Aggregate effects of imperfect tax enforcement

Aggregate Effects of Imperfect Tax Enforcement (Job Market Paper) Miguel Robles ∗ University of California Los Angeles - UCLA November 17, 2006 Abstract I study an economy in which the government is not able to perfectly enforce tax compliance among operating firms and compare it with one in which perfect enforcement is attainable. I develop a competitive general equilibrium model where imperfect tax enforcement may affect aggregate outcomes through two mechanisms. First, it may distort firms’ optimal output level as long as the probability of avoiding tax compliance is related to the firm’s size. Second, poor tax enforcement may lead to a low provision of the public goods that complement firms’ productivity. The results for a calibrated version of the model suggest that in economies with tax enforcement problems aggregate output might be reduced by 12%. I also conclude that sizeable aggregate effects can be obtained only when the public goods mechanism is at work. JEL Codes: E26,H41,K42,L11. Keywords: Tax enforcement, Public goods, Informal Sector, Size distribution of firms. ∗ [email protected]. I would like to thank Hugo Hopenhayn, Hanno Lustig, Lee Ohanian and Mark Wright for excellent guidance and suggestions. I also received helpful comments from Christian Hellwig, William Zame, Harold Cole and seminar participants at the UCLA Macroeconomics and IO Proseminars. All errors are my own. 1 1 Introduction I study an economy in which the government is not able to perfectly enforce tax compliance among operating firms and compare it to a similar one where perfect enforcement is attainable. I develop a competitive general equilibrium model where imperfect tax enforcement may affect aggregate outcomes through two mechanisms. First, it may distort firms’ optimal output level as long as the probability of avoiding tax compliance are related to the firm’s size. Second, poor tax enforcement may lead to a low provision of public goods that complement firms’ productivity. The results for a calibrated version of the model suggest that in economies with tax enforcement problems aggregate output might be reduced by 12% and TFP by 9%. I also conclude that sizeable aggregate effects can be obtained only when the public goods mechanism is at work. In this paper I take the ability of a government to enforce tax compliance as an exogenous feature of the economy. I also restrict the tax system to rely completely on taxation of corporate profits. Thus the exercise here is to compare two economies that are identical except for the capacity of the government to enforce tax compliance among firms. Are aggregate equilibrium outcomes (output, total factor productivity, average firm size and wages) among these two economies different? If they are then by how much?. Those are the questions I try to answer in this paper. There is empirical cross-country evidence that a strong system of legal enforcement is correlated with economic development either directly (Knack and Keefer [26]) or through the development of the financial system (La Porta et al. [30]). Here I investigate a particular channel by which legal enforcement may affect economic performance, namely through the capacity to enforce tax compliance. In any economy 2 taxes are necessary to raise revenue for the government and that revenue allows its operation and provision of public goods and services. Here I consider the case of a benevolent government that runs a neutral tax system 1 and that maximizes aggre- gate consumption. However distortions may arise because of constraints in the set of policies available to the government. In particular the government is constrained by its tax enforcement technology. One implication of a government’s lack of enforcement capacity that has received some attention in the economic literature is the emergence of the informal sector, understood as the set of economic agents that do not comply with government regulations and taxes. A general view is that informality arrives as the response of the private sector to excessive or distorting taxes and regulations (De Soto [10], Maloney [29], Rauch [31]) and in that sense it has the effect of reducing potential negative aggregate effects of government-imposed distortions over the economy. Other studies emphasize the role of the informal sector as a source of inefficiencies. Loayza [27] develops a simple growth model where the presence of the informal sector negatively affects growth by reducing the availability of public goods. Amaral and Quintin [1] study a competitive model in which commitment problems are introduced in the economy due to the presence of the informal sector. Recently De Paula and Scheinkman [5] have developed a competitive model with two stages of production that highlights the role of value added taxes in transmitting informality which in turn affects firms’ size decisions and the capital labor-mix. A series of case studies by McKinsey [23][22] [13][8][25] suggests that the informal economy allows for the existence of less efficient firms and therefore contributes to lower the overall productivity of the economy. 1 The tax system considered here is one such that in an economy with perfect tax enforcement and no public goods does not introduce any distortion in the economy 3 Governments that face tax compliance problems are usually not able to generate sufficient fiscal revenues, which may translate into a low provision of public services and goods. This link between enforcement capacity, the informal sector and provision of public goods has been studied by Loayza [27]. Furthemore the contribution of public infrastructure to output and productivity has been studied by Aschauer [2], Lynde and Richmond [28], Gramlich [18], Garcia-Mila et al. [16]. In this literature it is widely accepted that public infrastructure and aggregate productivity are positively correlated and the work by Fernald [14] has been an important contribution to empirically establish causality. He shows a positive effect from roads to US productivity. At the theoretical level it is well known the growth model of Barro and Sala-i-Martin [7] with congestible public goods. At the empirical level Gordon and Li [17] have recently documented that tax revenue as a fraction of GDP is surprisingly low in developing countries compared with developed ones. I take this as an indicator of relatively low enforcement capacity in the developing world. Furthermore it is a well known fact that the presence of large informal sectors is an important characteristic of the developing world. Schneider and Klinglmair [35] have estimated that the average size of the shadow economy over 1999-2000 in developing countries is 41%. Additionally, using cross-country data I find the following evidence (see Appendix): • Fact 1a: Provision of public services is negative correlated with the size of the informal economy. • Fact 1b: Provision of public services is positively correlated with tax revenues (as % of GDP) • Fact 2: Provision of public services is positive correlated with GDP per capita, 4 TFP and output per worker. • Fact 3: GDP per capita, output per worker and TFP are negative correlated with the size of the informal economy. At the firm level it is a well established regularity that informality is negatively correlated with firm size. Moreover, the McKinsey case studies show convincing evidence that informal firms are less productive than formal ones. The model I develop in this paper emphasizes all of these empirical regularities. I build on a modified version of the Lucas span of control model (Lucas [33]) with a fixed labor supply. I introduce public goods into the model and, following Barro and Sala-i-Martin [7] and Loayza [27] consider the case of congestible public goods. The government’s tax enforcement technology enters the model as a probability that a firm’s profits are seized if it does not pay taxes; this probability is increasing in firm’s output. In doing this I follow De Paula and Scheinkman [5]. The mechanics of the model are straightforward: when the tax enforcement technology is not perfect some firms may find optimal not pay taxes and face an incentive to reduce their output level to keep a low probability of being caught by the authorities. This reduces aggregate labor demand, which calls for a lower equilibrium wage which in turn facilitates the operation of low productivity firms. A second mechanism works through the availability of public goods. The fraction of firms that in equilibrium decides not to pay taxes (informal sector) is higher the poorer the tax enforcement technology. In this way tax revenue and the provision of public goods as well as the overall productivity of the economy are negatively affected. As far as I know this is the first paper to consider the inclusion of public goods in a version of the Lucas’ span of control model. Another novel feature of this model is that I explicitly introduce a non-distorting tax system in a model that endogenously generates informal firms. In 5 calibrating the model I use the observed size distribution of firms in the US to back out the distribution of idiosyncratic productivity (or managerial talent). In order to calibrate the contribution of the public goods to the economy I use the observed corporate tax rate by assuming it is the one that maximizes aggregate consumption. In the next section I introduce the model. In section 3 I explain the calibration strategy. Section 4 discusses the results and in section 5 I provide some final comments. 2 The Model I set up a model to study an economy in which the government has limited capacity to enforce corporate tax compliance. I first describe a model in which the government has full tax enforcement capacity, and then I introduce the notion of an imperfect tax enforcement technology. 2.1 Setup I consider a one period economy 2 populated by a mass one of households. This is a one good economy and each household has an endowment of cb units of the good. There are two individuals in each household, a worker and an entrepreneur. All workers across households are identical in that they provide the same quality of la- bor services. In contrast, entrepreneurs are characterized by a parameter θ which indicates their idiosyncratic quality in entrepreneurship or managerial talent. I use θ 2 I do not introduce dynamics since here I am not interested in intertemporal distortions. One can think of the one period economy described here as the steady state of a dynamic model with entry and exit of firms, where the exit rate will be given by those firms that are caught not paying taxes 6 as an index for households and entrepreneurs since there is no other source of heterogeneity across households. θ is distributed according to a cumulative density function (cdf) G (θ ) with G (0) = 0 and G ′ (θ ) > 0 for θ ∈ (0, ∞). An entrepreneur that employs l workers, k units of capital and has access to ρ units of public goods produces output equal to y = h(ρ)θ f (k, l )γ , where f (.) is homogenous of degree 1 and 0 < γ < 1. The parameter γ determines the degree of diminishing returns to scale of the production process, ρ represents the amount of public goods available to each production unit while h(.) is a strictly increasing and concave function. Capital is provided from outside the economy in infinite supply at a rental price r. Within household θ the only decision maker is the entrepreneur. Her objective is to maximize household’s consumption c, which is a linear combination of both members’ consumption. Entrepreneur θ faces the following decisions for given prices w and r, access to public goods ρ and tax rate τ: the quantity of labor supply from the household’s worker, whether or not she runs a production unit, and if she runs a production unit how much labor l and capital k to hire. Since worker’s income can only add to household’s consumption it is optimal that the worker inelastically supply his unit of labor. The rest of the analysis takes this optimal decision as given. Thus, the problem of entrepreneur θ is: max x ∈{0,1},k ≥0,l ≥0 c (1) s.t. c = x (1 − τ )(h(ρ)θ f (k, l )γ − wl − rk − ce) + w + cb where ce is a fixed cost, x is the decision to run a firm (x = 1 if she decides to run a firm, 0 otherwise) and τ is the tax rate on firm’s profit. I break (1) into two problems. First, entrepreneur θ maximizes profits as if she were to run a production unit or a 7 firm 3 : max(1 − τ )(h(ρ)θ f (k, l )γ − wl − rk − ce) (2) k,l Denote the solutions to (2) as l (θ ) and k(θ ). I show in the Appendix that l (θ ), y(θ ) = h(ρ)θ f (k (θ ), l (θ ))γ , and π (θ ) = y(θ ) − wl (θ ) − rk (θ ) − ce are increasing in θ and that: π (θ ) = (1 − τ )((1 − γ)y(θ ) − ce) Also notice that the optimal capital-labor ratio is independent of the entrepreneur’s quality θ as well as of τ and h(ρ). Second, given profits π (θ ) entrepreneur θ decides whether or not to run a firm: max x (1 − τ )π (θ ) + w (3) x ∈{0,1} Denote the solution to (3) as x (θ ). As long as π (θ ) < 0 it is optimal not run a firm and x (θ ) = 0, while if π (θ ) > 0 then x (θ ) = 1. Since π (θ ) is increasing in θ there exists a marginal entrepreneur θ0 who is indifferent between running a firm or not. I assume x (θ0 ) = 1. Therefore x (θ ) = 1 if θ ∈ {θ ≥ θ0 }, otherwise x (θ ) = 0. In this economy there is a government that can only tax firms’ profits 4 and uses tax revenues to finance the provision of public goods. In particular the government announces a tax rate τ such that every firm is supposed to pay τ fraction of its profits as taxes. Consider first the perfect enforcement case, where the government has the 3I use these terms interchangeably is not my goal here to study optimal taxation issues. I choose this tax system because of its neutrality. In a version of this model with no public goods and perfect tax enforcement (version I use as a benchmark) taxing profits is fully neutral, in other words the equilibrium is independent of the tax rate). 4 It 8 ability to enforce tax compliance of all operating firms. Total tax revenues are then, T = τΠ = τ (1 − γ)Y − τc where Π = R∞ 0 Z ∞ 0 x (θ )dG (θ ) x (θ )π (θ )dG (θ ) is aggregate profit and Y = gregate output. R∞ 0 (4) x (θ )y(θ )dG (θ ) is ag- Each unit of tax revenue is transformed into one unit of a public good. I call ρ the total amount of public goods per unit of output: ρ= T Y (5) Each firm has access to ρ units of the public good 5 and their contribution to a firm’s production process is given by h(ρ), where h has the following properties: h′ (θ ) > 0 and h′′ (θ ) < 0. Thus, public goods are essential, in the sense that as ρ approaches to zero the output of every operating firm also approaches to zero 6 ; in addition, they are subject to decreasing returns. 2.2 Equilibrium Given a cdf G (θ ), a tax rate τ and a rental price for capital r, an equilibrium in this economy is an allocation of capital and labor across operating plants {k(θ ), l (θ )}; operating decisions { x (θ )}; a quantity of public goods available to each operating firm ρ; and a price w all satisfying the following conditions: 5 This is the case of a public good that is rival but not excludable and therefore it is subject to congestion. Barro and Sala-i-Martin [7] argue that this kind of public good apply to highways and other transportation facilities, water and sewer systems, courts and domestic security 6 I avoid indeterminacies restricting τ to be strictly positive and less than 1 9 1. k(θ ) and l (θ ) solve (1) for any θ ∈ {θ : x (θ ) = 1} 2. 1 = 3. ρ = R∞ 0 x (θ )l (θ )dG (θ ) τΠ Y Denote an equilibrium for a given tax rate τ as ξ (τ ) = {{k (θ )}, {l (θ )}, { x (θ )}, ρ, w}. Assumption 1. E ( θ z | θ ≥ θ0 ) θ0z = b, where b is a constant bigger than 1, z is a finite positive number and θ0 > 0 7 . Proposition 1. Given assumption 1 and a tax rate τ ∈ (0, 1) there is a unique equilibrium wage w and a unique cutoff value θ0 such that π (θ0 ) = 0, π (θ ) > 0 for θ > θ0 , and π (θ ) < 0 for θ < θ0 2.3 No Public Goods Case Consider an economy with no public goods such that h(ρ) = 1. This will be the Lucas span of control model without the occupational choice margin. I show in the Appendix that if ξ (τe) is an equilibrium for a given tax rate τ then it is also an equilibrium for a different tax rate τe, where τ and τe are in (0, 1). In other words in the absence of public goods and under perfect tax enforcement taxing profits is fully neutral. I am interested in the neutrality of the tax system due to the following. Suppose the tax system were distortive and at the same time the government had almost no capacity to enforce tax compliance, then this would be equivalent to having a very low effective tax rate and therefore almost no tax distortion in the economy. I explicitly want to avoid this positive effect on efficiency of a poor government’s tax enforcement capacity. 7 This assumption is satisfied by a Pareto distribution with parameter α and 0 < z < α 10 2.4 Imperfect Tax Enforcement Consider now the case in which the government has limited ability to enforce tax compliance. In this case an entrepreneur θ must decide whether to run a firm and comply with tax payments, or to run a firm without paying taxes 8 or simply not to run a firm. In making that decision firms take into account the probability of getting caught by the government if not paying taxes. If a firm is caught then its profits are seized. The perfect enforcement case analyzed above can be understood as a particular case in which a firm always get caught. I label a firm that decides not to pay taxes as informal and conversely one that does decide to pay taxes as formal. I model the probability of getting caught by the government as increasing in output. Thus, the higher the production level of a firm the higher the probability of getting caught. Even though I take this probability as an exogenous feature of the model it is not difficult to justify a government that puts more effort in enforcing tax compliance of big firms than small ones. This idea has been used by De Paula and Scheikman [5]. For convenience I focus on the probability of not getting caught, which accordingly is decreasing in output. I denote the conditional probability of not getting caught as P(”not getting caught”|y) and I use P(y) as a shorthand. P(y) satisfies the following conditions: P(0) = 1 and P′ (y) ≤ 0. Definition 1. A tax enforcement technology is a probability P(y) that a firm does not get caught if it does not pay taxes. P(y) is a better tax enforcement technology than Pe(y) if P(y) ≤ Pe(y) and P(y) < Pe(y) for at least some y. All entrepreneur’s decisions are made at the beginning of the period. At the end of the period all households will be in one of two mutually exclusive states: caught(σc ) 8 Here I consider the case of a discrete decision, namely paying taxes or not paying at all. I’m currently working on a continuous version where firms decide how much to pay as taxes 11 or not caught (σnc ). Denote x F as the decision to run a formal firm and x I as the decision to run an informal firm. Entrepreneur θ maximizes expected consumption and solves the following program: max x I ∈{0,1},x F ∈{0,1},k,l s.t. P(y)c(σcn ) + (1 − P(y))c(σc ) (6) c(σnc ) = x F (1 − τ )(y − wl − rk − ce) + x I (y − wl − rk − ce) + w + cb c(σc ) = x F (1 − τ )(y − wl − rk − ce) + x I (0 − ce) + w + cb y = h(ρ)θ f (k, l )γ x I + x F 6= 2 One can break this problem into three problems: the first under the assumption entrepreneur θ has decided to run a formal firm, the second under the assumption she has decided to run an informal firm, and the third problem is the decision between running a formal firm, an informal firm or simply not running any type of firm. For the first of these problems entrepreneur θ solves the same program as in (2). I relabel l (θ ) as l F (θ ) and k(θ ) as k F (θ ), and define y F (θ ) and π F (θ ) in the obvious way. Now take the decision of running an informal firm as given. In this case the entrepreneur θ solves the following program: max [ P(y)(y − wl − rk ) − ce] + w + cb k,l s.t. (7) y = h(ρ)θ f (k, l )γ Define l I (θ ) and k I (θ ) as the solutions to (7). Also y I (θ ) = h(ρ)θ f (k I (θ ), l I (θ )) and 12 π I (θ ) = P(y I (θ ))(y I (θ ) − wl I (θ ) − rk I (θ )) − cb Proposition 2. Given prices w and r, a tax rate τ ∈ (0, 1), and a quantity ρ of public goods per firm y I (θ ) ≤ y F (θ ). Proof Let C (y; θ ) be the corresponding cost function for an entrepreneur with quality θ. P(y) is non increasing in y and y F (θ ) maximizes y − C (y, θ ). Therefore if ye > y F (θ ) then: P(y F (θ ))(y F (θ ) − C (y F (θ ), θ )) > P(ye)(ye − C (ye, θ )) Q.E.D. Now consider the decision between being formal or informal or not running a firm. Entrepreneur θ decides to be formal if π F (θ ) > π I (θ ) and π F (θ ) ≥ 0. In this case x F (θ ) = 1, otherwise x F (θ ) = 0. If π I (θ ) ≥ π F (θ ) and π I (θ ) ≥ 0 then the optimal decision is to run an informal firm, and x I (θ ) = 1, otherwise x I (θ ) = 0. If x F (θ ) + x I (θ ) = 0 then it is optimal not run a firm. As before, the provision of public goods is fully funded by tax revenues 9 . However tax revenues are now provided only by formal firms, such that T = τΠ F = τ (1 − γ)YF − τe c Z θ x F (θ )dG (θ ) R∞ where YF = 0 x F (θ )y F (θ )dG (θ ) is the aggregate output of the formal sector, and R∞ Π F = 0 x F (θ )π F (θ )dG (θ ) is the aggregate profit of the formal sector. ρ is the total amount of public goods per unit of aggregate output. Notice that ρ can be expressed as c YF τe ρ = τ (1 − γ ) − Y 9I R θ x F (θ )dG (θ ) Y (8) assume that if a firm is effectively caught the corresponding seized profits cannot be used to finance public goods. For simplicity I assume those seized profits are destroyed. 13 The first term in (8) shows how as the relative importance of the formal sector declines so does the available amount of public goods for each firm, either an informal or formal one. An equilibrium in this economy, given a cdf G (θ ), a tax enforcement technology P(y), a tax rate τ and a rental price for capital r, is an allocation of capital and labor across operating plants in the informal sector {k I (θ ), l I (θ )}, an allocation of capital and labor across operating plants in the formal sector {k F (θ ), l F (θ )}; operating decisions { x I (θ ), x F (θ )}; a quantity of public goods available to each operating firm ρ and a price w, all satisfying the following conditions: 1. k I (θ ) and l I (θ ) solve (5) for any θ ∈ {θ : x I (θ ) = 1} 2. k F (θ ) and l F (θ ) solve (1) for any θ ∈ {θ : x F (θ ) = 1} 3. 1 = 4. ρ = R∞ 0 x I (θ )l I (θ )dG (θ ) + τΠ F Y R∞ 0 x F (θ )l F (θ )dG (θ ) I name an equilibrium for a given tax rate τ and a given tax enforcement technology P(y) as ξ (τ, P) = {{k I (θ )}, {k F (θ )}, {l I (θ )}, {l F (θ )}, { x I (θ )}, { x F (θ )}, ρ, w}. 2.5 The Government In this economy the government collects revenue from taxing firms’ profits with a common tax rate τ. There is no other tax system. All tax revenues are automatically converted into public goods. The government is endowed with a tax enforcement technology P(y). I will consider two cases for determining the tax rate τ. First, I consider a government that exogenously inherits a tax rate that cannot change. I 14 label this as the passive government case and will allow me to study an economy with a sub-optimal provision of public goods for a given tax enforcement technology. Second, I consider a benevolent government that chooses a tax rate τ in order to maximize aggregate consumption. I call this the active government case. Notice that the only choice available to the government is the selection of τ; it takes the competitive behavior of firms as given. The government solves the following program max C τ ∈(0,1) (9) subject to C= Z ∞ 0 x I (θ )π I (θ )dG (θ ) + Z ∞ 0 x F (θ )π F (θ )dG (θ ) + w + cb {{k I (θ )}, {k F (θ )}, {l I (θ )}, {l F (θ )}, { x I (θ )}, { x F (θ )}, ρ} ∈ ξ (τ, P) 3 Calibration Strategy I choose f (k, l ) to be Cobb-Douglas such that f (k, l ) = kα l 1−α and where α captures the capital income’s share out of aggregate labor and capital income. I follow the conventional choice of 1/3 for α. Following Atkeson and Kehoe [4] I choose 0.85 for γ, the diminishing returns to scale parameter. This implies that in an economy with perfect tax enforcement total variable costs represent 15% of aggregate output. I also follow Atkeson and Kehoe [4] in choosing 4% for the interest rate r. I calibrate G (θ ), the distribution of entrepreneurs’ quality, by looking at the observed size distribution of firms in the US economy. I assume that the US government runs a perfect tax enforcement technology and set h(ρ) as given and equal to 1. In other 15 words I revert to a modified version of the Lucas’ span of control model with perfect tax enforcement and no public goods, which I use as the benchmark model. According to Axtell [6] the observed size (number of workers) distribution of firms can be parameterized as a Pareto distribution with parameter δ. I show in the Appendix that if in the benchmark model θ follows a Pareto distribution with parameter δ 1− γ then the size distribution of firms also follows a Pareto distribution with parameter δ. Therefore I use for G a Pareto cumulative density function (cdf) with parameter δ 1− γ and support [θ M , ∞). Call L0 the number of workers in the smallest operating firm and L the average firm size. Using L0 = 2 and the observed average firm size L = 21.8 I am able to pin down δ 10 and θ M . I set the initial household endowment cb to be equal to ce, so any household is at least able to pay for the fixed cost ce if decides to run a firm. The fixed cost ce is arbitrarily set equal to 1. The calibration strategy is summarized in Table 1. I do not need to choose a tax rate for the benchmark model since the endogenous variables are independent of τ. Next I specify the h(ρ) function, the function that determines the importance of public goods in the model. I consider the following specific form: h(ρ) = σρη 0 < η < 1, σ > 0 Thus, public goods are essential and are subject to diminishing returns. In order to calibrate the parameters σ and η I take the following steps: 1. I compute the equilibrium for the benchmark model calibrated to the US economy. Call it ξ US 2. I add public goods to the benchmark model and assume that the government 10 If x follows a Pareto distribution with support [ x0 , ∞) and parameter β > 1 then 16 E( x ) x0 = β β −1 Table 1: Calibration of the Benchmark Model P(y) = 0 Perfect tax enforcement h(ρ) = 1 No public goods f (k, l ) = k α l 1− α Cobb-Douglas α = 1 3 Capital share of output net of profits γ = 0.85 Atkeson and Kehoe [4] L0 = 2 Axtell [6] L = 21.8 ge = δ Axtell [6] L0δ L δ +1 Observed size distribution of firms fits a Pareto distribution cb = ce ce = g(θ ) = δ ( θ M ) 1− γ 1−γ 1−δ γ +1 θ δ = 1.10101 θM = 0.99228 initial household endowment 1 fixed cost, arbitrarily chosen Calibrated parameters: δ Consistent w/ US size distribution of firms ge( L) L= δ δ −1 L 0 l ( θ0 ) = 2 Endogenous variables: θ0 = 1.5102 w = 1.8889 17 chooses a tax rate τ ∗ that maximizes aggregate consumption C. Call the corresponding equilibrium ξ (τ ∗ )US 3. I find the parameters σ and η such that ξ US = ξ (τ ∗ )US . The underlying reasoning is that observable allocations and prices for the US economy that allow for the calibration of the benchmark model, are consistent with a provision of public goods close to its optimal level. In terms of the model the two conditions to pin down parameters σ and η are: h(ρ(t∗ )) = 1 (10) ∂C ∗ (τ ) = 0 ∂τ (11) I show in the Appendix that when adding public goods to the benchmark model then ρ(τ ) = τ (1 − γ)(1 − δ −1 δ ). The ideal estimation for τ ∗ would be spending on public goods as a fraction of GDP in the US. Given the lack of that information I take two approaches. First I rely on the observed tax rate on profits. US C-corporations profits are subject to four basic tax rates: 15%, 25%, 34%, and 35%. However any taxable income above $75, 000 is subject to a tax rate of 34% or higher (39% being the highest). In the model I consider a flat tax schedule such that any unit of profit is taxed at the same rate. There is also a difference between the definition of profits in the model and what typically is considered taxable income of corporations in the US and abroad. While taxable income of corporation profits typically reflects total revenues less payments to factors of production other than those financed by shareholders, in the model payments to all factors of production are subtracted from revenues. Therefore the observed tax rate should enter the model with an upward correction. I call ω to the fraction of 18 capital provided by non-shareholders, then the adjusted tax rate τ and the observed tax rate τe are related according to: τ (1 − γ)y = τe(y − (1 − α)γy − ωαγy) or τ = τe 1 − γ(1 − α(1 − ω )) 1−γ I consider the observed tax rate to be 34% and for ω I use the average ratio of liabilities to the sum of equity and liabilities for US firms. This average is 37.1% 11 and the corresponding adjusted tax rate is 74.4%. At this adjusted tax rate government expenditures in public goods reach 10.13% of aggregate output. This number is approximately one third of total government expenditures in the US (which represent 30% of GDP). Second, I assume that one quarter of government expenditures in the US are devoted to the provision of public goods. This delivers a more conservative estimate for the adjusted tax rate, 55%. Table 2 summarizes the calibration strategy for h(ρ; η ) Finally I introduce the probability that a firm does not get caught if it does not pay taxes, P(y). I choose the following functional form: P(y) =      1 if y ≤ ȳ ³ y ´φ     if y > ȳ y with 0 < φ < 1. Figure 3 shows the shape of this function. I set the parameter y to be slightly above the output level of the smallest firm in the benchmark model 12 . This 11 I would like to thank Daisuke Miyakawa who computed this ratio for me using COMPUSTAT is 0.5 above. This addition is needed to prevent that some very small firms would prefer to 12 It 19 Table 2: Calibration of public goods function h(ρ) h(ρ ; σ, η ) = σρη τ ∗ based on observed tax rate τe = 34% observed tax rate ω = 0.371 Average τ∗ = 74.4% adjusted tax rate σ = 1.31142 calibrated parameter η = 0.11843 calibrated parameter ρ = 10.13% ratio public goods expenditures to output liabilities equity+liabilities , (compustat) τ ∗ based on expenditures in public goods ρ = 7.5% Assuming expenditures in public goods = 14 GUS τ∗ = 55% adjusted tax rate σ = 1.24275 calibrated parameter η = 0.08387 calibrated parameter functional form provides a closed form for y I (θ ): y I (θ ) = where      y F (θ ) ³     1− φ 1−γφ if θ ≤ θ ´ γ 1− γ y F (θ ) if θ < θ à µ ! γ ¶ 1− γ 1 1 − α 1− α ³ α ´ α y F (θ ) = γ ( h ( ρ ) θ ) 1− γ w r 1 θ = y− F (y) It is important to notice that the ratio between y I (θ ) and y F (θ ) is decreasing in φ for φ ∈ [0, 1] 13 . Take the case of a government fully incapable of enforce tax compliance become formal instead of informal 13 This is for θ ∈ ( θ, θ ) 1 20 P(y; y, φ) ✻ 1 P(y; y, φ0 ) P(y; y, φ1 ) ✲ y y Figure 1: Tax enforcement technology P(y) (φ = 0), in this case firms do not need to hide through reductions in their output levels, and will choose the same output as if they were formal. Also notice that the tax rate τ does not affect the ratio y I (θ ) . y F (θ ) I leave the parameter φ free, which allows me to introduce different levels of tax enforcement capacity into the model. As φ moves from zero to one the tax enforcement technology improves. 4 Results In the previous section I showed the choices for the functional forms and parameters implied by the model, except for the parameter φ of the tax enforcement technology P(y). An extreme case is φ = 0 which implies no tax enforcement at all so that any firm can avoid paying taxes without getting caught. As φ increases, conditional on output level, the probability that a firm gets caught raises 14 and therefore according 14 This holds for any output level bigger than y 21 to definition 1 the tax enforcement technology improves. Since I am interested in the potential effects of a poor tax enforcement technology I solve the model for different values of φ and investigate for differences in aggregate output, measured total factor productivity, equilibrium wage rate, the size of the informal sector and the average size of firms. The discipline on how far φ can increase is given by the model’s prediction of the size of the informal economy. I use as a benchmark the perfect enforcement case in which all firms comply with tax payments. I consider two cases for the government’s behavior. One case assumes a benevolent government which given a tax enforcement technology, chooses a tax rate τ ∈ [0, 1) such that aggregate consumption C is maximized. I call this the active government case. I also consider a passive government case in which the government exogenously inherits a tax rate that cannot adjust. In addition I study the effects of an imperfect tax enforcement technology in the absence of public goods. In the model as the tax enforcement technology gets worse, ceteris paribus, more firms decide to become informal and tax revenues and the provision of public goods per unit of output decrease. This in turn reduces all of the operating firms’ productivity. In the absence of public goods this effect is not present. However, given a poor enforcement technology the incentives for a firm to become informal are still at work. A subset of firms may optimally choose not to pay taxes and may reduce their input demands relative to the perfect enforcement case. This puts into work a general equilibrium effect that distorts the allocation of resources across firms. I call this the no-public-goods case. Since a benevolent or active government would like to minimize the effects of its imperfect tax enforcement it will trivially choose a zero tax rate, therefore I only consider the no-public-goods case under the assumption of a passive government. 22 In Table 3 I summarize the results for different cases. In particular in this table the public goods function h(.) has been calibrated so that the optimal provision of public goods is 10.13% of aggregate output and is consistent with τ = 74.4%. As is exhibited in the top panel of Table 3, I consider a tax enforcement technology P(y) with parameter φ = 0.15. In order to give an interpretation to this number I compute the probability that a firm in the percentile 90 of the size distribution of firms 15 gets caught if it does not pay taxes. This probability is 26.1%. At this level of tax enforcement capacity and assuming a benevolent government the model predicts an output level 12.1% below the perfect enforcement case and an informal sector that accounts for 30.3% of aggregate output. Moreover, observed TFP and wages are 8.6% and 12.7% below the benchmark case. According to Schneider [34] more than 50% of countries out of a sample of 144, mainly developing economies, exhibit similar or higher levels of informal output. At this level of tax enforcement capacity the government finds it optimal to set the tax rate at 46%, well below the optimal tax rate of the benchmark case. Reducing the tax rate positively affects the consumption of households running formal firms. Also, ceteris paribus, it reduces tax collection and therefore the provision of public firms. However, this effect is compensated by a reduction in the numbers of firms that choose to become informal such that tax collection and provision of public goods increase. In addition I find a selection effect that reduces the number of operating firms by 7.8% relative to the benchmark. These are firms that would be operating if it were not for the fact that the government operates a poor tax enforcement technology. Two counter effects are at work, first since wages are lower than in the benchmark this will lead to entry of low productivity firms (negative selection), second since the provision of public goods is scarce and negatively affects all firms’ productivity this will lead to the exit of low productivity firms (pos15 This corresponds to the size distribution of operating firms under the benchmark case 23 itive selection). The empirical results show that the second effect dominates. When I solve the model for the same tax enforcement level and under the assumption of a passive government the effects on output, TFP and wages are very similar. However the size of the informal sector is much higher, 56.6% 16 . Also the selection effect is lower, accounting for an exit of 4.2% of firms relative to the benchmark model. When considering the no-public-goods case the negative effects of the poor tax enforcement technology over output, TFP and wages are drastically reduced. Now aggregate output is less than 1% below the benchmark, the effect on TFP is negligible and wages are lower by 2%. However now there is a negative, instead of a positive, selection effect. The size of this effect is 8.67% which means there are this many additional operating firms relative to the benchmark, which would not have operated with perfect enforcement . Since in this case the effect of public goods provision on productivity is not present, the only mechanism at work comes through lower wages. What I conclude from these results is that in the model sizeable effects of a poor tax enforcement technology arise only under the presence of the public goods mechanism. In a broader sense this points to the idea that distortions that mainly affect the left tail of the size distribution of firms will have limited aggregate effects unless there is a more direct feedback effect over the entire distribution beyond the standard general equilibrium effects. In the middle and bottom panels of Table 3 I consider less severe tax enforcement inefficiencies. The model predicts lower levels of informality (24% and 14%) and, as expected, smaller reductions - albeit still significant - in aggregate output, TFP and wages. Equilibrium aggregate output is reduced by 10% and 7% respectively. Also notice that in the absence of public goods a low tax enforcement capacity can deliver 16 Countries at this level of informal sector size are Nigeria, United Republic of Tanzania, Zimbabwe, Peru, Azerbaijan Panama, Bolivia and Georgia 24 Table 3: Aggregate Effects of Imperfect Tax Enforcement Optimal τ = 74.4% in benchmark model Aggregate Output (Y) TFP wage θ0 Informal Output (%) Perfect Enforcement Benchmark model 3.333 1.361 1.888 1.510 0.00 Imperfect Enforcement (φ = 0.15) - Active Government (τ = 0.46) 2.931 1.244 1.648 1.527 30.29 - Passive Government 2.929 1.245 1.636 1.519 56.57 3.317 1.361 1.853 1.493 56.57 No Public Goods (φ = 0.20) - Active Government (τ = 0.49) 3.006 1.267 1.689 1.521 23.53 - Passive Government 3.049 1.283 1.700 1.508 44.54 3.314 1.361 1.848 1.491 44.54 No Public Goods (φ = 0.30) - Active Government (τ = 0.53) 3.103 1.296 1.745 1.514 13.58 - Passive Government 3.178 1.322 1.771 1.497 28.15 3.307 1.360 1.843 1.489 28.15 No Public Goods φ: Firm in percentile 90 gets caught with probability.: 0.15 0.20 0.30 26.13% 33.22% 45.43% 25 equilibrium output, TFP and wages that are higher than in an economy with a better enforcement technology. That occurs because at low levels of enforcement informal firms have almost no incentive to reduce their output levels relative to the output they would have chosen under perfect tax enforcement. At the same time the effect of informal firms over provision of public goods and overall productivity is not present. In Table 4 I show some predictions of the model for firm size and distribution of the labor force across formal and informal firms. Since by construction the tax enforcement capacity of the government is higher among large firms the model predicts that informal firms belong to left tail of the size distribution of firms while formal ones belong to the right tail. Also the model predicts a discontinuity in the size distribution of firms such that no firms of intermediate size will be observed. This feature of the model matches qualitatively with the fact known as ”missing middle”, according to which in less developed countries the size distribution of firms is characterized by a smaller number of medium size firms relative to what it is observed in developed countries. Tybout [37] has documented this fact for the manufacturing sector. In the model the range of the ”missing middle” as well as the ”mean missing size” decrease with tax enforcement capacity. Similar predictions applies to the informal sector. When considering a low tax enforcement technology (top panel Table 4) the informal sector is populated by firms with less than 108 workers. And the ”missing middle” is in the range of firms with 108 to 128 workers. Tybout [37] shows evidence of a ”missing middle” in the range of 20 to 100 workers for a country like Mexico where the size of the informal sector is 30%. Therefore it seems the model predicts a too high average ”missing middle”. In part I claim this is because the model does not incorporate another well studied fact of informal (small) firms in LDC countries, namely that they are significantly more labor intensive than formal ones (Amaral and 26 Quintin [1], Tybout [37]). One can think that this empirical evidence can be matched to the model by adding size dependent financial frictions (as in Quintin [1]) or a probability of detection increasing in capital (as in De Paula and Scheinkman [5]). I leave these modifications as candidates for future improvements of the model and for now simply claim that those modifications will reduce, at least, the lower bound of the ”missing middle” predicted by the model. As one consider better tax enforcement technologies (middle and bottom panels of Table 4) the model’s prediction for the ”missing middle” looks closer to the empirical evidence. The model also does well in predicting a negative correlation between the fraction of the labor force in the informal sector and the level of tax enforcement capacity. In Tables 7 and 8 in the Appendix I show results under the assumption that aggregate consumption is maximized when expenditures on public goods represent 7.5% of the aggregate output in the benchmark economy. This implies a tax rate on profits of 55%. With respect to the case where the optimal tax rate equals 74.4% the aggregate effects of imperfect tax enforcement are reduced. When analyzing a low tax enforcement capacity (φ = 0.15) the result is that aggregate output is lower by 6.3%, observed TFP by 4.4% and wages by 6.9%, where all figures are relative to the benchmark model; and the size of the informal economy reaches 24.38%. At this level of lack of enforcement an active government will set the tax rate at 39%, 16 points lower than the corresponding optimal tax rate for the benchmark economy. I confirm also that without public goods in the model it is very difficult to obtain sizable effects on output, TFP and wages. In the absence of public goods the reduction on equilibrium aggregate output generated is lower than 0.5%. 27 Table 4: Labor Force and Firm Size ( τ = 74.4% in benchmark ) Mean Size Informal sector LF (%) Min Mean Formal Max Min – 2.00 Perfect Enforcement Benchmark model 21.8 0.00 – – Imperfect Enforcement (φ = 0.15) Active government 23.68 29.76 2.29 7.13 108 128.73 Passive government 22.78 55.94 2.31 12.74 17, 395 20, 705 No pub goods 20.11 55.94 2.04 11.25 15, 361 18, 284 (φ = 0.20) Active government 22.96 22.89 2.24 5.42 41 53 Passive government 21.63 43.65 2.22 9.45 1, 550 1, 982 No pub goods 19.91 43.65 2.04 8.70 1, 426 1, 823 (φ = 0.30) Active government 22.34 14.45 2.18 3.56 12.80 18.68 Passive government 20.49 26.95 2.13 5.57 110.02 166.67 No pub goods 19.69 26.95 2.05 5.35 105.74 160.18 LF: labor force φ: Firm in percentile 90 gets caught with probability.: 0.15 0.20 0.30 26.13% 33.22% 45.43% 28 5 Final Comments In this paper I investigate the economic implications of technological differences in the government side of the economy. In particular I study economies that have access to the same set of resources and productive technologies, but where governments are endowed with different technologies to enforce tax compliance. My implicit assumption is that while private technologies can freely flow across economies, this is not the case for some government-related technologies. The paper shows that a government’s lack of tax enforcement capacity may have implications for aggregate output, total factor productivity, wages and the size of the informal economy. I mainly exploit two mechanisms through which government’s lack of tax enforcement capacity distorts the economy. First, provided that the government operates a tax enforcement technology that is more efficient in detecting large firms’ evasion than small ones, firms face an incentive to reduce their optimal output level. Second, a poor tax enforcement technology may lead to a fiscal problem and low provision of productive public goods. The results for a calibrated version of the model suggest that in economies where the government has limited capacity to enforce tax compliance, aggregate output is lower by 12% and TFP by 9% relative to an economy where the government can perfectly enforce tax compliance. Moreover, poor tax enforcement suggests an informal sector that accounts for 30% or more of aggregate output. Based on the numerical predictions of the model I also conclude that sizeable aggregate effects can be reached only when the public goods mechanism is at work. This suggests that any competitive equilibrium model that introduce a friction distorting mainly economic decisions on the left tail of the size distribution of firms (small firms) 29 will not be able to deliver sizeable aggregate effects unless it also incorporates an externality or feedback effect (beyond standard general equilibrium effects) on the right tail of the distribution (large firms). This is because distorting small and medium size firms decisions is equivalent to a distortion that only binds a small share of the economic activity. 30 References [1] Pedro S. Amaral and Erwan Quintin. A competitive model of the informal sector. Journal of Monetary Economics, 53(7):1541–1553, October 2006. [2] David Alan Aschauer. Journal of monetary economics. Is public expenditure productive?, Volume 23(Issue 2):Pages 177–200, March 1989 1989. [3] Asli Demirg-Kunt and Vojislav Maksimovic. Law, finance, and firm growth. Journal of Finance, 53(6):2107–2137, December 1998. [4] Andrew Atkeson and Patrick J. Kehoe. Modeling and measuring organization capital. Journal of Political Economy, 113(5):1026–1053, October 2005. [5] Aureo De Paula and Jos A. Scheinkman. The informal sector. Working Paper, July 2006. [6] Robert Axtell. Zipf distribution of u.s. firm sizes. Science, 293:p.1818–20, 2001. [7] Barro, Robert J. and Sala-I-Martin, Xavier. Public finance in models of economic growth. 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Series Public Policy, 45:12962, 1996. [28] Lynde, Catherine and Richmond, J. Public capital and total factor productivity. International Economic Review, 34(2):401–414, may 1993. [29] William Maloney. Informality revisited. World Development, 32(7):1159–78, 2004. [30] Rafael La Porta, Florencio Lopez de Silanes, Andrei Shleifer, and Robert W. Vishny. Law and finance. Journal of Political Economy, 106(6):1113–1155, December 1998. [31] James E. Rauch. Modelling the informal sector formally. Journal of Development Economics, 35(1):33–47, January 1991. [32] Restuccia, D. and R. Rogerson. Policy distortions and aggregate productivity with heterogeneous plants. Unpublished paper, Arizona State University, 2003. [33] Jr. Robert E. Lucas. On the size distribution of business firms. Bell Journal of Economics, 9(2):508–523, Autumn 1978. [34] Friedrich Schneider. The size of the shadow economies of 145 countries all over the world: First results over the period 1999 to 2003. IZA Discussion Papers 1431, Institute for the Study of Labor (IZA), December 2004. [35] Friedrich Schneider and Robert Klinglmair. Shadow economies around the world: What do we know? CREMA Working Paper Series 2004-03, Center for Research in Economics, Management and the Arts (CREMA), January 2004. [36] Schneider, Friedrich and Enste, Dominik H. Shadow economies: Size, causes, and consequences. Journal of Economic Literature, 38(1):77–114, mar 2000. [37] James R. Tybout. Manufacturing firms in developing countries: How well do they do, and why? Journal of Economic Literature, 38(1):11–44, mar 2000. 33 A Appendix: Empirical Evidence Table 5: Facts 1a and 1b. Cross-country evidence (Correlation, Number of observations) Informal Tax Tax Sector Revenues Compliance (% GDP) (% GDP) -0.5154* 0.3934* 0.3726* 136 106 43 -0.4448* 0.1906* 0.3222* 136 106 43 Days to resolve 0.2396* -0.2328* -0.4517* a debt dispute 143 95 43 Cost to resolve 0.1998* -0.2185* -0.3609* 143 95 43 Paved Roads per capita Paved Roads per Km2 debt dispute * 10% significance level; GPD(1+informal size) was also used DAtA: Schneider [34], Caselli [9], World Development Indicators Ross Levine, ”Financial Structure and Economic Growth: A Cross-Country Comparison of Banks , Markets, and Development” Data 34 Table 6: Facts 2 and 3. Cross-country evidence (Correlations, Number of observations) GDP per Y/L TFP capita Paved roads TFP H adjusted 0.7302* 0.7806* 0.7059* 0.5694* 162 100 99 90 0.5321* 0.5936* 0.5453* 0.4896* 162 100 99 90 Days to resolve -0.2841* -0.3119* -0.2641* -0.1159 a Debt Dispute 142 100 99 89 Cost to resolve -0.3531* -0.3944* -0.4327* -0.4295* 142 100 99 89 -0.6889* -0.7396* -0.7103* -0.6041* 135 96 95 85 per capita Paved roads per Km2 Debt Dispute Informal Sector (% GDP) * 10% significance level; GPD(1+informal size) was also used; H: Human capital DAtA: Schneider [34], Caselli [9], World Development Indicators Ross Levine, ”Financial Structure and Economic Growth: A Cross-Country Comparison of Banks , Markets, and Development” Data 35 B Appendix: Numerical Results Table 7: Effects of Imperfect Tax Enforcement Optimal τ = 55% in benchmark model Aggregate Output (Y) TFP wage θ0 Informal Output (%) Perfect Enforcement Benchmark model 3.333 1.361 1.888 1.510 0.00 Imperfect Enforcement (φ = 0.15) - Active Government (τ = 0.39) 3.122 1.301 1.758 1.516 24.38 - Passive Government 3.174 1.318 1.781 1.508 37.99 3.322 1.361 1.864 1.499 37.99 No Public Goods (φ = 0.20) - Active Government (τ = 0.41) 3.170 1.315 1.785 1.512 17.62 - Passive Government 3.229 1.334 1.812 1.504 28.03 3.320 1.361 1.863 1.498 28.03 No Public Goods (φ = 0.30) - Active Government (τ = 0.44) 3.232 1.333 1.824 1.510 7.77 - Passive Government 3.289 1.351 1.848 1.500 14.83 3.319 1.360 1.865 1.499 14.83 No Public Goods φ: Firm in percentile 90 gets caught with probability.: 0.15 0.20 0.30 26.13% 33.22% 45.43% 36 Table 8: Labor Force and Firm Size ( τ = 55% in benchmark ) Mean Size Informal sector LF (%) Min Mean Formal Max Min – 2.00 Perfect Enforcement Benchmark model 21.8 0.00 – – Imperfect Enforcement (φ = 0.15) Active government 23.17 32.65 2.20 7.63 141.74 158.25 Passive government 21.61 37.40 2.12 8.11 358.30 426.47 No pub goods 20.65 37.40 2.02 7.74 342.32 407.45 (φ = 0.20) Active government 22.43 23.92 2.15 5.54 41.41 49.29 Passive government 21.17 27.32 2.09 5.87 77.72 99.34 No pub goods 20.58 27.32 2.03 5.70 75.57 96.60 (φ = 0.30) Active government 21.83 7.38 2.07 2.30 4.07 6.17 Passive government 20.85 14.10 2.04 3.24 11.72 17.76 No pub goods 20.66 14.10 2.03 3.21 11.62 17.60 LF: labor force φ: Firm in percentile 90 gets caught with probability.: 0.15 0.20 0.30 26.13% 33.22% 45.43% 37 C Appendix: Statements and Proofs of Propositions C.1 Output is increasing in θ By assumption f (k, l ) is homogenous of degree 1 in k and l. Therefore it can be expressed as: f (k, l ) = lf(k) where k = kl . Cost function: min wl + rk k,l s.t. y = h(ρ)θ (lf(k))γ First order conditions: w = λh(ρ)θγ(lf(k))γ−1 (f(k) − fk (k)k) r = λh(ρ)θγ(lf(k))γ−1 fk (k) (12) (13) Combine (12) and (13): w f(k) −k− = 0 fk (k) r Given standard production function conditions fk (k) > 0 and fkk (k) < 0 and using the implicit function theorem, k is increasing in wr k=k ³w´ r Labor demand for a given output level y : l= µ y h(ρ)θ ¶1 γ 1 f(k) (14) Express total variable cost wl + rk as l (w + rk). Use (14) and get the cost function: C (y) = µ y h(ρ)θ 38 ¶1 γ w + rk f(k) (15) Use the first order condition: 1 = C ′ (y) (16) to get the optimal output decision for a given θ : y(θ ) = µ γf(k) w + rk ¶ γ 1− γ ¶ γ 1− γ 1 ( h ( ρ ) θ ) 1− γ Since 0 < γ < 1, y(θ ) is increasing in θ. An alternative expression for y(θ ) is: y(θ ) = µ γfk (k) r 1 ( h ( ρ ) θ ) 1− γ C.2 Labor demand is increasing in θ (16) implies: 1 y= γ µ y h(ρ)θ ¶1 γ w + rk f(k) (17) Plug (17) and y(θ ) in (14) and get optimal labor decision: l (θ ) = µ 1 w + rk ¶ γy(θ ) Since y(θ ) is increasing in θ then l (θ ) is also increasing in θ. C.3 Profits From (18): l (θ )(w + rk) = γy(θ ) Therefore, π (θ ) = (1 − τ )((1 − γ)y(θ ) − ce) Since y(θ ) is increasing in θ then π (θ ) is also increasing in θ. 39 (18) C.4 Proof of Proposition 1 π ′ (θ ) > 0 and π (0) < 0 ⇒ ∃ a unique θ0 such that π (θ0 ) = 0 and: x (θ ) = Zero profit condition:    0 if 0 ≤ θ < θ0   1 if θ0 ≤ θ < ∞ Condition π (θ0 ) = 0 implies ce 1−γ y ( θ0 ) = and µ γfk (k) r ¶ γ 1− γ (19) 1 ( h ( ρ ) θ 0 ) 1− γ = Use (19) and Assumption 1 to express ρ as: ce 1−γ (20) 1 ρ = τ (1 − γ)(1 − ) b where b is a constant greater than 1. Apply the implicit function theorem to (20) and get w = w1 (θ0 , h(ρ)) (21) which is an increasing function in θ0 . By inspection it is easy to see that as θ0 → 0 also w → 0. Labor market equilibrium: Express 1= Z ∞ 0 as 1 = l ( θ0 ) x (θ )l (θ )dG (θ ) Z ∞ µ ¶ 1−1 γ θ θ0 θ0 40 dG (θ ) Plug l (θ ) and (19) in this expression: 1= µ 1 w + rk ¶ γe c 1−γ Z ∞ µ ¶ 1−1 γ θ θ0 θ0 dG (θ ) (22) Multiply and divide (22) by 1 − G (θ0 ) and use Assumption 1: 1= µ 1 w + rk ¶ b(1 − G (θ0 )) (23) Apply the implicit function theorem to (23) and get: w = w2 ( θ 0 ) (24) which is a decreasing function of θ0 . By inspection it is easy to see that as θ0 → 0 then w → ∞. (21) and (24) intersect once. Then there is a unique θ0 that satisfies w1 ( θ 0 ) = w2 ( θ 0 ) > 0 C.5 Benchmark model’s equilibrium is independent of τ In the benchmark model h(ρ) = 1. Therefore (21) becomes independent of τ. (24) is also independent of τ. Therefore any pair {w, θ0 } that satisfies (21) and (24) is independent of τ. C.6 Pareto Distribution of θ Let’s assume that employment l follows a Pareto distribution with parameters δ and L0 . The corresponding density function is: f (l ) = δl0δ l δ +1 41 l ≥ l0 In the standard model employment l is related to θ according to: 1 l (θ ) = Aθ 1−γ A>0 Then, I can infer a density function for θ according to: ¯ ¯ ¯ ∂l (θ ) ¯ ¯ g(θ ) = f (l (θ )) ¯¯ ∂θ ¯ and l0 = l (θ0 ) After replacing terms: δ 1− γ δ 1− γ θ0 g(θ ) = δ θ 1− γ +1 which is a Pareto density function with parameters θ0 and 1−δ γ . A property of the Pareto distribution is that one can interpret g(θ ) as the density function of θ conditional on θ > θ0 . And by the same property I can call δ ge(θ ) = 1− γ δ 1− γ θ M δ θ 1− γ +1 θ M < θ0 the unconditional density function of θ with parameters θ M and δ 1− γ C.7 Public Goods Express ρ as: ρ = τ (1 − γ ) − y ( θ0 ) R∞ ceτ Since G (θ ) is the cdf of a Pareto distribution with parameters θ M and 42 (25) 1 θ 1−γ dG (θ ) ) ( θ0 θ0 (1− G (θ0 )) δ 1− γ then, View publication stats R ∞ ³ θ ´ 1−1 γ θ0 θ0 R∞ dG (θ ) = θ0 1− G ( θ0 ) δ = 1δ−−γ1 1− γ = δ−δ 1 plug this expression and y(θ0 ) = c 1− γ δ 1 1− γ δ 1− γ 1− γ θ 0 δ +1 θ 1− γ δ −1 δ −1 1− γ 1− γ θ 0 δ −1 +1 θ 1− γ ³ ´ θ θ0 R∞ θ0 dθ dθ in (25) to get ρ = τ (1 − γ)(1 − 43 δ−1 ) δ (26)