Intetempoal Macoeconomics Genot Doppelhofe* May 2009 Fothcoming in J. McCombie and N. Allington (eds.), Cambidge Essays in Applied Economics, Cambidge UP This chapte eviews models of intetempoal choice consumption demand and labou supply. We discuss optimal decisions by individuals at the micoeconomic level and the implications fo the aggegate economy. The chapte descibes the equilibium in a maket-cleaing neoclassical model and analyses effects of poductivity and govenment secto shocks on optimal decisions by consumes and wokes. Pedictions fom the benchmak neoclassical model ae contasted with altenative theoies and aggegate data fo the US and UK. JEL Code: D9, E2, E3, E2, E24, E62, H30 Depatment of Economics, Nowegian School of Economics and Business Administation (NHH), Helleveien 30, N-5045 Begen, Noway; email: G.Doppelhofe@nhh.no. I thank Xavie Sala-i-Matin, whose macoeconomics lectues at Columbia Univesity povided inspiation and guidance to the mateial in this chapte, Donald Robetson fo helpful discussions and the deivation of the Keynes-Ramsey ule, and seveal cohots of students at Cambidge fo helpful feedback. I thank Bhaavit Agawal fo excellent eseach assistance, the Economics Depatment at UW-Madison fo thei hospitality while completing this chapte and Tinity College fo financial suppot. All eos ae mine.
Table of Contents Intoduction... 2 Static model of consumption and leisue choice... 2 2. Substitution and income effect... 3 3 Intetempoal Consumption Choice... 5 3. A two peiod model... 5 3.2 Intetempoal substitution and income effects... 7 3.3 Intetempoal optimisation and the Eule equation... 8 3.4 Consumption functions in multi-peiod models... 0 3.4. Modigliani s life cycle hypothesis (LCH)... 0 3.4.2. Fiedman s pemanent income hypothesis (PIH)... 2 3.5 Role of uncetainty and Hall s andom walk hypothesis... 4 3.6 Depatues fom classical consumes... 5 4 Intetempoal Labou Demand and Supply... 8 4. Labou Demand... 8 4.2 Intetempoal labou supply... 8 4.3 Empiical Evidence of Labou Supply... 20 5 Equilibium in goods and labou maket and poductivity shocks... 20 5. Aggegation... 2 5.2 Poductivity shocks... 22 5.2. Pemanent poductivity shock... 22 5.2.2 Tempoay poductivity shock... 22 5.2.3 Theoetical pedictions and stylised facts... 24 6 Govenment... 24 6. Effect of govenment spending... 25 6.. Pemanent change in govenment spending... 26 6..2 Tempoay change in govenment spending... 27 6..3 Effects of govenment spending in Keynesian Model... 28 6.2 Budget Deficits and Ricadian equivalence... 28 6.3. Taxes... 29 7 Concluding Remaks... 3 Refeences... 32 2
Intoduction This chapte gives an oveview of the liteatue on intetempoal macoeconomics. We eview models of individual, optimising behaviou by consumes and wokes, discuss implications fo aggegate vaiables and contast the theoetical pedictions with empiical facts fo the US and UK economies. The advantage of using an optimising model is that we gain a bette undestanding of the aggegate economy. Instead of simply postulating ad hoc macoeconomic elations, we undestand the micoeconomic foundations of aggegate vaiables such as consumption and labou supply. Stating fom micoeconomic foundations ensues intenal consistency of ou macoeconomic models. The dawback of a micofounded appoach is that going fom individual to aggegate behaviou equies stong assumptions. We stat by analysing the neo-classical model as a benchmak. Consumes choose optimally how much to consume and save ove time given the path of income and the inteest ate. Wokes supply labou (demand leisue) each peiod, given the path of wages and the inteest ate. Pefectly competitive fims hie labou at the maketcleaing wage and we can deive the equilibium in goods, labou and bond makets. Next we can analyse the effects of poductivity shocks and changes in govenment spending and tax decisions on the model. We contast the pedictions of the model with empiical obsevations and discuss implications of altenatives theoies. Fo efeence, we list the following key aggegate quately data fo the United States aveaged ove the 959-96 peiod (fom Bao 997, Table.). Components of GDP Shae of GDP Standad Coelation Deviation with GDP GDP.000 0.07.00 Consumption 0.589 0.008 0.83 Investment 0.203 0.065 0.93 Govenment spending 0.226 0.07 0.02 Othe vaiables Employment - 0.00 0.8 Woke hous - 0.05 0.88 Output pe employee - 0.0 0.84 Real wage - 0.0 0.48 Real Inteest Rate - - 0.23 This chapte is oganised as follows: Section 2 biefly descibes a simple static model of consumption and leisue choice. Section 3 discusses models of intetempoal consumption choice. Section 4 intoduces intetempoal supply and demand fo labou and discusses empiical evidence. Section 5 deives the equilibium in the neoclassical model and analyses the effect of poductivity shocks on optimal choices. Section 6 intoduces the govenment secto into the model and section 7 concludes.
2 Static model of consumption and leisue choice Conside fist the optimal demand fo consumption and leisue in a highly simplified static model of an economy. A epesentative individual has pefeences ove consumption and leisue ae epesented by a utility function () u( c, l) whee c epesents individual 2 consumption and l the supply of labou. The total time endowment is nomalised to and (-l) epesents leisue. We assume that utility inceases in both consumption and leisue, epesented by positive maginal utilities u c >0 and u -l >0, and that the atio of maginal utilities u c /u -l falls as the consumption/leisue atio ises. The pefeences can be epesented by convex indiffeence cuves shown in Figue. The individual pefes moe consumption and moe leisue (less wok) and modeate amounts of both compaed to extemes of only one of them. Fo each unit of labou l the consume eans a wage w measued in tems of the homogenous final good 3. The set of feasible choices (budget constaint) is theefoe given by (2) c = wl + b whee b is the stock of wealth which is independent of labou income. The budget constaint is shown as upwad sloping line in figue with slope w. Inceases in initial wealth b coespond to upwad paallel shifts of the constaint. The poblem fo the consume is to maximise the utility function () subject to the budget constaint (2). Fomally, we can set up the Lagangian and find the optimal choices of consumption and leisue max u( c, l) + λ c, l [ wl + b c] The fist ode conditions ae u c =λ and u -l =wλ, whee λ is the Lagange multiplie associated with the budget constaint (2). Combining those two conditions implies that at the optimum the maginal ate of substitution (MRS) between leisue and consumption equals the wage ate MRS u u l, c = l = c w Aggegation of pefeences acoss individuals equies vey estictive assumptions; see Deaton (992) and section 5. below fo a discussion. 2 Thoughout this chapte the notation convention is that lowecase denotes individual and uppecase aggegate vaiables. Economy-wide pices such as the inteest ate ae witten as lowecase. 3 This chapte abstacts fom money and theefoe does not discuss the ole of nominal igidities (sticky pices o wages). Fo an excellent discussion of such issues see Blanchad (990). 2
The optimal choice of consumption c * and labou supply l * is shown in Figue. Gaphically the optimum can be found at the point whee an indiffeence cuve is tangent to the budget constaint. c wl bette-off egion c * 0 l * l Figue : Optimal choice of consumption and labou supply (leisue) 2. Substitution and income effect Conside next the effect of changes of the wage ate on the optimal choice of consumption and labou supply. Suppose that the wage ate inceases to w > w which coesponds to a steepe slope of the budget constaint in Figue 2. The change in wages can be decomposed into substitution and income effect. The substitution effect fo consumption and leisue demand is associated with a change in the elative pice of labou, the wage ate. We analyse the effect of an incease in the wage ate to w in the neighbouhood of the oiginal optimal choice (c *,l * ). We assume eal income to be unchanged so that the initial choice is still feasible 4. Figue 2 shows the new optimum (c,l ) whee a highe indiffeence cuve is tangent to the dashed line obtained by otating the budget constaint counte- 4 Altenatively, we could define the substitution effect as change in demand esulting fom a change in elative pice while keeping the consume equally well-off (on the same indiffeence cuve). Fo small changes these two definitions coincide. 3
clockwise aound the initial point (c *,l * ). Intuitively, the substitution effect induces the consume to wok moe (enjoy less leisue) and incease consumption. The income effect (sometimes called wealth effect) is due to the highe wage at evey level of labou supplied l. Figue 2 shows the paallel shift fom the dashed line to the new (solid) budget constaint. Note that both lines shae the same steepe slope w. The income effect implies moe demand fo both consumption and leisue which ae usually assumed to be nomal goods (movement fom c,l to c **,l ** ). The total effect of an incease of wages is the combination of substitution and income effects: Fo consumption, the two effects einfoce each othe and imply an ise fom c * to c ** in Figue 2. Fo leisue (and labou supply) substitution and income effect wok in opposite diections and the oveall effect is ambiguous depending on the elative stengths of substitution and income effects. c w l wl c ** c c * 0 l * l l ** l Figue 2: Substitution and income effect on consumption and labou esulting fom wage incease 4
Empiically we obseve inceased levels of consumption and falling labou supply (measued by weekly hous) as economies ae developing and wages ae ising. Fo example, the aveage numbe of weekly hous in manufactuing has fallen fom between 55 and 60 in 890 to 42 in 996 in the United States and has fallen fom 60 in 850 to 44 in 994 fo the UK (see Bao 997, pp. 76-77). Simila secula declines of wok hous can be obseved in othe counties ove time, howeve the elationship appeas to become weake at high levels of income (weekly hous have not fallen in the US ove the last 50 yeas). 3 Intetempoal Consumption Choice 3. A two peiod model Conside next the optimal choice of consumption ove time. Suppose that individuals live fo two peiods, (denoted by subscipts and 2). Each peiod, individuals ean income y and chose how much to consume c o altenatively save by puchasing bonds b paying a constant 5 inteest ate. The budget constaint fo the two peiods ae y + b ( + ) = c y 2 0 + b ( + ) = c 2 + b + b 2 whee b 0 denotes the level of assets at the beginning of the individuals life and b 2 at the end of its lifetime. The budget constaints fo each peiod can be ewitten altenatively as intetempoal budget constaint (4) y2 c2 b2 y + + b0 ( + ) = c + + + + + which states that the pesent value 6 of lifetime income and initial wealth equals the pesent value of lifetime spending. Individuals live fo 2 peiods and ae assumed to cae only about thei own consumption in peiods and 2 (we will elax this assumption in section 3.5.2). Theefoe, optimal assets in the final peiod of individuals life will not be positive, that is b 2 0, as long as consumes ae not satiated. On the othe hand, individuals ae not allowed to leave behind debt, so b 2 0. In equilibium each geneation is theefoe bon with zeo wealth (b 0 =0) and the intetempoal budget constaint theefoe simplifies to (5) y2 y + + c2 = c + + The intetempoal budget constaint (5) is shown in Figue 7 3 as the black solid line that passes though the endowment point (y,y 2 ) and has slope (+). Consumes can 5 Thoughout this chapte we assume the inteest ate to be fixed and known. In a moe ealistic model, the inteest ate is detemined in capital makets and changes ove time. Fo a discussion of vaiable inteest ates see fo example Obstfeld and Rogoff (996, pp. 76-78). 6 Given inteest ate the pesent value at time 0 of an amount X t t peiods in the futue is defined as PV(X t ) X t /(+) t. The tem /(+) t can be viewed as the elative pice of X t in tems of cuent goods. 7 Such figues intoduced by Iving Fishe (907) ae know as Fishe diagams. 5
chose any combination of fist and second peiod consumption (c,c 2 ) on this line and the optimal choice depends on pefeences ove consuming in the fist o second peiod of thei life. We assume diminishing maginal utility of consumption and individuals pefe to smooth consumption between time peiods. Indiffeence cuves in Figue 3 ae theefoe negatively sloped and convex to the oigin. Figue 3 shows examples of diffeent types of individuals:. Save: consumes ae elatively patient, they save in peiod by chosing c * <y and consume thei income and savings in peiod 2 giving c 2 * >y 2. 2. Boowe: consumes ae elatively impatient, they chose c * >y and boow in peiod which they epay by saving in peiod 2 and hence c 2 * <y 2. 3. Autakist: individuals consume at the endowment point and ae neithe saving no boowing, c * =y and c 2 * =y 2. c 2 slope (+) c 2 * =y 2 0 c * =y c Figue 3: Optimal consumption choice between peiods and 2 How does the optimal consumption choice espond to changes in income? Notice that individuals want to smooth consumption ove time (because of diminishing maginal utility of consumption) and they can boow and lend feely at inteest ate. Consumption each peiod is theefoe a function of the pesent value of lifetime income, PV(y) y + y 2 /(+) in equation (5). An incease in income in one peiod only, say y, leads to less than popotional inceases of consumption each peiod. Howeve, consumption moves popotional to changes in the pesent value of income. 6
3.2 Intetempoal substitution and income effects Next conside the effect of a change in the inteest ate on optimal consumption choice. A ise in the inteest ate to > implies an incease of the elative pice of cuent (peiod ) elative to futue (peiod 2) goods which induces the consume to substitute away fom cuent consumption towads futue consumption. The intetempoal substitution effect leads to a fall in cuent consumption and incease in futue consumption (incease in savings). Figue 4 shows the intetempoal substitution effect by moving fom the initial point (c *,c 2 * ) to the new optimum at (c,c 2 ). Notice that the dashed budget constaint with slope (+ ) passes though the initial point (c *,c 2 * ) which is still feasible. The intetempoal income effect associated with a ise of the inteest ate depends on whethe the consume is a save o boowe in the fist peiod. Figue 4 shows the case of a save in peiod. In this case the highe inteest ate aises lifetime income (because of highe etun on savings) and the intetempoal income effect is positive leading to moe consumption in both peiods. Fo a boowe the intetempoal income effect of a ise in inteest ates is negative (due to highe debt epayment) leading to less consumption in both peiods. c 2 slope (+ ) c 2 ** c 2 c 2 * slope (+) y 2 * 0 c c c ** y c Figue 4: Intetempoal substitution and income effect of inteest ate incease on optimal consumption choice in peiod and 2 The total effect is a combination of intetempoal substitution and income effects. Fo a save, the two effects einfoce each othe and imply highe consumption in peiod 7
2, but have opposite signs, hence ambiguous total effect on consumption in peiod. The new optimal point is (c **,c 2 ** ) in Figue 4. A boowe on the othe hand educes c since both effects einfoce each othe, but the oveall change is ambiguous fo c 2. Fo the aggegate economy only the intetempoal substitution effect is elevant since thee is no net saving o boowing in equilibium (see section 5 below). Aggegate consumption demand (denoted by supescipt d) depends theefoe negatively on the inteest ate and positively on the pesent value of aggegate income C d (, PV ( Y ),...) ( ) 3.3 Intetempoal optimisation and the Eule equation We can analyse the consume s poblem moe fomally. Suppose that individuals have additively sepaable pefeences ove fist and second peiod consumption: ( + ) (6) max u( c) + βu( 2) c, c 2 c whee the peiod utility function u(.) is stictly concave u >0, u <0 and β is the subjective discount facto measuing the degee of impatience of the individual. We assume that the individual discounts futue utility with 0<β<. The poblem of the consume is to maximise lifetime utility (6) subject to the intetempoal budget constaint (5). Fo example, we can use the substitution method and substitute fo c 2 fom (5) into (6): maxu( c c [( + )( y c + y ] ) + β u ) The necessay fist-ode condition fo this poblem is also called an intetempoal Eule equation 8 (7) u ( c ) = ( + ) βu'( ) ' c2 The Eule equation detemines optimal consumption choice ove time. The left hand side of equation (7) is the maginal utility of consumption in peiod which measues how much the individual values one unit of consumption. If the individual saves one unit she would eceive (+) units of consumption next peiod which would incease lifetime utility by the maginal utility of consumption (discounted by β) on the ight hand side of (7). Altenatively, the Eule equation can be witten as βu'( c2 ) = u'( c ) + which equates the maginal ate of substitution between c and c 2 on the left hand side to the elative pice /(+). This optimality condition can also be seen in Figue 3 2 8 It is named afte the Swiss mathematician Eule who investigated dynamic systems moe geneally. 8
whee the indiffeence cuve epesenting the tade-off between cuent and futue consumption is tangent to the intetempoal budget constaint with slope (+). In ode to futhe undestand intetempoal consumption choice conside the following example of a powe utility function (8) θ c u ( c) = θ whee the paamete θ>0 contols the cuvatue of the utility function and /θ measues the constant intetempoal elasticity of substitution (IES) of consumption between time peiods. Utility function (8) is also called isoelastic utility function. The θ maginal utility of consumption in this case is simply u '( c) = c and the Eule equation (7) becomes θ c c = β ( + ( ) ) 2 / To intepete this condition, ewite the discount facto as β =/(+ρ), whee ρ>0 is the ate of time pefeence. Fo small changes in consumption c (c 2 -c ), the left hand side of this expession can be appoximated by ( 2 Δ θ θ c / c ) = ( Δc / c + ) + θ c / c Fo small values of and ρ the ight hand side of this expessions is appoximately equal to ( + ) /( + ρ) + ρ. Combining the two appoximations, the Eule equation (also known as Keynes-Ramsey ule) becomes Δc (9) = ( ρ) c θ The gowth ate of consumption Δ c / c is detemined by the diffeence between the inteest ate and the ate of time pefeence and the stength of the esponse depends on the intetempoal elasticity of substitution /θ. Hall (988) tests the empiical elationship between aggegate consumption and inteest ates and finds only a weak elationship between these two vaiables. Hall s estimates of the intetempoal elasticity of substitution (IES) ae shown in Table. The estimates depend on which asset is used to calculate undelying inteest ates. The fist thee ows of Table ae based on annualised postwa data of expected etuns based on inflation expectations fom the Livingston suvey 9. The point estimates ange fom 0.066 fo etuns on 400 S&P stocks to 0.346 fo US Teasuy bills. Howeve, the estimated standad eos ae so lage that the point estimates ae not statistically diffeent fom zeo. The estimates based on the sample stating in 924 and ealised etuns gives maginally significant, negative estimates of the IES. 9 The Livingston suvey is based on expectations of many vaiables by a panel of economists. 9
Table : Hall s (988) estimates of intetempoal elasticity of substitution (IES) Inteest ate based on IES estimate Standad eo Sample T Bills (expected) 0.346 (0.337) 959-83 Savings Accounts 0.27 (.330) 959-83 Stocks 0.066 (0.050) 959-83 T Bills (ealised) -0.40 (0.20) 924-40, 950-83 Beaudy and Van Wincoop (996) use panel data fo US States instead of nationwide data. Fo the peiod 953-9 they stongly eject the zeo estimate of the IES and find estimates close to one. Seveal studies have also used micoeconomic data to addess poblems of aggegation and lack of contol fo demogaphic and labou maket effects. Attanasio and Webe (993) and Blundell, Bowning and Meghi (994) find fo both the US and the UK estimates of the IES just below one. Bowning, Hansen and Heckman (999) suvey geneal equilibium models and implied empiical estimates of the IES using mico data (see fo example thei Tables 3. and 3.2). They ague that the IES paamete is often pooly detemined, and that thee is evidence vaies with demogaphic chaacteistics and wealth. 3.4 Consumption functions in multi-peiod models The analysis fo the two peiod case can be genealised to many peiods. The next two sections discuss the case of finite and infinite life-time. 3.4. Modigliani s life cycle hypothesis (LCH) In the 950s Fanco Modigliani developed with co-authos the life cycle hypothesis (LCH) to descibe consumption and savings behaviou ove individuals lifetime 0. Suppose that individuals live fo T peiods and each peiod t face a budget constaint y + b ( + ) = c + b t t t t Simila to the two-peiod case we can deive the intetempoal budget constaint saying that the pesent value of lifetime income and initial wealth equals the pesent value of lifetime spending y2 y y + +... + + ( + ) T T c2 c c + +... + + ( + ) T T + b ( + ) = 0 b + ( + ) T T 0 Hee we follow the simple vesion pesented in Modigliani s (986) Nobel lectue. 0
Since individuals have finite hoizons they leave behind no assets as bequests fo futue geneations and set b T to zeo. Fo simplicity, assume that individuals ean a constant labou income y until etiing at R yeas, but no moe labou income until the expected end of life at time T. We assume that individuals pefe a smooth consumption pofile ove the life-time. Figue 5 shows the allocation of consumption and assets ove the lifetime. Individuals accumulate wealth until etiement and daw down the stock of wealth until life ends to ensue a smooth path of consumption; moe fomally, the Eule equation (7) shows that individuals want to smooth maginal utility acoss time peiods. The life cycle model pedicts the following path of consumption and assets: befoe enty into the labou maket individuals should boow, they should accumulate savings while woking and dissave afte they etie. Figue 5 shows the esulting hump shaped path of wealth. wealth y c R T age Figue 5: Modigliani s life cycle hypothesis (LCH) The life cycle hypothesis has othe impotant implications. Consumption esponds little to tempoay changes in income and popotionally to pemanent changes; also the maginal popensity to consume out of cuent income depends on age. Some empiical obsevations seem at odds with the simple LCH model. Fist, young individuals consume too little compaed to expected life-time income. A high maginal popensity to consume could point to myopia o liquidity constaints (see section 3.6 fo a discussion). Second, consumption seems to fist incease and late fall in line with labou income which appeas at odds with consumption smoothing. Bowning and Cossley (200) ague that pecautionay savings (pudence) and demogaphic changes (childen in family) could explain these changes ove the woking life. Thid, the eldely dissave too little afte etiement and consumption
falls discetely at etiement. These obsevations might be explained by pecautionay savings motive o the impotance of bequests (see section 6.2 below). 3.4.2. Fiedman s pemanent income hypothesis (PIH) Suppose instead that geneations ae linked to each othe. Assume that each geneation lives fo one peiod and caes about its own utility and the utility of the next geneation: t = : t = 2 :... u( c ) + βu( c u( c 2 2 ) + βu( c ) 3 ) Lifetime utility of geneation bon at time is theefoe 2 (0) u( c ) + β u( c2) + β u( c3) +... = t= t β u( ct ) Since each geneations caes about the next, each geneation acts as if it was infinitely lived and had an infinite hoizon. Notice that each geneation discounts futue utility with discount facto β. Simila to the life cycle model pesented in the pevious section, we can combine the budget constaints faced by each geneation t =,2,.. into the intetempoal budget constaint t= ( y t + b 0( + ) = t + ) t= ct ( + ) t b + lim T ( + ) T T The left hand side shows the pesent value of cuent and futue income ove the infinite hoizon (discounted at elative pice ) plus initial assets. On the ight hand side is the pesent value of consumption spending plus a tem showing the pesent value of wealth as the hoizon is pushed towads infinity. Remembe that fo the two-peiod and life cycle model consumes wee not allowed to leave behind debts (b T < 0). In the infinite hoizon case this condition is eplaced by T lim /( + ) T b T 0 which states the pesent discounted value of assets must be non-negative and that debt must not gow faste than the inteest ate. Othewise the consume could finance infinite consumption by boowing eve inceasing amounts 2. Futhemoe, consumes will not find it optimal to accumulate savings at a faste ate than the inteest ate o else the pesent value of savings would be unbounded. The optimal consumption path theefoe satisfies the so-called tansvesality condition Fank Ramsey (928) viewed such discounting of futue welfae as ethically indefensible in nomative models. In a positive (desciptive) sense discounting would be consistent with selfish individuals pefeing cuent to futue consumption (see section 3.6 below fo discussion). 2 This condition is also known as the No-Ponzi-game condition afte the swindle Chales Ponzi who an a pyamid scheme in Boston in the 920s. 2
bt () lim 0 ( + ) = T T With condition () satisfied the intetempoal budget constaint becomes yt ct (2) + b + = 0( ) t ) ( + ) t= ( + t= The poblem fo an individual bon at time 0 is to maximize utility (0) subject to the intetempoal budget constaint (2). The condition descibing optimal consumption fo the infinite hoizon poblem is the familia intetempoal Eule equation 3 u' ( ct ) = ( + ) β u'( ct + ) The intuition is exactly the same as in the two peiod case: the maginal utility of consumption between peiods t+ and t equals the maket valuation /(+). Along the optimal path the consume is indiffeent to consume o save the maginal unit. Conside the special case of β=/(+). Then the Eule equation simplifies to (3) u ( c ) u'( c ) ' t = t+ which implies that consumption is constant ove time ct = ct + = c. Because consumes discount futue utility at the same ate as the maket inteest ate, consumes have no incentive to tilt the consumption path ove time and hence consumption is constant (emembe the Keynes-Ramsey ule in equation (9)). Substitute the constant level of consumption c in the intetempoal budget constaint (2) and solving gives: t= c yt = + b0 ( + t t ( + ) ( + ) t= which says that the pesent value of consumption equals the pesent value of total wealth which equals the pesent value of life-time income plus initial wealth. Each peiod t=,2, individuals consume the annuity value 4 of total wealth which Milton Fiedman (957) calls pemanent income y P p yt+ s (4) ct = yt + b ( + ) s t + s= 0 ( + ) ) t 3 It is staightfowad to extend the deivation of the intetempoal Eule fom the two peiod model (section 3.3) to the life-cycle model with finite T peiods. A fomal deivation fo the infinite hoizon case is beyond the scope of this chapte. Fo a discussion of this case see fo example Dixit (990). 4 Consuming the annuity value leaves total wealth unchanged ove time. Notice that we use t= /( + t ) = ( + ) / in deiving (4). 3
Income y can be decomposed into pemanent income y P and tansitoy income y T defined as (y - y P ). Accoding to Fiedman s pemanent income hypothesis (PIH) consumption demand esponds popotionally 5 to changes in pemanent income and not to at all to changes in tansitoy income. Fiedman s theoy delives stong empiical pedictions. In a deteministic envionment (with no shocks) consumption and cuent income would be unelated ove time. Futhemoe, thee would be a elationship between consumption and pemanent income when obseving individuals in a coss-section. This can explain the lowe maginal popensity to consume and highe popensity to save out of cuent income among individuals with below aveage pemanent income (fo example, the diffeence between White and Black population in the United States). 3.5 Role of uncetainty and Hall s andom walk hypothesis Next let us analyse the choice of intetempoal consumption when consumes face uncetainty ove futue income and consumption due to the pesence of andom shocks. Suppose that individuals maximise the expected value of lifetime utility t max E β u( ct ) c, c2,... t= whee E denotes mathematical expectations conditional on infomation available at time. The expected value equals the pobability weighted aveage of possible outcomes. The intetempoal budget constaint is still given by equation (2) stating that the ealised (ex post) pesent values of consumption equal total ealised wealth. The stochastic vesion of the Eule equation govening optimal consumption equates maginal utility of consumption in peiod and expected discounted maginal utility in peiod 2 [ u'( )] u' ( c ) = ( + ) βe c2 Hall (978) tests the stochastic vesion of life-cycle and pemanent income hypotheses theoy by making the following simplifying assumptions: Assumption Suppose the utility function is quadatic 2 u ( c) = c ac / 2, with a > 0 This assumption implies that maginal utility u' ( c) = ac is linea in consumption. A consequence of this assumption is that the individual consumption decision exhibits cetainty equivalence which implies individuals act as if futue consumption was at its conditional mean value and ignoe its vaiation. Assumption 2 Assume + = /β. 5 In Fiedman s (957), consumption is popotional to pemanent income c t d = k(,wealth, ) y t P. In equation (4) the facto of popotionality k equals since =β. Note that we abstact fom uncetainty about futue income (see section 3.6). 4
Assumption 2 ensues that consumes want to hold maginal utility and hence consumption constant ove time. See discussion afte equation (3) above. Unde these two assumption the stochastic Eule equation becomes (5) c = ( ) E c2 Equation (5) says that consumption follows a andom walk and the best estimate of next peiod s consumption is the cuent level of consumption. Intuitively individuals plan to smooth consumption ove time (assumption 2) and use all infomation available at time to make optimal foecasts of pemanent income. An implication of this theoy is that changes in consumption should be unpedictable and epesent only supise infomation (not known at time ). Hall (978) tests the theoy using postwa aggegate US data and finds that past levels of consumption and income have no pedictive powe fo futue consumption. Howeve, the pue LCH- PIH theoy fails because lagged levels of S&P stock maket pices help to pedict changes in aggegate consumption. Flavin (98) looks at the joint behaviou of aggegate consumption and income and finds that consumption exhibits excess sensitivity to anticipated changes in cuent income. On the othe hand Campbell and Deaton (989) find that consumption exhibits excess smoothness with egad to contempoaneous innovations to income. Notice that these two obsevations ae not mutually contadicting each othe since excess sensitivity efes to anticipated changes wheeas excess smoothness is with egads to unexpected innovations (see Abel 990 fo discussion). Hall s test of the andom walk esult elies on the maintained assumption of quadatic utility (assumption ) and ejection of a test by the data could imply that the undelying LCH-PIH theoies and/o the assumptions ae wong. Quadatic utility povides a good appoximation to pefeences if maginal utility is not esponding stongly to fluctuations in consumption and is close to the linea cetainty equivalence case. If maginal utility is vey non-linea on the othe hand, individuals stongly dislike uncetainty and will accumulate pecautionay savings. Seveal studies investigate the impotance of pecautionay savings empiically and conclude that the answe depends cucially on pefeence paametes, the distibution of eanings and the individual s age (see Attanasio, 999). 3.6 Depatues fom classical consumes This section looks at deviations fom classical (LCH and PIH) models of consume behaviou and discusses empiical evidence. Keynes (936) postulates a linea consumption function whee aggegate consumption demand depends linealy on cuent aggegate income: (6) C = α + λy 5
whee α is a constant and λ is the maginal popensity to consume which lies between zeo and one. Keynes thought of this function as epesenting a fundamental psychological law. Fiedman (957) stongly citicises Keynes and agues that consumption demand depends on pemanent income instead (see section 3.4.2 above). Howeve the classical theoies (LCH and PIH) ely cucially on individuals having access to cedit makets and being able to boow and lend feely at inteest ate to smooth consumption ove time. What happens if some consumes ae liquidity constaint? Suppose a faction of consumes ae liquidity constained. They would like to boow against futue income, but cannot use it as collateal. Such liquidity constained (hencefoth LC) consumes ae foced to consume at thei cuent income level (7) c LC = y Figue 6 shows that LC consumes would achieve highe utility by boowing against highe futue income and consume at (c *,c 2 * ), but the binding constaint makes this optimum unfeasible. c 2 -(+ ) y 2 = c 2 LC c 2 * -(+ ) 0 y = c LC c * c Figue 6: Intetempoal choice of liquidity constained (LC) consume What happens to the choice of LC consumes if the inteest ate inceases and the slope of the budget constaint in Figue 6 ises? The answe is that changes in inteest ate leave constained consumption demand (c LC ) unchanged. Consumption demand 6
of LC consumes depends only on cuent income as in equation (7), and changes in the inteest ate 6 o futue income ae not elevant. Suppose next that a faction λ of the population is liquidity constained and behaves as in (7) and the emainde (- λ) behaves like classical consumes fom section 3. with consumption function c d (, PV(y),..). Aggegate consumption is the weighted sum of the two tems Total LC d d (8) C = λc + λ) c = λy + ( λ) c (, PV ( )) ( y o witten moe compactly as Keynesian consumption function (6) α + C K λy whee α ( λ) c d (, PV ( y)) is not constant, but function of the inteest ate and λ measues the faction of LC consumes in the economy. Campbell and Mankiw (989) test the validity of Eule equation (9). As alteantive they allow fo the pesence of ule-of-thumb consumes whose consumption gowth is also affected by expected changes in income (compae this to Hall s (978) test) (9) Δ log( C + ) = α + σ + + λδ log( Y + ) + ε + t t Using 953-86 US data Campbell and Mankiw estimate the shae of ule-of-thumb consumes λ to be appoximately one half. Does this imply that 50% of the US population is liquidity constained? Not necessaily, since thee ae othe easons that consumption could depend on cuent income: these include life-cycle effects and pecautionay motives discussed ealie, deviations fom standad optimization (see below) o habit fomation 7. The estimate of σ is not statistically diffeent fom zeo indicating little coelation between expected changes in consumption and ex ante eal inteest ates. This finding is consistent with Hall s (988) esults shown in Table. Flavin s (98) findings of excess sensitivity of consumption to anticipated changes in cuent income ae consistent with the pesence of liquidity constaints. Hall and Mishkin (982) use panel data on food expenditues and find that appoximately 20% of US households ae liquidity constained. Hayashi (987) points out that even if consumes ae cuently not constained, the possibility of being constained in the futue shotens thei effective planning hoizon. Loewenstein and Thale (989) descibe seveal anomalies in intetempoal choice including time-inconsistent pefeences o nominal anchoing. Angeletos et al (200) ague that the intoduction of time-inconsistent pefeences though hypebolic discounting poduces can help to explain the elatively low levels of liquid assets (and lage cedit cad debt) ove the life cycle. Hypebolic consumes ae less able to smooth consumption which is consistent with Flavin s excess sensitivity esult. t t 6 Fo a sufficiently high inteest ate, liquidity constained consumes become unconstained and save in the fist peiod. We assume the liquidity constaint binds also afte changing the inteest ate. 7 See Attanasio (999) and Caoll (200) fo ecent eviews of the liteatue. 7
4 Intetempoal Labou Demand and Supply 4. Labou Demand Conside fist labou demand by competitive fims. Suppose that each peiod fims poduce final output using a poduction function (20) y = AF(l) whee F(.) exhibits positive, but diminishing etuns to labou, F (l)>0 and F (l)<0. Poductivity is measued by the paamete A>0 which epesents factos such as the weathe o available knowledge that incease output fo given pivate inputs. Fo any given labou input l an incease in poductivity A aises output in equation (20). Suppose that pefectly competitive fims maximise pofits maxπ l d s d d = y wl = AF( l ) wl whee l d epesents the fim s labou demand. The necessay fist-ode condition 8 implies that fims hie labou up to the point whee the maginal poduct of labou equals the eal wage ate, w * =AF (l). At this maket cleaing wage w * thee is full employment. An incease in poductivity A implies a highe maginal poduct of labou and hence the wage ate inceases (see also section 5.2 below). 4.2 Intetempoal labou supply Conside next the intetempoal choice of labou supply by individuals in a simple 2 peiod model (which extends staightfowad to seveal peiods). Household ean wage income and consume and save in peiods and 2. Pefeences ove consumption and leisue ae given by (2) max u( c, l) + u( c2, l2) c,l + ρ whee the peiod utility function u(c,l) has the same popeties as equation (). We assume lifetime utility to be additively sepaable between time peiod which implies maginal utilities ae not a function of consumption and leisue choices acoss time. The discount facto is given by β = /(+ρ), whee ρ is the ate of time pefeence. Assuming that individuals ean labou income in both peiods, the intetempoal budget constaint (IBC) equals (22) c + c2 = wl + w2l2 + + which implies that the pesent discounted value of lifetime consumption equals lifetime labou income, both discounted at the inteest ate. d 8 Optimal input demand hee is static and depends only on cuent inputs and pices. They would become dynamic if inputs hied in pevious peiods wee used to poduce output in equation (20). 8
The necessay fist-ode conditions 9 fo maximising lifetime utility (2) subject to the intetempoal constaint (22) ae ul ( l) (23) = w u ( c ) + (24) uc ( c ) = uc ( c2) + ρ c (25) ul ( l) + w = u ( l ) + ρ w l 2 2 Equation (23) epesents the static optimal tade-off between consumption and leisue analysed in section 2.. The consume equalises the maginal ate of substitution between consumption and leisue to the equilibium wage ate. Equation (24) is the same intetempoal Eule equation as equation (7) in section 3.3. Equation (25) is an intetempoal Eule equation fo leisue: at the optimum the atio of maginal utilities of leisue each peiod is set equal to the atio of wages and a facto depending on the inteest ate and the ate of time pefeence. What ae the implications of equation (25)? If the inteest ate inceases, individuals substitute out of wok tomoow into wok today. Individuals face a highe etun on saving today (peiod ) which inceases the incentive to wok today and incease lifetime income. Altenatively, the pesent discounted value of second peiod wages is lowe inducing individuals to wok moe today. Individual labou supply (denoted by supescipt s) is theefoe a positive function of elative wages w /w 2 and the inteest ate (, w /w,..) l s 2 ( + ) ( + ) and the esponse depends on the intetempoal elasticity of substitution (IES) of the labou supply (Lucas and Rapping 969). Notice that the Eule equation (25) can be obtained by substituting the static consumption/leisue tade-off (23) into the consumption Eule equation (24), so the thee optimality equations ae intedependent. Intuitively an optimal choice of leisue and consumption in each peiod and optimal consumption path ove time implies an optimal intetempoal labou/leisue choice. Mankiw, Rotembeg and Summes (985) test the validity of equations (22)-(25) fo aggegate US data and find that the estictions implied by the theoy ae stongly ejected by the data. These negative esults ae in line with ejections of tests of the intetempoal Eule equation (24) fo aggegate consumption data (see Attanasio 999). 9 Patial deivatives of utility with espect to consumption and leisue ae denoted by u c and u l u c / u / l, espectively. A good execise is to veify the fist-ode conditions (23)-(25), by setting up the Lagangian L = u(c,-l ) + u(c 2,-l 2 )/(+) + λ[w l + w 2 l 2 /(+) - c - c 2 /(+)]. 9
4.3 Empiical Evidence of Labou Supply Alogoskoufis (987a,b) finds that the esponse of labou supply to gowth in eal wages is moe impotant fo the numbe of employees than fo hous woked. Table 2 summaises the estimates of intetempoal elasticity of substitution (IES) of labou supply fo the United States and the United Kingdom using annual data and total employee numbes. Alogoskoufis (987b) agues that the IES estimates of labou supply appeas to be lowe than the static elasticity between consumption and leisue 20. Table 2: Alogoskoufis (987a,b) estimates of elasticity of labou supply * Elasticity to elative wages w /w 2 eal inteest ate US 948-82.9 0.74 (0.36) (0.8) UK 950-82 0.359 0.92 (0.25) (0.063) Alogoskoufis (987a), Table, ow A2 fo UK and Alogoskoufis (987b), Table, ow B5 fo US. Labou supply equation estimated by instumental vaiables using numbe of employees and aveage weekly eanings. Studies based on US micoeconomic panel data typically find much smalle (often statistically insignificant o negative) estimates of the IES of labou supply (see suveys by Pencavel 986, and Bowning et al. 999). Mulligan (995) citicises these estimates because they fail to distinguish between tempoay and pemanent changes of wages. Note that equation (25) pedicts that individual labou supply should espond only to tempoay changes Δ(w /w 2 ). Mulligan suggests to look instead at episodes of anticipated, tempoay high wages. Examples include labou supply esponses of taxi dives to changing demand conditions, agicultual wokes due to weathe conditions, seasonal vaiations, US labou supply duing wold wa II and othe tempoay episodes such as the constuction of a gas pipeline in Alaska (974-77) o the Exxon Valdez oil spill in 989. Mulligan finds estimates of intetempoal labou supply elasticities nea two. 5 Equilibium in goods and labou maket and poductivity shocks In this section we combine consumption demand and labou supply based on individual behaviou and deive the equilibium in the aggegate economy. Aggegate demand can be deived fom individual consumption demand by hoizontally summing up individual consumption demand at diffeent levels of the inteest ate 2 d d (26) Y = C (, PV ( Y ),...) ( ) Figue 7 shows a negatively sloped aggegate demand schedule when plotted against the inteest ate. Remembe that fo the aggegate consumption demand only the ( + ) 20 Compae equations (23) and (25) above. 2 We use of the fact that all consumes face the same economy-wide inteest ate. 20
intetempoal substitution effect of changes in the inteest ate is elevant since thee is no net boowing o saving in the aggegate (closed) economy. Similaly we can deive the aggegate supply schedule by summing individual labou supply at diffeent inteest ates. The aggegate supply function depends negatively on the inteest ate (shown in Figue 7) and positively on elative wages and poductivity: s s (27) Y = A F[ L (, w / w,...)] ( + ) ( + ) 2 ( + ) In the neoclassical model the inteest ate (the elative pice of goods between time peiods) ensues equilibium in goods and labou makets. Figue 7 shows the equilibium, whee aggegate demand fo goods equals aggegate supply at inteest ate * d * s * Y, PV ( Y ),...) = Y (, w /,...) ( w2 The bonds maket also cleas at * ensuing that thee is no net boowing o saving in the aggegate economy, B d =0. This is an example of Walas law that implies that when the goods and labou maket ae in equilibium though adjustment of elative pices (inteest ate and elative wages), the bonds maket has to clea too. Y s (, w /w 2,..) * Y d (,PV(Y),..) Y * Y Figue 7: Aggegate demand and supply and goods maket equilibium 5. Aggegation Deaton (992) wans of the vaious pitfalls when aggegating individual to aggegate behaviou. Poblems include () heteogeneity of pefeences, (2) aggegation of infomation available to individuals to fom conditional expectations of futue income and pices and (3) population heteogeneity because of finite lifetimes. Attanasio 2
(999) citicizes the pactice of using aggegate macoeconomic data to estimate micoeconomic (stuctual) paametes since these estimates ae likely to be systematically biased. Recent business cycle eseach theefoe often goes the opposite oute of imposing paametes estimated fom micoeconomic elations when calibating macoeconomic models. See the Pescott-Summes (986) debate fo a discussion of the pos and cons of this calibation appoach. 5.2 Poductivity shocks What ae the consequences of changes in poductivity on aggegate demand and supply and the inteest ate in equilibium. Let us analyse pemanent and tempoay poductivity changes in tun. 5.2. Pemanent poductivity shock Conside fist a pemanent poductivity shock -- poductivity inceases in all peiods fom today onwads, A = A 2 = >0. This case is shown in Figue 8.. The diect (mechanical) effect is an incease in output fo evey level of the labou supply and fo a given inteest ate which is shown as ightwad shift of the aggegate supply function. 2. The indiect effects (associated with economic decisions) ae as follows: - labou supply emains unchanged since elative wages ae the same; - consumption demand C d inceases popotionally with changes in cuent income, since income changes ae pemanent, ΔC d = ΔY The total effect combines diect and indiect effects: output inceases and the inteest ate emains unchanged. This is the case because individuals have no incentive to change intetempoal decisions leaving the elative intetempoal pice unaffected. Notice that supply and demand cuves shift by a simila amounts. 5.2.2 Tempoay poductivity shock Conside next a tempoay incease in poductivity, only A >0, futue poductivity A 2,A 3, emains unchanged. Figue 9 shows the combination of the two effects:. The diect effect is the same as fo a pemanent incease in poductivity above leading to a ightwad shift of aggegate supply (shift (a) in Figue 9). 2. The indiect effects ae as follows: - labou supply and hence aggegate supply incease because elative wages go up (w /w 2 )>0 (shift (b) in Figue 9); - cuent consumption and aggegate demand shifts by a small amount, ΔC d >0, due to an incease in tansitoy income. The total effect is a ise in output and a fall of the inteest ate. This is the case because individuals attempt to smooth consumption by saving some of the tempoay income inceases fo the futue. Notice that in equilibium the amount consumed changes much moe than the small shift of the consumption demand function 22. This can be explained by the combined effect of lowe inteest ates and highe labou income due to highe labou supply. The stength of the effect depends as usual on the IES (see Table ). 22 Distinguish between a shift of and a movement along the demand function. 22
Y s (, w /w 2,..) * Y d (,PV(Y),..) Y Y Y Figue 8: Effect of pemanent poductivity shock on output and inteest ate Y s (, w /w 2,..) (a) (b) Y d (,PV(Y),..) Y Y Y Figue 9: Effect of tempoay poductivity shock on output and inteest ate 23
5.2.3 Theoetical pedictions and stylised facts The model pedicts a countecyclical esponse of eal inteest ates. Fo US data (see table in section), inteest ates ae weakly pocyclical which can be econciled with the model by combining tempoay and pemanent poductivity changes. Real wages ae pocyclical which is in line with the stylised facts. Empiically, the model pedicts a pocyclical esponse of labou supply to changes in poductivity. Hansen and Wight (992) obseve that fo the US the numbe of hous fluctuates much moe than poductivity. Togethe with the small vaiation in wages this implies a high IES to econcile theoy and aggegate data. Altenatively, Hansen and Wight ague that the standad neoclassical business cycle model can be augmented with the following featues to poduce a bette empiical fit: (i) indivisible labou, (ii) pefeences that ae non-sepaable ove time, (iii) home poduction and (iv) the impotance of othe shocks such as govenment spending fo business cycles which we analyse next. 6 Govenment This section intoduces the govenment into the intetempoal model. The govenment secto (local, state and fedeal levels combined) is a lage pat of ou economies. Bao (997, p.439) shows a atio of total govenment expenditues 23 to GDP aveaged ove the 970-85 peiod of 35% fo the US and 44% fo the UK. Bao also agues that the size of the govenment secto stongly inceased in the United States ove the last 80 yeas which is in line with the long-un expansion of the shae of govenment to GDP in developed economies. The govenment spends and collects evenue facing the following budget constaint each peiod: g g s (28) G + V + B = T + ΔB + ΔM P t t t t t / Total spending on the left-hand side of (28) consists of spending on goods by the govenment G, tansfes payments V, and inteest payments on the stock of govenment bonds B g. Fo simplicity we assume that govenment bonds pay the same inteest ate as financial makets. Total evenue on the ight-hand side of (28) is given by evenue fom taxes T, issuing govenment debt ΔB, and changing the money supply Δ M s / P. To simplify the discussion, we ignoe evenue fom pinting money 24 and set ΔM s = 0 in equation (28). Also we do not discuss tansfes and set V=0 in (28), even though they ae vey impotant component of total public spending (exceeding 50% in the US). Lage pats of tansfes takes the fom of edistibuting though the social secuity system in the US (see also the chapte on Pensions in this volume) o between income goups in many Euopean counties 25. 23 Notice that total expenditues also include spending not counted as components of GDP, fo example tansfes payments. This explains the diffeence to the enty in the table of section. 24 This inflation tax is a elatively small component of evenue fo low inflation counties. 25 Such tansfes could distot seveal intetempoal decisions, fo example etiement age, pesonal savings o decision to take on a job. These issues ae howeve beyond the scope of this suvey. 24