Lecture notes in Macroeconomics

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1 Lecture notes in Macroeconomics Consumption & Saving Mario Tirelli Very preliminary material Not to be circulated without the permission of the author Contents 1. Introduction Traditional Keynesian view Post-war theories Consumption and saving under imperfect financial markets Consumption and saving under uncertainty and imperfect capital markets 15

2 1. Introduction In these short class notes we shall go through some benchmark theories of consumption and saving. In doing so we shall keep in mind a few main stylized facts: Consumption is the largest component of GDP (about % in most developed countries); Consumption is relatively stable function of current income: it is procyclical but it has a lower variance (it covarietes less than investments) with respect to GDP; saving is instead more volatile and might become negative when current income falls below some reference level; Consumption is sensitive to permanent or long lasting changes in income and wealth; The saving rate tends to be positively correlated with the GDP growth rate: countries with an higher average saving rate tend to have an higher accumulation rate and rate of growth of income. Consumption and savings are sensitive to changes in demographics (e.g. avg. population age, life expectancy, retirement age), family characteristics (e.g. number of children, presence of family networks), institutions (insurance market possibilities, pension and tax systems); 1.1. Traditional Keynesian view. Among the many insights found in Keynes General Theory (GT), one of the most relevant is the consumption function. Prior to Keynes, consumption had been viewed as a passive residual, that portion of income remaining after saving. Saving instead was the result of an intertemporal allocation decision that, for given resources, was essentially determined by the interest rate. Say law would then predict that national saving translates into investments, capital accumulation and output. Prices flexibility would then guarantee that output is demanded. Keynes contrasted this view observing that consumption decisions go beyond simple sacrifice payments. There are not many people who will alter their way of living because the rate of interest has fallen from 5 to 4 percent (GT, p. 94). Spending and saving are influenced by a host of objective attendant circumstance, subjective needs, psychological propensities and habits (GT, p. 91). Keynes recognized that people must consume to survive: for a man s habitual standard of life usually has first claim on his income (GT, p. 97). He proposed a fundamental psychological law of consumption (GT, p. 96). A law [...] upon which we are entitled to depend with great confidence both a priori from our knowledge of human nature and from the detailed facts of experience, [...] that men are disposed, as a rule and on average, to increase their consumption as their income increases, but not by as much as the increase in their income. That is to say, [...] (paraphrasing) if C is the amount of consumption and Y is income, dc/dy is positive and less than unity. Moreover, Keynesian economics first stressed the role of consumption as the main determinant of aggregate demand and affects the level of income. 1

3 According to Keynes view, the propensity to consume is seen as a relatively stable function. Short-run fluctuations of current (real, disposable) income produce changes in consumption of the same sign but of a much smaller amount. The rest of income variation translates in an adjustment of savings, which behaves as a superior good (one whose demand increase with income, with an elasticity above one) or a luxury good (a superior good that is purchased only by sufficiently high-wealth consumers). Of course, differently from superior or luxury goods, savings could be negative, helping stabilize consumption at economic downturns. It is worth reminding that Keynes analysis was essentially static, based on aggregate data and not rooted into the neoclassical theory of individual decisions. Moreover, Keynes attention was mostly in understanding the Great Depression, and in trying to shape up economic policy rules that could prevent it from happening again. Coherently with his view, and differently from earlier scholars, the increase of national saving is seen as potentially disruptive, because alternative to consumption. The possibility that saving would not translate into investment (a failure of the Say Law), accumulation and income growth is, for the first time, taken seriously as a potential important explanation of economic stagnation Post-war theories. Between 1952 and 1954, Richard Brumberg and Modigliani wrote two essays, 1 which provide the basis for the Life Cycle Hypothesis of Saving (LCH). Their purpose was to show that all the well established empirical regularities could be accounted for in terms of rational, utility maximizing, consumers, allocating optimally their resources to consumption over their life. In Modigliani s own words [Modigliani, 1986]: The hypothesis of utility maximization (and perfect markets) has, all by itself, one very powerful implication - the resources that a representative consumer allocates to consumption at any age, it, will depend only on his life resources (the present value of labor income plus bequests received, if any) and not at all on income accruing currently. When combined with the self evident proposition that the representative consumer will choose to consume at a reasonably stable rate, close to his anticipated average life consumption, we can reach one conclusion fundamental for an understanding of individual saving behavior, namely that the size of saving over short periods of time, like a year, will be swayed by the extent to which current income departs from average life resources. This conclusion is common to another influential contribution, Friedman s Permanent Income Hypothesis (PIH) (Friedman, 1957), 2 which differs from the LCH primarily in his model individual rational decisions are assumed by an infinitely lived consumer (or dynasty). Accordingly, the notion of life resources is replaced by that of permanent income, while the discrepancy between current and permanent income is labeled transitory income. 3 1 Utility Analysis and the Consumption Function: an Interpretation of Cross Section Data (Modigliani and Brumberg, 1954), and Utility Analysis and the Aggregate Consumption Function: an attempt at Integration (Modigliani and Brumberg, 1979) 2 Friedman, M., A Theory of the Consumption Function, Princeton University Press, Princeton, I. Fishers work (1924) was influential for both models. 2

4 In both theories, consumers decide how much to consume keeping in mind their future prospects. If no uncertainty is introduced in the model, its predictions are intuitively explained: concavity of the utility function implies a desire to smooth consumption over time; hence, the main motivation for saving is to smooth out fluctuations in income. Accordingly, consumption increases with current income only if that increase is a permanent one. In the case of the LCH model, the explicit consideration of retirement, that is a period in which income declines considerably, generates the main motivation for saving. Retirement represents one of the main sources of income variation; households accumulate wealth during the active period of theirs life to provide for consumption during retirement. Both of these theories, PIH and LCH, have a number of implications which have received ample empirical support, even with some occasional controversy (for a literature review, see Attanasio, 1999) 4. As with regard to Keynesian theory, Modigliani and Brumberg (1954: 430) concluded: The results of our labor basically confirm the propositions put forward by Keynes in The General Theory. At the same time, we take some satisfaction in having been able to tie this aspect of his analysis into the mainstream of economic theory by replacing his mysterious psychological law with the principle that men are disposed, as a rule and on average, to be forward-looking animals. We depart from Keynes, however, on his contention of a greater proportion of income being saved as real income increases [...] We claim instead that the proportion of income saved is essentially independent of income. This view had enormous impact. Consumption theories were recast in terms of life-cycle responses to expected lifetime or permanent income. As a result, they became ones of long run behavior; responses to current income changes were considered transitory and virtually ignored in the theoretical literature. Among the most important empirical predictions of the LCH is the positive correlation between the saving rate and the growth rate. This -among other things- allowed to explain why, in the post-war period, some of the richest countries like US had a rather low saving rate compared to poorer ones. Another important implication of LCH and PIH is Ricardian Equivalence. PIH says that the consumption only depends on the permanent income, while LCH says that it only depends on life-cycle resource. Thus any policy intervention (e.g. a tax cut or an increase of households subsidies) that is perceived by consumers as temporary, would not affect consumption A formal illustration of the Friedman and Modigliani s theory. If we now go back to our Ramsey economy, we can easily illustrate Friedman s Permanent Income Hypothesis. We shall comment more on Modigliani s when we will deal with economies with overlapping generations of individuals who are assumed to be finitely lived (OLG economies), as the LCH brings to fundamental different conclusions on many relevant aspects (em e.g. on the relationship between the growth rate and the saving rate). 4 Attanasio, O. P. (1999). Consumption. Handbook of Macroeconomics, 1,

5 Recall that equilibrium consumption dynamics are described by the Euler equation (E), which we have more explicitly presented as (E ). Rearranging terms, the latter can be written as, 5 c t+1 = 1 ( ) rt+1 θ c t σ(c t ) 1 + r t+1 This says that it is individually rational to postpone (anticipate) consumption as long as the interest rate is above (below) the rate of time preference, r > θ. At a steady-state requires, at all r t = r = θ (i.e. βr = 1) the individual experiences a constant consumption profile, achieving full-consumption smoothing. From sequential budget constraints, at all t, p t+1 a t+1 + c t+s y t+s = a t p t (and the NPG condition) we can derive the intertemporal budget constraint: at all t, p t+s (c t+s y t+s ) = a t p t s T where, a t denotes the initial asset/liability position and y the individual disposable income. Next, recall that p t+1 /p t (1 + r t+1 ) 1 and p t+1+s s 1 p t 1 + r t+1+k We can rewrite the IBC at t as follows, 6 ( s (c t y t ) + s T s T k=0 k= r t+1+k ) (c t+s+1 y t+s+1 ) = a t A the efficient steady-state is (c t+s, r t+1+s ) s T = (c, r) and r = θ. The present value (PV) of life-time consumption at t is, ( ) 1 s 1 + r c t+s = ss c 1 + r r (c ) s T P V (y t ) {}}{ 1 + r ( ) 1 s c = y t+s +a t r 1 + r c = r 1 + r [ P V (y t ) + a t ] This says that (when r = θ) current consumption is proportional to the PV of life-time income; which, measured at t, is defined as, P V (y t ) = ( ) 1 s y t+s 1 + r s T 5 The Euler equation is, u (c t ) βu (c t+1 ) = 1 + r, and β (1 + θ) 1. 6 Here we need that prices satisfy social transversality and the saving/borrowing plan a satisfies the NPG condition. 4

6 In general, one defines permanent income the sum of life-time income and current non-human wealth, y P t := r 1 + r [P V (yt ) + a t ] The fact that current consumption may be a function of life-time income is implied by two assumptions: i) individual rationality; ii) complete markets. By complete markets we mean that every kind of contract can be actually traded on some (competitive) market. In particular the individual can freely borrow against future income and any of his promise or obligation will be fully honored (no default is allowed). 7 Therefore, consumption is relatively stable with respect to expected fluctuations of current and future incomes, which are offset by changes in savings. Indeed, by definition, s t = y t c t using (c ), s t = y t r [ P V (y t ] ) + a t = yt yt P 1 + r This implies that if an anticipated negative shock hit future income, so that current income y t increases above P V (y t+1 ), savings s t are increased so as to prevent future consumption from falling that match relatively to today s. Similarly, if current income y t falls, while future income stays constant, s t is decreased to stabilize current consumption (i.e. to satisfy full-consumption smoothing). Remark 1.1 (A special case: θ = r = 0). For illustrative purposes, these theories are often presented assuming that individuals are perfectly patient (β = 1 or θ = 0). In which case the consumption and saving equations assume the following form, at all t, c t = P V (y t ) + a t, s t = y t (P V (y t ) + a t ) The derivation of the consumption and saving functions implicit in the LCH is similar; easier thank to the finite horizon of each individual consumer. We consider an individual that lives T 2 periods (say, years), works until the age of N > 0 and has a retirement period of M = T N > 0. For simplicity, we also assume that this individual is born with neither assets nor liabilities and has no bequest motifs (i.e. a 0 = a T +1 = 0). Since the Euler equation is the same in the two models, one has only to consider the life-time budget constraint. Continuing to consider a constant interest rate 0 r < 1, this is constructed from the per-period budget 7 The fact that obligations can actually be honored in a finite horizon is granted by the NPG condition, which we have used to derive the IBC from the sequential budget constraint. 5

7 constraint, which is, c 0 = y 0 a 1 c 1 = y 1 a 2 + a 1 (1 + r) c N = y N a N+1 + a N (1 + r) c N+1 = a N+2 + a N+1 (1 + r) c T = a T (1 + r) Iteratively, substituting into the first equation for the current saving decision, one obtains, T s=0 ( ) 1 s c t+s 1 + r N s=0 ( ) 1 s y t+s+1 = r As an illustration, let us follow Modigliani s example and consider (as in our Remark 1.1 above) a steady state with r = θ = 0, constant per-period consumption c and with an income process (y t ) that is also constant over time, y t = y for t N and equal to zero thereafter. Hence, P V (y 0 ) = N t=0 y t = Ny and, from the life-time budget constraint, c = N T y This says that per-period consumption is a fraction 1/T of life-time income N y. It implies that, { s = y c > 0, for all t N; s t = c < 0, in all remaining years (i.e. during the retirement period). Therefore, during the working years the individual accumulates wealth: his real assets in any period 0 t N are A(t) = ts = t(y c ) = t(m/t )y. A(t) is monotonically increasing with time up to A(N) (the aggregate assets accumulated by the people who are about to retire, in age N), which, substituting the consumption level, takes the form, A(N) = N s = N (y c ) = M {}}{ (T N) N y = (T N)c T Now, following Modigliani, if one assumes that there is neither productivity nor population growth, we can interpret the figure as illustrating the stationary distribution of consumption, income and wealth across a population of T individuals of different ages (from age 0 to T ), uniformly distributed on [0, T ]. In particular, people in their working age, from 0 to N form an aggregate wealth of A(N) = N(y c ), depicted in the first blue rectangle of base N. At the same time, retired people have on hand an accumulated wealth that equals their aggregate savings (T N)c (which, in this example, is again A(N)), represented by the second rectangle of base T N. So, at any date, the stock of wealth W is W = A(N) T/2, which graphically 6

8 y,c y c* 0 N T t correspond to the area of the triangle with base T. 8 wealth-to-gdp ratio W/Y is, This also implies that the aggregate W/Y = A(N)T/2(c T ) = (N T )c /2c = M/2 where Y c T ; namely, W/Y, being equal to M/2, depends on the retirement period only. This allows to say something on the average saving rate s: s W Y = W W W Y. Since in steady-state the growth rate of the economy g equals the growth rate of real wealth W, we have, an average saving rate that is constant and proportional to the length of retirement period M only, s = g M 2 We also have a long-run, average saving rate that is positively correlated with the growth rate of the economy. 9 It should now be clear that, although similar in spirit, the LCH is much more dependent on life-cycle characteristics than the PIH and this implies that the two models might lead to different empirical predictions. So, for example, according to PIH, Friedman said that an anticipated, persistent increase of productivity tends to depress the saving rate, because it produces a raise in permanent income relative to current and, according to the model, induces shift from future to current consumption (i.e. a drop in current savings); this will eventually reduce the path of wealth and perhaps produce a negative net-worth in the early life period. This prediction is however in contrast with the Modigliani s Life-Cycle theory and with the prevailing empirical evidence on cross-country data. According to the LCH, when growth is due to productivity, real wages increase imply that younger cohorts have larger life-time resources 8 By definition, in this example, the wealth is what is accumulated by all workers plus what is in the hand of the retired population: W = T 0 distributed over [0, T ]. Solving, st dt, where, recall that, the population is formed by T individuals uniformly W = (y c)[t 2 /2] N 0 + c[t 2 /2] T N = (y c)n[n/2] + c(t N)[(T N) 2 /2] = A(N)(N + M)/2 = A(N)T/2. 9 The previous analysis can be generalized to the case in which workers entering in the population, at age 0, receive a bequest from the retired who rich age T. Can you see this? Try to modify it and reproduce it in the figure. 7

9 than older individuals. This implies that, taking the saving rate as invariant with growth, the saving of the young are higher than the dissaving of the retired cohorts (this is called Bentzel effect). So, to have a reduction of aggregate wealth, we must observe a considerable intertemporal shift in favor of young age consumption (i.e. a considerable change in the saving rate of the young), the mechanism highlighted in the PIH model How do consumption and saving react to interest rate? Understanding how consumption and saving react to interest rate is important, for example, to assess the macroeconomic effects of shocks such as monetary policy interventions, changes in capital-income taxes, productivity shocks (affecting the marginal productivity of capital) etc.. For comparative statics, first, recall that the intertemporal elasticity of substitution (IES) measures the sensitivity of consumption growth to a change of the intertemporal rate of substitution (i.e. the individual willingness to pay for substituting future consumption for present consumption). It is defined as, E t := d(c t+1 /c t )/(c t+1 /c t ) d(mu t /MU t+1 )/(MU t /MU t+1 ) d ln(c t+1 /c t ) d ln(mu t /MU t+1 ) where MU t U(c 0, c 1,.., c t,..)/ c t is the marginal utility with respect to consumption in t and MU t /MU t+1 is interpreted as the willingness to pay in terms of t-consumption for a unit of additional consumption in t + 1. In the case considered in the text, utility is time additive and stationary (per-period utility u and the personal discount rate are constant). Hence, E t := d ln(c t+1 /c t ) d ln(mu t /MU t+1 ) = d ln(c t+1 /c t ) d ln(u (c t )/βu (c t+1 )) ln(u (c t )/βu (c t+1 )) = γ ln(c t+1 /c t ). Moreover, at an (interior) individual optimum, the Euler equation reads u (c t )/βu (c t+1 ) = R t+1. Therefore, the marginal willingness to substitute consumption over time coincides with the market interest rate and, E t = d ln(c t+1/c t ) d ln R t+1 Given all this, we can use the IES, E t, to measure the effect of a change in the interest rate on consumption growth, ln(c t+1 ln c t ). 10 Next, assume that u is of the CRRA class, with parameter γ > 0. The Euler equation now reads, ( ) 1 γ ct+1 = R t+1 β c t Taking logs, γ ln(c t+1 /c t ) = ln(βr t+1 ). Thus, differentiation the latest yields, E t = 1/γ, constant over time. 11 In the case of log-per-period utility (γ 1), a 1% increase in the interest rate determines a 1% increase in next period consumption relative to today s. 10 Geometrically, consider the positive plane of coordinates (ct, c t+1). Each ray from the origin has slope c t+1/c t. Consider the slope R of the indifference curve U(c t+1, c t) = U in any given point (the slope of the line tangent to the indifference curve at the point). The IES measures by how much the slope of the ray c t+1/c t changes with R. Do the graph yourself. 11 For stochastic consumption, γ is also the Arrow-Pratt measure of relative risk aversion. 8

10 Yet, for a complete comparative statics of the effect of R on consumption levels and saving, we need to consider the individual budget constraint and how this is modified by a change in the interest rate. As an illustration, consider a two-period model. The budget constraints of the second and first period, respectively, are c 1 = ar + y 1 and c 0 = y 0 a. The intertemporal budget constraint is, c 0 + c 1 R = y 0 + y 1 R a is the individual net-asset position and may be either positive (when the individual is saving), or negative (when borrowing). Solving for the asset demand, one finds, (a) a(r, y 0, y 1 ) = (βr)1/γ y 0 y 1 (βr) 1/γ + R Fix income (y 0, y 1 ) and consider only how saving varies with R, according to the function a(r). c 0 = y 0 a(r), c 1 = y 1 + Ra(R) An increase in R may or may not increase the individual asset position, if it does (a (R) > 0) we certainly observe a reduction of c 0 and an increase in c 1. However, it is possible that an increase in R reduces a (if a (R) < 0), increases c 0 and also c 1. In particular, the increase of c 1 occurs when the wealth effect is sufficiently strong, namely when the comparative statics is done at a point corresponding to a large asset position a > 0. Indeed, observe that, dc 0 dr = a (R), dc 1 dr = Ra (R) + a(r) The last term a(r) of the second expression if positive- captures a wealth-effect; so, even if saving does not react to interest rate, a (R) = 0, a change in R affects future consumption by changing the rent associated to wealth the asset position, positively if a(r) > 0 and negatively otherwise. To see how things work in a simpler case, assume that y 1 = 0, so that the individual has no income in the second period and is forced to save and hold assets in order to achieve a positive consumption in the second period. Equation (a) takes a simpler form, a(r) = s(r)y 0, s(r) := (βr t+1) 1/γ (βr t+1 ) 1/γ + R s(r) is the saving rate. It is easy to check the following: s < 0 γ > 1 s = 0 γ = 1 s > 0 γ < 1 γ = 1 is covered by the log-utility. In this case, substitution and income effects on consumption c 0 cancel out. This is the case in which an increase in the interest rate has only a marginal wealth effect that depends on the asset position a(r). Instead, γ < 1 corresponds to the case in which consumption demand c 0 has a substitution effect that prevails on the income effect: an increase in R represents an incentive to procrastinate consumption over time. If γ > 1 we can say the reverse, the income effect on c 0 prevails on the substitution. This last case is illustrated in the next figure, in which c c 0 denotes the compensated demand, cc 0 c 0 < 0 measures the 9

11 substitution effect and c 0 cc 0 > 0 the income effect.12 The overall effect on saving is negative: a a = (y 0 c 0 ) (y 0 c 0 ) = c 0 c 0 < 0. To illustrate this issue further, observe that the consumption functions can be written as follows, c 0 (R) = ( 1 1 ) (βr t+1 ) 1/γ, γ c 1 (R) = (y 0 c 0 (R)) R Hence, γ < 1 c 0 (R) < 0 c 1 (R) = Rc 0 (R) + (y 0 c 0 (R)) > 0 }{{} a>0 γ > 1 c 0 (R) > 0 c 1 (R) = Rc 0 (R) + (y 0 c 0 (R)) =? }{{} a>0! "! $ $%! #! & ' &!! # (! ) #! # income substit. If we go back to consider y 1 > 0, we now have to consider that an increase of R increases the return from saving and, in addition, reduces the present value of future income, y 1 /R. Graphically, as R increases, the budget line rotates clock-wise around (y 0, y 1 ); the vertical intercept y 1 + Ry 0 (date 1 maximal income) shifts up, but the horizontal intercept y 0 + y 1 /R (date 0 maximal income) shifts to the left. The latest is a wealth effect that was absent under y 1 = 0. To understand how this wealth effect works, first, observe that an individual chooses to be a lender whenever she picks a consumption bundle on the budget line, north-west of no-trade bundle (y 0, y 1 ), and a borrower if she picks a bundle south-est of (y 0, y 1 ). Clearly, a marginal increase of the interest rate makes the budget line rotates clock-wise, in a way that expand the budget set of lenders and contracts that of borrowers. Intuitively, lenders benefit from an increase in the interest rate because, ceteris paribus, this increases their return from lending (or saving); that is, the above wealth effect is positive. By an analogous reasoning, a 12 By definition, here, the compensated demand is obtained by an income compensation that, at the new interest rate R, is large enough to make feasible for the individual to achieve the original level of welfare, u(c 0) + βu(c 1). 10

12 marginal increase in R harms borrowers because it reduces their ability to repay future debt (y 1 /R falls), the above wealth effect is negative. How does dr affects c 0 and a, when y 1 > 0? The answer is c 0 (R) < 0 if the individual is a borrower, regardless the value of γ; while the sign of c 0 (R) is ambiguous for a lender if γ 1. A change in a can be computed using date-0 budget constraint, da = dc 0. Just, by reasoning, if γ 1 and the individual is a borrower, it must be that c 0 (R) < 0; now the wealth effect is negative and boosts up the substitution effect. If instead, the individual is a lender, the wealth effect is positive and could counterbalance the substitution effect, the sign of c 0 (R) is ambiguous. If γ < 1 and the individual is a borrower, income and wealth effect go in different directions and the sign of c 0 (R) is not obvious; however, computations show that it is again negative. If instead, the individual is a lender, the wealth effect is positive, it boosts the already prevailing income effect and results in c 0 (R) > 0. Analytically, first, show that individual demand for 0 consumption is, c 0 (R) = y 0R + y 1 (βr) 1/γ + R Then, computing its derivative and using the latest, one can write it in the following form, c 0(R) = y 0 c 0 (R) (βr) 1/γ + R c 0 (R) 1 γ γ[(βr) 1/γ (βr) γ + R] which makes clear why for borrowers c 0 (R) < 0, while for lenders things are trickier. As a last remark, you would like to consider that being a borrower or a lender is the outcome of an optimal decision, which depends on preferences (γ), on income (y 0, y 1 ) and on market opportunities R. More precisely, the individual is a borrower if and only if y 0 /y 1 (βr) 1/γ. It is interesting to notices that if βr > 1 individuals with an higher intertemporal elasticity of substitution 1/γ are more prone to be lenders/savers (i.e. they become borrowers only for a particularly low level of y 0 /y 1 ) Consumption and saving under imperfect financial markets. 11

13 In the stripped-down versions of the two models, agents where assumed to freely save and borrow on financial markets. In particular, in the LCH because the representative consumer is assumed to be either a young worker (earning positive income) or a retired one, the possibility to save was enough: consumption smoothing was accomplished uniquely by saving (or storing real resources) and dissaving (eating what had been stored before). In more general situations, such as those in which young individuals go to school prior to enter the labor market, the possibility to borrow becomes crucial; specifically, the underlined complete markets assumption would require that (in absence of some family collateral) young students are left free to borrow against future income (see, for example, the data on US median-household s net worth reported in the table). The implication of these more realistic assumptions are illustrated in the figure below age consumption assets income In this new context certain implications of the LCH change. For example the Ricardian equivalence proposition does not hold anymore: a borrowing constrained consumer would tend to increase consumption after a temporary increase in income, since this would now allow him to weaken his constraint. To see how the stripped-down LCH model changes when borrowing limits are introduced, consider the following. The consumer can only borrow at most b t+1 0 in each period t of his life. His per-period budget constraint is, at all t, c t y t + a t+1 a t (1 + r) = 0, a t+1 b t+1 with a 0 = 0 and y t = 0 for all N < t N. Forming the Lagrangian, he solves, max (c t,a t+1 ) T t=0 T β t u(c t ) t=0 T λ t [c t y t + a t+1 a t (1 + r)] + t=0 First order conditions yield, at all t, u (c t ) βu (c t+1 ) = λ t λ t+1 T µ t (a t+1 + b t+1 ) t=0 12

14 and λ t 1 + r with equality if a t+1 > b t+1 λ t+1 This implies that if in date t the borrowing limit is binding, u (c t ) βu > (1 + r) (c t+1 ) In words, a borrowing constrained consumer would be willing to pay an higher interest rate than the market in order to expand borrowing and current consumption; this does not contradict individual optimality since the borrowing constraint prevents him from doing so. Indeed, he can only consume (cfr. figure), c t c t y t + b t+1 + a t (1 + r) Graphically, at the individual optimum (the red dot in the figure) the IMRS exceeds the interest rate. In an efficient steady state, without binding borrowing constraints and β(1 + r) = 1, the individual would achieve full-consumption smoothing (i.e. u (c t )/u (c t+1 ) = 1 c t = c t+1 = c at all dates). With respect to this benchmark, suppose that the borrowing constraint binds at t only (a s b s holding with equality at s = t + 1 and strict inequality for all s > t + 1), still assuming β(1 + r) = 1, individual optimality yields, u (c t ) > u (c t+1 ) c t < c t+1 c by strict concavity of u( ). The strict concavity of u is responsible for the desire to smooth consumption.! "#$! %&'!(" )* " &+ "#$ &, " -%&'.! " Example 1.1. Assume that an individual is born with zero asset, a 0 = 0, and an initial income y 0 = ɛ < 1/2. His lifetime labor income is certain, but alternates between values of ɛ in even years (y 0, y 2, y 4,..) and (1 ɛ) in odd years (y 1, y 3, y 5,..). The individual can freely save, but cannot borrow (i.e. b t = 0 at all t T ). You can show that, if β(1 + r) = 1, the individual after experiencing a binding borrowing constraint at the initial date t = 0, immediately jumps 13

15 at full-consumption smoothing c = 1/2. Verify this by checking that such consumption profile is individually optimal at 1 + r = β 1. Do also compute saving and the asset position Some final remarks. Behind both theories, PIH and LCI, consumers decide how much to save and consume, rationally, forming expectations on their future prospects. In a deterministic setting (with no uncertainty) their predictions are quite straightforward: concavity of the utility function implies a desire to smooth consumption over time; consumption increases with current income only if that increase is a permanent one; otherwise, for temporary income fluctuations, consumption remains constant and saving (and wealth) is used to smooth out fluctuations in income. In the case of the life cycle model, the explicit consideration of retirement and the assumption on an hump-shape, lifetime income profile, represents the main motivation for saving. Indeed, under the LCI households accumulate wealth to provide for their consumption during retirement. An interesting implication of this model in its simplest incarnation is the way in which aggregate saving is generated. It is quite obvious that in a stationary life cycle economy with no growth (i.e. n, µ being zero) aggregate saving is zero: in every period, current young accumulate wealth that, while current old decumulate it. However, aggregate saving can be generated in the presence of growth; as the amount of resources available over the life cycle to younger generations is larger than that available to older ones. This introduces a relationship between aggregate saving and growth that Modigliani has stressed in several studies. Notice, however, that such a relationship depends on a number of factors including the life cycle profile, the way in which growth is generated and who benefits from it and so on. The PIH and the LCI were developed to provide an answer to several needs. First, by framing consumption decision within an intertemporal problem, immediately introduces dynamics into the picture. This gives the possibility of fitting some of the empirical facts that seemed at odd with the Keynesian consumption function, such as the difference between average and marginal propensity to consume in the short and long run. In addition, the introduction of dynamics is obtained in a theoretically consistent fashion which is appealing to economists. Obviously, the first empirical applications of the model were quite different from the studies of the last 20 years, mainly for the much more sophisticated treatment of uncertainty. The model, however, seemed to score a number of empirical successes. It accounts for differences between short run and long run responses of aggregate consumption to disposable income (or other variables). More generally, it was clear from the beginning that the model was able to generate very rich dynamic patterns for aggregate consumption and its response to disposable income. It could also explain the relationship between consumption and wealth and provide a rationale for the relationship between wealth-income ratios and growth. Indeed, as Modigliani has pointed out, the simplest version of the model is capable to generate an aggregate wealth to income ratio of 5 which is close to what this number is for the USA. Furthermore, the model also seemed able to explain some of the regularities observed in cross sectional data. 13 Just to mention one, Friedman showed how the permanent income model can explain the fact that black households seem to save, at each level of income, a larger fraction of their income than white households. 13 For more, see the discussion and references in, Attanasio, O. P. (1999). Consumption. Handbook of Macroeconomics, 1,

16 1.4. Consumption and saving under uncertainty and imperfect capital markets. Behind both theories, PIH and LCI there is the idea that individuals tend to smooth consumption over time. This clearly extend to cases in which income changes over agents lifetime are due to shocks. Negative shocks could be aggregate when personal income falls as a result of a macroeconomic downturn, or idiosyncratic (i.e. individual specific) when it is due to negative events such illness of a family member, or even the loss of ones job not due to cyclical conditions. When such shocks, positive and negative, are fully anticipated, a consumer essentially behaves as we have seen above. When shocks are not fully anticipated (i.e. the individual knows his income distribution, but cannot correctly anticipate the future realization of income), then there might be an additional component of saving, called precautionary, which is increasing in the consumer s degree of risk aversion. Although our all treatment is based on deterministic economies, a simple example can be introduced to illustrate this important point. Uncertainty is modeled assuming that an individual faces a life-time income profile, (y 0,..., y t,..., y T ) formed by a sequence of random variables. At t = 0, he knows the distribution of (y 0,..., y t,..., y T ) and he has the goal to maximize the expected value of his life-time income. The goal is, at all t, to maximize E T t t=0 βt u(c t ), where u > 0, u < 0 and lim c 0 u (c) = More precisely, at t = 0, the consumer solves, such that, at all t = 0,.., T, max E 0 T t=0 β t u(c t ) c t + 1 R a t+1 y t + a t, a 0 = 0, a T +1 0 This can be represented with the Lagrangian problem, { T [ max E T ( ) } 1 t 0 β t u(c t ) λ (c t y t ) a T +1] + γa T +1 (c t,a t+1 ) R t=0 whose first order (sufficient) conditions are, at every date t, t=0 (E) (IBC) u (c t ) = RβE t u (c t+1 ) T R t (c t y t ) R T a T +1 = 0 t=0 γa T +1 = 0 Clearly, at a solution a T +1 = 0. The Euler equation (E) says that next period expected consumption depends only on current period; precisely, the marginal utility behaves as a random walk The notation EtX t+1 is standard and indicates the expected value of a r.v. X t+1 computed conditionally on the information available at t (i.e. it denotes a conditional expectation). 15 The random walk model of consumption was introduced by Robert Hall: Hall, R. E. (1978). Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence. Journal of Political Economy, 86(6),

17 The implication of this relationship on consumption and saving changes depending on the class of per-period utility considered. If individuals have quadratic utilities, then marginal utilities are linear in consumption. This implies that consumption behaves as a random walk and saving has no precautionary component. If instead utilities are of other classes (e.g. CARA, CRRA, CES), then a precautionary motive emerges. We shall examine this next Quadratic utility: Hall s theory. Consider the case in which the economy has an efficient steady state, βr = 1. Moreover, assume that the per-period utility is quadratic, u(c) = c µ 2 c2 (µ > 0 but small ), whose marginal utility is linear, u (c) = 1 µc. 16 Hall s theory reads, (H) c t = E t c t+1 the individual wants to hold consumption constant over time. Next, use the budget constraint (IBC) and apply expectations (use also linearity of expectations), (IBC ) T R t E 0 c t = E 0 t=0 By the law of iterated expectations, T t=0 R t y t E 0 c t = E 0 E 1 c t =... = E 0 E 1 E t 2 E t 1 c t Now apply (H), sequentially, to the right hand side of the latest, Use the latest into (IBC), E 0 c t = E 0 E 1 E t 3 E t 2 E t 1 c t. = E 0 E 1 E t 3 E t 2 c t 1 = E 0 E 1 E t 3 c t 2 = E 0 c 1 = c 0 1 R (T +1) 1 R 1 c 0 = E 0 T R t y s More generally, letting T go to infinity and using βr = 1 (and β < 1), the left hand side of the latest equation reduces to (1 β) 1 c t. Thus, at all t, [ ] ( ) c t = (1 β) E t R s t y s + a t Therefore, ( ) is again consistent with the PIH and LCH: current consumption depends only on (the present value of) future expected income. Next, we can express c t in an even more s=t s=0 16 We want µ small relative to the levels of consumption the individual is likely to have so that u > 0. 16

18 intuitive form, as a function of c t 1 and an error correction term on the expected value of life-time income: [ ] c t c t 1 = (1 β) a t a t 1 + E t R (s t) y s E t 1 R (s (t 1)) y s = (1 β) = (1 β) = (1 β) = (1 β) = (R 1) [ [ [ [ s=t s=t 1 R(a t 1 + y t 1 c t 1 ) a t 1 + RE t R (s t 1) y s E t 1 R(a t 1 c t 1 ) a t 1 + Ry t 1 + E t R (s t) y s E t 1 (R 1)a t 1 Rc t 1 + RE t RE t s=t 1 s=t 1 R (s t) y s RE t 1 R (s t) (E t y s E t 1 y s ) s=t 1 s=t 1 s=t s=t R (s t) y s E t 1 R (s t) y s ] s=t 1 s=t 1 s=t 1 R (s t+1) y s ] R (s t+1) y s ] R (s t+1) y s ] where in the second to the last line we have used ( ) to substitute for Rc t 1 and carried out some simplifications. 17 In an easier-to-read notation, ( c) c t+1 = r [ E t P V (y t ) E t 1 P V (y t ) ] An individual changes consumption only if she receives news on her future income profile; that is if the individual realizes that she has made errors in forming expectations on her present value of life-time income. Instead, anticipated changes in income do not affect consumption and are absorbed by an adjustment of individual saving and of the net asset position. Most importantly, income variance does not affect consumption (only errors on the first moments count). This property is due to the particular form of the individual preferences: When u takes the linear form in consumption, u (c t ) = 1 µc t. the Euler equation (E), implies that, holding, if and only if, u (c t ) c t µc t = (E) E t (1 µc t+1 ) = 1 µe t c t+1 = u (E t c t+1 ) c t = E t c t+1 17 ( ) can be rewritten as, Rc t 1 = R(1 β)e t 1 = RE t 1 R (s t+1) y s R(1 β)a t 1 s=t 1 R (s t+1) y s E t 1 s=t 1 s=t 1 R (s t+1) y s Ra t 1 + a t 1 also using Rβ = 1. Then, this can be substituted back to obtain the last form. 17

19 This identifies a preference for the certainty equivalence; the decision under uncertainty is independent of the size of uncertainty, measured for example by the variance of income. There is a second important assumption, besides preferences, that explain this result: complete markets; namely, individuals can transfer resources from the future to the present by, for example, borrowing. Example 1.2 (Persistent shocks; income as an AR(1) process). Suppose labor income is representable by an AR(1) process; for all t, y t = γy t 1 + w t, 0 < γ < 1 w t is white noise with zero mean and variance σ 2. γ is a measure of the persistence of a shock. AR(1) is stationary; that is it a process whose distribution has moments (mean, variance, etc.) that are time-invariant. Under this assumption, forecast errors will be computed and then used to derive the implied consumer s behavior. To this end, notice the following, By induction, for all s t, {}}{{}}{ E t y t E t 1 y t = γy t 1 + E t w t γy t 1 E t 1 w t = w t {}}{{}}{ E t y t+1 E t 1 y t+1 = γy t + E t w t+1 γe t 1 y t E t 1 w t+1 0 w t = γ (y t E t 1 y t ) = γw t {}}{{}}{ E t y t+2 E t 1 y t+1 = γe t y t+1 + E t w t+2 γe t 1 y t+1 E t 1 w t+2 = γ (E t y t+1 E t 1 y t+1 ) = γ 2 w t E t y t+1+s E t 1 y t+s = γ s w t Substituting and carrying out some transformations, 18 ( ) 1 β c t = w t 1 βγ This says that the intensity with which consumption responds to an (unexpected) income shock increases in the persistence of the shock γ. When γ = 1, any income shock is permanent, c t = w t ; that is income behaves as a random walk, y t = y t 1 + w t, and so does consumption. Instead, when 0 < γ < 1, income shocks tend to shade away over time and c t < w t ; that is, consumption responds less than 1-to-1 to an income shock. This type of response is typical of consumption smoothing strategies and it is what we typically see in the data. If shocks are not too persistent, consumers react by reducing consumption less than proportionally. This is feasible because they can insure against unexpected shocks, for example, by running down 18 which yields the result. c t = (R 1) = R 1 R s=t 1 R (s t) (E ty s E t 1y s) R s γ s w t = (R 1)βw t s=0 0 s=0 0 (βγ) s 1 = (R 1)β 1 βγ wt

20 saving or borrowing through market channels (e.g. family). banks) or through social networks (e.g. The previous example gives an idea on how to empirically test the theory. Campbell and Deaton (1989) were the first to stress that, because equations ( ) and ( c) can be used to derive the relationship between changes in consumption and innovations to the process generating income, the relationship between the volatility of consumption (or permanent income) and that of current income depends on the stochastic properties of the process generating the latter. 19 In particular, if labor income is stationary, as in the example, the theory is confirmed: consumption is less volatile than current income. If, instead, labor income is difference stationary, consumption (and permanent income) will be more volatile than current income. 20 Intuitively, this result follows the fact that if labor income is not stationary, current innovations are persistent and will therefore imply a permanent revision of permanent income Utilities exhibiting prudence : precautionary saving. Quadratic utilities imply that marginal utility is linear in consumption. This, in turn, yields Hall theory: consumption behaves as a martingale. Again, this is a conceptually strong proposition, as it says that current consumption depends only on the expectation of future consumption, not on its variance. Moreover, because the marginal utility of consumption is also equal to the marginal utility of wealth, λ t = β t u (c t ), the marginal utility of wealth is also linear in consumption, hence in permanent income (i.e. wealth + PV of lifetime, labor income). In words, individuals with same preferences but different wealth experience the same marginal benefit from an increase of wealth. 22 Suppose, instead, that u is strictly convex (i.e. u > 0) and consider an economy that is at the efficient steady state, with βr = 1. Then, by Euler equation and Jensen s inequality, u (c t ) = E t u (c t+1 ) > u (E t c t+1 ) The individual is now more cautious and, since marginal utility is strictly decreasing in consumption (i.e. u < 0), he chooses, c t < E t c t+1 The reduction of current consumption and the consequent increment of saving, is what identifies a precautionary motive for saving. Notice that an individual with quadratic utility is risk averse (the quadratic has u = µ < 0); however, because quadratic utility implies certainty equivalence, the individual has no motive for precautionary saving. What is needed for precautionary saving is, prudence, u > 0. Just as the Arrow-Pratt degree of (absolute) risk aversion is measured by u /u, (i.e., the degree of concavity of u), the degree of (absolute) prudence is measured by u /u (i.e., the 19 Campbell, J., & Deaton, A. (1989). Why is consumption so smooth?. The Review of Economic Studies, 56(3), A process (yt) is different stationary if ( y t) is stationary. For example, a random walk y t = α + y t 1 + w t, w t w.n.(0, σ 2 ), is non-stationary; but y t = α + w t is stationary. 21 See Attanasio cit. for references and further discussion on the issue. 22 Here one has to be cautious. What we said is true only provided that both individuals are financially unconstrained. Thus, it is not so if the poor individual would like to consume more but is prevented from doing so because borrowing or liquidity constrained. 19

21 degree of convexity of marginal utility). The degree of risk aversion is important for the size of the required compensation for uncertainty, whereas the degree of prudence is important for how the households behavior is affected by uncertainty. The CARA function displays both risk aversion and prudence. The only less realistic feature of this form of utility is that it implies a risk premium that is constant with respect to wealth (or mean consumption). This is not really in line with data that see the risk premium declining with individual wealth; something that can be explained by the fact that wealthier individuals tend to have a decreasing degree of absolute risk aversion. However, as we discuss later, if we move to preferences which satisfy this last property we can still have prudence. More precisely, it is easy to show that if u > 0, u < 0, u > 0 is a necessary condition to have decreasing absolute risk aversion. 23 This condition is satisfied if, for example, u is of the CRRA class. Let us analyze the idea behind precautionary saving a little more in deep. 24 For expositional simplicity, consider an individual who lives only two periods, in which he earns income (y, y 1 ), and that can freely borrow and save at a zero interest rate (r = 0). Also assume that in the first period there is uncertainty on y 1, that Ey 1 = 0 and that per-period utilities are twice continuously differentiable, strictly increasing and strictly concave. The individual solves, At an individual optimum, c must satisfy, max u(c) + βeu(y c + y 1 ) c u (c) = βeu (y c + y 1 ) Notice that, certainty equivalence does not hold, u (c) = βeu (y c + y 1 ) > βu (y c + Ey 1 ) βu (y c) where the inequality follows from Jensen s inequality and u > 0 (i.e. u being strictly convex). Hence, by u < 0, there exists a ψ o > 0 such that Eu (y c+y 1 +ψ o ) = u (y c). This implies, that first period consumption is unaltered. More precisely, coming from a world of certainty with a consumption choice c, the individual who now faces an uncertain future income, receives a premium ψ o to compensate the effect of risk, ψ o is a compensating precautionary premium; ψ o has the effect of allowing him to keep on consuming c : 25 u (c ) = βeu (y c + y 1 + ψ o ) Analogously, we can determine a ψ > 0 such that Eu (y c + y 1 ) = u (y c ψ ); here ψ measures the premium the individual pays to eliminating risk and is called the equivalent precautionary premium. Now, the individually optimal consumption is c = c ψ, (Psi) u (c ψ ) = βeu (y c + y 1 ) 23 Just differentiate u (c)/u (c) with respect to c. 24 Interested readers can look up at Kimball, M. S. (1990). Precautionary Saving in the Small and in the Large. Econometrica, In a world of certainty, y1 = Ey 1 = 0 and the solution of the individual problem would is c > 0 s.t. u (c ) = βu (y c ). 20

22 Proposition 1. Let u be twice continuously differentiable with u, u > 0. Assume that y 1 is a random variable with zero mean and variance σ 2. Then, the equivalent precautionary premium takes the following form, [ u (y c ] ) ψ β σ2 2 u (c ) This is positive (respectively, zero) if and only if u > 0 and β > 0 (resp., u = 0 or β = 0). Consider equation (Psi) and take the Taylor approximation of each of its two terms around the individually optimal consumption obtained in absence of uncertainty (c, y c ). u (c ψ ) u (c ) + u (c )( ψ ) u (y c + y 1 ) u (y c ) + u (y c )(y 1 ) u (y c )(y 1 ) 2 Taking expectations of the latest and using the fact that c satisfies individual optimality, u (c ) = βu (y c ), Eu (y c + y 1 ) u (y c ) + u (y c )E(y 1 ) u (y c )E(y 1 ) 2 = β 1 u (c ) + σ2 2 u (y c ) Substituting the approximations in (Psi) yields the result. A similar derivation can be used to attain the compensating precautionary premium ψ o Consumption and saving when utilities are not quadratic. We have seen that when preferences are quadratic an explicit (or closed form) solution of the consumption (and saving) function is available. Yet, the derived function is such that income risk plays no direct role; only the means of unpredicted income innovations affect consumption decisions. The problem is that it is quite difficult to obtain analytic solutions when preferences are not quadratic. This implies that it becomes problematic to evaluate the propensity to consume in response to risk (income shocks). An exception is represented by very few studies, who assume particular classes of utilities and stochastic process of disposable, labor income. Among these, we shall briefly go through Caballero (1990, 1991) in which it is assumed that preferences are of the class of CARA. 26 Another notable case is when preferences are CRRA (having the more realistic properties of a decreasing coefficient of absolute risk aversion) and consumption innovation are log-normal. The rest of the literature essentially splits in two: studies who propose local approximations of the Euler equation, similar to that used above, under a series of other assumptions; 27 studies that are also based on the estimation of the Euler equation and on calibrated models (e.g. Zeldes, 1989) Caballero, R. J. (1990). Consumption puzzles and precautionary savings. Journal of monetary economics, 25(1), Caballero, R. J. (1991). Earnings uncertainty and aggregate wealth accumulation. The American Economic Review, See, for example, Skinner, J. (1988). Risky income, life cycle consumption, and precautionary savings. Journal of Monetary Economics, 22(2), ; or in its working paper version NBER No of the year Zeldes, S. P. (1989). Optimal consumption with stochastic income: Deviations from certainty equivalence. The Quarterly Journal of Economics, 104(2), A more recent, useful reference is Blundell, R., Pistaferri, 21