Maco Theoy B The Pemanent Income Hypothesis Ofe Setty The Eitan Beglas School of Economics - Tel Aviv Univesity May 15, 2015 1
1 Motivation 1.1 An econometic check We want to build an empiical model with eal data and check which of the two solutions, Autaky o CM, is moe likely to be the eal one. The Autaky solution was: c i t(s t ) = y i t(s t ) We apply log on both sides and get: log(c i t(s t )) = log(yt(s i t )) log(c i t+1(s t )) = log(yt+1(s i t )) log(c i t) = log(yt) i The CM solution was: c i t(s t ) = (α i) 1 σ j (α C j) 1 t (s t ) σ We apply log on both sides and get: log(c i t(s t )) = const + log(c t (s t )) log(c i t+1(s t )) = const + log(c t+1 (s t )) log(c i t) = log(c t (s t )) Fom hee we build the empiical model: We look at the null assumptions: Autaky: H 0 : β 1 = 1, β 2 = 0 CM: H 0 : β 1 = 0, β 2 = 1 log(c i t) = β 1 log(y i t) + β 2 log(c t ) Unsupisingly, both assumptions can be ejected! Neithe model is a good one. 1.2 Motivations fo saving We ae inteested in savings, and in modeling heteogenous levels of savings, because we want the model to yield a meaningful distibution of savings in the end, as opposed to a single value descibed by a epesentative agent. Since we want to model savings coectly we ae inteested in the motivations fo saving: 2
1. Intetempoal: β(1 + ) 1. In geneal tems, the HH s poblem satisfies: u (c t ) = β(1 + )u (c t+1 ) (in the case of uncetainty E t should be added). If β(1+) < 1 Loan (negative savings): This means that the individuals has a net loss fom tansfeing money to the next peiod. This is an incentive to consume cuent income and to loan money fom the futue to incease consumption now. If β(1 + ) > 1 Save: This means that the individuals has a net gain fom tansfeing money to the next peiod. This is an incentive to fogo consumption in the cuent peiod and to save pat of this income fo the futue. If β(1 + ) = 1 Indiffeence: In this case the individual is indiffeent bewteen loaning and saving. This is why we deive the 1 condition as an incentive fo saving (eithe positive o negative). 2. Boowing Constaint: In ode to smooth consumption the individual must boow/lend money. If we impose a limit to the amount that it is possible to boow, the individual has to take this into account (this is a cedit maket impefection). He does so by inceasing savings at othe times to miminize the possibility of eaching the limit when boowing is needed. We note that such a limit must be imposed if we ae eve to each an equilibium whee the makets clea, because if the agents can boow an infinite amount this will neve be possible. 3. Pecautionay Savings (Pudence): Hee we make an assumption on the popeties of u(c): That u > 0 (i.e. u is convex). This means that given a stochastic pocess with mean ȳ, if we incease its vaiance (i.e incease the measue of uncetainty) but maintain the mean, the individual will still want to incease his savings. In othe wods, inducing a mean peseving tansfomation on y t affects the individuals savings. 4. Othe: Life cycle, inte geneational/bequest etc. 2 PIH - Pemanent Income Hypothesis Assumptions: 1. β(1 + ) = 1 We want to neutalize this obvious incentive to save/lend, and see if the othe facets of the economy can geneate savings. 2. u(c t ) = b 1 c t 1 2 b 2c 2 t, u > 0 c b1 b 2, u < 0 b 2 > 0 We assume that the utility function is a paabola with a maximum. 2.1 The HH This is in essence an extension of the CM model, and so the individual s poblem is the same. 3
The HH s maximize the following: E 0 t=0 β t u(c t ) In CM, the sequential fomulation fo the RC was: a i t(s t ) + yt(s i t ) = c i t(s t ) + q t+1 (s t+1 )a i t+1(s t+1 ) s t+1 In a bond economy thee is by definition only patial insuance, so we have to limit the numbe of possible assets. We choose to limit it to a single asset which is not dependent on the ealization of histoy: a t+1 (s t ) a t+1. Similaly, we give this asset a pice that is not dependent on the ealization of histoy: q t+1 (s t+1 ) q. We get that the RC is: a t + y t = c t + q a t+1 Since q is the same fo all assets, it is also the pice fo the aggegate assets in the economy. It can theefoe be thought of the pice of assets, i.e. the inteest ate. Hee we fomulate the inteest ate so it satisfies: Ra t+1 = a t+2,r:= 1 1+. So q = 1 1+. We get: a t + y t = c t + 1 1 + a t+1 a t+1 = (a t + y t c t )(1 + ) c t = a t + y t a t+1 1 + (1) We now fomulate fully the HH s poblem: max E 0 t=0 βt u(c t ) s.t : c t = a t + y t at+1 1+ Inseting the RC into the poblem, we get: FOC s w..t a t+1 : max at+1 E 0 t=0 β t u(a t + y t a t+1 1 + ) β t u 1 (c t ) 1 + + E 0β t+1 u (c t+1 ) = 0 4
In peiod t we get: By assumption (1) of PIH we get: Fom assumption (2) of PIH we get: This dynamic is called a Matingale. u (c t ) = β(1 + )E t u (c t+1 ) u (c t ) = E t u (c t+1 ) 2.2 Solving fo the Consumption c t = E t (c t+1 ) (2) We use the soultion of the HH s poblem and plug it back into the RC. But fist we fomulate some expessions: At time t: c t = a t + y t at+1 1+ a t = c t y t + at+1 1+ At time t + 1: a t+1 = c t+1 y t+1 + at+2 1+ At time t: E t (a t+1 ) = E t (c t+1 y t+1 + at+2 1+ ) Now we plug these expessions into the sequential RC, equation (1), and get: c t = a t +y t 1 1 + [E t(c t+1 y t+1 + a t+2 1 + )] = a t+y t E t ( c t+1 1 + y t+1 1 + + a t+2 (1 + ) 2 ) c t + E t(c t+1 ) 1 + }{{} E t (c t+j ) (1 + ) j }{{} A = a t + y t + E t(y t+1 ) 1 + = a t + } {{ } E t ( ) (1 + ) j lim E t(a t+2 ) (1 + ) 2 }{{} j E t (a t+j ) (1 + ) j } {{ } B Fo optimality, we impose the tansvesality condition, which is: B = 0. We ecall the ationale fo this: If a t+j is high enough to yield B > 0 then this lowes the value of A, meaning it lowes the consumption steam. Since utility is equated with consumption in this model, this is obviously not an optimal path. If a t+j is low enough to yield B < 0 then this means that the individual has incued a debt which will neve be epayed. We assume that this can t happen since no lende will be willing to loan money which will neve be epaid. 5
So we get: E t (c t+j ) (1 + ) j = a t + E t ( ) (1 + ) j (3) Note: This is vey simila to the CM solution, which was: yt(s i t )Pt 0 (s t ) = c i t(s t )Pt 0 (s t ) t=0 s t t=0 s t Intuitively, we can see that in ou case it means that Pt 0 (s t ) 1 (1+). This j makes sense because we aleady defined q 1 1+, and so at time t, the pice of an asset which yields value 1 at time t = t + j should be ( 1 1+ )j. This is still not the final solution, since we can go futhe with the E t expessions. We note that fom (2) we get: E t+1 (c t+2 ) = c t+1 Taking E t on both sides: E t [E t+1 (c t+2 )] = E t (c t+1 ) Fom the law of total expectation: E t (c t+2 ) = c t Fom hee we see that E t (c t+j ) = c t j. Now we plug this into (3) and get: c t (1 + ) c t = = a t + := 1 + [a t + E t ( ) (1 + ) j E t ( ) (1 + ) j ] Aggegate Income {}}{ 1 + [a t + E t ( W t )] (4) }{{} total esouces We see that the esult hee is also that c t is some constant pecentage of the aggegate expected esouces. This means that even in PIH (no inte-tempoal savings motivation) thee is a level of insuance (savings) since the solution exhibits consumption smoothing ove time. Cetainty Equivalence: An impotant popety of this economy is that if we wee to solve the deteministic poblem (with y t not being stochastic) we would get an equivalent esult. Because consumption is dependent only on the expectancy and not the vaiance (i.e only on the fist moment) we could eplace the stochastic pocess ȳ t with a deteminisitc one which satisfies: E t ( ) t : ȳ t = (1 + ) j 6
If y t is i.i.d, fo example, then this would simply be ȳ t const. This eplacement would not affect consumption since the fist moment is identical (even if the second one is not). This is impotant because we would expect that the addition of uncetainty (vaiance) would cause the individuals to want to insue themselves beyond the levels of the deteministic case. Howeve, we emain with the same level of insuance. Looking at Assets: We notice that by looking at equation (1) we can get an expession fo the change in (o dynamics of) assets between times. Fom equation (1): a t+1 = (a t + y t c t )(1 + ), so: a t+1 = a t+1 a t = (a t + y t c t )(1 + ) a t = (y t c t )(1 + ) + a t (5) We ecall fom equation (4) that c t = into a t+1 we get: a t+1 = y t (1 + ) a t E t [ a t+1 = y t + y t y t E t [ (1+) j ] 1+ [a t + (1+) j ] + a t (1+) j ] a t+1 = y t E t [ We add and subtact 1 fom and expand: a t+1 = y t (( + 1) 1)E t [ E t [ (1+) j ] E t() (1+) ]. Inseting c j t (1+) j ] = y t + (1 + )E t [ a t+1 = y t + E t [ (1+) ] E j 1 t [ (1+) ] j We expand both sums and combine the expectancies: (1+) j ] a t+1 = y t +E t [ yt+1 1+ + yt+2 (1+) 2 +... y t+1 yt+2 1+...] = E t[ y t+1 + yt+2 1+ +...] Finally we get: a t+1 = E t [ ] (6) (1 + ) j 1 j=1 We seen an invese elationship between savings and income. This fomulation is intuitive since as if we get, fo example, a single peiod positive shock to the income then we would want to decease savings and aise consumption. (wealth effect). 2.3 Income (y t ) as a Random Walk: We now look at an altenative way of fomulating the stochastic natue of y t : 1. y t = y p t + u t 2. y p t = y p t 1 + v t Definitions: y t is the income at peiod t. 7
y p t is the pemanent component of income at peiod t. u t is a shock to the tansitoy componenent of the income at t. E t j (u t ) = 0 j 1. v t is a shock to the pemanent component of the income (called pemanent shock). E t j (v t ) = 0 j 1. The impotant distinction between v t and u t is that v t is elevant to evey j 0, wheeas u t is only elevant to y t. This become appaent though eplacement of one expession into the othe: y t+1 = y p t + v t+1 + u t+1 = y p t 1 + v t 1 + v t + u t+1 u t does not appea, but all v t j expession appea. In othe wods, v t pesists completely in y p t, since y p t = 1 y p t 1 + v t, whee the coefficient 1 means that even though v t is a tempoay shock it becomes a pemanent one. We deive some expessions which we will use late: y t 1 = y p t 1 + u t 1 y p t 1 = y t 1 u t 1 y t = y p t + u t = y p t 1 + v t + u t = y t 1 u t 1 + v t + u t At time t we get: E t (y t ) = y t 1 u t 1 + v t + u t E t (y t+1 ) = E t (y t u t + v t+1 + u t+1 ) = y t u t = y t 1 u t 1 + v t + u t u t = y t 1 u t 1 + v t Note that we take the expectation at time t fo a value even though it is known at time t. We do this fo ease of notation. At time t 1 we get: E t 1 (y t ) = y t 1 u t 1 + E t 1 (v t + u t ) = y t 1 u t 1 E t 1 (y t+1 ) = E t 1 (y t u t + v t+1 + u t+1 ) Using both of the expectency tems fo y t, we get the opeato: (E t E t 1 )(y t ) = v t + u t Using both of the expectency tems fo y t+1, we get the opeato : (E t E t 1 )(y t+1 ) = y t u t E t 1 (y t u t + v t+1 + u t+1 ) = E t 1 [y t u t (y t u t + v t+1 + u t+1 )] = E t 1 [v t+1 + u t+1 ] = v t Continuing the elationship, we get finally: (E t E t 1 )( ) = v t j 1 (7) We now look at the expession c t = c t c t 1. Fom (2) we get: c t = c t E t 1 (c t ) 8
Replacing c t accoding to the second line in (4) we get: c t = = 1 + [a t + 1 + E t ( ) (1 + ) j ] 1 + [a t + (E t E t 1 )( ) (1 + ) j E t 1 ( ) (1 + ) j ] = Using (6), we get: c t = = = 1 + [v t + u t + v t (1 + ) + v t (1 + ) 2 +...] 1 + [u t + v t(1 + ) ] 1 + u t + v t Intuitively, we get that the change in consumption is caused by the fact that we have leaned something about the state of the maket (at time t we aleady know v t and u t, wheeas at time t = 0 these ae only futue values with expectancy zeo). We see that the a shock to the pemanent component (v t ) causes an equivalent change in the consumption, and that a shock to the non-pemanent component causes only a patial change. Note: We get hee the same esult as befoe, meaning that thee is cetainty equivalence despite the fact that we have changed the stochastic natue of y t. Looking at assets: Hee we assume that y t+1 = y t + ɛ t+1, E(ɛ t ) = 0. So we get: y t+1 = y t+1 y t = y t + ɛ t+1 y t = ɛ t+1 E t ( y t+1 ) = E t (ɛ t+1 ) = 0. Now, fom (7): (1+) ] = [ E t j 1 j=1 (1+) ] = 0. j 1 a t+1 = E t [ j=1 The intuition fo this is that since y t is a andom walk, the individuals best estimate fo the futue is that the cuent shock, ɛ t, will be pemanent, since the dynamic of a andom walk is that the andom eo at t pesists fo evey time aftewads (this can easily be seen by ecusive eplacement using the expession fo y t ). The well known esult of a pemanent shock to income is that it is completely conveted into consumption, and so thee is no change in savings. 9
2.4 Income (y t ) as i.i.d vaiable: We define y t = ȳ + ɛ t, whee ȳ is some constant o aveage, and E(ɛ t ) = 0. We get: y t+1 = y t+1 y t = ɛ t+1 + ȳ (ɛ t + ȳ) = ɛ t+1 ɛ t E t ( y t+1 ) = E t (ɛ t+1 ɛ t ) = 0 E t (ɛ t ) = E t (ɛ t ) = ɛ t Similaly we get: E t ( ) = 0, j 2 Fom equation (6) we have: a t+1 = E t [ j=1 (1+) j 1 ] So we get: a t+1 = E t [ ɛ t ] = ɛ t This means that a t+1 is also a andom walk. The intuition fo this is that when y t is i.i.d a single peiod shock at time t suvives only one peiod. Since this is happening at time t, i.e cuently, and at no othe time, the individual wants to absob all of the change into the consumption, i.e decease consumption by the amount of the shock and save it (fom consumption smoothing motivations). We note that we ae still in PIH, i.e β(1 + ) = 1, but hee we do have intetempoal savings, despite the emoval of the inte-tempoal motivation. Hee the savings ae geneated by the fom of the uncetainty (y t as i.i.d). 2.5 PIH unde a boowing constaint ā: We have seen PIH in two diffeent cases: y t as a andom walk and as i.i.d. Random Walk Hee we saw that a t = 0. This means that a shock to the income is completely tanslated into savings, and not into consumption. We see that any stating value of a will emain constant acoss time. In this case the imposition of a boowing constaint ā will have no effect on savings. i.i.d Hee we saw that a t = ɛ t. This means that a shock to the income is completely tanslated into savings, and none into consumption. When a vaiable (in this case a t ) is a andom walk thee is a positive pobability that the vaiable will each any possible value in its ange. This means that unde a boowing constaint ā on the level of assets thee is a positive pobability that the individual will each this level. This means that at some point the individual will have to consume only accoding to his income, which may be vey small. This makes the individual incease savings at all peiods in ode to account fo this possibility. 10