Note that the welfare levels are very similar in both cases, which would be a consequence of the fact that the planner is optimizing in both cases with the same set of restrictions but the existence of an informal sector. Y The ratio of the product of the informal sector to the formal sector, Yinf (column 6 of Table 3) is difficult to reconcile with f empirical evidence that places the ratio close to 8% in United States and an average of 13% for OECD countries (Schneider and Enste, 2000;Schneider, 2002). One possible partial explanation is that in the real world taxes are not as predicted by this paper; in fact, they are generally positive (see Mendoza et al., 1994). In our context the optimal policy is to subsidize capital and possibly labor to induce firms out of the informal sector and into the formal one, increasing in this way production in the formal sector and reducing it in the informal one. As will be shown below, when evaluating our model using the tax rates observed in the real world instead of the ones derived optimally here we obtain a higher ratio of informal product to formal product. If the planner were able to choose the level of debt, it would choose to hold assets in order to finance a share of its expenditures. This asset accumulation would allow the planner to reduce welfare losses derived from distortionary taxation. Since our whole analysis is in steady state, bond holdings can be interpreted as initial conditions.20 Many works in the literature have shown that if the planner were able to choose the initial level of indebtedness, it would choose to hold assets in order to reduce welfare losses derived from distortionary taxation (see, for example, Lucas and Stockey (1983) and Ljungqvist and Sargent (2004), chapter 15). Note, however, that in our case the planner does not set taxes equal to zero but sets them as predicted by our theoretical findings. This is different from the standard case in the literature without uncertainty (or with complete markets) and commitment where if the government were able to choose initial assets it would choose a level of assets that allows it to finance its expenditures over time without needing distortionary taxation. We ask next how optimal taxes change if the government holds an amount of debt that is actually observed in the real world. This exercise is a natural approach given the interpretation of steady state bonds holdings as initial conditions.21 To do

19

Of course this implies that the before-tax and after-tax interest rates and wages differ, as indicated in the bottom of columns (3)–(6). Even if the analysis had not only been in steady state; the steady state debt levels resembles initial conditions in this type of models; see Debortoli and Nunes (forthcoming) for a recent discussion. 21 An example of an analysis of a Ramsey problem in steady state where debt is calibrated to match real values is Schmitt-Grohé and Uribe (2006). 20

51

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61 Table 4 Optimal taxes with positive government debt. (1) e

(2)

(3)

(4)

(5)

(6)

sk

sl

su

e Y

Y inf Yf

2.00 2.50 3.00

0.011 0.010 0.009

0.254 0.323 0.393

0.106 0.108 0.111

e

U

r%

~r%

5.92 5.93 5.93

5.99 5.99 5.99

2.00 2.50 3.00

3.379 3.195 3.011

(7) 1  G(A⁄)

(8) bss/Y

0.109 0.107 0.106

0.480 0.460 0.440

0.535 0.539 0.543

w

~ w

Proff YT

Profinf YT

10.880 10.892 10.904

8.116 7.370 6.624

9.02 9.03 9.04

0.153 0.193 0.233

%

%

3.52 3.49 3.46

this exercise we choose values of debt such that the ratio of debt to output is around 50%, which is a value observed in many countries (for example in the United States). Table 4 presents the results of this exercise. As can be seen the signs of the taxes are in accordance with the theoretical findings, which is not surprising since they imply that the sign of capital income and profits taxes do not depend on government’s asset position. However, in this case, the government has no assets to finance its expenditures and subsidies and consequently it taxes labor and corporate income. Note that the relative importance of the product obtained in the informal sector is higher in this case (around 10%) and that total output is lower in this case than in the case where the government is free to choose optimally its assets. This can be seen because the ratio of government expenditures to output is higher in Table 4 than in Table 3 and e is the same in both tables. In Table 5 we present a welfare comparison between the case where the planner chooses the level of assets (Table 3) and the case where a positive amount of debt is imposed (Table 4). Welfare is higher in the first case which is not surprising since in the second case we are imposing an additional restriction to the government. This welfare gain is higher the government’s expenditure. Column (2) and (3) of Table 5 present the level of consumption for the cases in Tables 3 and 4, column (4) refers to the implied increase in consumption. Columns (4), (5) and (6) present the same analysis but for the case of welfare instead of consumption. To give an additional piece of information in the welfare comparison we report in column (8) the increase in consumption that the representative consumer needs to receive in the case of Table 4 in order to reach the same level of utility as in Table 3 holding labor constant.

4.3.1. Quantitative effects on welfare of optimal policies In this section we present a quantitative exercise to see the welfare implications of following the optimal policies in our context. We compare two scenarios similar to Jones et al. (1993). In the first, we calculate the levels of welfare and consumption implied by the optimal taxes found in our numerical exercises presented above. In the second, we calculate welfare and consumption using our setup but instead using the tax rates and level of debt observed in the real world. We use the effective average tax rates for the United States reported in Mendoza et al. (1994) for our calculations, such that sk = 0.42, sl = 0.24 and su = 0.34 and, as before, a level of debt such that the ratio of debt to output is around 50%. The results are shown in Table 6. Since different values of the parameter h are used in the literature, we present the same exercise for two values of the homogeneity of the production function in the formal sector, h = 0.85 and h = 0.9. We keep the value of W as in Table 1. The first line of Table 6 presents the results of following the optimal policy when h = 0.85, while the second presents the estimations using the taxes and level of debt observed in the real world. The third and fourth lines present the same exercises as in lines one and two respectively but with h = 0.9. As can be seen in column 4, the number of firms in the formal sector is higher when optimal policies are followed; this is due to the subsidy to capital and because tax rates to labor and profits are lower in the optimal case than in the real world. Column 7 shows that the level of consumption is 33% higher in the optimal case when h = 0.85 while it is higher in 31% when h = 0.9 is considered. Welfare is around 16% higher if optimal taxes are set for both values of h, as can be seen in column 9. Column 10 in Table 6 labelled ‘‘factor’’ indicates by how much consumption must be increased in the case where taxes are at their real world values to reach the same level of utility obtained under optimal taxation. The results indicate that consumption would have to be increased by 14% and 16% when h = 0.85 and h = 0.9 respectively. This exercise is motivated by Jones et al. (1993), who present a welfare analysis of changing to the policies suggested by the Ramsey plan. Although they do this for different models and parameters than those used here, comparing their results with ours gives us some perspective. In their case, the coefficient by which consumption must be raised in order to bring utility under the current system up to the level attained in the Ramsey solution goes from 1.09 to 1.37 depending on the intertemporal elasticity of substitution in consumption.22 Our estimation is closer to Jones et al. (1993) when they use an intertemporal elasticity of substitution equal to two, which gives a coefficient of 1.15 which is close to our coefficient of 1.14 or 1.16. 22 They report a value of 3.92 for this coefficient in the case that the utility function is close to logarithmic. The results display high sensitivity to the magnitude of the elasticity.

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Table 5 Comparison Tables 3 and 4. (1) e

(2) C(3)

(3) C(4)

(4) DC%

(5) U(3)

(6) U(4)

(7) DU%

(8) Factor

2.00 2.50 3.00

8.17 7.80 7.43

7.54 6.98 6.42

8.35 11.77 15.78

3.42 3.27 3.11

3.38 3.19 3.01

1.21 2.20 3.31

1.018 1.024 1.046

Table 6 Effective taxes versus optimal taxes. (1) h

(2)

(3)

e Y

Y inf Yf

0.85 0.85 0.90 0.90

0.258 0.348 0.209 0.294

0.007 0.151 0.005 0.162

(4) 1  G(A⁄)

(5) C

(6) DC%

(7) U

(8) DU%

(9) Factor

0.500 0.480 0.500 0.400

5.673 4.379 7.435 5.644

22.81%

2.538 2.187 3.111 2.676

13.83% – 13.98% –

1.140 – 1.155 –

24.08%

Lastly, as mentioned above, note that the quotient of the products in the informal and formal sectors is higher when we consider the actual tax values rather than those suggested by the optimal policy. The values of 3% and 2% are higher than those reported in the first and third lines of Table 6, but still fall short of the values reported in the studies mentioned above. 5. Conclusions In this paper we have introduced heterogeneous firms and an informal sector that cannot be taxed to study optimal taxation. Firms differ in their productivity and have to decide in each period if they want to produce in the formal or informal sector after comparing expected profits in each alternative. Since the government cannot tax the informal activity, our environment is one of incomplete taxation (Correia, 1996). To finance its expenditure the government relies on capital income, labor, profits taxation and debt. The presence of heterogeneous firms and the option to produce in one of the sectors implies two kinds of possible distortions from taxes. They may affect the extensive margin decisions (i.e. whether or not to produce in the formal sector) and the intensive margin decisions (i.e. optimal allocation given the decision to produce in the formal sector). We have shown that the results depend on whether all firms decide to produce in the formal sector or the less productive firms decide to produce in the informal one. In the first case, there is no distortion in the extensive margin and the social planner will not tax capital, which replicates the Chamley (1986) and Judd (1985) result of not taxing capital income in the long run. Also, in this case, it is optimal not to tax labor and leave the tax on profits as the only source of fiscal revenues. This is because, in this case, the tax on profits does not distort any margin and it is optimal to tax only pure rents. However, when there are firms that are not so productive or when the informal sector is lucrative enough such that some firms prefer not to produce in the formal sector, it is optimal to subsidize capital. The sign of the tax on labor income is ambiguous depending on the distortions it creates in the labor supply. The intuition of the subsidy is related to the government’s inability to tax the firms in the informal sector. By subsidizing production costs, the government induces firms to produce in the formal sector, making it possible to tax their profits. That is, in the second best, the planner is partially completing the tax instruments by taxing firms that would not be taxed if they remain in the informal sector. In this respect, firm heterogeneity is key to the results. We also showed numerically that following a Ramsey policy rather than observed levels of taxation offers significant gains in welfare. Acknowledgments Superb assistance from Sergio Salgado is greatly acknowledged. We received helpful comments from Marco Bassetto, Sofia Bauducco, Eugenio Bobenrieth, Yan Carrière-Swallow, Javier Garcia-Cicco, Kenneth Judd, Michael Pries, Sergio Salgado, Klauss Schmidt-Hebbel, Felipe Zurita, the editor Theodore Palivos, various anonymous referees and seminar participants at several places. We also acknowledge financial support from FONDECYT, research Grant 1070962. Remaining errors are our own. Appendix A A.1. Derivation of government budget constraint used in Ramsey problem Since the production functions are homogeneous of degree h, the government budget constraint is:

slt wt

Z

Au

At

lit dGðAit Þ þ skt rt

Z

Au At

kit dGðAit Þ þ sut

Z

s

Au At

Ait ð1  hÞf ðkit ; lit ÞdGðAit Þ þ

btþ1 s  bt P et Rt

8t:

53

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

Further note that

slt wt

Z

Au

lit dGðAit Þ ¼ wt

At

Z

Au At

ft lit dGðAit Þ  w

Z

Au

lit dGðAit Þ

At

and

skt rt

Z

Au

kit dGðAit Þ ¼ r t

At

Z

Au

At

kit dGðAit Þ  ret

Z

Au

At

kit dGðAit Þ:

Substituting into the government budget constraint,

Z

Au

At

ft ðwt lit þ rt kit ÞdGðAit Þ  w

Z

Au

lit dGðAit Þ  ret

At

Z

Au At

kit dGðAit Þ þ sut ð1  hÞ

Z

s

Au

Ait f ðkit ; lit ÞdGðAit Þ þ

At

btþ1 s  bt Rt

8t:

P et

ð42Þ

Finally, since wtlit + rtkit = hAitf(kit, lit), we get the government budget constraint used in the Ramsey Problem. A.2. Optimality conditions for Ramsey problem in Eq. (28) The optimality condition with respect to lit is:

  ~ t gðAit Þ þ bt k1t Ait fl ðkit ; lit ÞgðAit Þ bt uh ðtÞ þ k3t uch ðtÞð~r t  dÞ  k4t uhh ðtÞ þ k4t uch ðtÞw           ~ t gðAit Þ þ bt k5t 1  sut ð1  hÞAt fl kt ; lt 1 Ait ¼ At þ bt k7it Ait fkl þ bt k2t sut ð1  hÞAit fl ðkit ; lit Þ þ hAit fl ðkit ; lit Þ  w   þ bt k8it Ait fll ¼ 0;

ð43Þ inf

and the optimality condition with respect to lit is:

 buc;h ðt þ 1Þ inf ~ t  þ k5t ðð1  aÞB0 Þ þ k6t ½B00 ðlinf Þ ¼ 0: þ k3t ½uhh ðtÞ  uch ðtÞw uh ðtÞ þ k1t B0 ðl ÞGðA Þ þ k3t uc;h ðtÞ þ 1 þ ret  d

ð44Þ

  ft ; ~rt evaluated in steady state are: The optimality condition with respect to ct ; At ; w

~ ¼ 0: ½c : uc ðc; hÞ  k1 þ ucc ðc; hÞ½k3 ð~r þ dÞ  k4 w ~ : k2 ½w

Z

Au 

ð45Þ

li dGðAt Þ  k4 uc ðc; hÞ þ k6 ¼ 0:

ð46Þ

A

½~r : k2

Z

Au A

ki dGðAt Þ  k2

b  k3 uc ðc; hÞ ¼ 0: 1 þ ~r  d 

ð47Þ

inf







~ þ k4 uhh ðl þ l ÞgðA Þ þ k1 ðð1  hÞA f ðk ; l Þ þ B þ ~r k ÞgðA Þ þ k2 ðh  su ð1 ½A  : ½uh  k3 uch ð~r  dÞ  k4 uch w 











~ ÞgðA Þ þ k5 ð1  su Þð1  hÞA f ðk ; l Þ  hÞÞA f ðk ; l ÞgðA Þ þ k2 ð~r k þ wl ¼ 0: A.3. Derivation of

ð48Þ R Au A

k7i ðAi fkk Þ þ k8i ðAi flk ÞÞgðAi Þ ¼ k5 ð1  su Þð1  hÞr gðA1  Þ

We derive here the expression used in the derivation of Eq. (34). Let us consider the first order condition of a firm that is not the marginal one and evaluate it in steady state:

  k1 b½r þ ð1  dÞgðAi Þ þ k2 b½su ð1  hÞr þ hr  ~r gðAi Þ þ b k7i ðAi fkk Þ þ k8i ðAi flk Þ gðAi Þ ¼ k1 gðAi Þ: In the case of the marginal firm the equivalent expression is: 



k1 b½r þ ð1  dÞgðA Þ þ k2 b½su ð1  hÞr þ hr  ~r gðA Þ þ k5 bð1  su Þð1  hÞA fk ðk ; l Þ1½Ai ¼ A  ¼ k1 gðA Þ: 8 The to k7 and   terms  related   k  disappear in this case because in the marginal firm’s case they are equal to zero  8  (k fkk  fkk ; ½Ai ¼ A  and k flk  flk ÞÞ. 7

54

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

Equating the last two expressions we get:

  k5   b k7i ðAi fkk Þ þ k8i ðAi flk Þ ¼ bð1  su Þð1  hÞA fk ðk ; l Þ1½Ai ¼ A ; gðA Þ and integrating them over all firms producing we obtain:

Z

Au 

A





k7i ðAi fkk Þ þ k8i ðAi flk ÞÞdGðAi Þ ¼ k5 ð1  su Þð1  hÞA fk ðk ; l Þ

1 ; gðA Þ

or equivalently

Z

Au 

A

k7i ðAi fkk Þ þ k8i ðAi flk ÞÞgðAi Þ ¼ k5 ð1  su Þð1  hÞr

1 ; gðA Þ

which is the expression employed in the paper. A.4. Derivation of Eq. (34) Integrating (29) overall firms involved in production in the formal sector and evaluating in steady state, we have:

k1

"

 Z Au   1 ð1  G At Þ þ k2 su ð1  hÞ Ai fk ðki ; li ÞdGðAi Þ þ ð1  dÞ  Ai fk ðki ; li ÞdGðAi Þ b A At  Z Au      Ai fk ðki ; li ÞdGðAi Þ  ~r 1  G At þh þ k5 ð1  su Þð1  hÞA fk ðk ; l Þ1½Ai ¼ A  A "Z u # Z u

Z

Au

A

þ

A

k7i Ai fkk dGðAi Þ þ

Adding and subtracting k2

R Au A

A

A

k8i Ai flk dGðAi Þ ¼ 0:

ð49Þ

   Ai fk ðki ; li ÞdGðAi Þ and dividing by 1  G At , we obtain:

2 3 R Au R Au

#  Ai fk ðki ; li ÞdGðAi Þ A f ðk ; l ÞdGðA Þ A f ðk ; l ÞdGðA Þ 1  A i i i i i i i i k k       þ ð1  dÞ  þ t  ~r 5 þ k2 4ðsu  1Þð1  hÞ A k1 A b 1  G At 1  G At 1  G At "Z u # Z Au A k5 1 7 8    ð1  su Þð1  hÞA fk ðk ; l Þ þ k A f dGðA Þ þ k A f dGðA Þ þ i kk i i i i lk 1  GðA Þ A i 1  G At A "R A u 

¼ 0:

ð50Þ

Since bð1 þ ret  dÞ ¼ 1 and using (7):

k1 ½r  ~r  þ k2 ½ðsu  1Þð1  hÞr þ k2 ½r  ~r þ þ

1 1  GðA Þ

"Z

Au

A

k7i Ai fkk dGðAi Þ þ

Z

Au

A

k5   ð1  su Þð1  hÞr 1  G At #

k8i Ai flk dGðAi Þ

¼ 0:

ð51Þ

Dividing by r:

k1 sk þ k2 ½ðsu  1Þð1  hÞ þ k2 sk þ

k5 1 ð1  su Þð1  hÞ þ 1  GðA Þ 1  GðA Þ

"Z

Au

A

k7i Ai fkk dGðAi Þ þ

Z

#

Au

A

k8i Ai flk dGðAi Þ

¼ 0:

ð52Þ

Using (33) in (52):

sk ¼

#

Z Au 

" Z Au ð1  hÞð1  su Þ k5 1 1 1 7 8 ½S  1  k ðA f ÞdGðA Þ þ k ðA f ÞdGðA Þ Y i kk i i lk i : i 1  GðA Þ r k1 þ k2 1  GðA Þ A i k1 þ k2 A

To simplify more this expression, plug in:

Z

Au 

A

k7i ðAi fkk Þ þ k8i ðAi flk ÞÞgðAi Þ ¼ k5 ð1  su Þð1  hÞr

obtained above to get Eq. (34) in the paper.

1 ; gðA Þ

ð53Þ

55

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

A.5. Derivation of Eq. (36) Integrating (35) overall firms involved in production, we have:

  ~ ð1  GðA ÞÞ þ k1 uh þ k3 uch ð~r  dÞ  k4 uhh þ k4 uch w þ k2 þ



su ð1  hÞ

Z

Au A

Z

Au

Ai fl ðki ; li ÞdGðAi Þ þ h

A

k7i ðAi fkl ÞdGðAi Þ þ

Z

Au A

Z A

Z

k8i ðAi fll ÞdGðAi Þ

Ai fl ðki ; li ÞdGðAi Þ

 ~  GðA ÞÞ þ k5 ð1  su Þð1  hÞA fl ðk ; l Þ Ai fl ðki ; li ÞdGðAi Þ  wð1

Au

A

Au 



¼ 0:

ð54Þ

Adding and subtracting k2

R Au A

Ai fl ðki ; li ÞgðAi Þ and dividing by (1  G(A⁄)), we obtain:

  ~ þ k1 uh þ k3 uch ð~r  dÞ  k4 uhh þ k4 uch w

R Au A

" # R Au Ai fl ðki ; li ÞgðAi Þ Ai fl ðki ; li ÞdGðAi Þ 2 u A ð s  1Þð1  hÞ þ k 1  GðA Þ 1  GðA Þ

# Ai fl ðki ; li ÞgðAi Þ k5   ~ þ þk w ð1  su Þð1  hÞA fl ðk ; l Þ  1  GðA Þ 1  GðA Þ

Z Au  Z Au 1 þ k7 ðAi fkl ÞdGðAi Þ þ k8 ðAi fll ÞdGðAi Þ  1  GðA Þ A A 2

"R A u A

¼ 0;

ð55Þ

or

k5 ð1  su Þð1  hÞw 1  GðA Þ

Z Au  Z Au k7i ðAi fkl ÞdGðAi Þ þ k8i ðAi fll ÞdGðAi Þ :

  ~ ¼ k1 w þ k2 ½ðsu  1Þð1  hÞw þ k2 wsl þ uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w þ

1 1  GðA Þ

A

A

ð56Þ

It follows that: l



s ¼

 ~ uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w 1 1  GðA Þ

"

k5

þ 1

#

k2 ð1  GðA ÞÞ

Z Au  Z Au k7i ðAi fkl ÞdGðAi Þ þ k8i ðAi fll ÞdGðAi Þ : k2 w





k1

A

k2

ð1  su Þð1  hÞ þ

1 k2 w

A

ð57Þ

Using (33) in (57):

sl ¼



 ~ uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w

þ

1 1  GðA Þ

 SY  1 ð1  su Þð1  hÞ SY k2 w

Z Au  Z Au k7i ðAi fkl ÞdGðAi Þ þ k8i ðAi fll ÞdGðAi Þ : A



k1



þ k2

A

ð58Þ

Note that using (45)–(48):

  ~ uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w k2 w



k1 k2

¼

RA

uhh Au li dGðAi Þ b ~ ð r  dÞ  uc w 1 þ ~r  d k2 w A R Au Z Au

~ uch A li dGðAi Þw uc ucc b ð~r  dÞ þ ki dGðAi Þ þ  2 uc w 1 þ ~r  d uc k A Z Au

ucc k6 u  u  ~ þ 2 h ucc  ch :  li dGðAi Þ w 2 uc w k uc A uh

þ

uch uc w

Z

Au

ki dGðAi Þ þ

ð59Þ

56

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

~ ¼ uh , we have: Since uc w

  ~ uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w k2 w



k1 k2

Z Au



Z Au ucc b ~ ð~r  dÞ þ ki dGðAi Þ þ li dGðAi Þ w 1 þ ~r  d uc k A A

Z Au Z Au

 uch b ~ ð1  sl Þ ð~r  dÞ þ ki dGðAi Þ þ li dGðAi Þ w þ 1 þ ~r  d uc A A RA uhh Au li dGðAi Þ k6 u h  uch  ð1  sl Þ þ 2 2 ucc   : uh w k uc uc

¼ sl

 2

h R  ~ A Further bð1 þ ret  dÞ ¼ 1 implies 1þb~rd ¼ bb. Let rcc ¼ uuccc ~c ; rch ¼ uuch c ; rðhÞ ¼ uhhuð1hÞ where ~c ¼ ~r Au ki dGðAi Þ þ bss b þ h h R Au ~ A li dGðAi Þ is total individual’s income -excluding firm’s transfers- in steady state. It follows that: w

  ~ uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w 2

k w



k1 k

2

uc

¼ sl

k

2

k6 uh  uch  ucc  : 2 u2 w k c

þ rcc þ rch ð1  sl Þ þ rhh ð1  sl Þ þ

ð60Þ

Replacing (60) in (58):

sl ¼ sl

uc k

þ rcc þ rch ð1  sl Þ  rhh ð1  sl Þ þ 2



 SY  1 k6 uh  uch  ð1  su Þð1  hÞ þ 2 2 ucc  SY u w k c

Z Au  Z Au 1 k7i ðAi fkl ÞdGðAi Þ þ k8i ðAi fll ÞdGðAi Þ  1  GðA Þ A A

uc ) sl 1 þ 2 þ rch þ rhh k

 SY  1 k6 uh  uch  1 1 ð1  su Þð1  hÞ þ 2 2 ucc  ¼ ðrcc þ rch þ rhh Þ þ þ 2  SY w k uc k w 1  GðA Þ þ



Z

Au A

k7i ðAi fkl ÞdGðAi Þ þ

Z

Au

A

 k8i ðAi fll ÞdGðAi Þ :

ð61Þ

This is Eq. (36) in the text. A.6. Derivation of Eq. (37) From the first order condition with respect to linf, Eq. (44), we find that:

~ ¼ k1 B0 ðlinf Þ  k5 ð1  aÞB0 ðlinf Þ  k6 B00 ðlinf Þ: ½uh  k3 uch ð~r  dÞ þ k4 uhh  k4 uch w

ð62Þ

Substituting this expression and (56) into (48):

k1 w þ k2 ðsu  1Þð1  hÞw þ sl wk2 þ

 1 1   5 u  k ð1  s ð1  hÞÞw þ  P l gðA Þ 1  GðA Þ 1  GðA Þ

inf







þ ½k1 B0  k5 ð1  aÞB0  k6 B00 l gðA Þ þ k1 ðð1  hÞA f ðk ; l Þ þ B þ ~r k ÞgðA Þ 











~ ÞgðA Þ þ k5 ð1  su Þð1  hÞA f ðk ; l Þ ¼ 0; þ k2 ðh  su ð1  hÞÞA f ðk ; l ÞgðA Þ þ k2 ð~r k þ wl

ð63Þ

where

P¼

Z

Au A

k7i ðAi fkl ÞdGðAi Þ þ

Z

Au

A

 k8i ðAi fll ÞdGðAi Þ :

Then, considering that aB(linf) = linfB0 (linf): 





k1 wl gðA Þ þ k2 ½ðsu  1Þð1  hÞwl gðA Þ þ k2 wsl l gðA Þ þ 



P





1  GðA Þ

l gðA Þ þ



k5  ð1  su Þð1  hÞwl gðA Þ 1  GðA Þ 







~ gðA Þ þ k1 ½1 þ ð1  su Þð1  hÞA f ðk ; l ÞgðA Þ þ k1~r k gðA Þ þ k2 ½su ð1  hÞ  hA f ðk ; l ÞgðA Þ þ k2 ½~r k þ wl 



inf

inf

inf

þ k5 ð1  su Þð1  hÞf ðk ; l Þ þ k5 ð1  aÞaBðl ÞgðA Þ þ k6 B00 ðl ÞÞl gðA Þ ¼ 0:

ð64Þ

57

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

Using ~r ¼ rð1  sk Þ and additional steps of algebra: 





P



ðk1 þ k2 Þ½wl þ rk gðA Þ þ k2 ½ðsu  1Þð1  hÞwl gðA Þ þ 



1  GðA Þ



l gðA Þ þ ðk1 þ k2 Þ½1 þ ð1  su Þð1 



 hÞA f ðk ; l ÞgðA Þ  ðk1 þ k2 Þr sk k gðA Þ þ k5 ð1  su Þð1  hÞf ðk ; l Þ þ inf

inf

k5  ð1  su Þð1  hÞwl gðA Þ 1  GðA Þ

inf

þ k5 ð1  aÞaBðl ÞgðA Þ þ k6 B00 ðl ÞÞl gðA Þ ¼ 0:

ð65Þ

It follows that:

" 

# k5    ð1  su Þð1  hÞwl gðA Þ þ k5 ð1  su Þð1  hÞf ðk ; l Þ 1  GðA Þ



ðk1 þ k2 ÞhA f ðk ; l ÞgðA Þ þ k2 þ þ

P



1  GðA Þ





inf



inf

inf

l gðA Þ  ðk1 þ k2 Þrsk k gðA Þ þ k2 ½su ð1  hÞ  hA f ðk ; l Þ  k1 B0 ðl Þl gðA Þ þ k5 ð1  aÞB0 ðl Þl

inf



inf

inf



þ k6 B00 l gðA Þ þ k1 ðA f ðk ; l ÞÞ þ k1 Bðl ÞgðA Þ ¼ 0:

ð66Þ

Rearranging this expression we get:

" 

# k5    ð1  su Þð1  hÞwl gðA Þ þ k5 ð1  su Þð1  hÞf ðk ; l Þ 1  GðA Þ



ðk1 þ k2 ÞhA f ðk ; l ÞgðA Þ þ k2 þ þ

P



1  GðA Þ inf







l gðA Þ  ðk1 þ k2 Þrsk k gðA Þ þ ðk1 þ k2 Þð1 þ ð1  su Þð1  hÞÞA f ðk ; l ÞgðA Þ þ k5 ð1

inf

inf

inf

 aÞB0 ðl Þl gðA Þ þ k6 B00 ðl Þl gðA Þ ¼ 0:

ð67Þ k5 S , 1GðA Þ Y

2

Since (33) implies k ¼ 

we have:

"

# k5  ð1  su Þð1  hÞwl gðA Þ þ ðk1 þ k2 Þð1 þ ð1  su Þð1 1  GðA Þ



ðk1 þ k2 ÞhA f ðk ; l ÞgðA Þ þ k2 þ 



 hÞÞA f ðk ; l ÞgðA Þ þ 



 hÞf ðk ; l Þ 

P

inf



1  GðA Þ

inf

inf

inf

l gðA Þ þ k5 ð1  aÞB0 ðl Þl gðA Þ þ k6 B00 ðl Þl gðA Þ þ k5 ð1  su Þð1

k5  ð1  su Þð1  hÞðSy  1Þrk gðA Þ 1  GðA Þ

¼ 0: ⁄

⁄

ð68Þ ⁄

⁄

⁄

Since hA f(k , l ) = wl + rk we get:

k5 P     ð1  su Þð1  hÞð1  Sy Þrk gðA Þ  su ðk1 þ k2 Þð1  hÞA f ðk ; l ÞgðA Þ þ l gðA Þ þ k5 ð1 1  GðA Þ 1  GðA Þ inf

inf

inf





 aÞaBðl ÞgðA Þ þ k6 B00 ðl Þl gðA Þ þ k5 ð1  su Þð1  hÞf ðk ; l Þ ¼ 0; 5

ð69Þ inf

inf inf

⁄

where k (1  a)aB(l )g(A ) comes from B0 (l )l (1  a)B(linf) = (1  su) (1  h)A⁄f(k⁄, l⁄) we get:

su ðk1 þ k2 Þð1  hÞA f ðk ; l ÞgðA Þ ¼

inf

= aB(l ) given the homogeneity of the production function. Since

k5 P    ð1  su Þð1  hÞð1  Sy ÞhA f ðk ; l ÞgðA Þ þ l gðA Þ þ k5 að1 1  GðA Þ 1  GðA Þ 



inf

inf





 su Þð1  hÞA f ðk ; l ÞgðA Þ þ k6 B00 ðl Þl gðA Þ þ k5 ð1  su Þð1  hÞf ðk ; l Þ ¼ 0:

ð70Þ

Dividing by (k1 + k2)(1  su) (1  h)A⁄f(k⁄, l⁄)g(A⁄), we finally obtain:









su k5 hð1  Sy Þ 1 1 P  inf inf ¼ 1 þaþ  l þ k6 B00 ðl Þl ; A gðA Þ ð1  su Þðk1 þ k2 Þð1  hÞA f ðk ; l Þ 1  GðA Þ 1  su k þ k2 1  GðA Þ which is (37) in the text.

ð71Þ

58

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

Appendix B. Numerical appendix B.1. Numerical method In our simulations we consider the long run steady state of the Ramsey equilibrium. To obtain our results we use the method developed in Schmitt-Grohé and Uribe (2006). Since the Ramsey problem is linear in the lagrange multipliers, the equilibrium conditions may be written as follows (see Schmitt-Grohé and Uribe, 2009)23:

Aðx; CÞ þ Bðx; CÞK ¼ 0;

ð72Þ

Cðx; CÞ ¼ 0:

ð73Þ

where x are the endogenous variables of the model and C are the parameters. A(x, C) and B(x, C) are, respectively, the Jacobian matrices of the objective function and of the equilibrium conditions of the Ramsey problem.The matrix C(x, C) contains the competitive equilibrium conditions and K are the Lagrange multipliers. To find the steady state of a competitive equi  ~ t ; sut , but there are librium we solve (73). Note that in our problem we have eight endogenous variables, ct ; kt ; bt ; lt;f ; lt;inf ; ~rt ; w only six equilibrium conditions in C(x, C). Moreover, one of these conditions does not give any information in steady state.24 This leaves us with five equilibrium conditions to determine eight variables. We make three guesses to match the number of equations and unknowns in C(x, C). In the case presented in the paper where bonds are calibrated, we have to make two guesses to match the number of unknowns to the number of equations since bonds are not determined endogenously in this case. Once we have determined the steady state allocation associated with a certain vector of guesses, we use a numerical minimization package in order to find the allocation that maximizes welfare, which is the Ramsey allocation. The specific computational steps are as follows: 1. 2. 3.

Make an initial guess for s = [(sl, sk, su)], the tax level of labor income, capital income and profits income. Conditional on the guess, s, solve the system in (73) to find the associated competitive equilibrium allocations, xss. Evaluate A(x, C) and B(x, C) using xss and obtain K using the MATLAB operator for OLS and construct the regression residuals as:

b ^e ¼ Aðx; CÞ þ Bðx; CÞ K and use them to define g(s) as

gðsÞ ¼ ^e0 ^e: 4.

Iterate on s until g(s) <  where  is a previously chosen convergence criterion. Then update of s until convergence is chosen using the steepest decent method.25

The result of this numerical procedure is a vector of xss which is the steady state of the Ramsey equilibrium. B.2. Calculation of the guess xjss In this section we explain how we solve the system of equations in C(x, C) in each iteration j using our guess for the three policy variables, sk, sl and su. Using the consumer’s intertemporal optimality condition evaluated in steady state we get:

~rjss ¼

1  ð1  dÞ: b

ð74Þ

Note that we need to find the labor market equilibrium for both, the formal and the informal sectors.We proceed iteraj tively. Given a fixed number of productivity levels (firms), first we guess the labor supplied in both sectors, lss , and the numj 26 ber of firms in the informal sector, N inf . Then we verify if the guess is correct. j j Let li;f and li;inf be each firm’s labor demand in the formal and informal sector respectively. Labor market clearing condition requires: j lss

0 j 1 N inf N X 1 BX j j C ¼ @ l þ li;f A: N i¼1 i;inf j i¼N

inf

ð75Þ

þ1

where N is the total number of productivity levels in the economy, each of them with equal probability of occurrence (1/N). 23

Since we are evaluating the steady state, in the notation that follows, we eliminate the time index. Note that in steady state 1b ¼ 1 þ ~rt  d, therefore, the intertemporal consumption Euler equation implies that uc = uc which is useless in order to determinate the consumption level. 25 See Judd (1998). 26 By difference we obtain the labor supply in the formal sector. 24

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

59

If a firm chooses to operate in the formal sector, its labor demand is given by27: j

li;f ¼

 1 ah1

1 ah1 Ai ah 1h ki 1h Ai ah 1h  a w 1h ¼ : r r li 1a r

ð79Þ

Adding each firm’s labor demand in (79) and using labor market equilibrium (75) we obtain:

2 j li;f

3

1

A1h 6 ¼ 4P i N

1 1h

j

i¼N inf þ1

Ai

  7 j j i ¼ 1; . . . ; N; 5 Nlss  lss;inf

ð80Þ

  j j where Nlss  lss;inf is the demand of labor in the formal sector which is the first term in (75). Similarly capital demand, if the firm operates in the formal sector, is: j

j ki;f

~r ss 1skss

¼

1  h1 ah1 j Ai ha li;f

ð81Þ

:

j

To see how lss;inf is distributed among firms in the informal sector we obtain each firm demand in that sector:



j

li;inf ¼

Ainf ð1  ainf Þ ~ w

a 1

inf

ð82Þ

:

Consequently, since all firms in the informal sector demand the same amount of labor, we have: j

lss;inf ¼ Njinf



Ainf ð1  ainf Þ ~ w

a 1

inf

ð83Þ

:

~ j , is computed as: After tax wages, w

 aÞh1  ~ j ¼ AN ð1  aÞhkNah lð1 w 1  sl;j N ss ; where the subindex N indicates the firm with the larger productivity level.28 Next we calculate each firm’s profits in both sector. To do this we use the firm’s labor and capital demands in the formal sector, Eqs. (80) and (81), and the firm’s labor demand in the informal sector, Eq. (82). Next we define an indicator function, 1i ðoperationÞj , equal to one when the firm decides to operate in the formal sector, and 0 if the firm is in the informal sector by using:

1i ðoperationÞj ¼ 1 





j;a j;1a h ð1su ÞAi ðk l Þ þ/ i;f i;f arctan p =2 þ 

p

;

ð84Þ j

where  > 0 is a parameter and / are the profits of a firm in the informal sector. Note that in the above indicator function ki;f j and li;f represent capital and labor demand if the ith firm operates in the formal sector. These demands are defined in (80) and (81). The parameter  determines the shape of the function, where the smaller is  the less smooth is the function. An example of this function is shown in Fig. 1. It shows the case of a firm that faces (su = 0.1, / = 1). In the figure we treat capital and labor demand as exogenous while in our model are endogenous. We treat them as such to provide a simple description. In the figure, Ai  1.1 is a threshold level: if the firm draws a productivity parameter larger than the threshold level, the firm’s after-tax profits are larger than profits in informal sector. Clearly, the smaller is  the more the function resembles an indicator function. Once we have calculated firms profits in both sectors we can obtain the number of firms that choose to produce in the informal sector, N⁄. 27

In fact, each firm’s optimality conditions are ah1 ð1aÞh

Ai ahki

li

¼ r;

ah ð1aÞh1

Ai ð1  aÞhki li

ð76Þ ¼ w;

ð77Þ

which imply:

ki a w ¼ : li 1a r Replacing Eq. (78) in (76) is (79). 28 We may have used other firm productivity since all firms in the formal sector face the same wage.

ð78Þ

60

R.A. Cerda, D. Saravia / Journal of Macroeconomics 35 (2013) 39–61

 Fig. 1. Indicator function, 1i ðoperationÞ ¼ 1 

arctan

ð1su ÞA ðka l1a Þh þ/ i

i;f i;f



p

þp=2

 . ki = 1.5, hi = 0.7, / = 1, su = 0.1, a = 0.36, h = 0.4.

Next we calculate the bonds hold in steady state using the government budget constraint:

! j N N N X X X 1 bss j j j j j wsl li;f þ r sk ki;f þ su ð1  hÞAi f ðli;f ; ki;f Þ þ ¼ e þ bss : ~r  d N 1 þ    i¼N þ1 i¼N þ1 i¼N þ1

ð85Þ

Using the clearing of the goods market, we obtain consumption in steady state:

cjss

     h N N 1ainf X a j 1a 1 X j j þ ¼ ð1  1i ðoperationÞÞAinf li;inf 1i ðoperationÞAi ki;f li;f N i¼1  i¼N þ1 ! N X 1 j  d 1i ðoperationÞki;f  e; N i¼N þ1

!

ð86Þ

where e is fiscal expenditure. Using this level of consumption in the individual marginal rate of substitution between consumption an leisure we obtain the effective total labor supply to be compared with the initial guess: j;updated

lss

¼H

 ðcjss Þ 1  /2 : ~j w /2

ð87Þ

Eq. (87) allows us to update our initial guess in labor supply. If

    j j;updated    j  max lss  lss ; N  Ninf  < #; where # > 0 is a convergence criterion, we have computed the candidate vector xjss . Otherwise, we go back to (74) and recomj;updated pute the steady state variables in the (j + 1)th iteration using lss and N⁄ as new guesses. Eqs. (74)–(87) and the tax guesses allow us to obtain our candidate vector xjþ1 ss in the (j + 1)th iteration. References Aiyagari, S.R., 1995. Optimal capital incom taxation with incomplete markets, borrowing constraints, and constant discounting. Journal of Political Economy 103 (6), 1158–1175. Abel, A., 2005. Optimal taxation when consumers have endogenous benchmark levels of consumption. Review of Economic Studies 72 (1), 21–42.

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