Chapter 3 Savings, Present Value and Ricardian Equivalence


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1 Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision, as it was done within the same peiod. We now tun to anothe impotant decision of households fo which thee is a time element namely, the household s saving decision. How much one saves today affects how much one consumes today and in the futue. Because ones decision today affects one s futue situation, the decision is intetempoal. This chapte consists of two pats. We begin this chapte by studying the household consumption/savings behavio. As we shall see, the optimization poblem and conditions ae vey simila to those developed in the pevious chapte on the household s labo/leisue decision. In studying this poblem, we define the notion of pesent value, and wealth. The second pat of the chapte analyzes the effect of a paticula type of tax change on consume s savings decision. This is what is known as Ricadian Equivalence. We will postpone the integation of these concepts into a dynamic geneal equilibium model until the next chapte. In doing so, we shall examine the US social secuity system and detemine if thee is anothe system that will not go bankupt and that can make both young and old bette off.
2 I. Micoeconomic Foundations Consume Theoy We etun to ou study of micoeconomics and the theoy of the consume. Cuently, we ae inteested in the choice of savings, namely, how much to eat today vesus how much to eat tomoow. Fo this pupose, we shall begin with an assumption that ou consume lives fo only two peiods, Peiods and, and that he/she deives happiness o utility fom consumption in Peiod and consumption in Peiod. Fo now, we will assume that the household does not deive any utility fom leisue. We will elax this assumption in the next chapte. Given these assumptions, the household utility function is denoted by U c, c ) whee c is the quantity of the good consumed in the fist peiod and c in the ( quantity of the good consumed in the second peiod. Indiffeence Cuves Given that the household s utility is defined ove two goods, we can still epesent it via indiffeence cuves and show the optimal saving choice gaphically. Figue shows the indiffeence cuves. On the hoizontal axis is Peiod consumption, c, and on the vetical axis is Peiod consumption, c.
3 Figue : Indiffeence Cuves c U U 0 U c As we saw in ou study of the household labo decision, each indiffeence cuve descibes all the combinations of consumption in peiod and consumption in peiod that gives the household a cetain level of utility. The consume, hence, is indiffeent between any of the consumption bundles along a given cuve. Again, as long as moe consumption of each good is desiable, the indiffeence cuve must be downwad sloping. Additionally, as long as people like vaiety, the indiffeence cuve, is convex, namely, bowed in towad the oigin. Budget Constaints Because the consume lives two peiods, he has two budget constaints. Let us assume that the consume does not begin his life with any inheited debt, and let us also assume that he must pay all his debt back befoe he depats the wold. Thus, in the fist peiod, 3
4 the consume is able to save o boow, and in the second peiod, he must pay back any loans plus inteest, o in the case that he saved, be paid back the pincipal and inteest. To gain some intuition, let us think of the consumption good in each peiod as con, measued in bushels/lites. Savings and boowing ae thus done in tems of bushels of con. In the fist peiod say you lend 0 bushels. In the second peiod, you ae epaid the 0 bushels plus inteest on those 0 bushels, also paid to you in bushels. Because the inteest payment is made in tems of con, it is a eal ate, not a nominal ate. To distinguish between boowing and lending, we use s > 0 to indicate savings, and s < 0 to indicate that an individual is boowing. We ae now eady to wite down the household s budget constaints in Peiod and Peiod. Peiod : Assume that the household has some labo income in the peiod measued in bushels of con. Because he does not value leisue (i.e., by assumption, leisue does not ente the household s utility function), he will supply his entie time endowment to the labo maket. In the pevious chapte we thought of a peiod epesenting a week. Hee we ae viewing a peiod of being fa longe. To make things simple as possible, let us assume that the household is endowed with one unit of time in each peiod. Let w be the wage ate of labo in Peiod measued in bushels of con. Thus, the consume s labo income in the fist peiod is just w. 4
5 The consume can do two things with this income: he can save (s) and consume con, (c ). (If he eats moe con than his income, then his savings is negative and is boowing.) The Peiod budget constaint is thus c + s. () w Peiod : In Peiod, the household again supplies labo to the economy and eceives labo income. The wage ate in Peiod, is w. This is the numbe of bushels of con the household eceives in the second peiod fo woking an hou in that peiod. The household also has some savings and inteest on the savings he that he can use fo consumption in the second peiod. We shall denote the net inteest ate by the lette,. This is in eal tems, so it is the numbe of bushels a lende must pay you fo each bushel he boows fom you. Thus, if you lent s= in the fist peiod, you would have (+) s bushels of con in you savings account in the second peiod. In the second peiod, the household will eat all of his income, as he does not live past that peiod. In light of this, the second peiod budget constaint is: c w + ( ) s () + Pesent Value, Wealth, and the Intetempoal Budget Constaint We now pesent the consume s maximization poblem via the indiffeence cuve diagam. We have depicted household pefeences fo consumption (con) in Peiod and consumption (con) in peiod via the indiffeence cuves. To chaacteize the optimal savings decision, we must add in the budget constaints. Recall in the pevious chapte, that thee was only one budget constaint shown in the diagam. With a Peiod 5
6 lived household, thee ae two such budget constaints. The question is how we should poceed in light of the fact that thee ae two budget constaints. The tick is to convet the two single peiod constaints into a single budget one, which is called the intetempoal budget constaint. This can be done in thee steps. Fist, fom Equation (), we solve fo savings: c s + w + (3) Equation (3) is next substituted into Equation () so as to eliminate the savings tem fom the fist peiod budget constaint. This gives us c c + + w + w The final step is to eaange tems as follows c w c + w + (4) + + Equation (4) is the consume s intetempoal budget constaint. It says that the pesent value of consumption expenditues ove the consume s life is equal to the pesent value of its time endowment. The pesent value of the consume's time endowment is what economists define as a peson s wealth. The concept of pesent value is just as it name suggests. In effect, pesent value convets the value of a commodity in the futue in tems of today s consumption. In this two peiod model, the pesent value of Peiod consumption is its quantity equivalent value today in Peiod consumption. To undestand this bette, conside the following savings 6
7 option. You could save one bushel of con today, in which case you would eceive (+) bushels of con tomoow. Altenatively, you could save just enough today so that tomoow you would bushel of con in you savings account. To do this, you would need to save s = +. If you saved exactly this amount, then s(+) would leave you with exactly bushel of con in you bank account in the next peiod. If you think about this a little moe, what we have just done is computed the pice of a bushel of con in Peiod measued in Peiod bushels. That is to say, we can buy bushel of con to be deliveed tomoow at the pice of + bushels today. This is the elative pice of the Peiod good in tems of the Peiod good. This elative pice is the key to concept to pesent value. If we take c bushel of con and divide by +, what we obtain is the numbe of bushels of con today that those c bushels ae woth. Its c pesent value is +. If we had a 3 peiod wold, then we would divide consumption in peiod 3 by (+). Moe geneally, we can convet a quantity of any good o income in peiod t into its pesent value by dividing by (+) t. 7
8 Notice that if we denote p =, and use V to denote a peson s wealth, namely, + V w w +, the intetempoal budget constaint given by equation (4) can be ewitten + as p c + pc V. (5) This is just what we had befoe, except c was leisue in the pevious chapte. This is infomative because it shows that the intetempoal maximization poblem is no diffeent fom the static one we studied ealie. The same utility maximization conditions will apply. We now show this gaphically again. Utility Maximization Fo the gaphical pesentation of the household s optimization, it is useful to solve fo c in the Equation (4). This is c ( + c. (6) + ) w + w ( ) This is just a staight line with yintecept (+)w +w and slope (+). This is gaphed in Figue. The yintecept is the amount you could eat in Peiod if you ate nothing in Peiod. If you ate nothing in peiod, you would save you entie wage income, w, and have (+)w in you bank account tomoow. Similaly, the xintecept is w +w /(+) and it epesents the total amount that you could consume in Peiod if you consumed nothing in Peiod. This is just you wealth. Think of it as follows: you can eat all of you income today, w, as well as obtain a loan fom the bank which you could use to buy some of the good in Peiod. The maximum amount you could boow is such that you could pay back with you wage in Peiod, w. This maximum loan is w /(+). 8
9 Figue 3: Budget Constaint (+)w +w c (+) w +w /(+) c Utility Maximization As befoe, maximizing utility means eaching the highest indiffeence that the household can each given its budget constaint/line. The highest indiffeence cuve that can be eached is the one that is just tangent to the budget line. In Figue 4, this indiffeence cuve coesponds to U. The household could cetainly affod to buy the consumption bundle given by point A 0 which is on U 0, but clealy he is not optimizing in this case. 9
10 Figue 4: Utility Maximization c A 0 U U 0 U c Recall, that the slope of a function o cuve at a point is the slope of the tangent at that point. The tangent just happens to be the budget constaint so that the slope at the utility maximizing point is p /p =(+). Theefoe, the slope of the indiffeence cuve at the utility maximizing consumption bundle is equal to the numbe, p /p. By definition, the slope is the change in the dependent vaiable associated with a change in the independent vaiable. Hee, the dependent vaiable is utility and thee ae two independent vaiables. The indiffeence cuve U epesents all the consumption bundles fo which U = U c, c ). Along U, we can change c and c with no change in utility. The ( change in total utility along this indiffeence cuve is U c c + U / c c 0. (7) / = This just says that the change in total utility equals the change in c times the change in utility fom changing c, U / c, plus the change in c times the change in utility fom changing c, U / c. The ight hand side of equation (7) is zeo because we ae moving 0
11 along the same indiffeence cuves, so the level of utility does not change along a given indiffeence cuve. Reaanging equation (7) we find that the slope of the indiffeence cuve, c / c, is c / c. (8) c = U / c / U / Since we now have the slope of the indiffeence cuve, and we know that it must equal the atio of the pices, p /p, it follows that the utility maximizing bundle, c *, *) satisfies ( c + = U / c c. (9) / U / The left hand side is what is efeed to as the maginal ate of substitution in exchange as it is the ate at which the household can exchange good fo good. The ight hand side is efeed to as the maginal ates of substitution in consumption, as it is the ate the consume is willing to exchange the two goods so as to keep his utility constant. In what follows, it will be moe useful to flip the numeatos and denominatos of Equation (9). This is + = U / c / U / c. (0) An example Logaithmic Utility We will now use a specific example to illustate via algeba the optimal savings decision. The utility function fo ou consume is U ( c, c ) = log( c) + β log( c ) ()
12 The Geek lette β epesents a numbe between 0 and. It is efeed to as the discount facto, and it is meant to captue the psychological fact that people value today ove tomoow. Fo example, given the option, would you athe have a slice of pizza today o a slice of pizza tomoow? Most people would pick the slice of pizza today. With this utility function, the maginal ate of substitution in consumption is, U / c U / c c = β c. () Thus the utility maximization condition is c β c + =. (3) Using this utility maximizing condition with the intetempoal budget constaint, we can deive the household's savings function. To do this we eaange the above expession so that c β c =. (4) + We know fom the diagam that the household maximizes utility by being on the budget constaint. Thus, the intetempoal budget constaint holds with equality, namely. c c w + = w Thus, substituting Equation (4) into the intetempoal budget constaint above we aive at w c ( + β ) = w + (5). +
13 Next, we use Equation (5) to solve fo c. This is c = + w β w + +. In wods, the consume eats a constant faction /(+β) of its wealth in the fist peiod. We can now detemine the household's savings. Savings ae defined as the diffeence between peiod income and consumption, c. Thus, s = w c = w w + β w β + = w + + β w ( + β )( + ) (6). We can gaph this as a function of the inteest ate. As the inteest ate inceases, the amount of savings inceases. Notice in the special case that the consume has no second peiod income its savings function is independent of the inteest ate. In this case, the substitution and income effects completely cancel each othe out. II. Govenment Finance and Ricadian Equivalence Ultimately, we shall encompass the saving decision studied above in a geneal equilibium model. This we will do in the next chapte. Fo now, we can use some of the concepts we developed hee to deive a few insights about the effect of govenment taxes on consumption and savings behavio. What do govenments do? They spend. They tax. If they have a shotfall, namely tax eceipts ae smalle than expenditues, they boow. In the opposite case, namely a suplus, they can lend. Thee is also anothe impotant way govenments make up a deficit pinting money. While this has and continues to be an impotant way actual 3
14 govenment finance deficits, it is not something we ae not eady to study at this point in the couse. The poblem is that ou model is a eal one all tansactions ae done in tems of the economy's output, and pices ae denoted in output as well, so thee is no ole fo money in this wold. Nevetheless, we can still lean an impotant lesson about govenment finance in this simple eal, twopeiod wold. In a two peiod wold, the govenment like the consume can boow o lend in the fist peiod. We will denote the govenment s boowing o lending with the lette, B. Like the consume the govenment cannot boow o lend in the second peiod, as the wold ends at that point. Thus, like the household, the govenment has two budget constaints, one fo the fist and anothe fo the second. In each peiod it buys some of the economy's output (con). We denote these puchases by G and G espectively. It also taxes the household in the economy. In contast to the last chapte, we shall conside a diffeent type of tax hee a lumpsum tax. With this tax, the govenment simply takes a fixed amount of income away fom the consume in each peiod. This is in contast to the maginal tax ate that we studied ealie, whee the amount of taxes collected depends on the numbe of hous the consume woks. Let T and T denote the lumpsum taxes the govenment imposes on the consume. The govenment s budget constain in the fist peiod is G + B = (7) T and its budget constaint in the second peiod is G = ( B + T (8) + ) 4
15 With these taxes, the budget constaint of the consume in peiod becomes c + s w T and the budget constaint in peiod becomes c w + + ) ( s T. As we did befoe, we can convet the two single peiod budget constaints of the consume into an intetempoal budget constaint. With the lump sum taxes, the consume s intetempoal budget constaint is c w T c + w T +. (9) + + We can pefom the same steps to aive at an intetempoal budget constaint fo the govenment. This is G T G + = T +. (0) + + The intetempoal budget constaint of the govenment implies that the pesent value of govenment spending must equal the pesent value of its tax eceipts. Note that if the govenment has a deficit in the fist peiod, (i.e. G > T ), then Equation (0) implies that it must have a suplus in the second peiod (i.e. G < T ). Anothe impotant implication of this is that if the govenment lowes lumpsum taxes in the fist peiod (i.e., T =), then the second peiod tax must go up by exactly T (+). 5
16 Heein lies the key to the notion of Ricadian Equivalence. By definition, Ricadian Equivalence efes to the neutal effect of change in lump sum taxes today on the consume s consumption choices in both peiods. Why is this the case? In the above example of a one bushel of con tax cut in the fist peiod, the govenment must aise the tax in the second peiod by (+). In the consume s intetempoal budget constaint, we have c c w [ T + ( + ) T ] + w ( T T ) This educes to c c w T + w T + + +, which is the same as befoe. In othe wods, the effect of the tax decease in peiod has no effect on the household s wealth. The intetempoal budget constaint is the same, and thus the optimal consumption choices ae the same as befoe. While consumption in each peiod is unaffected, will savings be unaffected unde the altenative tax plan? 6
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