pdfs.semanticscholar.org€¦ · Presumptive Income Taxation and Costly Tax Compliance1 Christian R. Jaramillo H. Department of Economics University of Michigan 611 Tappan St. Ann

Presumptive Income Taxation and Costly Tax

Compliance1

Christian R. Jaramillo H.Department of Economics

University of Michigan611 Tappan St.

Ann Arbor, MI 48109-1220Email: [email protected]: (734) 255 0929

March 19, 2004

1I thank Joel Slemrod, James Hines Jr, Daniel Silverman, Robert Axelrod, FrederickLehmann, Peter Katuscak, Naomi Feldman and Robert Gazzale for their useful commentsand suggestions. All remaining errors are mine.

Abstract

This paper uses a Stackelberg game to analyze the problem of taxation by a benevolentsocial planner in the face of costly tax compliance. The social planner minimizes dead-weight loss subject to a marginal benefit of tax revenues. I ask whether it is optimal inthis case to introduce an alternative presumptive income tax, and I find conditions forits optimal tax rates.

I model the production and tax return filing decisions of risk-neutral firms facing ahybrid tax system that allows them to choose between a self-assessed tax on profits and apresumptive income tax. Anybody can choose to file under the presumptive regime, butin order to file under the self-assessed regime, a taxpayer must keep costly accountingbooks. The cost of bookkeeping differs across firms.

The model implies that, in the presence of costly tax compliance, a benevolent socialplanner with a given marginal value of tax revenue can always enhance social welfare ifshe allows taxpayers the alternative of filing a presumptive income tax return. In addi-tion, the optimal marginal tax rate on profits may fall below one, even if the individuallabor decision is binary.

1 Introduction

In spite of being quite widespread in developing countries, presumptive taxation hasreceived very little attention in the public finance literature. With few exceptions,the available analyses are based on collections of case studies and focus mainly on theadministrative rationale for such taxes, but they lack an explicit formal model of the taxstructure or the taxpayer’s problem. Perhaps the one context where they are consistentlytaken into consideration is tax reform, but even then only as a compromise solution.1

Moreover, the term “presumptive tax” is applied to several tax constructs that usepresumption of income but imply quite different problems for the taxpayer and the taxcollector.

Broadly speaking, a presumptive income tax (P-tax for short) is a tax based on somemeasure of economic activity that proxies for taxable income, rather than on taxableincome itself. For instance, it may be assessed on the basis of a firm’s inventory ofoutput, of some input of the production process or of gross sales over a period of time.In any case, the aim of the tax authority is to estimate the taxable income of the wholeeconomic activity at hand. Its intent is not to tax the particular measure chosen nor toextract any economic rents associated with factors of production. In this paper I analyzein detail the optimal implementation of a presumptive income tax for the taxation ofsmall firms with compliance costs.

The model I use is a Stackelberg game with a benevolent government and competitiveprofit-maximizing firms as players. The government plays first and the firms second. Thefirms utilize one variable input to produce an output good. The production functionis identical for all firms and exhibits decreasing marginal returns to the variable input.The output market is in partial equilibrium, and the game solution concept is NashEquilibrium.

The particular tax structure of interest is an optional presumptive income tax, OPTfor short. The firms must first choose their level of output. Then they face a tax systemthat allows them to choose between a standard tax return and a presumptive incometax return. The standard tax (S-tax) is linear, it is levied on profits and requires self-assessment. The presumptive income tax (P-tax), also linear, is based on the observablevariable input of production. Anybody can choose to file under the presumptive regime,but in order to file under the standard regime, a taxpayer must keep costly accountingbooks. The firms in the industry have different bookkeeping costs, but these are privateinformation.

The benevolent government minimizes a social loss function given a shadow value ofmarginal tax revenue λ. It announces its tax policy parameters in advance of the firms’production decision.

I find that the equilibrium outcome of the game with an OPT always includes somefirms filing a P-tax return. Even in the absence of labor supply distortion on the intensivemargin, the optimal S-tax rate in an OPT system can less than one. However, it is verylikely one.

1Yitzhaki [25] and Andreoni [2] use formal models in their treatment. For analyses of the successof presumptive taxes in different countries, see Rajamaran [15] and Tanzi [18]. For a discussion ofpresumptive taxes within the tax reform context see Bird [7], [8], [9], Thirsk [23] and Musgrave [14].

1

1.1 On presumptive income taxes

P-taxes are potentially valuable whenever direct measurement of the tax base is difficultfor the government, a problem common in, but by no means exclusive of, developingeconomies. Consequently, the exact shape of a given P-tax construct depends on thefeatures of the tax base it applies to.

A familiar example of a P-tax is the Alternative Minimum Tax (AMT), which actsas a lower bound on the tax liability and is usually applied to large firms in developingcountries. Nevertheless, the most developed P-tax systems are probably the tachsiv inIsrael and the French forfait. Under the tachsiv, the tax collector has detailed guidesshe uses to assess the income of small businesses. These guides are updated on a regularbasis and cover a wide range of trades. They rely on easily observable external indicatorsas the base for estimation of income: the number of tables and seats in a restaurant, acount of employees or the location of the business are fair examples.

The forfait, on the other hand, is used in France and parts of francophone Africa.It applies to farms, enterprises and professionals whose gross receipts are below certainamounts. For farmers, the forfait uses an estimate of land productivity by region andcrop type to impute a taxable income. For other activities, the tax collector has regionalor national monographs describing in detail the gross profit margins and other relevantcharacteristics of each trade. Based on this information, the tax collector assigns a taxliability to the taxpayer, who can accept or contest it. If accepted, the tax paymentremains fixed for two years.2

The accounting requirements are not completely absent in either of these systems, asthe taxpayer must be able to prove that she qualifies for the presumptive regime. Thisis likely to be true for any presumptive tax: the compliance burden on the taxpayer isreduced but not eliminated. Also, both the tachsiv and the forfait allow the taxpayerto contest the assessed tax liability. Bird [8] emphasizes the necessity of this feature,without which the system may appear too inflexible and the taxpayer has no poweragainst potential abuses of authority by the tax collector.

The tachsiv and the forfait also illustrate the pitfalls of a large presumptive taxsystem. They both require specialized technical research and field surveys to updatetheir respective assessment guides, so the burden on the government is still considerable.Moreover, the taxpayers tend to view these presumptive systems as the default for theactivities covered, even though standard income assessment is available as an optionin most cases. These features are the result of the large range of activities covered bythese two systems and the fact that they have been in place for a long time (Taube andTadesse [22]). This need not be the case for presumptive tax regimes of a more limitedscope.

In view of the motivation and the examples above, it would them seem that P-taxesare a second-best solution at most. This perception is not necessarily correct. Tanzi[18] points out the paradoxical fact that P-taxes are viewed in the policy literatureas a compromise option when actual taxable income is not observed, while the theoryof optimal taxation favors the idea of an ability-to-earn tax as a first-best solution.Since some P-taxes are arguably measures of potential rather than actual income, theyresemble ability-to-earn based taxes and may in this sense be more efficient than thesecond-best taxation of actual income.

2For further discussion of the merits and disadvantages of presumptive income taxation, refer toBird [8] and [9, Ch. 8] and Goode [12, Ch.5]. Tanzi [18], Taube and Tadesse [22] and Rajamaran [15]also provide excellent examples of actual presumptive tax systems. Lapidoth [13] describes the tachsivin detail.

2

As a consequence of their use in a variety of areas, it is difficult to characterize com-mon features of P-taxes. Several classifications have been attempted in the literature,but perhaps the most useful ones treat P-taxes in light of the particular problem ad-dressed by the tax rather than according to its formal implementation. Alternatively, forefficiency analyses it is the type of presumptive measure used that is most enlightening.I give here some examples of these criteria, starting with the latter.

According to the type of presumptive measure of income they use, Bulutoglu [10]classifies P-taxes in

• Income estimation through proxies for inputs or other quantities that have a directbearing on the economic activity at hand (e.g. dough consumption at the pizzaparlor).

• Indirect income estimation through external indicators.

• Asset taxes (wealth taxes).

• Gross turnover taxes, levied on the revenues rather than the profits (e.g. receipttaxes).

The choice of presumptive measure has implications for the distortionary effects ofthe tax that must be weighed against any gains in ease of implementation and admin-istration. Although it is difficult to pick one of these measures as best, the consensusin the literature is that gross turnover taxes tend to be particularly distortionary andencourage avoidance activities. Beyond this, the tax administrator must make her as-sessment on a case by case basis. In any event, the designer of an optimal P-tax mustconsider issues of horizontal and vertical equity, the choice of instruments available tothe tax administration, administrative and compliance costs, administrative fairness andtransparency.

Bulutoglu also provides an example of a classification according to the objective ofthe tax:

• Income estimation in hard to measure activities which don’t allow for an easycross-check of self-assessed tax liability, like self-employed doctors and lawyers.

• Evasion curbing, for instance to avoid transfer pricing by multinationals.

• As audit pointers, to determine the allocation of auditing resources.

• Small business taxation, to alleviate the burden of compliance.

This is by no means an exhaustive list. Indeed, in a sense all taxes are to somedegree presumptive in practice, as instances of asymmetric information are common inthe interaction between the tax administration and the taxpayer.

My main concern in this paper relates to the use of presumptive income taxation inthe last bullet item, alleviation of compliance costs. The taxable income proxy I modelis an input of the production process.

3

Businesses with high tax compliance costs It is a common ocurrence in develop-ing countries that small, especially individually owned enterprises, lack the resources tokeep organized records of their economic activity. Small grocery stores, fast-food kiosksand street vendors, small service enterprises and other such economic agents comprise alarge part of the so-called “informal sector”. These businesses are often family-operatedand have very low profit margins. They are called informal because, for the most part,they do not exist as firms nor file tax returns–even if they are required to do so by law.While many such enterprises would undoubtedly fail to pay taxes anyway, the burden oftax compliance is very often a strong disincentive for those who are inclined to complywith the tax requirements.

At the same time, the tax administration has very limited audit and enforcementresources. The risks of tax evasion under these circumstances are low enough that manysmall enterprises opt not to file a tax return. They become what’s commonly called“ghosts”. If there are enough ghosts, the enforcement efforts of the tax administrationare likely to be ineffective, which in turn makes evasion even more appealing. It isimportant to note that these are in part taxpayers who might be willing to file, evenwith low risk of penalty, if their compliance costs were lower, but they just lack theresources to meet the usual accounting standards the law asks for.

Any solution in to this problem must start with an effort to reduce the compliancecosts for the prospective taxpayer. Its first objective should be to increase the num-ber of tax returns filed. Just increasing the number of tax returns will allow the taxadministration to get better information on tax filers, regardless of the actual revenuecollection. This should lower the cost of future enforcement and eventually lead to lowerexpected return to evasion efforts.

Presumptive income taxes can help in a case like this. Their use in this particularcontext has been disputed, however, on the grounds that it may cause erosion of thestandard tax base. This erosion may occur if firms that would file tax return underthe existing regime experience a lower tax liability under P-taxes. Indeed, Bird [7]suggests this should be a primary concern in the implementation of tax policy basedon presumption of income. Thus part of the aim of this paper is to find conditionsthat guarantee appropriate incentives to taxpayers and minimize unnecessary tax baseerosion.

Given the strategic aspects inherent in evasion, a model that considers only its directrevenue and distortionary effects may be insufficient. At the end of this paper, I brieflydiscuss the case where evasion enters the government’s objective function explictly. Thisis a departure from the standard tax literature. In theoretical models, evasion is typ-ically a concern only insofar as it directly affects the tax revenue.3 There are at leasttwo reasons for treating it differently. First, there are negative externalities attached toevasion that go beyond revenue collection and have implications for the general percep-tion of legitimacy of the authority and the stability of the political system (Bird [8]). Inparticular, the perception of fairness is fundamental to encourage compliance with thelaw. For instance, in the context of tax collection as a game between taxpayers, one canview evasion as a player’s equilibrium response to other players who are evading. Thus,a critical mass of tax evaders may erode the willingness of the citizens to comply withthe law to the point where it is beyond the resources of the administration to enforcetax collection. At this point, it becomes optimal for the remaining citizens to evade,and the system collapses.4

3For a model with individual preferences over tax compliance, see for example Cowell [11, Ch.4 and6].

4For an empirical analysis of the extent of this effect refer for example to Andreoni, Erard andFeinstein [3]. Slemrod [17] has a concise survey of the related literature.

4

A second direct cost of evasion was already mentioned above. It arises when oneconsiders the value of information gathering for the tax administration. Other thingsequal, it is advantageous for the tax collector to have a potential taxpayer volunteerinformation about herself, however limited it may be, because the future monitoringcosts of this particular taxpayer are likely to be lower–it will be easier to raise revenuefrom her. A ghost is thus denying this information to the tax administration, whomight stand to gain from inducing this taxpayer to file, even if no tax liability obtainsthis period. This cost of tax evasion would be best analyzed separately in a dynamicsetting, but in the static model I use it enters the welfare function directly.

2 A model of tax regime choice

The model is a Stackelberg game with two types of players, the benevolent government,who plays first, and a population of small risk-neutral firms indexed by i, who playsecond. Each firm is owned and operated by an agent who supplies her labor. The laborchoice of the agent is binary, and she can guarantee herself a wage w, her outside option.

Government The government aims to minimize the deadweight loss in the market athand subject to a marginal benefit of tax revenue. Its problem is

min{T}

L− λR (1)

where L is the deadweight loss from the tax system, including market distortions andcompliance costs, R is the total tax revenue and λ is the shadow social value of an extradollar of government funds. {T} stands for the available tax instruments.

To tax the firms in the market, the government chooses the tax rates for two types oftaxes, a linear tax on profits t (hereafter referred to as standard tax, or S-tax) and a linearpresumptive tax on the production input θ (P-tax). The P-tax involves no costs to firmsbeyond the tax liability. In contrast, because taxable income is not directly observable,the S-tax requires firms to keep books and thus has an additional compliance cost. Thiscompliance cost is not observable by the government.

Finally, the government cannot force a choice of tax regime on any firm. Rather,each firm files taxes under one of the two regimes, and it is free to choose which one(hence the name Optional Presumptive Tax, OPT for short).

Firms Each firm in this model chooses input K to produce a non-tradable consumptiongood. The technology used to produce the output good, given by g (K), is the same forall firms.5 They face identical output price p and factor price r, both taken as given byall firms, so their pre-tax profit is π (Ki; p) = pg (Ki) − rKi. Note that the labor wageis not taken into account for this profit.

Firms are required by law to pay taxes Ti to the government. Each individual firmchooses the level of Ki and a tax return filing regime to maximize its net-of-tax profitπ (Ki; p) − Ti. Finally, each firm can guarantee itself a payoff equal to the wage w ifthey leave the market.

With regard to tax compliance firms are heterogeneous. The hybrid tax system allowseach firm the choice between two filing regimes, standard or presumptive. Under the

5The production function is assumed to have diminishing marginal returns, so g′ (x) ≥ 0 and g′′ (x) <0 for x ≥ 0, and g (0) = 0.

5

standard (or self-assessed) regime, firms pay taxes on their pre-tax profit at a constantmarginal rate: TS

i = tπ (Ki; p), which they take as given. However, in order to be able tofile such a standard tax return, a firm needs to keep accounting books from the beginningof the fiscal period. The cost of keeping those books, given by ci, varies across firms. Itis private information. Thus, whenever a firm i chooses to file a tax return under thestandard regime, its net-of-tax profit is given by Y S

i = (1− t) [π (Ki; p)− w − ci].

The other tax filing option allowed under the OPT system is the presumptive regime.In this case, the tax liability of any given firm is assessed directly on the chosen input“size” Ki at a constant marginal rate θ, taken as given by all firms, TP

i = θKi. All firmsare eligible for this tax regime, regardless of their keeping books or not. Firms that filea P-tax return have a net-of-tax profit of Y P

i = π (Ki; p) − θKi− δici, where δi = 1 iffirm i carried accounting books, δi = 0 otherwise.6

Let the compliance cost ci > 0 define a population density function f (c) of firms, suchthat f (c) dc is the fraction of firms whose compliance costs lie in the interval [c, c + dc).Assume f (c) = 0 for c > cmax. Let also F (c) be the fraction of firms with compliancecosts ci < c and normalize the maximum possible number of firms F (cmax) = N .

The firms’ problem is then

maxKi,status,δi

Yi (2)

where Yi stands for net-of-tax profit of firm i and status, the filing status of the firm,can take the values (standard filer [S], presumptive filer [P ]).

Market Demand To simplify the government’s objective function, I make the follow-ing assumption:

Assumption 1 Individual preferences are defined over the output good and a numeraire,and they are quasilinear in the numeraire.

In this case, the deadweight loss of taxation is identical with the decrease in totalsurplus in the market. Let the market demand schedule be given by X (p) with X ′ (p) <0.

2.1 The standard, self-assessed income tax problem

It is useful to consider first the government’s problem without the option of a presump-tive tax system. I call this the pure S-tax system. In order to avoid confusion in latersections, I denote the solutions to this problem by the superscript S . The reader isreferred to the Appendix A for an overview of the nomenclature.

Suppose the government is constrained to use a self-assessed linear tax on profits.In this case, each firm chooses, for any given price p, an input level Ki that maximizes(1− t) [π (Ki, p)− w − ci].

maxKi

Y Si = (1− t) [π (Ki, p)− w − ci] (3)

6In the absence of uncertainty no firm will choose δi = 1 if it plans to file a P-tax return. I introducethis choice parameter here, however, to emphasize the point for later discussion.

6

The first order condition for the optimal factor input level KSi is identical for all

firmsg′ (Ki) =

r

p(4)

Note that (4) does not depend on the tax rate t. This is a consequence of the discretelabor choice: unless the owner of the firm decides to abandon the industry altogether,her labor supply is fixed. Thus, there are no distortionary effects of the S-tax on thechoice of input level of those firms that produce.

The effect of the S-tax is felt through the equilibrium price in the market. Each firmperceives both the wage and the compliance cost as fixed costs that must be covered.In effect, the compliance cost induces heterogeneity in the firms’ fixed costs, which areraised from w to w + ci. A number of firm owners will close their businesses and themarket supply schedule will shift left, increasing the equilibrium price. This shift isdiscrete in nature, present even if t = 0.7

It is useful to define, for later use, the following function

Definition 2 Let γ be a function such that γ ◦ g′ : x −→ x. Thus γ = (g′)−1 is theinverse of g′ (·). It follows that γ′ (·) = 1

g′′(γ(·)) < 0.

Whether a firm actually stays in the taxed market depends on the implicit fixed costsw + ci. Let the solution to (4) be denoted kS

i (p) = kS (p) = γ (r/p) and write the exitrule

Exit if ci > ρS , file an S-tax return otherwise (5)

whereρS (p) = pg

(kS

)− rkS − w (6)

Thus, for any given price, only firms with compliance costs below a certain threshold ρS

will stay in the industry. The market supply schedule is upward sloping and independentof the tax rate:

ZS (p) = NF(ρS

)g

(kS

)(7)

Fig.1 decomposes the determinants of ZS . Suppose for a moment that all firms haveci = 0, and the tax rate is t = 0. In that case, the market supply Z = Ng

(kS

)is given

by OABB′′ on the right diagram. The wage w acts as the only fixed cost. As long asthe equilibrium price is above the break-even price, all N firms are in the industry.

If a S-tax rate t > 0 is set, but compliance costs are still zero, there is no effect onthe supply schedule: it is still given by OABB” on the right diagram.

Finally, suppose that a new fixed cost afflicts the firms, but this time it is heteroge-neous and given by ci. As with the introduction of t > 0, the supply level of those firmsthat stay in the business is the same as before. However, the break-even threshold isdifferent for different firms. If at a given price p a certain firm leaves the industry, thesupply curve gets an extra movement to the left. The result is depicted by the curveOACC ′ in Fig.1. Of course, regardless of t, if the output price is high enough, even thefirms with the largest compliance costs will stay in the business. The fraction of firms

7This is a direct result of modelling the compliance cost as a constant idiosincratic feature of the firmrather than as a function of the tax rate t. This choice is arguably a better reflection of the case of smallbusiness owners, but it is not adequate in other instances of presumptive income taxation involving largefirms or well-compensated professionals who may engage in avoidance rather than evasion activities.

7

A

Q

p p

q

B

C’B”

O

ATC

A’ B’C

NF(� S)g(kS)

Ng(kS)

ATC’

ATC”

FIGURE 1: Effect of a S-tax with compliance costs on the market supply schedule

Notes: The dashed line indicates the supply schedule in the absence of taxes, or if there is a S-tax but it

involves no costs of compliance. The solid line is the actual supply with a S-tax and compliance costs.

in business at any given price is given by the ratio ZS/Z. For example, at the price A′,ρS = A′C

A′B′ . Thus, OACC ′ converges to OABB′′ as p rises.

Two questions of interest can be answered at this point: What consequences does atax rate hike have on the equilibrium price and quantity? And on the number of firmsin the industry? One should expect no effect whatsoever, dpS

dt = dcS

dt = dQS

dt = 0. TheS-tax has no marginal distortionary effect beyond the fixed compliance cost.8

Equilibrium for a S-tax rate t requires a price pS that satisfies

X (p) = ZS (p) (8)

Provided that firms with sufficiently low compliance cost exist, a non-trivial equilibriumof the market is present. For simplicity, denote the solution to (8) by KS = kS

(pS

),

cS = ρS(pS

)and πS = π

(KS , pS

).

The social planner is aware of the equilibrium solution. She faces a social loss givenby:

L =∫ Q0

QS

[X−1 (Q)− Z−1 (Q)

]dQ + N

∫ cS

0

cf (c) dc (9)

Here(Q0, p0

)and

(QS , pS

)are the equilibrium quantity and price in the absence of

taxes and with tax t respectively.9 The first term on the right hand is the loss in totalsurplus, illustrated in Fig. 2 as the area of the triangle OSS′. The second term is thesum of compliance costs incurred by all producing firms.

8This result depends on the deductibility of the fixed costs from the taxable income, of course. Thismay be a tricky proposition regarding compliance costs, as they are likely to be in part non-monetary.

Therefore, if at least a part of the compliance cost is not deductible for tax purposes, there will besocial losses associated with increases in the marginal tax rate t.

9From the definition of the untaxed supply it follows that Z−1 (Q) = r

g′�

g−1�

QN

�� .

8

Z-1 (p)

Q

p

Q0QS

pS

NF(� S)g(kS)

S’

O

S

Z-1 (QS)

p0

FIGURE 2: Loss in total surplus

Notes: The area of the triangle OSS′ is the loss in total surplus of the economy. Together with the

compliance costs, it represents the excess burden of taxation in the market.

Tax revenue is given by

R(t) = tN[pSg

(KS

)− rKS − w]F

(cS

)− tN

∫ cS

0

cf (c) dc (10)

The first order condition for the government’s problem is

λdR

dt=

dL

dt(11)

(11) is the well-known requirement that the benefit of an extra dollar of tax revenuebe equal to the social loss incurred in obtaining that dollar. The right hand side isthe marginal social loss due to a raise in the tax rate t. Since the supply schedule isindependent of the S-tax rate, so is the social loss

dL

dt= 0 (12)

and the only effect of an increase in the tax rate is a redistribution of wealth from firmsto the government.

dR

dt=

R

t> 0 (13)

This is a familiar result: the marginal rate of a non-distortionary tax is to be increasedto the maximum possible, and all pure profits are to be taxed away. The only caveat tothis prescription is that the net social gain from the tax system as a whole be positive,i.e. λR − L > 0. Should this condition not be satisfied, it would be optimal to abolishthe S-tax altogether. Hereafter, I will assume that, given the choice between no tax anda pure S-tax system, the optimal social planner’s strategy involves the existence of theS-tax.

2.2 The optional presumptive tax problem (OPT)

I turn now to the problem where the firm can choose between the S-tax and a P-tax.10 Conceptually, the firm has three choices to make at two different points in time.

10It should be noted that the form of P-tax used here is necessarily a second-best solution. Throughoutthis paper, and in line with common assumptions of the optimal tax problem, I have assumed that the

9

First, it must choose whether or not to carry books. Second, it chooses its input levelKi. Later, when the pre-tax profit is there, it must choose a filing status, constrainedby its previous choice regarding bookkeeping. Formally, however, those choices aresimultaneous because there is no uncertainty in the outcome.

The firm’s problem is

maxKi,status,δi

Y Pi =

{(1− t) [pg (Ki)− rKi − w − ci] if S-tax filerpg (Ki)− (r + θ)Ki − w − δici if P-tax filer (14)

Obviously, the optimal solution in the absence of uncertainty will feature δi = 0 forall P-tax filers and δi = 1 for all S-tax filers. The optimal level of firm output Ki alsodepends on the tax regime the firm choses. If the firm chooses S-taxation, the FOC isidentical to the one in the previous section, given by (4).

g′ (Ki) =r

p

Under P-taxation, however, the FOC is

g′ (Ki) =r + θ

p(15)

As before, use Def. 2, to write the solution to (4) as kSi = kS (p) = γ (r/p) and the

solution to (15) as kPi = kP (p, θ) = γ

(r+θ

p

). Note that kP < kS .

Firm i’s choice between P-tax and S-tax yields the optimal rule

File P-tax if ci > ρP , file a S-tax return otherwise. (16)

The threshold is

ρP (p, θ, t) = pg(kS

)− rkS − pg(kP

)− (r + θ) kP

1− t− t

1− tw (17)

Proposition 3 (i) At any price p > 0 , the number of firms that file an S-tax returnis weakly lower under the OPT regime than under the pure S-tax regime. However, (ii)the total number of firms in the industry is always weakly larger than it was in the pureS-tax regime, and therefore so is the total number of tax return filers.

Proof. (i) If pg(kP

) − (r + θ) kP ≤ w, it’s better for any firm to exit the industryaltogether rather than filing a P-tax return. In that case, the firms effectively face achoice between an S-tax and exit, their optimal rule is (5) and the number of S-taxfilers is NF

(ρS

), the same as in the pure S-tax case. If, on the other hand, pg

(kP

)+

(r + θ) kP > w, the number of S-tax filers is NF(ρP

). Since pg

(kP

)+ (r + θ) kP >

w ⇐⇒ ρP < ρS , in this case there are strictly fewer S-tax filers than in the pure S-taxregime.

usual first-best solution involving lump-sum taxes is not feasible. In the absence of distortionary laborsupply effects, this limitation would not by itself rule out a first-best outcome with a S-tax. The fact thatthe firms in this market are heterogeneous with respect to the S-tax is what generates the deadweightloss of taxation.

In the presence of a P-tax, the inability to levy lump-sum presumptive taxes is also critical. Withrespect to the P-tax, firms are homogeneous. The social planner could achieve the first-best by extractingall profits from P-tax filers through a lump-sum tax. With labor supply fixed, this would render theS-tax non-distortionary so that t = 1 become again the optimal marginal rate.

10

A

Q

p p

q

B

B’

O

ATCA’ C

ATC’

MC’ MC

C”

C’

D

FIGURE 3: Introduction of an OPT

Notes: ATC is the cost curve of a firm with ci = 0. ATC′ depicts the costs of any firm that files a

P-tax return. OABB′ is the supply curve if there are no distortionary taxes. The S-tax shifts the curve

to OACD. The optional P-tax introduces a kink. The final supply curve is OACC′C′′.

(ii) From (i), whenever ρP < ρS , all firms file a tax return, either S or P. There willexist firms that exit only if pg

(kP

)+(r + θ) kP ≤ w, but in that case ρP ≥ ρS and thus

F(ρP

) ≥ F(ρS

).

The introduction of the OPT causes a kink in the supply schedule. At market priceswhere firms under the P-tax regime do not cover their implicit fixed cost (the wage w),the supply schedule is identical to ZS . At the price where pg

(kP

)+ (r + θ) kP = w,

however, two things happen. First, all firms that are outside the market now choose toenter, and the supply curve has a perfectly elastic segment of size N

[1− F

(ρP

)]g

(kP

).

Second, the marginal number of firms filing S-tax returns jumps to a lower positive valueas (16) becomes binding instead of (5). This is depicted in Fig. 3

The market supply is given by:

ZP (p, θ, t) =

NF(ρS

)g

(kS

)if ρP < ρS

N[F

(ρP

)g

(kS

)+ [1− F (α)] g

(kP

)]if ρP = ρS

N[F

(ρP

)g

(kS

)+

[1− F

(ρP

)]g

(kP

)]if ρP > ρS

(18)

with ρP ≤ α ≤ 1. The equilibrium occurs at a price pP (θ, t) satisfying

X (p) = ZP (p, θ, t) (19)

The equilibrium solution depends on t and θ. Denote KSP = kS(pP

), KP = kP

(pP , θ

)and cP = ρP

(pP , θ, t

). Also, let πSP = π

(KSP , pP

).

The government’s loss function has a form similar to (9)

L (θ, t) =∫ Q0

QP

[X−1 (Q)− Z−1 (Q)

]dQ + N

∫ cP

0

cf (c) dc (20)

where(Q0, p0

)and

(QP , pP

)are the equilibrium quantity and price in the absence of

taxes and with the OPT in place, respectively. The loss depends on θ, but it also depends

11

A

Q

p

O

C

C”

D

D’

C’

FIGURE 4: Two qualitatively different outcomes

Notes: The government’s optimal choice of θ and t may cause the market equilibrium to look like either

case, D or D′. It is never optimal to have the demand curve cross the supply curve to the left of point

C.

on t. Increases in the S-tax rate may a distortionary effect now because they inducesome firms to file a P-tax return instead of a S-tax return.

If ρP < ρS , the tax revenue is given by (10). Otherwise, it is

R(θ, t) = tN[πSP − w

]F

(cP

)− tN

∫ cP

0

cf (c) dc + θNKP [1− F (α)] (21)

with ρP ≤ α ≤ 1 if ρP = ρS , and α = cP if ρP > ρS .

In equilibrium only consider the two types of outcomes are possible. They are illus-trated in Fig. 4.

Proposition 4 Let θmax (p) ≡ pg(kP )−w

kP − r, the maximum P-tax rate that allows P-filers to break even at price p. Then, under the OPT, a government’s solution such thatpP = pS, and θ∗ = θmax

(pS

)is at least as good as the pure S-tax system.

Proof. Consider Fig. 5(a). Suppose that the initial tax system allows only S-taxfiling, so that the market supply is given by OAD. The introduction of an optionalpresumptive tax with a marginal rate θmax

(pS

)at the current market price pS changes

the market supply schedule to OACC ′C ′′. However, this has no effect on the equilibriumoutcome.

Proposition (4) also guarantees that, for a firm, filing a P-tax is at least as good asexiting the industry. It imposes a cap on the P-tax rate θ at any given market price p.In Fig.3, the plateau in the supply curve must occur at the pure S-tax equilibrium priceor below, A′ < pS . This corresponds precisely to the two cases depicted in Fig.4.

Proposition 5 In equilibrium, (i) ∂pP

∂θ > 0 if ρP = ρS; (ii) ∂pP

∂t > 0 if ρP > ρS, and∂pP

∂t = 0 if ρP = ρS.

12

A

Q

p

O

C

C”

D

C’E E’

E”

p’

(a)

A

Q

p

O

C

C”

D

C’E E’

E”

p’

(b)FIGURE 5: A decrease in θ

Notes: The market supply shifts from OACC′C′′ to OAEE′E′′ in both cases. In case (a), the effect

on the market output is determined exclusively by the demand side.

13

A

Q

p

O

C

C”

C’

C” ’

p’

FIGURE 6: An increase in the S-tax rate t.

Notes: The market supply shifts from OACC′C′′ to OACC′C′′′.

Proof. To see (i), consider a small decrease in the P-tax rate to θ < θmax

(pS

)in

Fig.5(a), keeping the S-tax rate t constant. The plateau in the supply schedule is nowlower, as the P-filers are able to break even at a lower price. The new market supply isthen OAEE′E′′ and the equilibrium price must fall.

For (ii), consider an increase in t as in Fig.6. The only effect is to drive firms to theP-tax regime, which lowers their output. This happens only if the P-filers make positiveprofits, so the upward-sloping part of the supply curve becomes steeper.

The government’s first order conditions are analogous to the ones in the pure S-taxcase.11

λ =∂L/∂t

∂R/∂tand λ =

∂L/∂θ

∂R/∂θ(22)

Let ∆P ≡ pP − Z−1(QP

). From (20), at the equilibrium

∂L

∂t=

[−∆P dX

dp+ NcP f

(cP

) ∂ρP

∂p

]∂pP

∂t+ NcP f

(cP

) ∂ρP

∂t(23)

∂L

∂θ=

[−∆P dX

dp+ NcP f

(cP

) ∂ρP

∂p

]∂pP

∂θ+ NcP f

(cP

) ∂ρP

∂θ(24)

The last term in each equation accounts for the compliance costs of the firms that chooseto become S-tax-return filers on the margin due to the direct incentive effects of changesin t or θ. Because ∂ρP

∂t < 0 and ∂ρP

∂θ > 0, the number of S-tax filers may increase as thedirect result of a decrease in t or an increase in θ.

The first two terms in (23) and (24) reflect the loss generated by the changes inequilibrium price. Increases in the equilibrium quantity–the first term in each equation–happen if the equilibrium price falls. They decrease the surplus loss of the market. Thesecond term within the price effect says that increases in the threshold for S-tax filingincrease the number of S-tax filers and compliance cost losses. Increases in price affectthe profits of P-filers and S-filers differently, so some firms switch regime on the margin.

11As in the case with only a S-tax, first order conditions are not sufficient to characterize the optimum,and second order conditions must be fulfilled to guarantee that

�QP , pP

�is indeed a minimum of the

social loss function. Moreover, multiple local optima may be present.

14

Because the sign of ∂ρP

∂p is ambiguous, there’s no way of telling the direction of this effectat the equilibrium.

The case where ρP > ρS is illustrated in Fig. 4 by the demand curve D′. Thederivation for the revenue yields in this case:

1N

∂R

∂t=

[[πSP − w

]F

(cP

)−∫ cP

0

cf (c) dc

]− θKP f

(cP

) ∂ρP

∂t

+[tg

(KSP

)F

(cP

)+ θ

[1− F

(cP

)] ∂kP

∂p− θKP f

(cP

) ∂ρP

∂p

]∂pP

∂t(25)

1N

∂R

∂θ= KP

[1− F

(cP

)]− θKP f(cP

) ∂ρP

∂θ+ θ

[1− F

(cP

)] ∂kP

∂θ

+[tg

(KSP

)F

(cP

)+ θ

[1− F

(cP

)] ∂kP

∂p− θKP f

(cP

) ∂ρP

∂p

]∂pP

∂θ(26)

As before, there are direct effects and equilibrium price effects in both (25) and (26).In both equations, the first term is a positive direct effect of the tax rate increase onall firms that maintain their tax filing status. In (25), this effect explicitly deducts thecompliance costs incurred by the firms. The second term is the loss in P-tax revenue dueto the direct effect of the tax rate change on the filing threshold. It is positive for a raisein θ or a drop in t, as both cause some firms to switch to the P-tax retun. The thirdterm in (26) accounts for the direct influence of a rise in θ on the production decisionof P-filers, and it is negative. There is no such effect of t in (25).

The last term in each equation is the equilibrium price effect. The price affects taxrevenue through several channels. The first two terms within the brackets reflect a risein tax collection from S-filers and P-filers respectively if the price rises. According tothe last term within the brackets, if the threshold ρP decreases due to a price change,the firms that switch to the P-tax may increase the tax revenue on the margin However,neither ∂ρP

∂p nor the direction of the change in tax liability can be unambiguosly signed.12

In the particular case where the P-filers just break even, the firms that switch to theP-tax go from contributing 0 to θKP . Even worse, while ∂pP

∂t > 0, the sign of ∂pP

∂θ isambiguous in equilibrium.

The first order conditions for an interior solution in this range are quite cumbersomeand can be found in the Appendix B. Is it likely that the outcome of the game involvesρP < ρS? I will argue presently that it is not. Rather, it is likely that the solution tothe government’s problem require ρP = ρS .

For ρP = ρS , shown by demand D in Fig. 4, one has somewhat simpler expressionsfor the changes in revenue:

1N

∂R

∂t=

[πSP − w

]F

(cP

)−∫ cP

0

cf (c) dc (27)

1N

∂R

∂θ=

[1− F (α)] KP + θKP g(KSP )g(KP )

f(cP

)∂ρP

∂θ +[1−

[1− KP g′(KP )

g(KP )

]F

(αP

)]θ ∂kP

∂θ +[

tg(KSP

)F

(cP

)+ θ [1− F (α)] ∂kP

∂p

− θKP

g(KP )

[1N

dXdp − f

(cP

)g

(KSP

)∂ρP

∂p − rpS F

(cP

)∂kS

∂p

]]

∂pP

∂θ

(28)

12Their tax liability increases as long as g′�kS�

>g(kS)−g(kP )

kS−kP .

15

The impact of a change in t on revenue is the same as the first term in the previouscase: an increase in t increases revenue from all existing S-filers. It is the only effectbecause the P-filers are breaking even and ∂pP

∂t = 0 in this range.

(28) describes both direct effects and price effects. An increase in θ increases therevenue from existing P-filers (first term). It also raises the break-even threshold forP-filers (second term, a negative revenue effect) and decreases their output (third term,also a negative effect).

Through the price, θ has an impact on the revenue from both S-filers and P-filers.A rise in θ increases the price and the break-even threshold for P-filers. This increase inprice raises revenue from previous S-filers (first term in the large brackets), and P-filers(second term). It lowers the equilibrium output in the market–the third term in the largebracket. This third term is positive if ∂ρP

∂p > 0, but its sign is ambiguous otherwise.

The first order condition in this case calls for θ to be increased if

0 ≤

[1−F (α)F (cP )

]KP +

[θKP

cP

g(KSP )g(KP )

− 1λ

]cP f(cP )

F (cP )∂ρP

∂θ +[1−

[1− KP g′(KP )

g(KP )

]F(αP )F (cP )

]θ ∂kP

∂θ +

tg(KSP

)+ θ

[1−F (α)F (cP )

]∂kP

∂p + θKP

g(KP )r

pS∂kS

∂p[∆P

λ − θKP

g(KP )

]1

NF (cP )dXdp −

[θKP

cP

g(KSP )g(KP )

+ 1λ

]cP f(cP )

F (cP )∂ρP

∂p

KP

g(KP )

(29)

and t to be increased if[

1λ (1− t)2

cP f(cP

)

F (cP )+ 1

]cP ≥

∫ cP

0cf (c) dc

F (cP )(30)

Here, I’ve replaced cP = πSP−w and ∂pP

∂θ = KP

g(KP ). The partial derivatives are evaluated

at the market equilibrium:

∂ρP

∂p = g(kS

)− g(kP )1−t

∂ρP

∂θ = kP

1−t > 0 ∂ρP

∂t = 0

∂kP

∂p =−g′(kP )pg′′(kP )

> 0 ∂kP

∂θ =g′(kP )

rg′′(kP )< 0 ∂kS

∂p =−g′(kS)pg′′(kS)

> 0

These two equations characterize the interior solutions to the government’s problemin this range.13

Condition (29) is generally ambiguous in its prescriptions. However, close inspectionprovides some guidance about when an OPT can improve on the outcome of a pureS-tax system:

Remark 6 Other things equal, the optimal solution with ρP = ρS involves a lowerθ∗ if (i) demand for the good is more elastic and the marginal surplus loss is large,∆P

λ > θKP

g(KP ), (ii) the factor elasticity of the production function

KP g′(KP )g(KP )

is higher,

(iii) λ θKP

cP <g(KP )g(KSP )

and 1 − t <g(KP )g(KSP )

(high compliance cost but low firm output

distortion), (iv) the marginal value of government funds is high andg(KP )g(KS)

< 1− t (highfirm output distortion).

13See note 11. See also Appendix B for calculations.

16

Consider a small decrease in the P-tax rate to θ < θmax

(pS

), keeping the S-tax rate

t constant. The plateau in the supply schedule is now lower. The new market supply isthen OAEE′E′′.

First, note that some firms will start to file a P-tax in equilibrium, since ∂cP

∂θ =g(KS)g(KP )

KP > 0 at θ = θmax

(pS

).

On the margin, tax revenue is likely–but not certain–to increase, provided the markethas a low equilibrium price. A few S-filers switch to file a P-tax return. They were justbreaking even but pay now a positive amount of tax θKP . Since there are no P-tax filersto begin with, the decrease in θ does not decrease previous P-tax collections. However,the P-filers must be breaking even still as the plateau in the market supply is now atthe new, lower equilibrium price.

Yet there are two sources of revenue loss. Tax collections from the remaining S-filers decrease. This effect is due to the lower market price, and it is proportional tothe number of firms currently filing S-tax returns. Also, because of the lower price,everybody decreases output. These two negative aspects are relatively small at lowmarket prices: there are few S-filers and firm output is likely to be less responsive (themarginal costs are likely to increase more slowly).

The excess burden, on the other hand, must decrease. The equilibrium quantity inthe market increases, so that distortion losses decrease, and the switch from S-filers tothe P-tax saves compliance costs. In short, a small decrease in θ from θmax

(pS

)to a

lower value is very likely to increase the welfare of this market.

Of course, as θ continues to drop, the firms that switch to the P-tax regime start tocause a decrease in S-tax collections–tax base erosion–, and the total P-tax collectionssuffer as well. On the margin this loss of revenue is larger the lower θ is, just because morefirms are affected. On the other hand, the marginal gains due to new firms entering themarket as P-filers are smaller. The new firms still increase P-tax revenue and decreasequantity distortions in the market. Even the compliance cost gains from the firms thatswitch are lower, as the remaining S-filers have lower ci than those who have alreadyswitched.

If θ falls enough, the market equilibrium will be eventually at the right kink in themarket supply, point C ′ in Fig.5(b) and θC′ in Fig.7. All firms have entered the marketat this point. Further decreases in θ will then decrease the equilibrium quantity andincrease the associated surplus loss. The only gains come from turning the compliancecost ρP of the marginal firm that switches into P-tax revenue. And even though P-filersstill break even at θC′ , that compliance cost does not translate fully into tax revenue forthe government because of the firm output distortion. This gain would have to outweighthe surplus losses and the revenue losses from existing P-filers for the optimal solutionhave lower θ.

The discontinuity in dR/dθ and dL/dθ at θC′ is illustrated in Fig.7 for given t.Because of these jumps, it is a priori very unlikely that that the optimal P-tax rate liebelow θC′ = θmax

(pP

)< θmax

(pS

). There are good chances that it is precisely θC′ . If

this is the case, the equilibrium quantity distortion is minimized under the OPT, and itis lower than under the pure S-tax.

Consider now the effects of a S-tax rate increase on the market supply schedule ZP ,depicted in Fig.6.

17

θmax

λ(dR/dt)

(dL/dt)

θ* (t) θθC’

FIGURE 7: Optimal θ for given S-tax rate

Notes: For θ < θC′ all firms have entered the market.

Proposition 7 Under the OPT, if the government’s optimal strategy is such that theP-tax filers just break even after tax, then the S-tax is not distortionary and the S-taxrate must be one, t∗ = 1.

Proof. To see this, notice that (30) is always satisfied: the right hand term is theaverage compliance cost of those firms that file a S-tax return in equilibrium. The lefthand side is higher than the maximum compliance cost among those firms.

If the equilibrium in this market does not obtain on the perfectly elastic part ofZP , almost anything can happen. For example, it may be optimal to shut down theS-tax system altogether, or to have both types of tax returns. Increases in t decreasethe equilibrium quantity in the market, so the rationale for such policy must come fromrevenue considerations. The equilibrium quantity effect of a change in θ is ambiguous.A decrease in θ directly lowers tax revenue from the existing P-filers but increases itthrough a rise in their firm output. Raising t raises the revenue from S-filers withoutdirectly affecting their output decision, but it drives some firms to switch to the P-tax.This in turn raises equilibrium price–which again helps tax collections.

Proposition 8 Under an OPT system, if the market supply is less than perfectly priceelastic at the equilibrium, then t∗ ≤ 1. Then (i) a necessary condition for some S-taxreturns to exist is t∗ < 1, and (ii) there exist equilibrium prices such that t∗ < 1 andboth types of tax retun are filed.

Proof. (i) If ZP is not perfectly price elastic, then θ∗ < θmax

(pP

)and the P-tax

filers have positive after-tax profits. For any S-tax returns to exist, they must be allowedsome after-tax profit. (ii) Moreover, for very high prices, the output distortion must bevery large in the P-tax regime, and the compliance costs must be negligible for somefirms. At that point, the government would find it optimal to lure those firms into theS-tax regime. By (i), this requires t∗ < 1.

Unfortunately, there is no clear result for “low” price ranges, precisely where an OPTis most interesting.

2.3 On evasion

If evasion is an available alternative to the firm, one must consider two new features ofthe tax system, both related to enforcement activities. The first one is the probability

18

of detecting evasion (which I call q), and the second one is the schedule of penaltiesapplied to evaders, denoted Φ (T ∗, T ), where T ∗ is the declared tax liability, as opposedto the true tax liability T . Both these parameters are determined by the government.

In this context, the firm’s problem is akin to a gamble. With probability 1 − q,the firm it and keeps all its undeclared income. If it loses, it must pay the tax plusthe fine. In general, there are two types of evasion available to a firm: non-filing andunderstatement. The difference between them is more than just quantitative: the extradegree of freedom that a risk averse evader has in the case of understatement is crucialin determining her filing choice, given a fine schedule. It allows her to separate theproduction decision from the size of the bet (i.e. the amount evaded) and fine-tune theevasion gamble. On the other hand, a firm that does not file a tax return (commonlycalled a “ghost”) faces a discrete, all-or-nothing decision. The difference becomes evenmore pronounced if the probability of audit for a ghost is different from that for a firmfiling a tax return.14

From the perspective of the social planner, the production distortions resulting froma ghost respond discontinuously to its policy parameters, while this is generally not thecase for understaters. However, both types of evasion are alike in one decisive aspect: forany evasion to exist at all, the expected return of the gamble must be positive. Indeed,Yitzhaki [25] shows that the amount of evasion chosen by a risk averter depends on thetax schedule through the net-of-tax income, but the decision to cheat hinges on the betbeing actuarially favorable (Allingham and Sandmo [1]). In other words, for any givenincome the government can dissuade potential evaders completely through proper designof a fine schedule and an audit probability, regardless of the shape of the tax schedule.

Some thought must be given to the fine schedule before we proceed. A well knownresult states that, if enforcement is costly, it is socially optimal to raise the penaltyfor cheating as far as possible, thus keeping the necessary probability of audit very low(Yitzhaki [25]). Alas, this is not a general result. Andreoni [2] shows that it does notalways obtain if the firms are liquidity constrained. As a practical matter, evasion finesare likely to politically constrained to be in certain ranges deemed reasonable, and theirdesign can take into account only a limited set of parameters. Moreover, it is difficultto carry out a policy analysis unless a particular type of fine schedule is assumed.

A consideration on the nature of the enforcement parameter q is also necessary here.If one takes q to be akin to a pure audit probability, and then assumes that any evasionwill be detected if a firm is audited, the only difference between auditing a P-filer and aself-assessed S-filer is the administrative cost of the audit procedure itself. On the otherhand, in the presence of ambiguity in the tax code, it may be difficult to determine theactual extent of the tax liability.15 An effective presumptive measure of tax liabilityin this case should be designed to address ambiguity, and the parameter q will thenembody both a probability of audit and a probability of detection of evasion given anaudit. In the extreme case of a perfectly observable measure of income, a tax code withclear wording can in principle eliminate any significant ambiguity regarding the true taxliability and q can become a pure audit probability.

In actual cases, accurate measures of income are not perfectly observable, and per-fectly observable features of the production process are only imperfect measures of in-come. Moreover, the observability of a measure isn’t determined exogenously: it is

14It also makes a difference whether the tax evasion penalty is assessed on the evaded tax liabilityrather than the undeclared income. Slemrod and Yitzhaki [16] point out that “An increase in [the taxrate] t has a substitution effect, increasing the relative price of consumption in the audited state of theworld, and thereby encouraging evasion, if the penalty is related to income, rather than tax avoided.”

15Such ambiguity may generate incentives to devote resources to avoidance. For example, one mightwant to consider “creative” accounting practices.

19

achieved by costly activities, both of the tax administration and the taxpayer. In thissense, the choice between pure presumptive income taxation and standard income tax-ation addresses a tension between the ideal and the feasible objectives of the tax law.The compromise reached will be reflected in the tax code, the audit costs incurred byboth the taxpayer and the tax authorities, and the compliance costs faced by the tax-payer. The choice of a particular income measure with a potential degree of accuracydoes not completely characterize the tax administrator’s policy, but the activities thatindicate the resources devoted to achieve observability must also be taken into account.A seemingly very accurate measure, one that would typically fall under standard filing,might in truth be closer to or at least less accurate than a presumptive tax regime if theeffort–and costs–of compliance and audit are too low.16

2.3.1 Adding ghosts to the model with an OPT

Suppose that in the model from section 2.2 the firm has the additional option of notfiling any tax return. It cannot, however, understate its tax liabilities if it files a taxreturn.

The firm’s income is not given but it is determined by the its input size decision. As aconsequence the input size decision takes place jointly with the choice of the prospectiveevasion gamble. The declared tax liability T ∗ is always truthful if the firm files a taxreturn (and zero if not). Any existing firm, regardless of its chosen filing status, getsaudited with probability q.

Also, assume that q is a pure audit probability for any existing firm and the evasionfine is proportional to the evaded tax, Φ = φT . The probability of detecting evasiongiven that a firm is audited is one. The audit costs to P-filers are negligible, and theaudit costs to self-assessed S-tax filers are included in the compliance costs ci.

Claim 9 Under an OPT system with risk-neutral taxpayers who can choose not to filea tax return, if the penalty schedule for evasion is linear on the tax evaded, Φ = φT ,then the government’s optimal probability of audit q is such that the expected payoff frombecoming a ghost is identical to the certain payoff of filing a P-tax return.

This claim reflects the fact that, under the assumptions, the problem of ghosts isformally the same as the OPT system. To see this, note that the payoff of the evasiongamble is Y − qT (1 + φ). If the government finds a ghost, it must assess its incomeY in order to find its tax liability T . Moreover, it must do so on the basis of limitedinformation and, more often than not, without the help of the taxpayer. Under theseconditions, if the OPT is properly designed, it must be the best guess about Y thatthe government can make.17 Thus T = θK and the evasion gamble pays in expectationY − qθK (1 + φ). It the firms are risk neutral, they will be indifferent between filing aP-tax return if q (1 + φ) = 1 but will evade taxes if q (1 + φ) > 1. In expectation, thegovernment will also be indifferent between evasion and a P-tax return.

The crucial point in this claim is that the audit must resort esentially to incomepresumption. In the absence of accounting books, the auditor cannot pretend to estab-lish the true tax liability under the standard tax regime. Moreover, it may be in thegovernment’s interest that this be so. To see this, consider the motivation that a ghostmight have who produces accounting books when audited. In this model, a firm knows

16Alternatively, one could think that, for a given taxpayer and a given measure of income, the measureis likely to become presumptive as the compliance cost decreases.

17This is true as long as there are no externalities of evasion, nor any direct welfare effects other thantax revenue losses.

20

from the beginning of the period whether it’ll file a standard tax return or not. The onlyreason why a ghost might have kept books must be that it wants to be able producethem if it gets audited in order to decrease its evasion fine. However, allowing this isnever sensible on the side of the government: it raises administrative costs of audit,weakly lowers revenue of the penalty and raises overall compliance costs, and decreasesthe firms’ incentive to file tax returns. Thus, it is optimal for the government to refuseany accounting books from a ghost that is being audited.18

2.4 An application to an externality in evasion

If evasion (e) in and of itself entails losses beyond those associated with tax revenue,the government’s problem must be modified accordingly. Consider a pure S-tax sys-tem where the firms can choose to become ghosts, in which case they are caught withprobability 1 − q and the evasion penalty is linear, Φ = φT . Suppose also that the taxliability T is assesed linearly on a presumptive basis. Such a tax system is an OPT forall practical purposes, and its effective P-tax rate is θe = qθ (1 + φ).

As analyzed in the previous section, in the absence of the externality the governmentis truly indifferent between the OPT and the pure S-tax with an audit program. Withthe externality, however, the government is strictly better off introducing an officialOPT that replicates the firms’ expected payoff of evasion. However, the firms’ payoff ofevasion is not the optimal OPT described in 2.3, as the government’s optimal parameterst∗ and θ∗ are different. How do they differ from the previous result?

Write the government’s problem as

min{T}

L + µ (e)− λE [R]

where µ is increasing in e, and there is uncertainty in the tax revenue. However, interiorsolutions for the optimal tax rate are determined by the first order conditions

λ =∂L∂t + dµ

de∂e∂t

∂E[R]∂t

=∂L∂θ + dµ

de∂e∂θ

∂E[R]∂θ

where e is the fraction of firms that produce but do not pay the S-tax. those. Exactlyhow this externality affects the choice of the optimal tax revenue depends on ∂e

∂t and ∂e∂θe

and the shape of R (θ, t).

If the equilibrium happens on the plateau of the supply curve, ∂e∂t = 0 and ∂e

∂θe< 0.

At the optimum, t∗ = 1 but the P-tax rate embodied in the optimal evasion fine ishigher than the P-tax rate in the legal OPT. This drives up the equilibrium price andincreases the market output distortion. There are fewer tax evaders, but the losses frommarket output distortions and compliance costs are higher.

If, on the other hand, the equilibrium is on the upward sloping part of the marketsupply, to the right of the plateau, ∂e

∂t = −f(cP

)∂cP

∂t and ∂e∂θe

= −f(cP

)∂cP

∂θe. In this

case, if t = 1 all firms choose to evade taxes, e = 1. An interior solution with t < 1is then guaranteed if µ′ (1) is high enough. ∂cP

∂t < 0, so t∗ is lower than without theexternality. The change in θ∗ is ambiguous because the sign of its effect on tax revenueis unknown.

18Additionally, a schedule of fines thus assessed has the advantage of less room for manipulationand increased transparency. This decreases the risk of perceptions of unfairness and makes it easier toadminister.

21

The effect of the uncertainty in revenue depends on whether ∂E[R]∂t ≷ ∂R

∂t and ∂E[θ]∂t ≷

∂R∂θ , so its direction is ambiguous in general.

These results imply that the introduction of a legal OPT that eliminates evasionentails an increase in the optimal S-tax rate whenever a negative externality of evasionis present (and provided t∗ < 1 to begin with). In addition, if the ghosts break even inexpectation before the OPT is introduced, the elimination of the externality will allowa lower P-tax than the one implicit in the evasion fine. P-filers will still break even withthe OPT, but the distortions in firm output will be lower. In consequence, when anOPT is implemented, the evasion fine–or the probability of audit–can be decreased andstill be able to deter potential evaders.

An additional channel of potential gains is, of course, the elimination of uncertaintyin the tax revenue. In this model, however, its not possible to establish the direction ofthis effect.

3 Concluding remarks

This paper illustrates the complexity of the issues that arise when presumptive incometaxation is used. Typically, these taxes are considered in order to decrease administrativeand compliance costs, but their benefits are potentially larger as information gatheringinstruments in a dynamic environment. Moreover, they can be used to curve widespreadtax evasion and avoidance by reducing enforcement costs, which is likely to generatepositive externalities in taxpayer compliance. However, careful analysis is required astheir effects in terms of efficiency are multiple and can be large, and the exact optimalpresumptive tax structure is ultimately linked to the characteristics of the targeted taxbase.

My model analyzes the implementation of an optional presumptive income tax sched-ule (OPT) with the purpose of inducing small firms with high compliance costs to filetax returns. In particular, the analysis focuses on the incentive effects of the OPT onthe firms’ economic activity and the government’s optimal welfare function.

The analysis suggests that an OPT is more likely to be able to improve on theoutcome of a pure self-assessed tax on profits in markets where (i) demand for the goodis very elastic and the market output distortions are large, (ii) the production functionhas high factor elasticity, (iii) there are high compliance costs for the marginal firm, and(iv) the firm output distortion is large. Moreover, it is likely that, at the optimum, allprofits need to be extracted from all firms in the market. In this case, all firms withpositive output are indifferent between the choice of tax regimes.

This is not, however, a general result. If the optimal solution is such that thefirms under the presumptive regime do have profits, the self-assessed tax rate becomesdistortionary and it may be less than one. The distortions in equilibrium quantity inthe market are not minimized in this case.

If the firms can choose not to file an income tax return, I find that the P-tax of theoptimal OPT regime should take the form of a legal alternative with the same expectedpayoff as evasion.

I argue that evasion may be viewed as involving a negative externality. This may berelated to strategic behavior from taxpayers in a static one-time game or to informationeffects in dynamic repeated games. If this is the case, the model shows that there are

22

added welfare gains from an OPT. A pure self-assessed tax system is strictly inferior toan OPT under these circumstances. Moreover, if the OPT is implemented, then the P-tax rate should be lower than the one implied by the evasion fine schedule. In contrast,the S-tax rate should be higher.

This paper provides only a small glimpse into the possibilities–and difficulties–ofpresumptive income taxation. The structure in this model is one common type of pre-sumptive tax system, closest in spirit to the tachsiv in Israel and the French forfait.However, presumptive income taxes come in a variety of flavors, with very differentstructures and target markets, and the parameters of interest vary accordingly. A goodcase has been that they have some desirable features of ability-to-pay taxes, and I showhere that they may also be desirable in the face of high compliance costs. Nevertheless,they are still viewed as a second-best compromise. Their extensive use calls for moreformal analysis of their effects and their robustness to both micro and macroeconomicenvironment conditions. Future avenues of research should include their performance inthe presence of uncertainty and the dynamic aspects of tax presumption, this last itembeing of particular interest in the context of tax reform.

References

[1] Allingham, Michael G. and Agnar Sandmo, 1972, Income Tax Evasion: A Theoretical Anal-ysis. Journal of Public Economics, Vol 1, No. 3-4, pp. 323-338.

[2] Andreoni, James, 1992, IRS as a Loan Shark: Tax compliance with borrowing constraints.Journal of Public Economics 49 pp.35-45.

[3] Andreoni, James, Brian Erard and Jonathan Feinstein, 1998, Tax Compliance. Journal ofEconomic Literature, Vol.36, Issue 2 (June), pp.818-860.

[4] Arrow, Kenneth J, 1970, Essays in the Theory of Risk Bearing. Amsterdam: North Holland.

[5] Auerbach, Alan, 1985, The Theory of Excess Burden and Optimal Taxation. Handbook ofPublic Economics, Vol.1, Ch.2. North Holland: Elsevier Science Publishers B.V.

[6] Auerbach, Alan, and James R. Hines Jr., 2001, Taxation and Economic Efficiency.Manuscript (February 2001).

[7] Bird, Richard M, 1970, Taxation and Development: Lessons from the Colombian Experience.Harvard University Press.

[8] Bird, Richard M, 1989, The Administrative Dimension of Tax Reform in Developing Coun-tries, in “Tax Reform in Developing Countries”. Duke University Press, 1989, pp. 315-346.Refer also to the final draft, April 1988.

[9] Bird, Richard M, 1992, “Tax Policy and Economic Development”. The Johns HopkinsUniversity Press, Baltimore and London.

[10] Bulutoglu, Kenan, Presumptive Taxation, in “Tax Policy Handbook”. International Mon-etary Fund, 1995, pp.258-262.

[11] Cowell, Frank A.,1990, “Cheating the Government”. Cambridge, Massachusetts: MITPress.

[12] Goode, Richard, 1984, “Government Finance in Developing Countries”. The BrookingsInstitution, Washington D.C.

[13] Lapidoth, Arye, 1977, The Use of Estimation for the Assessment of Taxable BusinessIncome. Amsterdam: International Bureau of Fiscal Documentation.

[14] Musgrave, Richard A, 1990, Income Taxation of the Hard-To-Tax Groups, in “Taxation inDeveloping Countries”. Fourth Edition, John Hopkins University Press, 1990.

[15] Rajamaran, Indira, 1995, Presumptive Direct Taxation: Lessons from Experience in De-veloping Countries. NIPFP, New Delhi. Manuscript (May 1995).

[16] Slemrod, Joel and Shlomo Yitzhaki, 2000, Tax Avoidance, Evasion and Administration.NBER Working Paper 7473.

23

[17] Slemrod, Joel, 2002, Trust in Public Finance. NBER Working Paper 9187.

[18] Tanzi, Vito, 1991, Potential Income as a Tax Base in Theory and in Practice, in “PublicFinance in Developing Countries”. Edward Elgar Publishing Limited, Ch. 13, pp.193-209.

[19] Tanzi, Vito, and Milka Casanegra de Jantscher, 1987, Presumptive Income Taxation: Ad-ministrative, Efficiency and Equity Aspects. International Monetary Fund Working PaperWP/87/54.

[20] Tanzi, Vito, and Howell Zee, 2000, Tax Policy for Emerging Markets. International Mon-etary Fund Working Paper WP/00/35.

[21] Tanzi, Vito, and Parthasarathi Shome, 1993, A Primer on Tax Evasion, in “Staff Papers”.International Monetary Fund, pp.807-828.

[22] Taube, Gunther and Helaway Tadesse, 1996, Presumptive Taxation in Sub-Saharan Africa:Experience and Prospects. International Monetary Fund Working Paper WP/96/5.

[23] Thirsk, Wayne, 1997, Overview: The Substance and Process of Tax Reform in Eight De-veloping Countries, in “Tax Reform in Developing Countries”. The World Bank.

[24] Gandhi, Sonia, 2000, Presumptive Income Taxation: An Overview. The World Bank:http://www1.worldbank.org/publicsector/tax/presumptivetax.htm

[25] Yitzhaki, Shlomo, 1987, On the Excess Burden of Tax Evasion. Public Finance Quarterly,Vol.15 No.2 (April) pp.123-137.

24

A Notation

Throghout this paper, the superscript S is used to denote the solutions to the standard taxproblem with linear taxation of profits. Similarly, the superscript P denotes solutions to theoptional presumptive tax problem. Therefore, in the following table α can take the values S, P .

Summary table

Firms There are N agents are therefore N possible firms, indexed by i.Ki : Firm i’s perfectly observable input.g (·) : Individual firm’s production function, common to all firms g′ > 0, g′′ < 0. For

convenience, I denote the inverse of the derivative of the production function by γ (·) = [g′]−1(·).

w : Payoff that each agent/firm can guarantee itself if it leaves the industry.πi = pg (Ki)− rKi : Agent i’s implicit labor wage before taxes and compliance costs.Y α

i : Net payoff to agent i after taxes and compliance costs are deducted.

Government λ : Marginal social value of government fundsL : Social loss due to taxation, including distorsionary effects and compliance costs incurred

by taxpayers.R : Total tax revenue collected by the government.

Market Demand and Supply X = X (p) : Aggregate market demand function,X ′ < 0.

Zα = Zα(p; θ, t) : Aggregate market supply schedule, with α = S, P .Z = Z (p) : Market supply schedule in the absence of taxes.

Tax system T αi : Agent i’s tax liability.

t : Constant marginal tax rate on profits under the standard income tax regime (S-tax)θ : Constant marginal tax rate on input K under the presumptive income tax regime (P-

tax).

Compliance costs ci : Bookkepping cost incurred by firm i in order to be able to filea S-tax return.

F (c) : Percentage of all possible firms that feature ci < c. Denote F ′ (c) = f (c).

Solution to the firm’s problem given output price Given S-tax rate t (andpossibly P-tax rate θ) and an equilibrium price p, the solution to the firm’s FOC is denoted

kα = kα (p; θ, t) : Solution to the firm’s first order condition for the optimal input levelunder tax regime α.

ρα = ρα (p; θ, t) : Threshold compliance cost for tax regime change or exit under tax regimeα.

Equilibrium solutions If the market equilibrium for S-tax rate t (and possibly P-taxrate θ) obtains at a price pα = pα (θ, t) and quantity Qα (θ, t), then

Kα ≡ kα (pα; θ, t) :cα ≡ ρα (pα; θ, t) :πα ≡ π (pα; θ, t) :In the OPT regime, a special case arises: KSP ≡ kS

�pP ; θ, t

�Miscellaneous

�p0, Q0

�: Equilibrium price and quantity in the absence of taxes.

∆α ≡ pα − Z−1 (Qα) : Marginal surplus loss in the market due to decreases in equilibriumquantity.

e : Evasion as a fraction of all possible firms.

25

B OPT: Calculations

B.1 ρP > ρS

In order to write the first order conditions purely as functions of the tax rates θ and t, dif-ferentiate the definition of ρP in (17) and use the firms’ first order conditions (4) and (15) toobtain

∂ρP

∂t= ∂ρP

∂t= − pg(kP )−(r+θ)kP−w

(1−t)2∂ρP

∂θ= KP

1−t∂ρP

∂p= g

�kS�− g(kP )

1−t

Also, from the definition of kP and kS :

∂kP

∂p=

−g′(kP )pg′′(kP )

> 0 ∂kP

∂θ=

g′(kP )rg′′(kP )

< 0

∂kS

∂p=

−g′(kS)pg′′(kS)

> 0 ∂kS

∂t= 0

Differentiate the equilibrium condition X (p) = ZP (p, θ, t) (19) at the solution to write:

∂pP

∂t=

∂ZP

∂t

X′(p)− ∂ZP

∂p

∂pP

∂θ=

∂ZP

∂θ

X′(p)− ∂ZP

∂p

Also, directly from ZP

∂ZP

∂p= N

hf�ρP� �

g�kS�− g

�kP��

∂ρP

∂p+ F

�ρP�g′�kS�

∂kS

∂p+�1− F

�ρP��

g′�kP�

∂kP

∂p

i∂ZP

∂θ= Nf

�ρP� �

g�kS�− g

�kP��

∂ρP

∂θ+ N

�1− F

�ρP��

g′�kP�

∂kP

∂θ∂ZP

∂t= Nf

�ρP� �

g�kS�− g

�kP��

∂ρP

∂t< 0

The FOC for θ under the OPT is:

λ∂R

∂θ≥ ∂L

∂θ

Use (26) and (24)

1

N

∂R

∂θ=

8<: �1− F

�cP�� h

KP + θ ∂kP

∂θ

i− θKP f

�cP�

∂ρP

∂θ+h

tg�KSP

�F�cP�

+ θ�1− F

�cP��

∂kP

∂p− θKP f

�cP�

∂ρP

∂p

i∂pP

∂θ

9=;∂L

∂θ=

�−∆P dX

dp+ NcP f

�cP� ∂ρP

∂p

�∂pP

∂θ+ NcP f

�cP� ∂ρP

∂θ

Replace and rearrange to obtain

0 ≤8<: �

1− F�cP�� h

KP + θ ∂kP

∂θ

i− � 1

λcP + θKP

�f�cP�

∂ρP

∂θ+h

1λ∆P 1

NdXdp− � 1

λcP + θKP

�f�cP�

∂ρP

∂p+ tg

�KSP

�F�cP�

+ θ�1− F

�cP��

∂kP

∂p

i∂pP

∂θ

9=;or, with equality

∂pP

∂θ= −

�1− F

�cP�� h

KP + θ ∂kP

∂θ

i− � 1

λcP + θKP

�f�cP�

∂ρP

∂θ

1λ∆P 1

NdXdp− � 1

λcP + θKP

�f (cP ) ∂ρP

∂p+ tg (KSP ) F (cP ) + θ [1− F (cP )] ∂kP

∂p

Similarly

λ∂R

∂t≥ ∂L

∂t

1

N

∂R

∂t=

8<:h�

πSP − w�F�cP�− R cP

0cf (c) dc

i− θKP f

�cP�

∂ρP

∂t+h

tg�KSP

�F�cP�

+ θ�1− F

�cP��

∂kP

∂p− θKP f

�cP�

∂ρP

∂p

i∂pP

∂t

9=;∂L

∂t=

�−∆P dX

dp+ NcP f

�cP� ∂ρP

∂p

�∂pP

∂t+ NcP f

�cP� ∂ρP

∂t

26

0 ≤8<:

h�πSP − w

�F�cP�− R cP

0cf (c) dc

i− � 1

λcP + θKP

�f�cP�

∂ρP

∂t+h

1λ∆P 1

NdXdp− � 1

λcP + θKP

�f�cP�

∂ρP

∂p+ tg

�KSP

�F�cP�

+ θ�1− F

�cP��

∂kP

∂p

i∂pP

∂t

9=;∂pP

∂t= −

h�πSP − w

�F�cP�− R cP

0cf (c) dc

i− � 1

λcP + θKP

�f�cP�

∂ρP

∂t

1λ∆P 1

NdXdp− � 1

λcP + θKP

�f (cP ) ∂ρP

∂p+ tg (KSP ) F (cP ) + θ [1− F (cP )] ∂kP

∂p

Substitute ∂pP

∂θfrom the differentiation of the equilibrium above into the optimality condition

∂ZP

∂θ

dXdp− ∂ZP

∂p

= −�1− F

�cP�� h

KP + θ ∂kP

∂θ

i− � 1

λcP + θKP

�f�cP�

∂ρP

∂θ

1λ∆P 1

NdXdp− � 1

λcP + θKP

�f (cP ) ∂ρP

∂p+ tg (KSP ) F (cP ) + θ [1− F (cP )] ∂kP

∂p

replace ∂ZP

∂θand ∂ZP

∂pand rearrange to obtain the first order condition

g′�KP�

∂kP

∂θ+�g�KS�− g

�KP�� � f(cP )

1−F(cP )

�∂ρP

∂θ

1

F(cP )1N

dXdp− [g (KS)− g (KP )]

f(cP )F(cP )

∂ρP

∂p+ g′ (KS) ∂kS

∂p+

�1−F(cP )

F(cP )

�g′ (kP ) ∂kP

∂p

=

−hKP + θ ∂kP

∂θ

i+�

1λcP + θKP

� � f(cP )1−F(cP )

�∂ρP

∂θ

1λ

∆P

F(cP )1N

dXdp− � 1

λcP + θKP

� f(cP )F(cP )

∂ρP

∂p+ tg (KSP ) +

�1−F(cP )

F(cP )

�θ ∂kP

∂p

Similarly for ∂pP

∂t

∂ZP

∂t

dXdp− ∂ZP

∂p

= −

h�πSP − w

�F�cP�− R cP

0cf (c) dc

i− � 1

λcP + θKP

�f�cP�

∂ρP

∂t

1λ∆P 1

NdXdp− � 1

λcP + θKP

�f (cP ) ∂ρP

∂p+ tg (KSP ) F (cP ) + θ [1− F (cP )] ∂kP

∂p

∂ZP

∂p= N

hf�ρP� �

g�kS�− g

�kP��

∂ρP

∂p+ F

�ρP�g′�kS�

∂kS

∂p+�1− F

�ρP��

g′�kP�

∂kP

∂p

i∂ZP

∂θ= Nf

�ρP� �

g�kS�− g

�kP��

∂ρP

∂θ+ N

�1− F

�ρP��

g′�kP�

∂kP

∂θ∂ZP

∂t= Nf

�ρP� �

g�kS�− g

�kP��

∂ρP

∂t< 0

the first order condition is

f(cP )F(cP )

�g�KS�− g

�KP��

∂ρP

∂t

1

F(cP )1N

dXdp− f(cP )

F(cP )[g (KS)− g (KP )] ∂ρP

∂p+ g′ (KS) ∂kS

∂p+

�1−F(cP )

F(cP )

�g′ (KP ) ∂kP

∂p

=

−��

πSP − w�− R cP

0 cf(c)dc

F(cP )

�+�

1λcP + θKP

� f(cP )F(cP )

∂ρP

∂t

1λ

∆P

F(cP )1N

dXdp− � 1

λcP + θKP

� f(cP )F(cP )

∂ρP

∂p+ tg (KSP ) + θ

�1−F(cP )

F(cP )

�∂kP

∂p

B.2 ρP = ρS (P-filers just break even)

As before, differentiate the definition of ρP in (17) and use the firms’ first order conditions (4)and (15) to obtain

∂ρP

∂t= 0 ∂ρP

∂θ= KP

1−t∂ρP

∂p= g

�kS�− g(kP )

1−t

and from the definition of kP and kS :

∂kP

∂p=

−g′(kP )pg′′(kP )

> 0 ∂kP

∂θ=

g′(kP )rg′′(kP )

< 0

∂kS

∂p=

−g′(kS)pg′′(kS)

> 0 ∂kS

∂t= 0

27

Differentiate the break-even condition for P-tax filers pg�kP�− rkP − w = 0 to write:

∂pP

∂t= 0 ∂pP

∂θ= KP

g(KP )

Denote α at the equilibrium by αP . Then,

∂αP

∂θ=

∂α

∂θ+

∂α

∂p

∂pP

∂θ

∂αP

∂t=

∂α

∂t+

∂α

∂t

∂pP

∂t= 0

and

g�KP�

f�αP� ∂αP

∂θ=

8<: −F�αP�

r+θpP

∂kP

∂θ− f

�cP�g�KSP

�∂ρP

∂θ+h

1N

dXdp− f

�cP�g�KSP

�∂ρP

∂p− r

pS F�cP�

∂kS

∂p

i∂pP

∂θ

9=;Replacing into ZP

ZP (p, θ, t) = N�F�ρP�g�kS�

+ [1− F (α)] g�kP��

if ρP = ρS

∂ZP

∂p= ∞

∂ZP

∂θ= Nf

�ρP�g�kS�

∂ρP

∂θ−Ng

�kP�f (α) ∂α

∂θ+ N [1− F (α)] g′

�kP�

∂kP

∂θ∂ZP

∂t= 0

The FOC for θ under the OPT is the same as before:

λ∂R

∂θ≥ ∂L

∂θ

Differentiate L and R:

1

N

∂R

∂θ=

8<: [1− F (α)]hKP + θ ∂kP

∂θ

i− θKP f (α)

h∂α∂p

+ ∂α∂p

∂pP

∂θ

ihtg�KSP

�F�cP�

+ θ [1− F (α)] ∂kP

∂p

i∂pP

∂θ

9=;=

8>>>>>><>>>>>>:[1− F (α)]

hKP + θ ∂kP

∂θ

i+h

tg�KSP

�F�cP�

+ θ [1− F (α)] ∂kP

∂p

i∂pP

∂θ

− θKP

g(kP )

8<: −F�αP�

r+θpP

∂kP

∂θ− f

�cP�g�KSP

�∂ρP

∂θ+h

1N

dXdp− f

�cP�g�KSP

�∂ρP

∂p− r

pS F�cP�

∂kS

∂p

i∂pP

∂θ

9=;

9>>>>>>=>>>>>>;=

8>>>><>>>>:[1− F (α)] KP + θKP g(KSP )

g(kP )f�cP�

∂ρP

∂θ+

�1−

�1− KP g′(KP )

g(KP )

�F�αP��

θ ∂kP

∂θ

+

24 tg�KSP

�F�cP�

+ θ [1− F (α)] ∂kP

∂p

− θKP

g(kP )

h1N

dXdp− f

�cP�g�KSP

�∂ρP

∂p− r

pS F�cP�

∂kS

∂p

i 35 ∂pP

∂θ

9>>>>=>>>>;∂L

∂θ=

�−∆P dX

dp+ NcP f

�cP� ∂ρP

∂p

�∂pP

∂θ+ NcP f

�cP� ∂ρP

∂θ

Replace and rearrange to obtain the FOC

0 ≤

8>>>>><>>>>>:[1− F (α)] KP +

�θKP g(KSP )

g(KP )− 1

λcP

�f�cP�

∂ρP

∂θ+

�1−

�1− KP g′(kP )

g(kP )

�F�αP��

θ ∂kP

∂θ

+

264 tg�KSP

�F�cP�

+ θ [1− F (α)] ∂kP

∂p+ θKP

g(kP )r

pS F�cP�

∂kS

∂p�∆P

λ− θKP

g(kP )

�1N

dXdp−�

θKP

g(kP )f�cP�g�KSP

�+ 1

λcP f

�cP��

∂ρP

∂p

375 ∂pP

∂θ

9>>>>>=>>>>>;Substitute ∂pP

∂θand divide through by F

�cP�

0 ≤

8>>>>>>>>>><>>>>>>>>>>:

�1−F (α)

F(cP )

�KP +

�θKP g(KSP )

g(KP )− 1

λcP

�f(cP )F(cP )

∂ρP

∂θ

+

�1−

�1− KP g′(KP )

g(KP )

�F(αP )F(cP )

�θ ∂kP

∂θ

+

2664 tg�KSP

�+ θ

�1−F (α)

F(cP )

�∂kP

∂p+ θKP

g(kP )r

pS∂kS

∂p�∆P

λ− θKP

g(KP )

�1

NF(cP )dXdp−�

θKP

g(KP )g�KSP

�+ 1

λcP

�f(cP )F(cP )

∂ρP

∂p

3775 KP

g(KP )

9>>>>>>>>>>=>>>>>>>>>>;28

0 ≤

8>>>>>>>>>><>>>>>>>>>>:

[1− F (α)] KP +

�θKP g(KSP )

g(KP )− 1

λcP

�f�cP� h

KP

1−t

i+�

1−�1− KP g′(KP )

g(KP )

�F�αP��

θ

�1

pg′′(KP )

�+2664 tg

�KSP

�F�cP�

+ θ [1− F (α)]

�−g′(KP )pg′′(KP )

�+ θKP

g(KP )r

pS F�cP� �−g′(KS)

pg′′(KS)

��∆P

λ− θKP

g(KP )

�1N

dXdp−�

θKP

g(KP )g�KSP

�+ 1

λcP

�f�cP�

∂ρP

∂p

3775 KP

g(KP )

9>>>>>>>>>>=>>>>>>>>>>;Similarly

λ∂R

∂t≥ ∂L

∂t

1

N

∂R

∂t=

8<:h�

πSP − w�F�cP�− R cP

0cf (c) dc

i− θKP f (α) ∂α

∂t

+htg�KSP

�F�cP�

+ θ [1− F (α)] ∂kP

∂p− θKP f (α) ∂α

∂p

i∂pP

∂t

=

"hπSP − w

iF�cP�−Z cP

0

cf (c) dc

#∂L

∂t=

�−∆P dX

dp+ NcP f

�cP� ∂ρP

∂p

�∂pP

∂t+ NcP f

�cP� ∂ρP

∂t

= NcP f�cP� ∂ρP

∂t

where I have already used ∂pP

∂t= 0 and ∂αP

∂t= 0. Replacing"h

πSP − wiF�cP�−Z cP

0

cf (c) dc

#≥ 1

λcP f

�cP� ∂ρP

∂t

29