Savings, Wealth and Ricardian Equivalence

Transcription

1 Savngs, Wealth and Rcardan Equvalence I. Introducton In the prevous chapter e studed the decson of households to supply hours to the labor market. Ths decson as a statc decson, beng done thn the same perod. We no turn to another mportant decson of households, but one n hch the decson s dynamc or ntertemporal as t nvolves a comparson across tme. Ths s the household s savng decson. Ho much one saves today affects ho much one consumes today and n the future. Ths chapter conssts of three parts. The frst s the mcreoconomc analyss of the household consumpton/savngs behavor. As e shall see, the optmzaton problem and condtons are very smlar to those developed n the prevous chapter on the household s labor/lesure decson. In studyng ths problem, e defne the noton of present value, and ealth. The second part of the chapter analyzes the effect of a certan types of taxes on the household s savngs decson. The man result s that tax changes of a lump sum varety have no real mpacts on the economy. Ths s hat s knon as Rcardan Equvalence. Recall that a lump-sum tax s fxed amount that s not affected by anythng the household does.

2 The analyss of ths tax change brngs the government and ts budget constrant nto play. Ths e study n thrd part of the chapter. In studyng the government s fnancng needs through tme, e ll arrve at an ntertemporal budget constrant. Ths has mportant mplcatons partcularly for the recent events n some European Monetary Unon countres such as Greece, Italy, Span, Ireland, Portugal, the so called PIIGS. The plght of the PIIGS s the subject of the thrd secton of ths chapter. II. Mcroeconomc Foundatons Consumer Theory We return to our study of mcroeconomcs and the theory of the consumer. Currently, e are nterested n the choce of savngs, namely, ho much to eat today versus ho much to eat tomorro. For ths purpose, e shall begn th an assumpton that our consumer lves for only to perods, Perods and, and that he/she derves happness or utlty from consumpton n Perod and consumpton n Perod. For no, e ll assume that the household does not derve any utlty from lesure. We ll relax ths assumpton n the next chapter. Gven these assumptons, the household utlty functon s denoted by U c, c ) here c s the quantty of the good consumed n the frst perod and c n the ( quantty of the good consumed n the second perod. Indfference Curves Gven that the household s utlty s defned over to goods, e can stll represent t va ndfference curves and sho the optmal savng choce graphcally. Fgure shos the

3 ndfference curves. On the horzontal axs s Perod consumpton, c, and on the vertcal axs s Perod consumpton, c. Fgure : Indfference Curves c U U 0 U c As e sa n our study of the household labor decson, each ndfference curve descrbes all the combnatons of consumpton n perod and consumpton n perod that gves the household a certan level of utlty. The consumer, hence, s ndfferent beteen any of the consumpton bundles along a gven curve. Agan, as long as more consumpton of each good s desrable, the ndfference curve must be donard slopng. Addtonally, as long as people lke varety, the ndfference curve, s convex, namely, boed n toard the orgn. Budget Constrants 3

4 Because the consumer lves to perods, he has to budget constrants. Let us assume that the consumer does not begn hs lfe th any nherted debt, and let us also assume that he must pay all hs debt back before he departs the orld. Thus, n the frst perod, the consumer s able to save or borro, and n the second perod, he must pay back any loans plus nterest, or n the case that he saved, be pad back the prncpal and nterest. To gan some ntuton, let us thnk of the consumpton good n each perod as corn, measured n lters. Savngs and borrong are thus done n terms of lters of corn. In the frst perod say you lend 0 lters. In the second perod, you are repad the 0 lters plus nterest on those 0 lters, also pad to you n lters. Because the nterest payment s made n terms of corn, t s a real rate, not a nomnal rate. To dstngush beteen borrong and lendng, e use s > 0 to ndcate savngs, and s < 0 to ndcate that an ndvdual s borrong. We are no ready to rte don the household s budget constrants n Perod and Perod. Perod : Assume that the household has some labor ncome n the perod measured n lters of corn. Because he does not value lesure (.e., by assumpton, lesure does not enter the household s utlty functon), he ll supply hs entre tme endoment to the labor market. In the prevous chapter e thought of a perod representng a eek. Here e are veng a perod of beng far longer. To make thngs smple as possble, let us assume that the household s endoed th one unt of tme n each perod. Let be the age rate of labor n Perod measured n lters of corn. Thus, the consumer s labor ncome n the frst perod s just. 4

5 The consumer can do to thngs th ths ncome: he can save (s) and consume corn, (c ). (If he eats more corn than hs ncome, then hs savngs s negatve and s borrong.) The Perod budget constrant s thus ( st Per BC) c s. Perod : In Perod, the household agan supples labor to the economy and receves labor ncome. The age rate n Perod, s. Ths s the lters of corn the household receves n the second perod for orkng an hour n that perod. The household also has some savngs and nterest on the savngs that he can use for consumpton n the second perod. We shall denote the net nterest rate by the letter,. Ths s n real terms, so t s the number of lters a lender must pay you for each lter he borrosu. Thus, f you lent s= n the frst perod, you ould have (+) lters of corn n your savngs account n the second perod. In the second perod, the household ll eat all of hs ncome, as he does not lve past that perod. In lght of ths, the second perod budget constrant s: ( nd Per BC) c ( ) s Present Value, Wealth, and the Intertemporal Budget Constrant We no present the consumer s maxmzaton problem va the ndfference curve dagram. We have depcted household preferences for consumpton (corn) n Perod and consumpton (corn) n perod va the ndfference curves. To characterze the optmal savngs decson, e must consder the budget constrants. In terms of depctng the optmal consumpton choces va ndfference curves ths presents a dlemma because 5

6 e typcally only have a sngle budget constran to plot. Wth a -Perod lved household, there are to such budget constrants. The queston s ho e should proceed n lght of the fact that there are to budget constrants. The trck s to convert the to sngle perod constrants nto a sngle budget one, hch s called the ntertemporal budget constrant. Ths can be done n three steps. Frst, from Equaton ( nd Per BC), e solve for savngs: c s () Equaton () s next substtuted nto Equaton ( st Per BC) so as to elmnate the savngs term from the frst perod budget constrant. Ths gves us c c The fnal step s to rearrange terms as follos (IBC) c c Equaton (IBC) s the consumer s ntertemporal budget constrant. It says that the present value of consumpton expendtures over the consumer s lfe s equal to the present value of ts tme endoment. The present value of the consumer's tme endoment s hat economsts defne as a person s ealth. The concept of present value s just as t name suggests. In effect, present value converts the value of a commodty n the future nto today s consumpton. In ths to perod model, the present value of Perod consumpton s ts quantty equvalent value today n Perod consumpton. To understand ths better, consder the follong savngs opton. 6

7 You could save one lters of corn today, n hch case you ould receve (+) lters tomorro. Alternatvely, you could save just enough today so that tomorro you ould have lters of corn n your savngs account. To do ths, you ould need to save s. If you saved exactly ths amount, then s(+) ould leave you th exactly lters n your bank account n the next perod. If you thnk about ths a lttle more, hat e have just done s compute the prce of a lter of corn n Perod measured n Perod lters. That s to say, e can buy lter of corn to be delvered tomorro at the prce of lters today. Ths s the relatve prce of the Perod good n terms of the Perod good. Ths relatve prce s the key to concept to present value. If e take c lters of corn and dvde by, hat e obtan s the c number of lters today that those c lters are orth. Its present value s. If e had a 3 perod orld, then e ould dvde consumpton n perod 3 by (+). More generally, e can convert a quantty of any good or ncome n perod t nto ts present value by dvdng by (+) t-. Note that ths assumes a constant nterest n all perods. If the nterest rate n the perods dffered, then e ould dvde by (+ )(+ ) (+ t- ). 7

8 Notce that f e denote p, and use V to denote a person s ealth, namely, V, the ntertemporal budget constrant gven by equaton (IBC) can be rertten as p c pc V. Ths s just hat e had before, except c as lesure n the prevous chapter. Ths s nformatve because t shos that the ntertemporal maxmzaton problem s no dfferent from the statc one e studed earler. The same utlty maxmzaton condtons ll apply. For the graphcal presentaton of the household s optmzaton, t s useful to solve for c n the Equaton (IBC). Ths s c ) ( ) ( c. (6) Ths s just a straght lne th y-ntercept (+) + and slope (+). Ths s graphed n Fgure. The y-ntercept s the amount you could eat n Perod f you ate nothng n Perod. If you ate nothng n perod, you ould save your entre age ncome,, and have (+) n your bank account tomorro. Smlarly, the x-ntercept s + /(+) and t represents the total amount that you could consume n Perod f you consumed nothng n Perod. Ths s just your ealth. Thnk of t as follos: you can eat all of your ncome today,, as ell as obtan a loan from the bank hch you could use to buy some of the good n Perod. The maxmum amount you could borro s such that you could pay back th your age n Perod,. Ths maxmum loan s /(+). 8

9 Fgure : Budget Constrant (+) + c -(+) + /(+) c As before, maxmzng utlty means reachng the hghest ndfference that the household can reach gven ts budget constrant/lne. The hghest ndfference curve that can be reached s the one that s just tangent to the budget lne. In Fgure 3, ths ndfference curve corresponds to U. The household could certanly afford to buy the consumpton bundle gven by pont A 0 hch s on U 0, but clearly he s not optmzng n ths case. 9

10 Fgure 3: Utlty Maxmzaton c A 0 U U 0 U c Mathematcally, the tangency pont s the one here the slope of the ndfference curve equals the slope of the budget lne,.e. the margnal rate of substtuton n consumpton equals the margnal rate of substtuton n exchange. Ths s U / c U / c p p In our to perod model, the prce of the good n perod s normalzed to and the prce of the good n perod s /(+). Thus, the optmzng condton above s (MRSC=MRSE) U / c U / c An example- Logarthmc Utlty 0

11 We ll no use a specfc example to llustrate va algebra the optmal savngs decson. The utlty functon for our consumer s U( c, c) ln( c ) ln( c) () You probably note that t looks dentcal to the utlty functon e used n the prevous chapter n studyng the labor/lesure decsons except that e have used the Greek letter β nstead of α to denote the parameter on the utlty from good number. Ths s on purpose. The parameter on the utlty assocated th good has a clear and dfferent meanng n the context of the dynamc model. Its value s restrcted to be beteen 0 and. It s referred to as the dscount factor, and t s meant to capture the psychologcal fact that people value today over tomorro. For example, gven the opton, ould you rather have a slce of pzza today or a slce of pzza tomorro? Most people ould pck the slce of pzza today on account of havng a certan level of mpatence. Indeed, one ay to thnk of β s a measure of ho patent a person you are. Wth ths utlty functon, the umargnal rate of substtuton n consumpton s, U / c U / c c c. Thus the utlty maxmzaton condton s c c. Usng ths utlty maxmzng condton th the ntertemporal budget constrant, e can derve the household's savngs functon. To do ths e rearrange the above expresson so that

12 c c. We kno from the dagram that the household maxmzes utlty by beng on the budget constrant. Thus, the ntertemporal budget constrant holds th equalty, namely. c c. Thus, substtutng condton obtaned from equatng MRSC th MRSE above nto the ntertemporal budget constrant above e arrve at c ) ( Solvng for c, c. In ords, the consumer eats a constant fracton /(+β) of ts ealth n the frst perod. Ths s the houshold s demand for frst perod consumpton. It s a functon of the age n the frst perod, the second perod and the real nterest rate. To determne household's savngs, e use the defnton of savngs, namely the dfference beteen frst perod ncome and frst perod consumpton, c. Thus, ) )( ( c s We can graph ths as a functon of the nterest rate. As the nterest rate ncreases, the amount of savngs ncreases. Notce n the specal case that the consumer has no second perod ncome ts savngs functon s ndependent of the nterest rate. In ths case, the

13 substtuton and ncome effects completely cancel each other out. Ths s a specal property of the log utlty functon. II. Government Fnance and Rcardan Equvalence Ultmately, e shall encompass the savng decson studed above n a general equlbrum model th proft maxmzng frms. Ths e ll do extensvely n the next chapter. For no, e can use some of the concepts e developed here to derve a fe nsghts about the effect of government taxes on consumpton and savngs behavor. What do governments do? They spend. They tax. If they have a shortfall, namely tax recepts are smaller than expendtures, they mght borro. In the opposte case, namely a surplus, they can lend. There s also another mportant ay governments cover a defctprntng money. Whle ths has and contnues to be an mportant ay actual government fnance defcts, t s not somethng e are ready to study at ths pont n the course. The problem s that our model s a real one- all transactons are done n terms of the economy's output, and prces are denoted n output as ell, so there s no role for money n ths orld. Nevertheless, e can stll learn an mportant lesson about government fnance n ths smple real, to-perod orld. 3

14 In a to perod orld, the government lke the consumer can Government Segnorage The ord Segnorage has ts roots n France, meanng the rght of the lord to prnt money. Segnorage n the days of commodty money equaled the dfference beteen the face value of the mnted cons and the cost of producng t. Wth the advent of fat money, segnorage has come to be used synonymously th the nflaton tax. Ths s loss n the purchasng poer of money that s assocated th an ncrease n the money supply. Effectvely, as the ncrease n the money supply causes a rse n the prce level, the amount of goods one can buy th ther nomnal ncome falls. Ths acts lke an ncome tax. It has the equvalent effect as an ncrease n an ncome tax hch reduces the value of your paycheck. Seen n ths lght, there s not much dfference n fnancng a current defct by prntng money or by borrong today and rasng ncome taxes n the future. borro or lend n the frst perod. We ll denote the government s borrong or lendng th the letter, B. Lke the consumer the government cannot borro or lend n the second perod, as the orld ends at that pont. Thus, lke the household, the government has to budget constrants, one for the frst and another for the second. In each perod t buys some of the economy's output (corn). We denote these purchases by G and G respectvely. It also taxes the household n the economy. In contrast to the last chapter, e shall consder a dfferent type of tax here- a lump-sum tax. Wth ths tax, the government smply takes a fxed amount of ncome aay from the consumer n each perod. Ths s n contrast to the margnal tax rate that e studed earler, here the amount of taxes collected depends on the number of hours the consumer orks. Let T and T denote the lump-sum taxes the government mposes on the consumer. The government s budget constran n the frst perod s (GBT ) G B T and ts budget constrant n the second perod s (GBT ) G ( ) B T Wth these taxes, the budget constrant of the consumer n perod 4

15 becomes c s T and the budget constrant n perod becomes c ) ( s T. As e dd before, e can convert the to sngle perod budget constrants of the consumer nto an ntertemporal budget constrant. Wth the lump sum taxes, the consumer s ntertemporal budget constrant s c T c T. We can perform the same steps to arrve at an ntertemporal budget constrant for the government. Ths s (IBC G) G T G T. The ntertemporal budget constrant of the government mples that the present value of government spendng must equal the present value of ts tax recepts. Note that f the government has a defct n the frst perod, (.e. G > T ), then Equaton (IBC G) mples that t must have a surplus n the second perod (.e. G < T ). Another mportant mplcaton of ths s that f the government loers lump-sum taxes n the frst perod (.e., ΔT =-), then the second perod tax must go up by exactly ΔT (+r). Heren les the key to the noton of Rcardan Equvalence. By defnton, Rcardan Equvalence refers to the neutral effect of change n lump sum taxes today on the 5

16 consumer s consumpton choces n both perods. Why s ths the case? In the above example of a one lter tax cut n the frst perod, the government must rase the tax n the second perod by (+r). In the consumer s ntertemporal budget constrant, e have c [ T ( r) T ] c ( T T ). Ths reduces to c T c T, hch s the same as before. In other ords, the effect of the tax decrease n perod has no effect on the household s ealth. The ntertemporal budget constrant s the same, and thus the optmal consumpton choces are the same as before. Whle consumpton n each perod s unaffected, ll savngs be unaffected under the alternatve tax plan? There are a number of mplct assumptons that e have made to arrve at ths result. The frst s that the household s not credt constraned. If he or she s lmted to ho much they can borro, then t mght be the case that an alternatve tmng of the lumpsum taxes ould affect the savngs and consumpton choces. Another mplct assumpton s that the person that receves the tax cut today s the same one ho pays the tax ncrease tomorro. If ths s not the case, then the ealth of ndvdual consumers n the economy ll change th the change n tax polcy, and hence the polcy ll have real effects. And of course, Rcardan Equvalence requres that taxes be lump sum so that they do not affect the relatve prces of goods and hence the utlty maxmzng 6

17 condtons of the households. In other ords, a change n the margnal tax rate of ncome or consumpton ll generally affect the equlbrum outcomes. III. Soveregn Debt Crses: the European Monetary Unon and PIIGS We no use the ntertemporal budget constrant of the government to make sense of the events n several European countres that belonged to the EMU. All of these countres experenced a crss n ther credt markets as prvate nvestors hestated at buyng ne bond ssues that ere needed to cover revenue gaps. The result as a huge ncrease n nterest rates on government debt. Ths as the market adjustment to prvate nvestor s unllngness to hold Greek, Italan, Portuguese, Irsh and Spansh Bonds. 7

18 Why ere prvate nvestors hestant about holdng PIIGS bonds? Clearly, prvate nvestors felt there as a reasonable rsk of default on these natons on these bonds. The fear s easly understood n terms of the government ntertemporal budget constran, that e developed n the case that soveregn natons lacked the ablty to prnt money. Recall, that f you do not have the ablty to prnt money, then current defcts must be met th future surpluses. Ths s the donsde of beng a member of a currency unon, such as the European Monetary Unon. Wthout ths ablty, people should expect that the government ll at some date be unable to pay back hat t oes. Prvate nvestors, seeng ths potental don the road, balk at the opton of buyng the bonds no. There are a number of ays to generate future surpluses. One ay s to rase hgher taxes n the future. Another s to reduce future outlays. Stll another ay s to mantan current tax rates and keep government outlays at ther current level, but gro the economy. That s to say, that f e kept the tax rate at 0%, the government s tax recepts ould ncrease just on account of groth n GDP. A good ndcator of ho bg a country s future surpluses must be s the sze of ts publc debt (measured as a percentage of GDP). The belo fgure plots the Debt to GDP ratos for the OECD countres n 0. Certanly, Greece, Italy, Portugal and Ireland are among the OECD countres th the largest debt to GDP ratos. Hoever, they are not all that dfferent from the Unted States, Belgum, France and Canada and none of these countres underent a debt crss. Span s actually loer n ts debt to GDP rato, havng 8

19 a rato that s average for ths set of countres. There s also Japan hch has a debt to GDP rato almost double that of Italy, Ireland and Portugal. So hat s gong on? Recall that the US, Canada and Japan are not part of any currency unon. As they can prnt money, they do not need future surpluses to meet current defcts. Hoever, there are those EMU countres th smlar Debt to GDP ratos such as France and Belgum that dd not experence a crss, so there s stll a mystery here. One possble explanaton s that despte these smlar debt to GDP ratos, France and Belgum are relatvely better at gettng a grp on ths problem. Ths could be ndcated by 9

20 smaller recent defcts. Hoever, the evdence does not suggest that ths s the case, at least for understandng Italy s stuaton. Whereas government budget defcts n Greece, Ireland and Span n 00 ere among the largest among OECD members as a percentage of GDP, Italy s as not. It as a mere 5 percent of GDP, smaller than France s and Belgum s. So hy dd Italy have a debt crss? Or perhaps, better posed, Why dd France and Belgum not have ones? On ths, most economsts thnk that Italy experenced ths crss 0

21 on account of havng a very lo future groth prospect. In 005, Italy as dentfed by The Economst as the real sck man of Europe on account of hat t sa as lo future groth due to the large amount of bureaucracy and hgh cost of dong busness. IV. Concluson Ths chapter has studed the fnal mportant mcro decson, the savngs decson. Ths s the last key buldng block of modern macro general equlbrum models. In the next to chapters e ll consder ths decson th the labor supply decson to study ssues n publc fnance. In the secton that follos, e ll use t to study busness cycles. Although e dd not consder the savngs decson n the context of a general equlbrum model, our study gave us some mportant nsghts nto the problems that several European countres have encountered recently. REFERENCES: PROBLEMS:. A consumer lves for to perods and receves labor ncome n both the frst perod of hs lfe and the second. Let be the age ncome n the frst perod, and let be the age ncome n the second. The household pays lump-sum taxes n the frst perod, Tx and lump-sum taxes n the second perod Tx. The consumer does not value lesure, but does value consumpton n perod and perod. The tme endoment of the consumer s unt of tme n each of the to perods. Suppose the consumer has the follong utlty functon ln( c ) ln( c)

22 a.) Suppose that β(+)=. Derve the household s demand for consumpton n perod, consumpton n perod. Descrbe condtons on the after tax earnngs n perods and that ensures that the household ould nether be a borroer or lender. b.) Suppose that taxes are zero n each perod. Contnue to assume that β(+)=. No suppose that =5. Fnd the optmal savngs for the consumer as ell as hs frst and second perod consumpton quanttes. c.) No suppose there s a borrong constrant hereby the consumer can at most borro an amount up to hs frst perod age ncome,. Sho va a dagram ho ths borrong constrant affects the budget constrant. Fnd the optmal savngs and consumpton for ths consumer faced th ths borrong constrant.. A to-perod lved government has expendtures G, and G, and lump-sum taxes Tx, and Tx. The consumer has the same utlty as above. He does not face any borrong constrant. He has age ncome and n each perod. The producton of the consumpton good s A h and A h n perods and respectvely. Assume as above that the consumer s endoed th unt of tme n each of the to perods. a.) Assume that G =G =Tx =Tx = 0, A =0 and A = and β=. Verfy that the equlbrum for ths economy s c =0, c = and r=0%. b.) No assume that G =G = 0, but Tx = 0 and Tx = -. Contnue to assume that A =0 and A =, and β=. Verfy that the equlbrum s stll c =0, c = and =0%. Ho much does the consumer borro/lend n the frst perod? c.) No assume that G =0, G =, and Tx = 0 and Tx =. Contnue to assume that A =0 and A =, and β=. What are the equlbrum values for c, c, and? Why doesn t Rcardan Equvalence hold here?