1 mact i – lecture i 7 th december 2009 mact i- december 7 th 2009 consumption and saving orazio p...

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MACT I – Lecture I

7th December 2009

MACT I- December 7th 2009

Consumption and Saving

Orazio P Attanasio

Key reference: “Consumption and Saving: Models of Intertemporal Allocation and

Their Implications for Public Policy” – joint with Guglielmo Weber, University of PaduaUn

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7th December 2009

How the life-cycle/permanent income model came about

“The fundamental psychological law … is that men are disposed, as a rule and on the average, to increase their consumption as their income increases, but not by as much as the increase in their income.” (Keynes)

This sentence was the basis for the static “keynesian” consumption function: C= a + bY + error

But: on macro data, the marginal propensity to consume is lower in the short run than in the long run (Kuznets paradox)

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7th December 2009


Also: saving rates change systematically with income

For instance: Friedman noted that groups of individuals with, on average, lower level of income (such as blacks) had higher saving rates than other groups with higher levels of average income (such as whites) at any income level.

Still true to this date. Pooling together US micro data on income and consumption over a number of year (CEX 1980-2003) we find that the median saving rate for blacks is 2.55% higher than for whites, after controlling for real income

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7th December 2009


Let us define the saving rate as:

where yd is disposable income, rent is housing rent and nd denotes expenditure on non durable goods and services.

A LAD regression on real disposable income and a dummy for blacks yields:

Median regression Number of obs = 1366058

Min sum of deviations 1539489 Pseudo R2 = 0.1840

------------------------------------------------------------------------------

s_rate | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

r_ny | .0035721 3.89e-06 917.90 0.000 .0035645 .0035797

black | .0255026 .0034703 7.35 0.000 .0187009 .0323043

_cons | .015157 .0017841 8.50 0.000 .0116602 .0186537

------------------------------------------------------------------------------

nd

ndrentydrates

)(_

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7th December 2009


Also: saving rates change systematically with changes in income

People whose income has increased save more than people whose income has decreased (Katona, 1949)

Modigliani and Brumberg (1954) and Friedman (1957) developed models of intertemporal allocation of consumption that could explain these stylized facts

In both models, consumption is a function of available resources over a long time horizon (life-cycle wealth or permanent income)

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7th December 2009

How the life-cycle/permanent income model works

The life cycle/ permanent income model offers a coherent explanation for the stylized facts.

Consumers have concave utility functions and therefore prefer smooth paths of consumption (over time and across states of the world) over variable ones. Only unanticipated changes in income that are perceived as permanent induce substantive changes in consumption. Expected and temporary changes to income should not induce a strong change in consumption.

The explanation of the stylized facts above boils down to the observation that a large fraction of the changes in income considered are temporary.

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7th December 2009

How the life-cycle/permanent income model explains the stylized facts

Kuznets paradox: short run fluctuations in disposable income are more likely to be dominated by the variance of temporary shocks that would be averaged out in the long run. That is why consumption is more responsive to income in the long run than in the short run

The saving rates of blacks is higher than that of whites, at any income level, because the permanent income of blacks is lower and therefore, conditioning on a common income level, one selects the blacks with higher level of temporary shocks that should, according to the permanent income/life-cycle model, be saved

Individuals with income increases are more likely to be affected by positive transitory shocks, that should be saved according to the model.

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7th December 2009

Violations of the theoretical predictions

Empirical criticisms of the life cycle model since Hall (1978) have pointed to deviations from the prediction that expected changes in income should not be incorporated into consumption.

These deviations can be classified into three groups: those that identify correlations between expected

changes in income and consumption at low frequencies (“income tracking”)

those that consider short run fluctuations linked to changes in earnings and income (“the retirement consumption puzzle”)

those that refer to short run fluctuations that are linked to ad-hoc payments non necessarily related to labour supply behaviour (“excess sensitivity to transitory receipts”).

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7th December 2009

Violations of the theoretical predictions: consumption tracks income

Carroll and Summers show that life cycle profiles of income and consumption track each other.

For many countries both income and consumption life cycle profiles are hump shaped, in that they increase during the first part of the life cycle to reach a peak a few years before retirement and decline afterwards

Also groups and countries that exhibit relatively ‘steep’ income profiles also exhibit relatively ‘steep’ consumption profiles.

We can replicate their findings using UK FES data (1978-2007), where individuals are grouped according to their education attainment

It is worth distinguishing between total expenditure and expenditure on non durable goods and services

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7th December 2009

Violations of the theoretical predictions: consumption tracks income - total expenditure

200

400

600

800

25 35 45 55 65 75 25 35 45 55 65 75

Compulsory Post-compulsoryLevels, By Educ Levels, By Educ

Income Expenditure

Age of Head

Graphs by educ

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7th December 2009

Violations of the theoretical predictions: consumption tracks income – non-durables

20

04

00

60

08

00

25 35 45 55 65 75 25 35 45 55 65 75

Compulsory Post-compulsoryLevels, By Educ Levels, By Educ

Income Consumption

Inco

me

an

d C

on

su

mp

tio

n

Age of Head

Graphs by educ

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7th December 2009

Violations of the theoretical predictions: retirement consumption drop

Consumption drops around retirement are documented for the UK (Banks, Blundell and Tanner, 1998), for the US (Bernheim, Skinner and Weinberg, 2001), and for Italy (Battistin, Brugiavini, Rettore and Weber, 2009)

Banks et al (1998) find that there is a cumulated consumption shortfall over the 60-67 age band, where most people retire, of around 10%.

Bernheim et al (2001) estimate a median drop of 14%, but higher drops for low wealth, low income replacement households..

Battistin et al (2009) estimate at 9.8 percent the part of the non-durable consumption drop that is associated with retirement (food expenditure falls instead by 14%).

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7th December 2009

Violations of the theoretical predictions: excess sensitivity to predicted income changes

Evidence that consumption reacts to income at the business cycle frequency – dominated by changes in earnings (and hours of work)

Recent work on how consumption reacts to changes in income that are predictable and driven by events that do not affect hours worked or labour force participation, such as tax refunds or other changes linked to administrative issues (Souleles, 1999, Parker, 1999, Hsieh, 2003, Browning and Collado, 2001, and Stephens, 2007).

Souleles, Parker, Stephens and, in part, Hsieh find that consumption reacts to changes in predictable resources available to consumers. Browning and Collado and the second part of Hsieh’s paper find that consumers do not respond to such predictable changes in resources.

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7th December 2009

Theoretical predictions on the cross-sectional evolution of consumption

The evolution of the cross section distribution of consumption – and income - can be informative about the relevant model that describes the data.

Deaton and Paxson (1994) note that if income has a unit root, in a basic life cycle model, the cross sectional variance of consumption increases over time. As innovations accumulate, the cross sectional distribution of consumption fans out with age.

Blundell and Preston (1998) show that the relative evolution of consumption and income inequality can be used to identify permanent and transitory income variances changes, if the market structure is sufficiently simple.

Evolution of second moments can shed light on the amount of risk-sharing – if the data are of sufficiently high quality!

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7th December 2009

The life-cycle model

The version of the model we consider is one in which a consumer unit maximizes expected utility over a finite interval subject to a set of constraints

tT

jjtjtjtjtt zCUEMax

0

),,(

jtjtjtjtjt cyRWW )1( *1

ijt

N

ijt AW

1

ijt

N

i

ijtjt RR

1

*

0TW

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7th December 2009


The discount factor, , is time varying – reflecting survival probabilities

The observable taste shifters variables, z, can be a choice variable (leisure) or not (demographics)

There are unobservable factors, v , that also affect utility The household behaves as a unit (Browning, 2000, and

Mazzocco, 2007, investigate saving decisions in a collective model)

There are N assets whose returns do not depend on the position taken by the household (short, long etc)

The household maximizes expected utility (Epstein and Zin, 1989, relax this assumption)

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7th December 2009


There is an end-point constraint on net worth that must hold with unit probability, that is the consumer has to die without debt, or pay back all her debt with probability one. This imposes important limitations to the ability to smooth consumption. If the income process is not bounded away from zero and the marginal utility of consumption tends to infinity at very low levels of consumption, then the consumer will never want to borrow in such a situation.

Even if the income process is bounded away from zero, the consumer will not want to borrow more than the present value of the lowest level of income. Similar considerations apply whenever the survival probability is less than one, if longevity risks cannot be fully insured.

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7th December 2009


Flexible versions of this model can explain all three the stylised facts discussed above.

In particular, we shall show that the hump in the age profile of consumption is due to the

interplay of demographics and prudence the excess sensitivity of consumption growth to income

growth can be due to the dependence of the marginal utility of consumption on leisure

the retirement consumption drop is due partly to adverse shocks inducing retirement, partly to more efficient shopping that is made convenient by the increased leisure time.

But: To explain excess sensitivity to some ad-hoc income changes one may have to appeal to liquidity constraints.

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7th December 2009


To prove this we need to work out the solution to the optimization problem.

Some features of the solution can be understood by looking at the first order conditions, others require the derivation of the consumption function, either analytically (in some special cases) or numerically.

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7th December 2009

The life-cycle model – closed form solution under certainty equivalence

If utility is quadratic, there is only one asset that yields a fixed return R such that R=1, then the first order condition w.r.t. consumption (the Euler equation) is

and consumption is a random walk (Hall, 1978).

for all variables W known at time t. The consumption function is (Flavin, 1981):

ttt CICE )|( 1

0)|( 111 ttttt WECC

)|(11 0

tktk

tt IyEr

rAr

rC

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7th December 2009

The life-cycle model – closed form solution under certainty equivalence

The Euler equation highlights the consumption smoothing properties of the solution, emphasized by Modigliani and Brumberg (1954).

The consumption function makes clear the other main implication of the model, first stressed in Friedman (1957): consumption depends on the present discounted value of future expected income. The interest rate plays the role of converting future resources to present ones and therefore constitutes an important determinant of consumption.

Another implication of the model is that in appraising the effects of a given policy, for instance a tax reform that affects disposable income, a distinction must be drawn between permanent and temporary changes (Blinder and Deaton, 1985, Poterba, 1988).

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7th December 2009

The life-cycle model – precautionary saving

The derivation of a closed form solution for consumption when certainty equivalence does not hold is possible in the case where the utility function exhibits constant absolute risk aversion.

Caballero (1991) shows that with certain finite life and CARA preferences the optimal consumption age profile is flat with no uncertainty, but increasing with income uncertainty. This change in the slope of the consumption profile is labelled as precautionary saving and requires prudence (U”’>0).

The presence and size of precautionary savings is a matter of great relevance for public policy, in so far as public insurance schemes covering such risks as unemployment, health and longevity should reduce the need for consumers to accumulate assets.

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7th December 2009

The life-cycle model – the Euler equation

Quadratic utility and CARA have implausible implications for intertemporal substitution and portfolio choice

Also, there are several assets and their interest rates are not fixed In general we can then work with a set of Euler equations:

where t is the Lagrange multiplier associated to the budget constraint at t and can be interpreted as the marginal utility of wealth.

tctU

tkttt IrE |)1( 11

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7th December 2009


Euler equations let researchers be agnostic about the stochastic environment faced by the consumer, the time horizon, the presence of a bequest motive or of imperfections in financial markets (as long as there is at least one asset that the consumer can freely trade), and frictions in other variables affecting utility, z. All relevant information is summarized in the level of the marginal utility of wealth.

The Euler equation is particularly useful from an empirical point of view because it allows estimating preference parameters and testing the validity of the model without having to solve the dynamic optimization problem for consumption or other variables jointly determined with consumption.

Estimation of the Euler equation requires observations covering a long period of time (Chamberlain, 1984, Hayashi, 1987).

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7th December 2009


The first paper to estimate a consumption Euler equation (Hall, 1978) was entirely devoted to testing the model, as it focused on the case of quadratic utility and a fixed interest rate such that R=1. Under these conditions, the random walk equation obtains and preference parameters are not identified.

The Euler equation implies that no variable known to the consumer at time t should help predict the change in consumption between t and (t+1) – an important implication of the intertemporal optimization model that has been rejected on aggregate and micro data alike (Jappelli and Pagano, 1989, Hall and Mishkin, 1982).

Also for this reason, more credible preference structures have been used.

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7th December 2009


The iso-elastic, or CRRA, utility function has been mostly used

where 1/ is the elasticity of intertemporal substitution.This implies the following Euler equation:

1

1)(

1CcU

1)1( 11

kt

t

tt r

C

CE

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7th December 2009

That can be estimated in its log-linearized form

as long as suitable instruments can be found.

This gives us an estimate of the elasticity of intertemporal substitution (EIS), a particularly important parameter for policy purposes: it tells us how the marginal rate of substitution between today and tomorrow’s consumption reacts to changes in the interest rate, keeping life-time utility constant.

kt

kttt rC 1111 )1ln(

1ln


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7th December 2009

The increase in the interest rate represents a decrease in the price of future consumption relative to current consumption, and this induces a ‘substitution effect’ of a decrease in current consumption and a commensurate increase in current saving. This is counteracted by an ‘income effect’ since with a higher interest rate a given target level of future consumption is achieved with less saving. As noted by Summers (1981), wealth effects concerning the amount that expected future incomes are discounted reinforce substitution effects and also lead to a decrease in consumption or increase in saving when the interest rate goes up.

Ultimately, which of these forces dominates depends on preference parameters and is, therefore, an empirical issue, that depends on the size of the EIS.

The life-cycle model – the EIS

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7th December 2009

Hall (1988) claims that the EIS is close to zero. But a low response of consumption growth to the real interest rate could obtain if some consumers are liquidity constrained, or if the error term correlates with the part of the interest rate explained by the instruments. Attanasio and Weber (1993, 1995) stress aggregation bias can be responsible for such correlation. They use cohort data: when they focus on cohorts of individuals who are least likely to be liquidity constrained, and control for changes in taste shifters, they estimate an elasticity around 0.8 using both UK and US data.

Scholz et al. (2006) address the issue of how well the life cycle model predicts wealth holdings: their model fits best when EIS = 0.67. Engelhardt and Kumar (2007) use differences in employer’s matching rates in 401(k)s and its effect on participation to identify the EIS, and obtain a point estimate of 0.74.

The life-cycle model – the EIS

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7th December 2009

Hall (1988) claims that the EIS is close to zero. But a low response of consumption growth to the real interest rate could obtain if some consumers are liquidity constrained.

Campbell and Mankiw (1990 and 1991) find consumption growth reacts to expected income growth => either ‘rule of thumb’ consumers, or liquidity constraints (or leisure in the utility function!)

If, in addition to the non-bankruptcy constraint, one imposes some exogenous and more stringent limits on the amount people can borrow, it is possible that consumers will be constrained in a given period.

The Euler equation becomes an inequality – consumption may track income when the constraint is binding

Even when it is not binding, the properties of the solution change (the time horizon gets shorter – Hayashi 1987)

The life-cycle model – (exogenous) liquidity constraints

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7th December 2009

Explaining the violations: consumption tracks income


If there is productivity growth, pooling individuals born in different periods provides misleading information on the age profile

For this reason, we normally group people in year-of-birth cohorts.

Using FES data on income and non-durable consumption, we get the following picture – income and consumption are still hump shaped in a similar way

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7th December 2009

Violations of the theoretical predictions: consumption tracks income – cohort analysis

05

00

10

00

20 40 60 80 20 40 60 80

Compulsory Post-compulsoryCohort profiles Cohort profiles

Income Consumption

Age of Head

Graphs by educ

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7th December 2009



A similar point had been made by Thurow (1969) – Heckman (1974) pointed out that leisure could explain it (wages in middle age are higher – so individuals work more and earn more. They consume less leisure and compensate by spending more on goods and services)

More generally, our analysis suggests that we should control for demographics.

Simplest way is to deflate income and consumption by the number of “equivalent adults”

On FES data, we obtain the following:

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7th December 2009

Violations of the theoretical predictions: consumption tracks income – cohort analysis

10

02

00

30

04

00

50

0

20 40 60 80 20 40 60 80

Compulsory Post-compulsoryPer-capita, by cohort Per-capita, by cohort

Income Consumption

Age of Head

Graphs by educ

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7th December 2009


Tracking is even less visible here In general, if utility is specified as

where Z includes leisure and family composition variables, excess sensitivity of consumption growth to income growth disappears, and so does tracking

Attanasio, Banks, Meghir and Weber (1999) address this issue by solving the life-cycle model under uncertainty (“Humps and Bumps” paper), based on estimated age and demographic profile of four education groups.

1

)( 1t

Z

t

Ceu

t

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7th December 2009

Explaining income tracking: high school graduates (Attanasio et al, 1999)

.81

1.2

1.4

1.6

20 30 40 50 60age

consumption income

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7th December 2009

Explaining income tracking: college graduates (Attanasio et al, 1999)

.51

1.5

22.5

20 30 40 50 60age

consumption income

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7th December 2009


Attanasio et al. (1999) find that the increase in household size early in life, and decrease past age fifty, can explain why consumption age profiles are hump-shaped, in apparent contradiction of the consumption smoothing implications of the life-cycle theory.

Precautionary savings alone would instead imply a peak in consumption quite late in life

The interaction between demographics and prudence explains why the peak in consumption occurs later in life than the peak in household size

This interaction generates consumption-income tracking until age 45 for four different education groups when labour income is uncertain.

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7th December 2009

Simulating the model: high school graduates (Attanasio et al, 1999)

.6.8

11

.21

.4c1

20 30 40 50 60 70age

baseline

.6.8

11

.21

.4c2

20 30 40 50 60 70age

no demographics

.6.8

11

.21

.4c3

20 30 40 50 60 70age

no uncertainty

.6.8

11

.21

.4

20 30 40 50 60 70age

comparison

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7th December 2009

Explaining the violations: retirement consumption puzzle

The recent literature has focused on estimating how consumption levels change around retirement. The existence of a consumption fall around retirement is documented for the UK, the US and Italy and is known as the retirement consumption puzzle.

Retirement implies an increase in leisure time, with improved shopping and home production opportunities – a drop in expenditure may not translate into a drop in utility (Aguiar and Hurst, 2005 and 2007). Also, work-related expenses are no longer needed

Retirement may be the result of adverse health shocks – a drop in utility may not violate the life-cycle model predictions (Smith, 2006, Blau, 2008)

Retirement may also induce grown children to leave home (Battistin et al, 2009)

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7th December 2009

Explaining the violations: retirement consumption puzzle

Once preferences are correctly modelled, home production is taken into account, and attention is focussed on those who retire at the expected age, then the drop in (per-capita) food spending and total spending around retirement does not imply a violation of the model prediction that consumers smooth marginal utility over time.

As Hurst (2008) recently put it, we should no longer talk about the retirement consumption puzzle, rather about “the retirement of a consumption puzzle”.

This literature suggests that we should pay attention to the composition in consumption: consumption of work-related goods falls with age, consumption of leisure-intensive goods increases.

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7th December 2009

Explaining the violations: excess sensitivity to leisure-unrelated income changes

Recent papers have estimated non-zero effects of predictable tax changes (such as tax rebates, social security withholding tax) (Parker, 1999, Souleles, 1999, Shapiro and Slemrod, 2003, Johnson et al, 2004).

However, consumption does not appear to react to other anticipated income changes (Browning and Collado, 2003, Hsieh, 2003).

Consumption reacts to predicted changes in disposable income, if these changes are relatively small (Browning and Crossley, 2007). These small changes tend to be tax rebates or other predictable changes in tax-related payments – maybe consumers deep down distrust the government, and are surprised when they receive what they are due.

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7th December 2009

Explaining the violations: excess sensitivity to leisure-unrelated income changes

Recent papers report evidence in favour of a liquidity constraints interpretation.

Stephens (2008) shows that consumption reacts to the repayment of vehicle loans, more for young individuals, who are more likely to be liquidity constrained. Agarwal, Liu and Souleles (2007) investigate credit cardholders’ response to the 2001 tax rebates: most people first increased repayments, but then the young and those initially close to their credit card limit started spending more. The eventual rise in spending could be attributed to the operation of liquidity constraints.

Hsieh, Shimizutani and Hori (2008) find that Japanese consumers’ response to a spending coupon program tailored to families with children and the elderly was highest among those with low wealth.

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7th December 2009

Solving the model: excess sensitivity and excess smoothness

Under quadratic utility, we can derive the closed form solution of the optimization problem, and relate the error term in the Euler equation to the error term in the income process.

Assume labour income is given by

then the following relation holds (Flavin 1981) :

tt ayLA )(

11 1)

1

1(

tt r

rC

rA

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7th December 2009


If there is positive serial correlation in the first differenced process of income, then consumption growth should vary more than income growth over time.

Flavin (1981) and Campbell (1987) test the cross equation restrictions that arise in the quadratic utility case in the VAR representation of income and consumption

Campbell and Deaton (1989) and West (1988) use the same structure to propose a test that links the innovation to permanent income to consumption.

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7th December 2009


Campbell and Deaton (1989) present evidence that aggregate consumption is ‘excessively smooth’ in that it does not react enough to news about income.

Because the model for earnings is characterized by a unit root plus persistence in first differences, the model implies consumption changes should reflect the permanent income innovation more than one-to-one.

Then over the business cycle consumption should be more volatile than income. But in actual aggregate data consumption is smoother than income.

Hansen Roberds and Sargent (1991) propose a test of the intertemporal budget constraint that can be used to shed light on market structure.

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7th December 2009

Solving and simulating the more general model

Deaton (1991) derives the infinite life problem – and shows that liquidity constraints rarely bind in equilibrium.

Carroll (1992) covers the case of finite lives, and shows that if consumers are sufficiently impatient and their labour income is subject to both permanent and temporary shocks, they set consumption close to income, at least until they are in their forties. The model with impatient consumers under labour income uncertainty has been labelled “the buffer stock model”, because saving is kept to the lowest level compatible with the need to buffer negative income shocks.

Carroll’s buffer stock model can provide a rationale for the income tracking of consumption

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7th December 2009

Solving and simulating the more general model

Hubbard, Skinner and Zeldes (1994 and 1995) show instead how precautionary motives interact with the insurance properties of social security in the US.

Attanasio et al. (1999) – later refined by Gourinchas and Parker (2002) - clarifies the role played by age-related changes in demographics and the hump-shape age profile of labour income in generating income tracking for relatively young consumers

In recent years a number of papers have solved the model to ascertain the role played by health risks, longevity risk, business risk etc.

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7th December 2009

Solving the model: interest rate changes (and taxation)

Simulation results are also useful to study how consumption reacts to changes in the interest rate, an important topic for both monetary and fiscal policies.

Attanasio and Wakefield (2008) simulate a life cycle model to understand the importance of the elasticity of intertemporal substitution to determine the size of the reaction of savings to changes in the interest rate. They first simulate a model of a single consumer with an isoelastic utility function (they take EIS=1 as baseline), no bequest motive and a stochastic income process calibrated on UK data.

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7th December 2009

Solving the model: changes in life cycle asset profile induced by an interest rate rise from 2% to 2.5%

-0.5

0

0.5

1

1.5

2

2.521 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

Age

Asse

ts..

Baseline EIS = 1/2 EIS = 1/4

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7th December 2009

Solving the model: changes in life cycle asset profile induced by an interest rate rise from 2% to 2.5%

Attanasio and Wakefield also explore the role of demographics, and find they reduce responses to interest rate changes

-0.5

0

0.5

1

1.5

2

2.5

21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

Age

Ass

ets.

.

Baseline Family

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7th December 2009

TO DO LIST

In the next lecture, we shall look at the role played by markets

We shall consider models with complete markets (full insurance), and with endogenously incomplete markets

These can be classified in imperfect information models and imperfect enforceability models

We shall also look at some other stylized facts the standard model (even with market imperfections) cannot explain

Finally, we shall review models that can in principle explain these stylized facts: hyperbolic discounting and temptation, habits, lack of financial literacy.

1 mact i – lecture i 7 th december 2009 mact i- december 7 th 2009 consumption and saving orazio p...

Documents

income level

income people

income increases

permanent income slide

real income slide

real disposable income

lower level of income

higher saving rates