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Probability and Statistics for Econometrics

by: Mrs. Carmelo Deckow

Probability and Statistics for Econometrics ECON 203A

Marketplace > University of California - Los Angeles > Economcs > ECON 203A > Probability and Statistics for Econometrics
Mrs. Carmelo Deckow
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Date Created: 09/04/15
Chapter 2 Fiscal Policy In the United States expenditures of federal state and local governments make up about 35 of GDP In many European countries the scale of government activities is even larger In order to nance these expenditures the government has to levy taxes on its citizens or it is forced to issue government debt In either case the sheer size of the required receipts suggests that taxation government debt and government expenditures have a large ime pact on the economy In this chapter we are going to analyze the the role of government expenditures in the economy Throughout most of the analysis we will not be concerned with the question why the government is spending as much as it does or what exactly it is doing with its money Rather we will take government expenditures as given and ana lyze the effects of different ways of nancing these expenditures on the economy We will consider the effects of different income tax schedules the relative merits of income con sumption and capital taxation and the effects of government de cits and government debt 21 Introduction and Terminology I will start with a discussion of income taxes Personal income taxes and social security taxes are the major source of income for the US federal government In the US as in other countries income tax rates differ with income The marginal tax rate is the tax that has to be paid on one additional dollar of income In contrast the average tax rate is given by total tax average tax rate total income We talk about a progressive income tax if the marginal tax rate increases with income whereas a regressive income tax has marginal rates declining with income A at tax is in between progressive and regressive taxation The marginal tax rate is constant and 19 therefore equal to the average tax rate Finally a lumpsum tax is a tax that is independent of anything that can be in uenced by the taxpayer especially his income For example a uniform tax of 100 on each American citizen would be a lump sum tax So would be a tax that is determined by a random number generator it matters that the tax cannot be in uenced by the taxpayer but it does not need to be identical for everyone 40 Marginal Tax Rate 3 30 9 5 20 Average Tax Rate I 10 O l I 0 100 200 300 Figure 21 US Federal Income Tax Taxable Income in 1000 Marginal Tax in Percent 0 24 15 24 58 28 58 121 31 121 263 36 gt263 396 Table 21 US Federal Income Tax Schedule The US federal income tax is a progressive tax Table 21 displays the marginal tax rate schedule and Figure 21 shows the marginal tax rate together with the average tax Note that because the marginal rates are increasing the average tax rate is always below the marginal rate In addition to the federal government many states also levy income taxes For example Illinois has a 3 percent at tax The federal government raises social security taxes which are used to pay for special government programs Currently 62 percent of income up to 63000 have to be paid for Social Security and 145 for Medicare The same amount has to be paid directly by the employers so that the actual tax rate is 124 and 29 respectively Corporations pay an income tax of 35 to the federal government The taxation of corpo rate income can lead to double taxation if a corporation transfers its pro ts as dividends 20 to shareholders the shareholders have to pay personal income tax on the dividend as well As a result the total tax on corporate pro ts can be as high as 60 percent Governments often do not raise enough taxes to pay for all government expenditures The budget de cit is de ned as the difference between all government expenditures and all receipts The de cit has to be nanced by borrowing Therefore the budget de cit in a given year is equal to the increase in government debt in that year Of course the government has to pay interest on outstanding debt The core de cit or primary de cit is de ned as the budget de cit minus interest payments on outstanding debt or core de cit Spending interest payments receipts The core de cit is an interesting variable because it indicates whether receipts were suf cient to pay for the actual expenditures within a year not counting interest payments which in effect are payments for spending in earlier years If receipts are bigger than exe penditures we speak of a budget surplus and if the budget de cit or surplus is just zero the government has a balanced budget 22 Data Source Percent of Total Personal Income Tax 47 Social Security Taxes 34 Corporate Income Tax 12 Excise Taxes 4 Custom Duties 1 Other 2 Table 22 US Federal Receipts by Source Table 22 breaks down the receipts of the US federal government into its main compo nents Personal income taxes and social security taxes make up more than 80 of the federal budget In contrast custom duties which in earlier years used to be the main source of revenue for the federal government provide only 1 of federal receipts Fig ure 22 shows how the importance of different income sources of the federal government changed over time Recently social security taxes were the fastestegrowing component of federal receipts State and local governments rely on a different mix of taxes Both on the state and local level sales taxes are a major source of revenue and for communities the property tax is also important Table 23 shows the relative importance of different spending programs in the US fed eral budget With 45 of the total transfers are the most important spending component 21 100 v pgf Federal Reserve Social insurance levies a 1 I u 39 gt 395 a m a 50 4 39 OI 39 73 39 Corporate pro ts 3 quot 3 39 39 392 a H 39 a a I 1quot l j E 40 1 t I 2 8 A 2 2 t 5 Individual income tax etc 7 20 x 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Figure 22 Receipts of the Federal Government Use Percent of Total Transfers 45 Defense 18 Interest 14 Grant Aid 12 Other consumption 9 Subsidies 2 Table 23 US Federal Spending 1996 Transfers are direct payments to citizens The largest components are Social Security pen sions and Medicare and Medicaid bene ts Defense comes in second closely followed by the interest payments on accumulated government debt with 14 Grant aid are pay ments to states that are intended for speci c purposes especially in the areas welfare and education Other consumption includes the cost of federal agencies and administra tions With 2 of the total subsidies make up a relatively small part of the budget The projected total receipts of the federal government in 1999 are 17427 billion dollars With a projected GDP of about 81 trillion dollars this implies that the size of the federal government is about 215 of GDP Including state and local governments total US government spending is about 35 of GDP As Figure 23 shows government spending as a fraction of GDP has been higher in the past Within World War II government purchases alone not counting transfers etc amounted to 45 of GDP The graph also shows that while government purchases as a fraction of GDP have been relatively stable after the war transfer payments have increased steadily Figure 24 shows that the US government is relatively small if compared to other indus 22 World War II 19411945 Government outlays percent of GDP I U1 l 4O GOVERNMENT PURCHASES 35 30 25 Korean War 1951 1953 TRANSFER PAYMENTS 5 39 NET INTEREST PAYMENTS v I l 01940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 Figure 23 US Government Spending Table 161 Government Spending in Seven Industrialized Countries Percentage of GDP 1990 Country Central Government All Government Italy 37600 48100 France 205 463 Canada 222 440 Germany 147 426 United Kingdom 299 381 United States 174 346 Japan 139 262 Source OECD National Accounts 19781990 All data are from 1990 except those for the United States which are from 1989 Data for the quotCentral government in the United States Japan Germany France and the United Kingdom exclude social security payments but these payments are included in the All Government cate gory Data for Central Government in Canada and Italy include some social security benefits Figure 24 Government Spending 23 Federal budget de cit 6 1 4 n D 0 0 a 2 c 0 U h 3 I 0 l 2 De cit less interest paid on debt 4 lTTilllllllllllllllllll lllllllllll 1960 I 965 I970 I975 I980 1985 1990 I995 FTGURE 20 US BUDGET DEFICITS WITH AND WTTHOUT INTEREST PAID ON THE DEBT TOAD 100A Figure 25 The Federal Budget De cit trialized countries In most Western European countries the government share is above 40 in Sweden it is close to 60 However one has to keep in mind that a large part of government spending consists of transfers especially in contries with a high government share A high government share does not imply that the government takes an important role in the production of the national income because pure redistribution also raises the government share Figure 25 shows the federal budget de cit since 1960 Two sharp rises in the de cit stand out One in the 1970s which was related to the oil price shocks and the ensuing recession and another one in the 1980s which was related to the tax cuts and spending programs by the Reagan administration Because the difference between the de cit and the core de cit is given by interest payments the gap between the two rose when in the 1980s government debt and therefore interest payments sharply increased The core de cit turned into a surplus in the early 1990s and not yet visible on the graph recently also the overall de cit turned into a surplus Figure 26 shows the debt of the US government as a fraction of GDP since the founding of the union The initially outstanding debt from the Revolutionary War was mostly paid off by 1830 The next three debt spikes were all war related The government debt reached a peak of about 100 of GDP at the end of World War II Afterwards relative to GDP debt declined until the Reagan administration 24 I I 0 World War II I 00 90 80 7o Keyean War 60 50 Great Depression Percent of GNP 40 Post Revolutionary War 30 World War 1 Vietnam War Buildup Civil War 20 39IO I I I I I I I I I 1800 1820 1840 1860 1880 1900 1920 1940 I960 I980 FIGURE 202 THE USi DEBTTOINCOME RATIO IN HISTORICAL PERSPECTIVE Source Congressional Budget Office from material cited in james R Bart9 and Stephen 0 Morrell A Primer on Budget Deficits Federal Reserve Bane of Atlanta Economic Review August 1982 DRVMcGrawHill Macroeconomic Database Figure 26 Government Debt in the Long Run 25 Net public debt as of GDP 60 40 20 r oy 20 Belgium Italy 39 Canada Netherlands I Spain 39 Germany Austria France United States L39 Denmark i Iceand Sweden Australia r 1990 quot quot and m 999 1 1orecast Norway 0HNP DECO Figure 27 Government Debt 26 Figure 27 shows total government debt as a fraction of GDP for a number of industrie alized countries Both Belgium and Italy have outstanding debt in excess of GDP The Norwegian government on the other hand is a net creditor This re ects the accumue lated income from oil in the North Sea The Norwegian oil reserves are projected to run out soon and the Norwegian government is preparing for that moment by running large surplusses as long as the oil money keeps owing 23 Policy Analysis In this section we will examine the effects of different ways of nancing government expenditures We will begin with an analysis of taxation Afterwards the analysis will be extended to account for government de cits and public debt Income Taxes With the exception of lumpesum taxes which are hardly ever used for polictical reasons all taxes are distortionary This means that apart from transferring wealth from taxpayers to the government taxes affect the economy in a way that leads to inef cient outcomes The main reason is that taxes distort price signals We know from the First Welfare Theoe rem that outcomes in a market economy without taxation are ef cient This ef ciency is reached because optimization on the part of rms and consumers causes marginal rates of substitution to be equalized across consumers and marginal rates of substitution to be equalized to marginal rates of transformation in production This end is achieved through the price system For example consumer maximization implies that the wage equals the marginal rate of substition between consumption and leisure and rm optimization ime plies that the wage equals the marginal product of labor Since there is just one wage for rms and consumers in consequence the marginal rate of substitution between con sumption and leisure equals the marginal product of labor and outcomes are ef cient Now imagine that an income tax is introduced With the income tax the marginal rate of substitution will be equal to the afteretax wage while the marginal product of labor will be equal to the wage before taxes Firms and households see quot a different wage thus the marginal rate of substitution no longer equals the marginal product of labor Therefore outcomes are inef cient While this basic distortion effect cannot be avoided the govern ment can choose which goods or actions are to be taxed at which rate Below we will analyze the effects of different forms of taxation We will start with the income tax Consider a simple example with one consumer and one rm The consumer has the utility function 11031 l 27 and the rm has the linear production function f 1quot 1 I will rst compute the equilibrium when there is no government In this case the cone sumer39s budget constraint is simply a ml and the maximization problem is mlax2W 1 which gives the rsteorder condition 01quot The rm maximizes pro ts d d mlax l 39wl which gives w 1 As long as the wage equals one any labor demand yields the same pro t zero for the rm Since w 1 we also get that l 1 and labor demand equals one as well since the labor market has to clear Now a government steps into the world and for unknown reasons intends to spend an amount g lwill compare two possibilities to nance g a lumpesum tax and a proportional income tax With the lumpesum tax t the budget constraint of the consumer is c wl t The new maximization problem is mlax ZVwZS t 1 which gives the rsteorder condition w 1 7 0 le t l or t l w i w 28 From the rm39s problem we know that the wage is still going to satisfy w 1 The government39s budget constraint is 9 t spending has to equal taxes We therefore get 1 1 g Consumption is given by cwl t1g tl thus consumption is still the same after the advent of the government The consumer simply works longer to satisfy the needs of the government In this economy the marginal rate of substitution between consumption and leisure still equals the marginal product of labor The laboreleisure choice is not distorted As an alternative the government can choose a proportional income tax 7 to nance g In this case the budget constraint of the household is c 1 wl and the maximization problem is E 1 which gives the rsteorder condition xl T39UJ10 W or Z8 1 le Since we have w 1 from the rm39s problem labor supply is given by l 1 7 Thus the higher the tax the lower the labor supply of the consumer Notice that the re sult was opposite with the lumpesum tax there the consumer worked more to satisfy the needs of the government The results are different because a proportional income tax effectively lowers the wage as seen by the consumer Since the tax drives a wedge be tween the wage paid by the rm and the afteretaX wage as received by the consumer the incentives to work are distorted The marginal rate of substitution between consumption and leisure no longer equals the marginal product of labor which leads to inef ciencies 29 We still have to determine how high the tax needs to be set to pay for g The government budget constraint is given by g T39wl Plugging in the optimal labor supply and w 1 this is g 71 7 This gives a quadratic formula for 7 with the solutions 1 l 7722 E g Thus for a given 1 there are actually two different tax rates that satisfy the government budget constraint 1 will come back to this point later To see how the different tax schemes affect the utility of consumer assume as a numerical example that g 025 Under a lumpesum tax the necessary tax is t 25 and we have 1 125 c 1 Plugging these values into the utility function we see that the utility of the agent is given by 075 Under the proportional income tax from the above formula we see that the required tax is 7 5 Using the formula for labor supply we get that labor supply is l 05 and consumption therefore a 1 025 The resulting utility is 05 Since utility numbers have only an ordinal interpretation you should interpret the absolute differences The point is that utility is lower under proportional income taxation This is a general result proportional income taxes distort the laboreleisure choice while lumpesum taxes do not Lumpesum taxation always gives higher utility to the consumer With the proportional tax for a given amount of government spending there can be two different tax rates giving the same revenue The reason is that with a proportional tax revenue might actually fall even while tax rates rise This is because consumers respond to higher taxation by working less ie they substitute leisure for labor after the price change If this substitution effect outweighs the effect of higher tax rates revenue will fall In our example tax revenues R are given by R T39wl 71 7 7 72 Figure 28 shows tax revenues as a function of the tax rate Indeed if the tax rate exceeds 05 revenues start to fall This phenomenon is also known as the Laffer Curvequot after the economist who rst pointed out the possibility that higher taxes could actually lower revenue From the point of view of a benevolent government you never want to be on the down ward sloping part of the Laffer curve In that region it is possible to lower taxes give more utility to the consumers and at the same time get higher tax revenues which is an unambiguous Pareto improvement Whether actual tax systems ever place us on the 30 03 025 02 015 Revenue 01 005 0 01 02 03 04 05 06 07 08 09 1 Tax Figure 28 The Laffer Curve downwardisloping portion of a Laffer curve is an empirical question There is some evi7 dence that total tax revenue from high income groups actually increased after the Reagan tax cuts in the 1980s which would indicate the presence of a Laffer curve In the past marginal income tax rates in the US could be as high as 91 which almost surely leads to smaller revenue among highiincome individuals In Sweden even today marginal taxes can be as high as 80 Lowering this tax rate would provide incentives for people to work more and therefore pay more taxes If Sweden lowered its top income tax rate the fall in revenue would certainly be very small because of these substitution effects and it is possible that revenue would actually increase Capital Taxation So far we have only considered a tax on labor income In the real world consumers have income both from labor and capital In this section we are going to explore the relative merits of capital taxation As an example consider a model with an in nitelyrlived consumer For simplicity I will assume that there is no labor capital is the only factor of production In this world the government has a choice between taxation of consumption expenditures and capital income The consumer maximizes a discounted sum of utility from consumption max 6 lnct 50 The government levies a proportional tax 7 on consumption expenditures and a tax 7 on income from capital Given these taxes the ow budget constraint of the consumer at 31 time t is given by 1 Twit kt111 707 th Notice that only the interest on savings but not the principal is taxed Savings are de noted as km since savings are used as capital in the production function The production function is given by 35 The rm39s problem is to maximize pro ts in each period IngaxAkt uh 39t The rsteorder condition for this problem gives us an expression for the interest rate 7 The production function is linear in capital and does not change over time Capital does not depreciate in this model thus including undepreciated capital total output available in period t is given by 1 kt Output can be used for consumption for savings or for government expenditures The resource constraint is therefore given by it km Tacit TkAkt l Here we use already n for the interest rate This can also be written as the familiar breakdown of output into consumption investment and government expenditures 5 km k5 Tact TkAkt Akt V EH W V C5 It Gt Y5 In this model the government has access to two different taxes and the question is which of the two or which mix should be used to nance government spending This question can be settled by solving for the equilibrium of the economy and then comparing the effects of the two taxes The Lagrangian for the utility maximization problem of the household is given by L Z 3 lnc At11 7k7 tlkt 1 796 kc1 50 The rsteorder condition with respect to at is given by 35 76 0 it 32 and the rsteorder condition with respect to km is t t111 70TH 0 Isolating the A5 and AM we get 3 At T 1 TJQ 2391 and 11 nn1 22 AMI Using 21 and n1 in 22 we get 35 Tltt1 3 1l 7913 T 1 1 TOA Or L 31 1 MA at The last equation determines the growth rate of consumption c 1c as a function of parameters and taxes Notice that the consumption tax canceled and does not enter the equation while the capital income tax has a negative effect on the growth rate of con sumption The reason is that the capital income tax distorts the intertemporal margin By taxing capital income investing in capital becomes less attractive compared to immediate con sumption The consequence is that the capital stock grows slower than it would have otherwise which after some time leads to permanently lower consumption Of course the growth rate of consumption is not a suf cient measure of welfare in this economy but it can be shown that it is indeed optimal to set the capital tax to zero and use only consumption taxation This result holds in a variety of models and also when capital in come taxes are compared to taxes on labor income It is particularly harmful to distort the intertemporal margin because of the negative effects on capital accumulation Even when no lumpesum taxes are available it turns out that it is better to use no capital income taxes at all and raise all revenue with taxes on consumption or labor income Budget De cits and Government Debt So far we have assumed that the government balances the budget every period ie there are no budget de cits and no government debt In the real world governments make libe eral use of debt to nance expenditures In the United States the total outstanding gov ernment debt is about 34 trillion dollars or about 45 of GDP In the media government 33 debt receives a lot of attention and is often portrayed as a major problem One concern is that government debt leaves the burden of nancing current government expenditures to future generations Another fear is that government debt crowds outquot private in vestment in the sense that private savings that are spent on government bonds are not available for private investment While some of these points appear convincing on the surface a more careful analysis casts doubts on their validity For example while it is true that future generations will have to pay for government debt it is also true that most government debt is in hands of American people Future generations will inherit govern ment bonds from their parents Paying off bonds that they own themselves is not a net burden to future generations Also while government debt absorbs some private saving government debt also means that taxes are lower than they would have been if the bud get were balanced Lower taxes imply higher savings and therefore private investment is not necessarily crowded out We can see that it is possible to argue both for and against negative effects from govern ment debt without a clear guidance which view is correct To understand government debt better we need a formal model to organize these different thoughts The approach I will take is to present a model in which government debt indeed does not matter ie whether government expenditures are nanced via taxes or via debt does have any real consequences By doing this we can spell out which assumptions are needed to argue that government debt is irrelevant By examining the necessary assumptions we can then determine to which degree debt matters in the real world How do we construct a model in which debt does not matter The main feature that we need is that consumers perfectly anticipate that current debt will have to be paid by taxes in the future This anticipation effect will induce the consumers to do just enough extra savings to accommodate the government debt David Ricardo was the rst economist to point out that debt is irrelevant if taxpayers anticipate the future taxation implied by budget de cits For that reason models in which debt does not matter are also said to exhibit Ricardian Equivalencequot We need three assumptions to build a model in which Ricardian equivalence holds First consumers need to be in nitely lived Second capital markets need to be perfect ie there is a single interest rate for borrowing and lending and there are no borrowing restrictions Third the government must be able to levy lumpesum taxes lwill present a simple version of such a model in which the government spends a constant amount 1 each period This assumption is just for simpli cation it is not important for the main result The government can use lumpesum taxes 75 to nance expenditures or it can issue government bonds ml which will have to be paid back in the subsequent period I assume that at the beginning of time there is no outstanding debt ie 0 0 For technical reasons we also need to assume that borrowing cannot be extended without limit ie there has to be some number B such that 5 g B for all t Since B can be arbitrarily large this is not a restrictive assumption For simpli cation I assume that the interest is a constant 7 in every period The budget constraint of the government at time t is then 34 given by g 1 7 75511 On the left hand side are the expenditures which consist of new spending 1 and payments plus interest on government bonds 5 that were issued in the previous period On the right hand side are receipts which stem from lumpesum taxes n and newly issued debt n1 Even if the government uses debt to nance expenditure at some point it still will have to pay off the debt Another way of stating this is that the present value of taxes has to equal the present value of expenditures no matter which exact sequence of taxes and bond issues is used To see this formally we can solve the budget constraint at time 1 for 1 to get 71 g 2 1 r Plugging this expression into the time 0 budget constraint notice that 0 0 we get T gm g70170 r 01quot 1 1 TTlb2 17 quot 70 17 17 Solving the period 2 budget constraint for 2 yields 7 I 2 2 J 3 r Using this expression we get 171 71 7 771 772 7 11 112 17 11 112 11 Going on in the same way for n periods we get 1 1 I 1 7 7 quotit 4 1 Myquot 1 My 1 HM As 11 tends to in nity the last term tends to zero and we get M8 1 00 1 1 Myquot 1 1 H c The leftehand side is the present value as of time 0 of total government expenditures in all periods while the righthand side is the present value of all taxes Thus regardless of 35 the use of government debt the present value of taxes has to be equal to a given number the present value of expenditures Let us now turn to the representative household The household has preferences de ned over in nite streams of consumption The exact form of the utility function is irrelevant for this analysis Let us assume that the household has an income of wt in period t This income can be used either for consumption at or savings 3511 In addition the household has to pay the lumpesum tax n The time t budget constraint is given by it 3H1 wt 1 798 n The household has no initial assets ie so 0 Depending on the use of government debt the sequence of taxes n differs We need to show that the decisions of the household will not be affected by the exact path of taxes as long as the present value of taxes is constant To show this we will proceed in a similar way as above The time 1 budget constraint can be solved to give 0182 wl71 81 17 Plugging this into the time 0 budget constraint gives 82 0 1 1 UJO l 1 17 7 T 17 M l701r71 As above we can now solve the time 2 budget constraint for 32 and plug in again Cone tinuing for n periods in this way gives 1 s 1 1 A 7 7 4 1 at 1 m 1 w 39L 1 7957 As for the government we have to assume that there is some upper bound for household borrowing and lending In that case the second term on the leftehand side converges to zero and we get 1 1 1 ZWQZM39wt gw 50 50 Thus the present value of consumption expenditures has to equal the present value of in come minus the present value of taxes We know from the government budget constraint that the present value of taxes equals the present value of government expenditures Use ing this the household budget constraint can be written as 00 1 00 1 00 1 50 50 36 Notice that this budget constraint is the only actual constraint that the household is face ing For any consumption sequence that satis es the equation above there is a savings sequence that satis es the periodebyeperiod budget constraints Also notice that neither taxes nor government debt enter this budget constraint The timing of taxes or the use of debt do not matter only the present value of expenditures is important Since debt does not even enter the decision problem of the only household in this economy it does not have any real consequences The optimal consumption allocation will be completely unaffected by the use of government debt The reason is that the household has perfect foresight and anticipates that any current debt will be paid off later Therefore if more debt is issued the household will raise savings by exactly the same amount in order to offset any effects on consumption How realistic are these results Do they imply that government debt does not matter in the real world We can nd the answer for this question by examining the assumptions that were needed to get the result The rst assumption is that people live forever Obi viously this assumption is not literally satis ed in the real world What really matters however is that taxpayers care about future taxes regardless how far in the future they are to be paid That it is true of people who live forever but it is also true if people have nite lifetimes but care for their children A dynasty of nitely lived people where the utility of each child enters the utility of the parent is for our purposes identical to a single person who lives forever As an example assume that the government considers lowere ing taxes today by 1000 The additional government debt will be paid off 50 years later by raising taxes at that time In response to this change an in nitely lived person would raise savings by 1000 today and earn just enough interest to pay the extra taxes 50 years from now A person who dies within the next 50 years but cares about his child will also raise savings by 1000 and leave a higher bequest so that the child can pay the extra taxes in 50 years A nitely lived consumer who does not care about children however will simply consume more The important question therefore is whether people care about their children in a suf cient amount The evidence on this is not quite conclusive but it is certainly the case that parents care and leave bequests to their children The second assumption is that capital markets are perfect Realeworld capital markets are not perfect However what matters here is that the borrowing rate for the government equals the lending rate for the households Since households are free to buy the very bonds that the government uses to nance its spending these interest rates are at least potentially the same To be sure there are many other capital market imperfections in the real world but they are not relevant for Ricardian equivalence Finally we need to assume that the government has access to lumpesum taxation Clearly this assumption is not satis ed in the real world Almost all actual taxes used by govern ments are not lumpesum and therefore distortionary This implies that government debt matters to the degree that distortions will vary over time depending on the path of gov ernment debt and taxes Given this discussion what can we say about the effects of government debt To the 37 extent that people look into the future and care about their children the effects of govern ment debt are probably small or at least much smaller as portrayed in the media High government debt does imply higher taxation in the future however and since realelife taxes are distortionary this taxation will have negative effects We saw earlier that the distortion caused by taxes increases with the level of taxation In order to minimize dise tortions it is therefore best to keep tax rates as low as possible Over time it is best to keep tax rates stable From the point of view of optimal policy it is therefore best to use government debt to distribute tax distortions evenly over time Consider the situation of a government that has a large expenditure today like a war If this expenditure had to be nanced by taxes alone tax rates and therefore distortions would have to be very high By using government debt it is possible to distribute the tax distortions over a longer period of time That is just what many governments have done in wartimes in the past Consider Figure 26 which shows US government debt as a fraction of GDP Each major war the Revolutionary War the Civil War World WarI and World War II was nanced to a considerable amount by issuing debt that was paid of in subsequent years This pole icy led to smaller tax distortions during the war relative to a policy that would nance all expenditures by taxes Summing up we see that government debt does not matter in a world with people who have perfect foresight perfect capital markets and lumpesum taxation In the real world taxes are distortionary and therefore government debt matters because in in uences the time path of taxation and therefore distortion An optimal policy would be designed such that tax rates are roughly constant over time in order to keep distortions low However the main conclusion is that debt does not matter much if compared to the effects of gov ernment spending In other words it is much more important how much the government spends as opposed to how it nances the expenditures The same is true for households If Joe Average decides to buy a car it does not make much of a difference whether he pays cash or credit If he decides to pay cash he is still free to get a loan from the bank for other expenditures and if he pays credit he is still free to save more to be prepared for future payments On the other hand it is very important which car he buys regarde less of nancing buying a Mercedes S 600 will have a much bigger impact on his future consumption opportunities than buying a Honda Civic Social Security So far we have taken government expenditures as given and concentrated on the ques tion how the government should nance expenditures In this section we will link the analysis of spending and nancing in connection with one of the most important govern ment programs the pension system Most industrialized countries are projected to run into serious dif culties in nancing public pensions within the next 30 years or so In fact it is clear that most pension systems will have to be reformed if they are to survive In this section we will be mostly concerned with comparing two different modes of nancing pensions the fullyefunded system and the payeaseyouego system 38 In a fullyefunded pension system the contributions of each cohort are invested in bonds or equity The bene ts for each cohort are drawn from the returns on investing the contri butions of that same cohort At any point in time the social security system owns a capital stock that is large enough to cover the future obligations of the system In contrast in a payeaseyouego system contributions are paid out to currently retired people immediately The system does not maintain a capital stock The bene ts of each cohort are paid by the contributions of future cohorts Consequently the solvency of a payeaseyouego pension system crucially depends on the number of people who pay contributions relative to the number of people who receive bene ts Fertility rates life expectancy immigration lae bor force participation rates and the retirement age are all important variables that affect the solvency of the system In contrast in a fullyefunded system the return on the in vested contributions is important while the system is relatively immune to demographic change To start we will analyze pension systems from an individual perspective That is we take wages and taxes as given and examine the effects on the behavior of an individual Later we will extend the model to analyze the aggregate and longerun effects of different pension systems Consider a world with overlapping generations of people who live for two periods each At each point in time there are young people who were just born and old people who were born in the previous period The young people work and receive a xed income 35 and the old people are retired There is a perfect credit market with interest rate 7 The income grows at a constant rate g so that we have 31M 1 DZu and the number Ni of people who are born in period t grows at rate n NM l 701 Consumption of generation t when young will be denoted if while consumption of the same generation when old is denoted cf In other words superscripts refer to the time of birth while subscripts denote the current period The utility function of a member of the timeet generation is given by ucf cg lnc lnc 1 For simplicity we abstract from discounting In other words the consumer values both periods of life equally As a benchmark case we will rst determine the savings of a young person at time t under the assumption that there is no social security system at all In this case the budget constraint in the rst period of life is given by if 75 9m 39 where at is savings of a member of generation t and the budget constraint in the second period is cg 1 The consumer does not work when old and there is no social security system therefore all income is from returns on own saving Plugging these constraints into the utility function yields the following maximization problem mgax lnyt 35 ln1 The rsteorder condition for this problem is i 0 35 85 85 which gives the solution 21 2 85 Thus regardless of the interest rate young workers will save half of their income Pluge ging the optimal savings back into the budget constraint gives us the optimal consump tion values 7316 577 2 at 7 1 My 51 T We are now going to contrast this outcome to the two different pension schemes Under the fullyefunded pension system contributions are invested in the credit market We will assume that there is a proportional payroll tax 7 to nance pensions Since the pension system has access to the same credit market with interest rate 7 the contribue tions of a young worker with income gt are 735 while the oldeage bene ts are 1 7 7 y5 Therefore the budget constraints are now given by 7 1 T 8h and cg 1 7 l 797 The new maximization problem is mgax ln1 7 stln1 7 st 731m 40 The rsteorder condition for this problem is 1 1 7 1 715 85 85 73 which gives the solution 8525 7315 2 Thus the young worker simply lowers his own saving by the amount of the social security contribution This makes intuitive sense In a fullyefunded system social security is iden tical to forced savings It is optimal to just offset the forced saving in order to realize the same optimal consumption values Since the worker just offsets social security savings consumption is unchanged 35 of E and 5 7 1 My 11 f Thus fullyefunded social security and no social security are identical in this model Of course this outcome depends on some simplifying assumptions that we made For exame ple in real life the time of death is unknown Social security provides insurance against the possibility of a long life because bene ts are always paid up until the end This in surance motive does not play a role in our model since everyone dies at the same time anyway Let us now move to the payeaseyouego social security system Contributions are still given by 735 Bene ts however no longer depend on interest rate 7 since the contributions are not invested lnstead every retiree gets a share of the contributions of the young generae tions The retirement bene t 5 for a person born at time t is given by total contributions A Mrym of generation t 1 divided by the number of people born at time t Nt17yt1 1V 5 Since income and population grow at constant rates we have NMNt 1 n and gm 1 gyt Using these expressions we can express the retirement bene ts as a function of the income when young 5 1 n 1 nt Thus the actual return on the social security contributions is 1 n 1 g Depending on the parameters this return may be higher or lower than the interest rate 1 7 which de termines the return in the fullyefunded system Given the retirement bene t the budget constraints of the consumer are given by 7 1 T 8h 41 cg 1 st 1 n 1 nt The new maximization problem is mfx ln1 7 35 ln1 st 1 n 1 n5 The rsteorder conditions for this problem is 1 17 7 1 7 85 1 798 1 n 1 g7yt T 0 which gives the solution 7 391 T T 2 gt 20 tyt 85 The solution is identical to the one we had before apart from the last term The last term depends on the difference between the return on social security 1 n 1 g and the return on savings 1 7 If the return on social security is greater savings decline relative to the fullyefunded system Plugging the optimal savings into the budget constraints we get the following optimal consumption values 2 2l7 t 7 75 7535 5 1 7 yt 1 n 1 g 1 7 51 T 763 Thus if we have 1n1ggt1r consumption is higher in both periods compared to the fullyefunded system All generae tions could bene t to a switch to a payeaseyouego system because growth in population and wages is so high that it is more ef cient to have future generations pay for pensions instead of saving for them Unfortunately in the real world returns on social security usually turn out to be lower than market interest rates In the United States the average annual return on social see curity payments for current high school graduates is estimated to be around 17 This compares to an average real interest rate of 25 on longeterm bonds and about 8 on investments in SampP 500 stocks Given these returns the young generations should prefer a fullyefunded system since it will give them higher lifetime consumption and utility 42 Old people on the other hand would prefer to keep the payeaseyouego system because they are the ones pro ting now However our analysis so far is not yet sophisticated enough to determine which pension system is better We worked out how an individual consumer will adjust savings in the two different systems taking incomes and interest rates as given In the real world we have to allow for the possibility that the social see curity system might affect aggregate savings and investment which in turn would have an effect on incomes and interest rates An analysis that accounts for investment is more complicated however and therefore we will have to simplify the model in other dimen sions Consider a model in which people live for two periods They work when young but they do not consume and they consume when old but do not work Every period a new young generation is born and for simplicity I assume that all generations are of equal size A consumer born at time t has the following preferences over labor 5 when young and consumption cm when old u it1 lt ZxCM lb The consumer has to pay a proportional social security tax 7 on labor income in the rst period Since there is no consumption in the rst period all afteretaX income will be saved 85 1 wtlt In what follows 5 will denote retirement bene ts per dollar earned in the rst period Dee pending on the pension system 5 will take different values Consumption in the second period is then given by CM 1 085 f t39wtlt Using the last two equations in the utility function leads to the following optimization problem mlax 2 1 n1 1 7 btlwtlt it The rsteorder condition for this problem is l1 7 t11 Tltl wt 1 0 which gives lg 1 n1 1 7 btlwt 23 and st 1 7 1 ml 1 7 M1552 24 43 There is a single rm in the economy that produces the consumption good using the production function 35 2 V ktlt The pro temaximization problem is given by Zxktlt nkt wilt The rsteorder conditions imply that factor prices equal marginal products I 07 25 wt E 26 V 15 Equations 23 to 26 characterize equilibria under any social security system I will now analyze what this implies for the longerun behavior of the economy under the two different pension system and I start with the fullyefunded pension system In the fullyefunded system both the savings of the household and the social security contributions are invested The capital stock in the next period therefore equals the sum of the two km 1 Twtlt Twilt wilt 27 Since the social security contributions are invested they are used by the rm as capital and earn return n1 The return 5 on social security is therefore given by 5 1 7 1T 28 Using 28 in 23 gives ls 1 7 t11 T 1 7 t17l wt 1 7 139w Summing up equilibrium outcomes under a fullyefunded pension system are character ized by the following four equations CM wilt 29 ls 1 0755 210 44 l n 7 211 wt 4 212 t We are interested in the longerun steady state solution to this system in which all varie ables are constants lg l kt k n 7 and wt w Using 211 and 212 in 29 and 210 reduces the system to two equations k k jz 213 and z k CWa j 214 Equation213 now gives k l and using this in 214 gives 1 2 Thus both capital and labor input are equal to 2 under the fullyefunded system and both wage and interest rate are equal to 1 Of course these numbers do not mean anything until we compare them to outcomes under an alternative arrangement Let us therefore examine the payeaseyouego system We will concentrate on a steadyestate solution In a steady state all young generations have the same income Since social security contributions are transferred directly to re tired people and all generations have the same size the bene t per retiree equals the contribution per worker Relative to labor income the bene t is given by 5 7 Therefore labor supply is given by ls 1 7 t11 T Tl wc Since the social security bene ts are not saved and invested the capital stock is made up of private savings only kt1 1 wtlb Using the last two equations together with our expressions for the wage and the interest rate yields the following system kt11 wtlu 215 45 ltl1 7 t11 T Tl wu 216 H mm m mamp We are interested in the longerun steady state solution to this system in which all vari ables are constants lg l kt k n 7 and wt w Using 217 and 218 in 215 and 216 reduces the system to two equations k 1 z 219 l 1 4M7 220 Equation 219 gives and k 1 Zz Using this in 220 yields QOH 7U ampU 1 1 1 21 7 Consequently the capital stock is given by k 1 7W 21 7 Thus both the capital stock and labor input are lower under the payeaseyouego system which translates into lower consumption and utility for all generations The payeaseyoue go system is inferior because it lowers aggregate savings which in turn leads to a lower capital stock Up to this point our results indicate that a fullyefunded system has a number of advan tages compared to a payeaseyouego system It leads to higher savings and investment which is bene cial for the economy in the long run However the generation of initial 46 retirees would suffer during a switch to a fully funded system so from a welfare perspece tive it is not clear whether one should stick with a payeaseyouego system that is already in place There are some aspects to the social security system that were not discussed so far but may be also important when evaluating different systems First of all our results hinge on the assumptions that the young and the old do not care for each other When we discussed Ricardian equivalence we found that government debt does not matter much when people are altruistic The same applies in the world of social security If people are altruistic different pension schemes still matter in as much as they lead to tax distortions but the total effects on aggregate savings would be much reduced A main disadvantage of the fullyefunded system that was not discussed so far is that it relies on stable nancial markets If events like a war or a hyperin ation wipes out all assets of the system generations that were just about to retire experience a large welfare loss Wars and hyperin ations used to be common in many industrialized countries until about 50 years ago which partially explains why we observe a payeaseyouego system in most of these countries On the other hand a payeaseyouego system is illeprepared to handle demographic change like rising life expectancy and falling birth rates Changes of this kind pose a major challenge for most industrialized nations Given the demographic developments that can be expected in the next century it will most likely be impossible to continue payeaseyouego pensions in the way they are run now The large babyrboomer generations together with high lifeeexpectancy will lead to a large number of retirees while low birth rates imply a small number of working people who will have to pay for the pensions In the short run a transition to a fullyrfunded system will be infeasible due to the high cost for currently retired people The likely solution will be a transition to a mixed system with increasing reliance on private savings together with a reduction of bene ts in the payeaseyouego system 24 Outlook Our analysis of Fiscal Policy led to some strong and some weak conclusions We have good reason to believe that high marginal tax rates hurt consumers and should therefore be avoided wherever possible Taxation of capital gains inhibits capital accumulation which leads to larger welfare losses than other forms of taxation It is therefore advisable to keep interest rates on capital gains low Our results concerning the effects of govern ment debt were more ambiguous Under a certain set of assumptions government debt does not matter at all However the necessary assumptions are not satis ed in the real world As a rule of thump a debt policy that minimizes tax distortions over time seems advisable Finally we found that a fullyrfunded pension system is superior to a payeasr youego system as long as nancial markets can be expected to be stable in the future A transition to a fully funded system is costly for retired people however and therefore a mix of the two systems might be an appropriate compromise 47


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