THE ROLE OF GOVERNMENT IN THE CREDIT MARKET. Benjamin Eden. Working Paper No. 09-W07. September 2009

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1 THE ROLE OF GOVERNMENT IN THE CREDIT MARKET by Benjamin Eden Working Paper No. 09-W07 September 2009 DEPARTMENT OF ECONOMICS VANDERBILT UNIVERSITY NASHVILLE, TN

2 THE ROLE OF GOVERNMENT IN THE CONSUMER LOANS MARKET Benjamin Eden * Vanderbilt University September 2009 Te paper analyzes a government loan program tat complements money. Te focus is on administrative costs and te difference between te collection tecnologies available to te public and te private sectors. Among te questions addressed are: te optimal monetary and credit policy and te desirability of money substitutes. Te paper describes te loan program as an integral part of te tax system and comments on some tax reform ideas. Key words: Government loans, Tax reform, Tax Evasion, Collection Tecnology, Friedman Rule, Samuelson Rule, Money Substitutes JEL codes: E42, E52, E51, E58, H20, H21, H26 * I would like to tank Robert Driskill, Maya Eden, Boyan Jovanovic and Ayse Sapci for useful comments and discussions on an earlier draft of tis paper.

3 2 1. INTRODUCTION Te recent crisis as led to government intervention in te consumer loans market. For example, te Home Affordable Modification Program is aimed at elping about 9 millions Americans stay in teir omes by using government funds to lower teir montly payments to less tan 31% of income. 1 To evaluate tis program we need to understand wy te parties to te lending contract did not write a clause tat limits te payment to 31% of income, ex ante. More generally, to evaluate government intervention in te credit market, we may want to understand te collection tecnology tat is available to eac sector. We may start from te observation tat te government collects income contingent payments tat are not backed by collaterals, as in te case of income tax. Private lenders typically collect payments on loans tat are not contingent on income and are backed by collaterals. Tere tus seems to be a difference in te collection tecnology between te two sectors. Tis difference in collection tecnologies is discussed in Friedman (1960) celebrated book on Capitalism and Freedom. Friedman starts from te observed difference between te rate of return on an investment in uman and pysical capital. He ends wit a proposal of a government loan program as a remedy to tis market failure. In page 105 in te 40 t anniversary edition Friedman says: A governmental body could offer to finance or elp finance te training of any individual wo could meet minimum quality standards. It would make available a limited sum per year for a specified number of years, provided te funds were spent on securing training at a recognized institution. Te individual in return would agree to pay to te 1 ttp://

4 3 government in eac future year a specified percentage of is earnings in excess of a specified sum for eac $1000 tat e received from te government. Tis payment could easily be combined wit payment of income tax and so involve a minimum of additional administrative expense. Te argument tat led to tis proposal is of particular relevance to our question. Friedman stresses te ig costs of administering income contingent loans by te private sector given te freedom of individuals to move from one place to anoter, te need to get accurate income statements and te long period over wic te contracts would run. He ten continues to say tat: Tese costs would presumably be particularly ig for investment on a small scale wit a wide geograpical spread of te individuals financed. Suc costs may well be te primary reason tat tis type of investment as never developed under private auspices (page 103). He concludes tat: Government intervention migt terefore be rationalized on grounds bot of tecnical monopoly insofar as te obstacle to te development of suc investment as been administrative costs, and of improving te operation of te market, insofar as it as been simply market frictions and rigidities. Friedman (1960) focuses on te failure of te private sector to finance investment in uman capital. Apparently, Friedman did not see any problem in te operation of te market for investment in forms of capital tat can be used as a collateral. Te recent crisis puts a question mark on tis assumption. Indeed one of te argument in support of te Home Affordable Modification Program mentioned above is tat getting people out of teir omes is inefficient because evicted omes may loose as muc as 30% of teir

5 4 value. 2 In general, tere are bankruptcy costs tat arise in part because of te mismanagement of te collateral by te bank. Moreover, tere is a perfectly legitimate demand for uncollateralized loans to smoot consumption. Te credit card type loans are not a satisfactory solution to tis problem given te very ig rates of interest on suc loans. In general, if tere is a difference in collection tecnology between te two sectors tere may be room for bot sectors to operate and utilize teir relative advantages. I consider ere a proposal to create a government body tat will make income contingent loans for any purpose witout te need to ceck collaterals or to ceck weter te borrower satisfies minimum quality standards and use te loan to attend a recognized institution. I now turn to te question of weter Friedman s argument for a limited involvement in te credit market applies to te larger role I ave in mind. I start wit te tecnical monopoly argument. Friedman says tat te private sector as a problem in collecting uncollateralized obligations because of te freedom of individuals to move. Te advantage of te government ere is tat people ave an incentive to report teir location to te government in order to get various benefits like social security, medicare and unemployment benefits. Indeed te government can make tese benefits contingent on paying te income contingent obligations. In addition admission of cildren to public scool requires proof of residence in te scool district. Tus, I see no special education aspect to tis argument and it seems to justify a larger role for te government. Friedman (1960) also mentions te need to get accurate income statement as a reason for te absence of private income contingent loans. Presumably, te suggested 2 Vacant properties are routinely vandalized and broken into by tieves wo will scavenge watever was left beind, especially copper wiring and pipes to sell as scrap. In te first seven weeks of tis year, te Providence police took reports of at least 57 tefts or attempted tefts of copper. Mark Arsenault, Rode Island News, May

6 5 government body will ave access to te IRS records. A private lender on te oter and does not ave an easy access to tese records. Again, tis argument applies to uncollateralized loans in general. Te advantage of te government in making consumer loans is particularly strong wen considering intergenerational loans from old to young people. To see tis point consider te case in wic agents in an overlapping generations (OG) economy live for two periods and are endowed wit a unit of te consumption good in te second period of teir life. In suc an economy trade between generations is not possible even if we introduce money into te system because te young ave noting to give to te old. I sow tat tis is a problem in more realistic settings in wic te young wants to borrow rater tan lend (as in te standard OG model). Te paper is organized as follows. I start wit a discussion of te Hall-Rabuska flat tax proposal and use it to outline my own proposal. I ten use a finite orizon economy to better understand te government s advantage in collecting income contingent payments and intermediating between people of te same age. Te main part of te paper follows. Tis is an overlapping generations model tat is used to address questions about te optimal interest rate(s) and te way a government loan program can be integrated into te tax system. 2. THE HALL-RABUSHKA FLAT TAX PROPOSAL Hall and Rabuska (HR, 1995) propose to simplify te tax system by aving two marginal tax rates: zero up to te family allowance level and ten 19% for all income above tis level. 3 Tey say tat a consumption tax or a value added tax will be simpler but tey reject it because it is not progressive. 3 Altig et al. simulated te effects of te flat tax proposal and oter proposals.

7 6 A variation on te HR proposal is to ave a value added tax wit a rebate for lowincome ouseolds. Alternatively, te government may extend loans to young individuals and allow ouseolds wit low income to partially default on teir loan payments. Tis form of rebate is better tan standard rebate for two reasons. First, it solves a problem in te credit market. Second, it seems more fair because te present value of te rebates depends on te individual s earnings during is entire life. Since te value added tax is a tax on all wages (including te family allowance) its base is larger tan te flat-tax base. According to te numbers in HR Table 3.1 te value added tax base is larger tan te flat rate tax base by 52%. Te government may terefore impose a 19% value added tax and use a tird of its revenues to finance loans defaults by ouseolds wit low permanent income. For example, te government may require a loan payment of 31% of income if income is less tan a critical level and a full payment if income is above te critical level. (Tus, payment max 0.31(income),0.31( ) ). We may tink of te amount 0.31( ) as a full payment and of any amount less tan tat as partial payment. Te ceiling and te amount of te loans sould be set in a way tat te total default level is not greater tan a tird of te VAT tax revenues. Note tat agents tat consistently pay teir loan in full will pay teir entire debt before agents tat pay only partial amounts and only agents tat consistently earn low income will die wit debt to te government. Tere are several practical problems tat need to be addressed. Outmigration: I assume tat te government loan program will be extended only to US citizens. Wat sould we do about citizens wo take te loan and immigrate? Tis is not a new problem. An individual can get te benefits of public education and ten coose to immigrate witout letting te country tat made te investment reap te benefits. Tis brain drain problem is severe for some countries but not for te US tat does not suffer

8 7 from outmigration. But we sould keep tis problem in mind so as not to make te initial loan too ig. Design of te loan limits: Practical consideration may want to limit te amount of loan payment to say 31% of income (as in te current policy proposal). Tis implies a limit on te amount of te loan and as we sall see, a tension wit te Friedman (1969) objective of satiating te economy wit liquidity. Business loans: Te proposed loan program does not impose restrictions on te use of te funds. Te loan may tus be used to finance current consumption, education, investment in ousing, or to start a business. Evans and Jovanovic (1989) found tat wealtier people are more likely to become entrepreneurs because tey are less liquidity constrained. Tis is te apparent rationale used by te U.S. Small Business Administration to provide loans and loan guarantees to small business for start-up and expansion. Similar programs are present in oter countries. Our loan program relaxes liquidity constraints and may allow for poor people to become entrepreneurs. It may also incorporate or substitute existing programs aimed at facilitating start-ups. Te possibility of using part of te government loan for a business project rater tan for consumption smooting, is likely to mitigate moral azard and adverse selection problems. Self control issues: Traditional economics assumes tat once agents become adults tey gain self-control. Beavioral economics argues tat some people never become adults in tis sense. Beavior tat looks like lack of self-control is endogenous to some degree. For example, in te present system a young adult wo drives an expensive car may do it in an attempt to transmit a signal about te wealt of is parents or is own ability to make money. Once everyone gets a loan, suc a beavior may signal te lack of self-control. In any case, self-control is likely to be a problem and te loan program sould address it by making loans in various stages (say, a certain amount at age 20, an

9 8 additional amount at age 30 and so on) and by offering options to buy annuities and oter commitment devices. 3. INTERMEDIATION BETWEEN PEOPLE OF THE SAME AGE Enforcing income contingent loan contracts requires te verification of income. Townsend (1979), Williamson (1987) and Bernanke and Gertler (1989) studied loan contracts under te assumption tat income verification is costly. Here I depart from tis literature in assuming tat instead of running an audit directly, a lender wo wants to verify income obtains a costly court order tat allows access to te borrower s internal revenue service (IRS) records. Te government as an advantage because it as costless access to its own IRS records. Te use of te IRS records in bankruptcy court cases leads to a negative external effect imposed by private lenders on te IRS: A borrower wo plan on going bankrupt may coose to underreport is income to te IRS as a way of seltering some income from te lender. Tis makes te IRS job of collecting taxes more difficult relative to te case in wic IRS records cannot be used in bankruptcy cases. 4 Te government may mitigate tis external effect and economize on bankruptcy costs if it uses its advantage to lower te borrowing rate witout canging te amount of te loan. To illustrate, I assume a two periods (t 1,2) single good endowment economy. Tere are two types of agents: Borrowers (indexed b) and Lenders (indexed l). Te 4 Here I assume tat te lender bears te cost of going to court but in reality bot te lender and te borrower bears some costs. Te borrower often seeks te protection of te court and may be required to present IRS records to support is case. Te external effect occurs in many oter instances. An individual wo plans to get divorce may coose to underreport is income to cut on alimony and cild support. An individual wo applies for a tuition reduction in private scool is often required to present is IRS records. Te IRS will release te records if te individual explicitly ask for it. A law tat proibits te use of IRS records may make te IRS job easier but is not consistent wit te broader objective of supplying a public good.

10 9 number of borrowers is te same as te number of lenders and is normalized to 1. Borrowers get no endowment in te first period. In te second period tey participate in a lottery: A fraction of te borrowers get 1 unit of te consumption good and a fraction 1 get zero. Lenders get an endowment of 1 unit in te first period. Tey also participate in a lottery tat determines teir second period income: A fraction 1 of te lenders receives an endowment of 1 unit and a fraction receives zero. Tus te aggregate endowment in eac period is 1 unit. Te realization of te endowment is costlessly observable only to te agent tat receives it. Eac agent reports is income to te government (IRS). Te agent may report an income of 1 or zero and te report may be different from is true income. Te government uses te reports to redistributes income. Te (after tax) income in period t is t for an agent tat reports 1 and t for an agent tat reports zero, were 0 t t 1. Te government spends v t units on verification of income and we assume tat tis amount is sufficiently ig so tat all agents coose to report teir true endowment. Te determination of te government s expenditures on endowment verification will be discussed sortly. I start wit te case of private intermediation. Tere are banks tat, at t 1, take deposits and make loans. Te gross interest rates are R l on deposits and R b on loans. Te bank can verify income by obtaining a costly court order tat gives access to te individual s IRS records. Te bank offers te following loan contract. Te borrower gets b units at t 1. At t 2, te borrower may coose to declare bankruptcy. If e does not declare bankruptcy e pays to te bank br b 2 units. If e declares bankruptcy te bank get a court order and verifies is reported income. If te bank finds out tat te reported endowment is zero, te borrower pays br b 2 units were 0 1 is te fraction of te debt paid by te truly bankrupt borrower. If te bank finds out tat te reported endowment is one, te borrower pays te full debt of br b units. Since only borrowers wo realize low

11 10 income will benefit from declaring bankruptcy, I assume tat only low income borrowers declare bankruptcy. Tere are costs to te bank for verifying income (getting a court order) and terefore te bank does not recover te full amount paid by te bankrupt borrower. It is assumed tat te bank recovers only br b units were 0. Te difference ( )br b is te bankruptcy costs paid by te bank. Te bank makes zero profits and terefore: (1) R b (1 ) R l Note tat since 1, R b R l. Tus, te bank carges a iger borrowing rate to cover te losses from bad loans. I now turn to te trut telling constraints tat determine te amount spent by te government on income verification. I assume tat te agent can report to te IRS eiter 1 or 0. Avoiding detection wen under-reporting (i.e., reporting 0 wen true income is 1) is possible but costly. Te cost of avoiding detection is given by te strictly increasing and continuous function e(v t ) were v t is te amount spent by te government on income verification. Furtermore, I assume tat te fines wen getting caugt are ig and terefore agents eiter report trutfully or underreport and perfectly conceal income. 5 In te first period only lenders ave a reason to conceal teir income. A lender wo conceals gets 1 1 e(v 1 ) units. If e reports trutfully e gets 1 units. He will report is endowment trutfully if 1 1 e(v 1 ) 1 or: (2) e(v 1 ) Te tax enforcement literature seems to focus on te optimal way of spending te IRS resources on enforcement. For example, Scotcmer (1987) focus on te audit decision and Reinganum and Wilde (1988) focus on te enforcement decision wen an audit as already been conducted. I did not find papers tat focus on te ways tat tax payers spend resources on avoiding auditing and minimizing te enforcement probability wen an audit takes place. In te real world tis can be done by iring lawyers and accountants and by spending income in a way tat cannot be easily observed.

12 11 In te second period borrowers wo get an endowment of 1 ave te strongest incentive to conceal and terefore I focus on teir cost-benefit calculation. If te ig income borrower cooses to conceal, e will ave 1 2 br b e(v 2 ) units. If e reports trutfully e will ave 2 br b. He will report trutfully if: 1 2 b R b e(v 2 ) 2 br b or: (3) e(v 2 ) (1 )br b I assume tat te government spends te minimum amount of resources tat will make agents report trutfully and terefore (2) and (3) old wit equality. Under tis assumption we can solve for te verification costs: (4) v 1 e 1 (1 1 1 ) f (1 1 1 ), v 2 f (1 )br b, were te inverse function f is strictly increasing. Tis ties te amount spent by te government to te incentives to conceal income. Note tat te government needs to spend more resources on verification in te second period if te debt payment br b is larger, because te bank uses te information collected by te IRS in te case of bankruptcy. Tis is te external effect of te bank on te IRS mentioned earlier. I will not attempt a full description of te economy ere and treat (, ) as exogenous variables. Our model is similar to Williamson (1987) wo sows tat te optimal contract offered by te bank is a special case of wat I assume ere (wit 2 br b ). Williamson sows tat in is model tere are two types of equilibrium: Wit credit rationing and witout credit rationing. For te point tat I want to make it does not matter weter te loan market clears or not and I will allow for bot cases. I assume tat lenders and borrowers ave te same expected utility function u(c 1 ) u(c 2 ) were C t is consumption at time t, u is monotone and strictly concave and 0 1 is a discount factor. I use b s and b d to denote te supply and demand for bonds and define equilibrium as a vector ( ˆ 1, ˆ 2, ˆ 1, ˆ 2, v ˆ 1, v ˆ 2, R ˆ l, R ˆ b, b ˆ d, b ˆ s, b ˆ ) suc tat (1) and (4) are satisfied and (a) b ˆ s solves te lender s problem: max b s u( ˆ 1 b s ) (1 )u ˆ 2 b s R ˆ l u ˆ 2 b s R ˆ l ;

13 12 (b) b ˆ d solves te borrowers problem: max b d u( ˆ 1 b d ) (1 )u ˆ 2 b d R ˆ b u ˆ 2 b d R ˆ b ; (c) Te equilibrium amount of loans is: b ˆ min{ˆ b s, b ˆ d }; (d) Markets for goods are cleared: ˆ 1 ˆ 1 1 ˆ v 1, ˆ 2 ˆ 2 1 ˆ v 2. Government loans program: I start from an equilibrium interest rates ( R ˆ l, R ˆ b ) and assume tat te government creates a bank tat keeps te lending rate at te initial level R ˆ l but reduces te borrowing rate to R b R ˆ b. I assume tat te demand for bonds at te lower rate R b is: b d b ˆ d. I also assume tat te government offers te same size loan of b ˆ units. (Tis is likely to result in additional rationing because min{ˆ b s, b ˆ d } min{ˆ b s,b d }). In te second period, te government collects te full payment from borrowers wo report an endowment of one and a fraction of te loan from borrowers wo report an endowment of zero. I assume tat te loan program is self-financing: (5) R b (1 ) R ˆ l Comparing (1) to (5) and using, leads to: R b R ˆ b as we ave assumed. Note tat te government program completely crowds out private lending because private banks cannot make money at te lower borrowing rate. Since te borrowing rate is now lower te incentive to underreport is lower and terefore te amount spent on endowment verification is lower. Formally, since R b R ˆ b we get: (6) v 2 f 1 ˆ 2 ˆ 2 (1 )br b v ˆ f 1 ˆ 2 2 ˆ 2 (1 )b ˆ To see tat a Pareto improvement is possible note tat if te government does not cange te amount of resources spent on income verification, it will improve te welfare of borrowers wo now pay less interest for te same amount of loan. Tis is possible because of te elimination of bankruptcy costs ([ ]ˆ b ˆ R b ). Lowering te income R b

14 13 verification effort to v 2 will free additional resources and will allow for furter improvement in welfare. In tis example, te government can improve matters because it as better access to its own IRS records. I now turn to intermediation between generations tat may be possible only in te presence of a government loan program. 4. INTERMEDIATION BETWEEN PEOPLE OF DIFFERENT AGES I consider an overlapping generations economy in wic te government may ave an advantage even if income can be costlessly observed. Tis advantage will arise in a steady state in wic young agents want to borrow from old agents. Tis is not te case in te standard overlapping generations (OG) model. In te standard model, agents live for two periods and get income only in te first. In tis setting trade between generations is not possible in te absence of government intervention, but te required intervention is minimal. Samuelson (1958) as sown tat money can solve te problem: Wen every young generation wants to save and is willing to accept money, te old may use te money tey saved in te first period of teir life to buy goods from te young. Te money solution will not work wen agents get income only in te second period of teir life, because te young ave noting to give to te old. More generally, it will not work in any (realistic) setting in wic agents live for a finite number of periods and want to borrow at te beginning of te life cycle. Wat will work in tese cases is a government loan program (or negative money ). To illustrate, I assume a single non-storable good OG economy in wic agents get an endowment of 1 unit in eac of te two periods of teir life. As in te previous section, te utility function of te representative agent is: u(c 1 ) u(c 2 ), were C t is consumption at age t, 0 is a discount factor and u is a single period utility function wit te standard properties (u' 0,u'' 0).

15 14 I start from te problem of a planner wo wants to maximize welfare in te steady state. Since at eac period tere is 1 unit per agent and tere are two representative agents (young and old) te resource constraint is: (9) C 1 C 2 2 were C 1 is te (steady state) consumption of te young and C 2 is te steady state consumption of te old. Te planner s problem is: (10) max C1,C 2 u(c 1 ) u(c 2 ) s.t. C 1 C 2 2 Te first order condition tat an interior solution to tis problem must satisfy is: (11) u'(c 1 ) u'(c 2 ). Figure 1 illustrates te planner s solution. Te endowment (autarky point) is at point A wile consumption is at point B. Te intuition is as follows. From te planner s point of view te sadow price of consumption to te young in terms of consumption to te old is 1: If e wants to give an additional unit to te young e must take it from te old. In te steady state tis is also te sadow price of consumption in te first period of te representative agent s life in terms of consumption in te second period of is life. Decentralization: Te planner can implement te allocation by making an offer to lend and borrow at a zero interest rate. Te debt of te old agents to te government is C 1 1, were ( C 1, C 2 ) is te solution to te planner s problem. In te steady state te government collects tis debt and transfer it to te young as a loan. Te debt of te old generation is positive if C 1 1 as in Figure 1B and is negative oterwise, as in Figure 1A. Using money: Money can play a role if te young want to save wen te interest rate is zero. To see tis point, assume tat te government promises to buy and sell any amount of te good for 1 dollar per unit. In tis case, a young agent may sell 1 C 1 0 units for M 1 C 1 0 dollars. Consumption in te next period is equal to te amount of money accumulated in te first period plus te second period endowment: C 2 M 1.

16 15 Substituting M 1 C 1 in te last equation leads to: C 1 C 2 2. Te consumer can terefore coose any consumption pair (C 1,C 2 ) tat satisfies: (12) C 1 C 2 2 and 1 C 1 0. Wen te non-negativity constraint is not binding, as Figure 1A, te consumer will coose te planner s solution ( C 1, C 2 ). Money works in tis case. But wen te consumer wants to coose a point to te rigt of te endowment point, as in Figure 1B, te non-negativity constraint is binding and introducing money will not elp. In tis case a Government lending program is required. Negative money: We may tink of government loans as negative money. From a matematical point of view, tere is little difference between te positive money oldings tat occur wen C 1 1 and te negative money oldings tat occur wen C 1 1. But te involvement of te government is different. In te first case we need te government only for te initial step of introducing money: In te steady state, te old will simply give te dollar bills (wic says tat te government owes te owner of tis bill, 1 unit of consumption ) directly to te young in excange for goods. To make negative money work we need te government involvement in all periods: Te government must collect te debt from te old and transfer it to te young as a loan (because te young will not directly accept pieces of paper tat say Te owner of tis paper owe te government 1 unit of consumption ) in excange for goods. We may conclude tat if te society wants to maximize welfare in te steady state it must ave a government loan program. Te case for suc a program is muc weaker if we consider te transition to te steady state. Wen te economy starts at t 0 at te autarkic point A, te only way of getting to point B in Figure 1B is to tax te generation born at t 1. A Pareto improvement is not possible unless te government as an advantage in collecting income contingent payments and people wo belong to te same generation are different, as in te previous section.

17 16 C 2 2 B 1 A 1 2 C 1 A: Consumption is to te left of te endowment point C A B 1 2 C 1 B: Consumption is to te rigt of te endowment point Figure 1

18 17 5. THE MODEL I now attempt to integrate and expand on te special cases tat we ave discussed. Te previous section made an analogy between money and te government loans program and argued tat a loan program may be necessary for trade between generations. Here I intend to furter explore te monetary connection asking weter te interest on government loans sould be iger tan te rate on government bonds and te rate of return on olding money. A related question is weter te government loan program sould be self-financing. (As is apparent from equation (5), te self-financing requirement may lead to a difference between te lending and te borrowing rate). As in te discussion of te Hall-Rabuska proposal, I model te loan program as part of te tax system and similar to section 3, I focus on income verification issues. Similar to Sargent and Wallace (1982), I use legal restrictions to distinguis between government bonds and money and as in teir seminal paper, I find tat efficiency is acieved wen all interest rates are te same. But unlike teir paper, ere I assume tat te government must use distortive taxes to raise revenue. Te results ere are also in line wit te literature on te Friedman rule. But unlike tis literature, ere I focus on enforcement issues rater tan on labor market distortions. I consider a single good overlapping generations model. Tere are N types of agents. One agent from eac type is born every period and lives for T periods (1 T ). I focus on a steady state in wic age matters but te calendar date of birt does not. I will terefore talk about agent (of type) witout specifying is date of birt. Types ave different income and may ave different utility function. Type agent receives at age t an endowment (income) of Y t units of te good. His (strictly quasi concave and differential) utility function is: U (c 1,...,c T ), were c t is is age t consumption. Tere are two government-backed assets: indexed money (real balances) and indexed bonds. A unit of indexed money represents an obligation of te government to

19 18 excange it for a unit of consumption. In equilibrium te government will not excange money for goods. Our money is terefore like fiat money and te promise to excange it for goods is a way of committing to a real rate of return on money (or te inflation rate). balances. An indexed bond represents an obligation to excange it for a unit of real Tere is a government owned bank tat provides loans and accepts deposits. (Tere are no private banks.) Te government bank offers two types of accounts: Cecking and savings. Agents keep indexed money in teir cecking account and indexed bonds in teir savings accounts. Te government makes te distinction between bonds and money by proibiting te writing of cecks on savings accounts and requiring a trip to te bank to cange te evolution of balances in tese accounts. Tis is similar to regulation Q tat proibits paying interest on many cecking accounts. Te (real) interest rate on loans is r. Te interest rate on saving accounts is r b and te interest rate on cecking accounts is r m, were r m r b r. Te government can coose te real interest rates: r,r b,r m. I distinguis between reported and actual income: Y t is actual income and Y t t is reported income at age t, were t is te misreporting magnitude. Under-reporting occurs wen t 0. Tere are also reasons for over-reporting income (coosing t 0) in our model. I use y t (Y 1 1,...,Y t t ) to denote reported income up to age t. At age t 0, te agent gets a loan of L 0 units tat is directly deposited to is cecking account. Te agent may coose to take additional loans during is life. At age t 1 e can get a loan of L t units subject to te constraint: (13) 0 L t LL t (d t 1,y t ) for 1 t T and L T 0, were te loan limits LL t depend on te debt to te government at age t 1 (end of period debt = d t 1 ) and on reported income istory.

20 19 A trip to te bank is required wenever te agent does not elect te default option. In te case of loans, te default option is L t LL t : If te agent does not go to te bank, te amount L t LL t will be automatically deposited in is cecking account. Tere is also a default option in te contract tat governs te savings account: If te agent does not go to te bank te amount in is savings account will be b t (1 r b )b t 1, were b t is te end of period amount. Te agent may coose b t (1 r b )b t 1 if e goes to te bank. 6 As in te Baumol-Tobin model, tere is a fixed cost t, for a trip to te bank. At age t te agent pays te cost: (14) tc t (L t,ll t,b t,b t 1 ) t if L t LL t or b t (1 r b )b t 1 and tc t (L t,l t,b t,b t 1 ) 0 oterwise. Te loan payments are contingent on te accumulated debt and reported income istory. A t years old consumer wo owes te government (1 r)d t 1 beginning of te period) and reported y t will pay: (15) LP t (d t 1,y t ) (1 r)d t 1. units (at te Te loan payments (LP) are an automatic deduction from te individual's cecking account. In wat follows I assume a special case of (15): (16) LP t (d,y ) min rd d t 1 t 1 t t 1 T t 1, (Y t t ). Under (16), te agent pays te interest and a fraction of te principle if tis payment is less tan a fraction 0 1 of is (reported) current income. Oterwise te agent pays a fraction of is current income. As was said in te introduction, te ome affordable modification program cose Te debt to te government evolves according to: (17) d 0 L 0 ; d t (1 r)d t 1 LP t (d t 1,y t ) L t 6 We may view te savings account contract as an incomplete long-term contract tat allows for renegotiation as in Hart and Moore (1988). Balances accumulate interest automatically until a trip to te bank is made and te renegotiation option is exercised.

21 20 Te government imposes income and consumption taxes. Te consumption tax rate is 0. Te income tax payment (TP) depends on reported istory and is given by te weakly increasing function: (18) TP t (y t ). Te government keeps tab on te accumulated value of te tax payments: (19) J 0 0 ; J t (1 r)j t 1 TP t (y t ) J Note tat under (19) te government does not collect taxes from people wose accumulated income tax payments ave reaced a ceiling J. Following Alt (1983) I abstract from te cost of collecting consumption taxes and focus on te cost of collecting income contingent (taxes and loan) payments from individuals, due to te cost of auditing income, reporting income and underreporting income discussed in Maysar (1991) and Slemrod and Yitzaki (2002). As before, I assume tat te government spends v per period for auditing income but ere I allow for equilibrium in wic misreporting occurs. Te amount tat te consumer as to spend to avoid detection depends on te magnitude of misreporting and on te government verification efforts and is given by te function e t ( t,v). Avoiding detection wen under-reporting is costly: e t 0 for all t 0. But over-reporting is not costly: t e t ( t,v) 0 for all t 0. Te penalty wen ceating and getting caugt is proibitive so tat all agents coose to spend te amount required to avoid detection. In some cases we may view underreporting as analogous to working less tan wat te agent will coose to work in te absence of taxes. But tis analogy is not complete because ere te incentives to underreport do not increase wit te consumption tax rate. Tere is also a fixed cost, z t 0, tat occurs even if income is reported trutfully, because of te need to complete te required forms and to keep records. Agents may avoid tis cost by not filing te tax forms. Wen agents do not report income tey are treated as if tey reported te igest income in teir age group: Y t max {Y t }. In tis

22 21 case te misreporting magnitude is: t Y t Y t 0. Te amount spent on reporting and underreporting income is terefore: (20) X t (d t 1,J t 1, t,v) 0 if Y t t Y t ; X t (d t 1,J t 1, t,v) e t ( t,v) z t oterwise. I now turn to describe te evolution of assets. At t 0, agent as: (21) m 0 d 0 ; b 0 0 Te total value of assets evolves according to: (22) b t m t Y t (1 r m )m t 1 (1 r b )b t 1 L t LP t (d t 1,y t ) TP t (y t ) (1 )c t tc t X t Te consumer's problem is: (23) max U (c 1,...,c T ) s.t. (13)-(22). L t,c t,m t,b t, t 0 Note tat a consumer tat wants to consume along a pat tat is different from is net income pat may coose to old money to economize on trips to te bank. I use g to denote government spending on items oter tan income verification and G g v to denote total government spending. Te market clearing condition is: N T N T (24) G c t Y t tc t X t, 1 t 1 1 t 1 Te government budget constraint is: N T N T (25) G L t TP t (y t ) LP t (d t 1,y t ) c t 1 t 0 1 t 1 Note tat we allow for deficit in te loan program: Te total amount of loans granted, N T N T L t, may be different from te total amount of loan payments, LP t. 1 t 0 1 t 1 In equilibrium, consumers solve (23) and (24) - (25) are satisfied. To define equilibrium in detail, note tat government policy is a coice of interest rates, (r,r b,r m ), loans limits ( LL t ), a ceiling on lifetime income tax payments ( J ), a verification effort (v ), and payment functions (LP t,tp t ). I denote te government policy by

23 22 (r,r b,r m,ll 0,...,LL T 1,J,v,LP 1,...,LP T,TP 1,...,TP T ). Consumers coose (L t,c t,m t,b t, t ;t 1,...,T, 1,...,N). Equilibrium is tus a vector (, ) suc tat (a) given, (L t,c t,m t,b t, t ) solves (23) for all, (b) te market clearing condition (24) and te government budget constraint (25) are satisfied. I now focus on equilibrium in wic tere is enoug liquidity and all interest rates are equal to zero. Te equality between te interest rates is te Friedman rule. Te level of zero is te optimal interest rate in Samuelson (1958), Diamond (1965) and Cass and Yaari (1966) tat is obtained wen restricting attention to te steady state. Diamond (1965) sows tat te social planner will set te steady state interest rate at a level equal to te rate of population growt (n) even wen te marginal product of capital is different from n. Here we ave te special case: n 0. I evaluate te allocation of consumption by te following planner's problem: N (26) V (x) max U (c 1,...,c T ) c t s.t. c t x, 1 N T 1 t 1 were 0 are te weigts tat te planner assigns to type and x is te amount tat te planner can allocate to consumption. Te first order conditions for an interior solution to (26) require: (27) U t U 1, for all t and, were U t U (c 1,...,c T ). 1 c t Condition (27) insures tat a planner cannot increase welfare if e takes a unit from age 1 type agent and transfers it to age t type agent. Similarly, a type agent wo can lend and borrow at a zero interest rate will coose a consumption pat suc tat (27) will be satisfied and it is not possible to increase utility by reducing consumption in age 1 and saving it until age t. It follows tat under (27) it is not possible to increase te welfare of an individual agent by canging te allocation of consumption over is lifetime witout canging te present value of is consumption. Tis is te condition for intertemporal efficiency.

24 23 I now turn to caracterize a Friedman-Samuelson equilibrium defined as follows. A Friedman-Samuelson (FS) equilibrium is an equilibrium in wic te loan limits LL t are large (and not binding) and r m r b r 0. Proposition 1: A FS equilibrium as te following properties: (a) Te consumer cannot do better tan elect L t LL t and b t 0; (b) We can write te consumer s budget constraint in te following present value form: T T (28) (1 ) c t d T J T (Y t X t ); t 1 t 1 (c) We can write (25) as: N N T (29) G (J T d T ) c t 1 1 t 1 (d) Eiter (24) or (25) is redundant {(26) (24) (25) and (28) (25) (24)}; (e) A cange in te payment functions (LP t,tp t ) tat does not cange te total payments J T d T, is neutral. (f) Te condition for intertemporal efficiency (27) is satisfied and tere are weigts N T 0 suc tat welfare in equilibrium is: V G (Y t X t ). 1 t 1 Te proof is in te Appendix. Part (a) says tat te agents can do no better tan coosing te default option and using te cecking account only. Tis is because tere is no reason to go to te bank wen r m r b r. Te following two parts says tat once we adopt te Friedman rule, te economy looks like a standard non-monetary economy wit a perfect market for borrowing and lending: Part (b) says tat wen tere is enoug liquidity, te consumer can coose out of a single budget constraint tat limits te present value of expenditures and part (c) says tat from te government budget point of view, only te total net payment made by eac type ( J T d T ) is relevant. Part (d) says tat eiter te market clearing condition or te government budget constraint is redundant. Part (e) combines te results in (28) and (29) to sow equivalence of alternative policies

25 24 wit te same net payments. Part (f) says tat te allocation of resources devoted to consumption is efficient. I now turn to sow some additional claims about te equilibria in Proposition 1 tat igligt te role of te option not to report income and te ceiling on lifetime taxes. Proposition 2: A FS equilibrium as te following additional properties: (a) An individual wo pays is loan in full and reaces te maximum lifetime tax payments ( d T 0 and J T J) will not report is income (coose te option of not filing and aving is income recorded as Y t max {Y t }); (b) A Pareto improvement is possible if no one reaces te ceiling J and te type wit te igest lifetime tax payments, * argmax J T, under-reports income and pays is loan in full ( d * T 0). Part (a) says tat an individual wo reaces te ceiling and pays is loan in full will not report income. To see tis claim note tat an individual wo invests resources in underreporting or even true reporting and reaces te tax ceiling can increase is consumption by coosing te no reporting option and taking a loan tat is large enoug to finance te ig tax payments wen young. Part (b) is a direct consequence of (a). It says tat a Pareto improvement is possible if * cooses to under-report income. To sow tis claim, note tat wen te igest paying type cooses to under-report income, it must be te case tat no one reaces te ceiling J J * T. Te government can improve matters by lowering te ceiling and setting: J J * T. In tis case * will coose not to report is income and increase te present value of is consumption by te amount e spent on avoiding detection and filing reports. Te increase in consumption of type * agents implies an increase in

26 25 consumption tax revenues tat permits a decrease in te consumption tax rate and increase in te welfare of all types. 7 8 Individuals wo reac te ceiling face a zero marginal tax rate. Proposition 2 is terefore related to te result in Mirrlees (1971) and Stiglitz (1981). Tey considered a single period economy and sow tat in te normal case efficiency requires zero marginal tax rate on te ig ability individuals. Teir model focus on te labor supply coice wile ere I focus on collection costs. 6. DEVIATIONS FROM THE FRIEDMAN RULE Te FS equilibrium requires tat all interest rates are te same and are equal to zero. We may distinguis between two rules : (a) Te Samuelson rule tat requires an interest rate equal to te rate of population growt (zero in our case) and (b) Te Friedman rule tat requires zero nominal interest rate or more generally equality among te various interest rates (r r b r m ). Te Samuelson rule was discussed in section 4. Here I discuss te Friedman rule asking weter r m r r b 0 can improve matters. We sowed tat te allocation in te FS equilibrium is intertemporally efficient. It may still be possible to increase welfare by taxing real balances provided tat te tax on real balances reduces administrative costs witout aving an adverse effect on te distribution of income. I now argue tat tis is not likely to be te case. I follow da Costa and Werning (2008) wo use te mecanism design approac to argue tat inflation tax makes sense only if it aids te redistribution of income from te 7 Oter ig-income types may also coose not to report teir income and pay more (income and consumption) taxes. Tis reinforces te argument. 8 Having a ceiling on lifetime tax payments may lead to a large improvement in some countries. In Israel for example, residents must pay tax on teir worldwide income wile nonresidents pay only on income generated in Israel. As a result some people wo want to live in Israel are very careful not to spend more tat 180 days per year and satisfy some oter non residency requirements. A ceiling on lifetime income may solve te problem for tese people wo will be able to coose residency witout tax considerations.

27 26 more to te less productive. Tey study a direct mecanism in wic eac individual reports is type and te tax e pays (or te transfer e gets) is determined by is report. Formally, tey study te problem of a planner wo faces incentive compatibility constraints in addition to te resource constraint. Te incentive compatibility constraints insure tat individuals will not ave an incentive to misreport teir type. In tis context, taxing real balances can elp if it makes misreporting more difficult. Tat is, if te more productive individual wo pretends to be less productive and reports a lower income will want to old more money tan te individual e pretends to be. But in teir model te opposite is true. da Costa and Werning terefore concludes tat zero nominal interest rate is optimal. Tis result olds in a variety of models. A partial list includes, Kimbroug (1986), Correia and Teles (1996) and Cari, Cristiano and Keoe (1996). I now turn to apply tis line of reasoning. I sow in te Appendix te following Claim. Claim 1: Wen te average (per period) demand for real balances is proportional to average consumption, a tax on real balances is not optimal: It is possible to make a Pareto improvement by replacing te tax on real balances wit a tax on consumption. Te intuition is as follows. Wen real balances are proportional to consumption a tax on real balances can be viewed as a tax on consumption tat as te undesirable effect of inducing trips to te bank. Replacing te tax on real balances wit an additional tax on consumption will terefore free resources and allow eac individual to increase is consumption by te amount e spends on trips to te bank. In our model, income tax is used to cange te distribution of income from te distribution tat will prevail if only consumption tax is used. A tax on real balances can aid in te distribution of income if it is progressive. Tis is not te case in a Baumol- Tobin type model in wic te elasticity of te demand for money wit respect to

28 27 consumption is less tan unity. Since a tax on money is regressive, we may acieve a Pareto improvement by eliminating it, increasing te revenues from consumption tax and reducing te revenues from income tax THE OPTIMAL VERIFICATION EFFORT: AN EXAMPLE I now consider an example tat illustrates te determination of te verification effort v. I assume te special case: N 1, Y t 1, T 10. I assume tat tere are no costs of reporting true income ( z t 0) and terefore te representative agent will never over report is income. Tere is a standard proportional income tax. Te income tax rate is 0 1 and te tax payments at age t are: (1 t ). Te cost of underreporting for te representative consumer is: e v 2. Te agent cooses to minimize te sum of underreporting efforts and tax payments. He tus solves: min e v 2 (1 ). Te first order condition for tis problem requires: (30) 2v Te government takes (30) as given and seeks to minimize te amount of resources spent on tax collection subject to its budget constraint. It tus solves: min v, e v v 2 v s.t. (1 )T g v and (30). 9 Te assumption tat te cross sectional elasticity of te average olding of real balances wit respect to consumption is less tan unity is tus critical for our argument. To build some intuition, it may be useful to consider two individuals wo face te same cost of going to te bank. Individual 2 consumes in eac period twice te amount tat individual 1 consumes. Individual 2 can go to te bank exactly te same number of times as individual 1 and draw twice te amount of money in eac visit. If e does tis e will old on average twice te amount of money and will pay twice te amount of inflation tax (or more accurately nominal interest rate tax ). But since te cost of going to te bank is fixed individual 2 will coose to go to te bank more often tan individual 1 and terefore is olding of money will be less tan twice te amount eld by individual 1.

29 28 Using x v and for te Lagrangian multiplier, we can write te first order conditions for te government s problem as follows. (31) 1 1 ( 1 2)x 2 T ( 1 4)x 2 (32) ( 1 2)x T(1 x) gx (33) Tx ( 1 2)x 2 T 1 From (31) and (32) we get te following quadratic equation: (34) x 2 T (4T 2)x 4T 0 Te positive solution to tis equation is: x = Note tat (30) implies tat (T) ( 1 2)x(T) does not depend on g and (33) implies tat v x g Tx ( 1 2)x 2 T 1 is proportional to g. Furtermore, e ( v 2 1 4)x 2 g Tx ( 1 2)x 2 T 1 is proportional to g. Table 1 illustrates, assuming tree levels of government spending. Te level of underreporting remains at 0.51 regardless of te tax rate. A iger tax rate (induced by iger g) leads to iger verification efforts and iger underreporting efforts. Te fraction of income spent on tax collection (v Te)/T increases wit g. Table 1: A numerical example g v e (Te v)/t Tis example illustrates te difference between te standard analysis of tax distortion and te analysis ere tat focus on administrative costs. Here iger taxes do not cange reported income and may actually lead to an increase in measured GDP.

30 29 8. A BROADER PERSPECTIVE In economies tat last forever, we may ave private Ponzi scemes tat compete wit te government money creation activities (tat can also be described as a Ponzi sceme). For example, one can establis a firm tat owns virtually noting and offer to sell it to te public. If te public believes tat te stocks of te firm will ave te same rate of return as te money offered by te government, agents will be willing to excange te stocks for money or for goods and te stocks will indeed earn te expected rate of return. (See Karaken and Wallace [1981] for a similar argument in an international setting.) Sould we allow te private sector to compete wit te government by creating money substitutes, or sould we impose regulations aimed at discouraging money substitutes? Tis is an old question tat at least in my mind, as not been resolved. But as te current crisis illustrates, it is still igly relevant. I now offer a discussion of tis important policy issue, in an attempt to put te proposal to institute a government loan program in te broader context of capital market regulations. I start wit te difference between te social and te private value of financial innovations. To illustrate tis difference, I consider an economy in wic gold is te only asset and gold as no intrinsic value. A financial innovator comes up wit te idea of a bank. He issues claims on 10 ounces of gold for eac ounce of gold tat e as in is safe. Te claim says tat te older of te claim can excange it for one ounce of gold at a certain location (bank) at any time. Of course, tere is a well known problem of bank runs tat arises because tere are more claims tan gold in te safe, but for some reason people discount te possibility of bank runs and tey treat te claims on gold as perfect substitutes for gold. Te financial innovator sells te claims for a castle wort 10 times te amount of gold e as and retires. Te idea catces on and as a result te money supply is increased by a factor of 10. Te original innovators make seigniorage type