Two Perod Models Econ602. Sprng 2005. Lutz Hendrcks The man ponts of ths secton are: Tools: settng up and solvng a general equlbrum model; Kuhn-Tucker condtons; solvng multperod problems Economc nsghts: Rcardan equvalence. Equvalence of compettve equlbrum and plannng problem Math background: Lagrangean and Kuhn-Tucker (see lecture notes on mathematcal methods). The strategy of ths secton s to start wth exceedngly smple models. We wll then add more and more complcatons as we go along. Ths wll make the early models look somewhat slly, but t allows us to buld up the complexty step by step nstead of plungng rght nto a full-blown model. 1. Statc Models Ths secton begns our analyss of general equlbrum models. To keep the envronment as smple as possble, we start wth an economy that lasts for only one perod. Such models are obvously not very useful, but we wll see that the methods used to characterze ther equlbra carry over to much more complcated envronments. We begn by descrbng the model elements: Agents: There are N dentcal households. For now, there are no other agents (frms, government, ). Preferences: Households value consumpton of two goods accordng to a utlty functon u(c 1, c 2 ). Of course, u s strctly ncreasng n both arguments and (to be able to use standard optmzaton technques) strctly quas-concave (words n ths font are defned n the math notes). Margnal utlty goes to nfnty as consumpton goes to zero: lm c 0 u ( c1, c2) =. The subscrpts on functons (u 1, u 2 ) denote partal dervatves. Ths ensures that the household consumes postve amounts of both goods. Technology: The technology s trval: each agent receves endowments of the two goods (e 1, e 2 ). There s no producton. Endowments cannot be stored. Markets: Agents trade goods n a market, where everyone behaves as a prce-taker. There are no fnancal assets. The prces of the two goods are p 1 and p 2. That s t: nothng else s needed to descrbe the economy. In a sense, ths s already too much. Pursts would not want to prescrbe that agents behave as prce-takers. Instead, they would derve t from the fact that N s large. Recall the steps we have to go through n order to characterze equlbrum: 1
1. Solve the problem for each agent, takng prces as gven. Fnd the decson rules. Here the only agents are households. 2. Impose market clearng to determne equlbrum prces and quanttes. 1.1 Household problem Consder the problem facng one of the N households. It takes as gven market prces for the two goods, p 1 and p 2, and the endowments t receves, e 1 and e 2. The choce varables are c 1 and c 2. The only constrant s a budget constrant p 1 e1 + p2 e2 = p1 c1 + p2 c2. We can normalze the prce of one good to one (numerare), so we set p 1 = 1 and smply call the relatve prce of good 2 p = p 2 / p 1. The household then solves max u ( c 1, c 2) s.t. e 1 + p e2 = c1 + p c2 What exactly s a soluton to the household problem? One way of statng a soluton s as a vector of quanttes (c 1, c 2 ) for gven prces and endowments. An alternatve that s often more useful s to derve optmal choces as decson rules. The household s decson rules wll be of the form choce varables = f(state varables). The state varables are all relevant varables the household takes as gven: p, e 1, e 2. 1 The choce varables n ths case are c 1 and c 2, so we need to fnd consumpton functons. To derve the decson rules set up a Lagrangean: Γ = u( c1, c2) + λ ( e1 + p e2 c1 p c2) For ths partcular problem t would actually be easer to substtute the constrant nto the objectve functon and solve the unconstraned problem max u( e1 + p e2 p c2, c2), but the Lagrangean s nstructve. The frst order condtons are (1) Γ c = u ( c, c ) λ 0 / 1 1 1 2 = (2) Γ c = u ( c, c ) λ p 0 / 2 2 1 2 = The multpler λ has a useful nterpretaton. It s the margnal utlty of c 1, but more mportantly t s the margnal utlty of relaxng the constrant a bt,.e. the margnal utlty of wealth. The soluton to the household problem s then a vector (c 1, c 2, λ) that solves the FOCs together wth the budget constrant. Ths was, of course, a bt loose: the soluton s really a trple of functons. In partcular, we can wrte the decson rules as c p, e 1, e ). ( 2 1 Ths s actually a bt more complcated than t sounds. For example, why not add another rrelevant varable to the state vector, such as the poston of Jupter relatve to Neptune? If ths sounds lke a slly dea, take a look at the lterature on sunspots. The economcs lterature, of course, not the astronomy lterature. 2
Tp: Always explctly state what varables consttute a soluton and whch equatons do they have to satsfy. You should have a FOC for each choce varable and all the constrants. Make sure you have the same number of varables and equatons. Later on, ths wll make t easer to assemble the equatons needed for the compettve equlbrum. At ths pont, t s typcally useful to substtute out the Lagrange multpler. Take the rato of (1) and (2) to obtan (3) u 2 / u1 = p. Ths s the algebrac expresson of the famlar tangency condton: margnal rate of substtuton equals relatve prce. You have seen the graph wth ndfference curves tangent to budget constrants many tmes before. Now the soluton s a par (c 1, c 2 ) that satsfes (3) and the budget constrant. If we assume log utlty, u = log( c 1 ) + βlog( c2), ths can be solved n closed form: 1/ c 1 = λ, β / c 2 = p λ. Therefore, (3) becomes β c 1 = p c2. Substtute both nto the budget constrant to solve for λ: e + p = 1/ λ + β / λ λ = ( 1+ β) / W, 1 e 2 where W = e1 + p e2 s total wealth. Therefore, c p, e, e ) = W /(1 + ) and p c p, e, e ) = W β /(1 + ). 1( 1 2 β 2( 1 2 β Tp: Ths s a pecular (and often very useful) feature of log utlty: the expendture shares are ndependent of p. The reason s exactly the same as that of constant expendture shares resultng from a Cobb-Douglas producton functon: unt elastcty of substtuton. Tp: Recall that takng a monotone transformaton of u doesn t change the optmal polcy functons. In partcular, we can replace u by e same consumpton functons. 1.2 Market Clearng u = e β ln( c ) + ln( c2 ) 1 = c c 1 β 2. Convnce yourself that ths yelds exactly the There are two markets (for goods 1 and 2). Each agent supples the endowments e and demands consumpton c n those markets. Why sn t there just one market where agents exchange good 1 for good 2? It s better to thnk n terms of 2 markets n whch goods are traded for unts of account. I don t lke to use the word money here because there s no such thng n ths economy. The market clearng condton s aggregate supply = aggregate demand. Aggregate supply s smply the sum of ndvdual supples: 3
N h S = =1 e = N e where the second equalty follows from the fact that all agents are dentcal. Smlarly, aggregate demand s found by summng consumpton demands over households. To be pedantc, and nconsstent h wth what we dd above, let s wrte consumpton of household h as c. Then N h h D = =1 c = N c Market clearng therefore requres N e = N c or e = c. Ths s not surprsng: all agents are dentcal and therefore do not trade. More nterestng s to fnd the market clearng prce. The key s that each agent could trade any quantty at that prce, but chooses not to. The market clearng prce satsfes c = ( p, e1, e2) e. 1.3 Defnton of Equlbrum A compettve equlbrum s an allocaton ( c h ; h = 1, K, N; = 1,2) and a prce p that satsfy: 1. The h c satsfy the household optmalty condtons (FOC and budget constrant). 2. The two goods markets clear (e = c ). Now we count equatons and varables. We have 2 N consumpton levels and one prce. These satsfy 2 N household optmalty condtons and 2 market clearng condtons. However, Walras law tell us that one market clearng condton s redundant. Ths was more pedantc that we would usually want to be. Gven that all households face dentcal h problems, we would usually mpose from the outset that c = c for all h. Note that we could add the household s Lagrange multpler to the lst of varables. Then we would also have to add another equaton. We would do so by defnng the household optmalty condtons n 1. as 2 FOCs plus one budget constrant. Ths makes no dfference. We can do whatever s more convenent. In the log example, the prce s determned by p e = W β /(1 + ). 1.4 Insghts 2 β The method used to solve ths model carres over to more complcated ones. 1. Frst, derve condtons that characterze the soluton to each agent s problem, takng prces as gven. Ths typcally nvolves a number of FOCs and constrants. 2. State the market clearng condtons. 4
3. Make sure the number of unknowns equals the number of ndependent equatons, keepng n mnd that Walras law renders one market clearng condton redundant. 4. Solve. The rest s ether just algebra or smply ntractable. It s typcally useful to wrte out the defnton of equlbrum farly carefully: a lst of varables ( ) that satsfy Make sure the number of varables s the same as the number of equatons. It s also useful to be careful about the state varables: what are the gvens that we need to know n order to solve an agent s problem. These typcally nclude prces, endowments, asset holdngs, etc. Is t slly to have a model n whch nobody trades because all households are dentcal? It depends on the applcaton. The man reason for studyng these models s that they are tractable (all households can be dentcal; we can study a representatve household). Whether anythng s lost by makng that assumpton depends on the problem one s nterested n. 5
2. An Intertemporal Model Nothng prevents us from renterpretng the prevous model as a two-perod model. Assume that there s only one physcal commodty, but there are two dates (1 and 2). The utlty functon s the same as before [wrte t as u c ) + βu( ), but t s not essental that t be separable]. The good s not storable. ( 1 c2 How then can agents trade? They obvously need to trade ntertemporally. There are two possble arrangements. Frst, there may be markets at date 1 at whch agents can buy and sell goods at all future dates (n ths case only at date 2, but there could of course be more dates). Ths s called the Arrow-Debreu setup. In ths example, t means there s a market n whch I can sell goods today n order to receve unts of account, whch can then be used to buy goods for delvery tomorrow. Here, the prce p has the nterpretaton gvng up p goods today buys one unt tomorrow. Note that the equlbrum descrpton s exactly the same as n the one perod model. Whether the goods refer to dfferent physcal commodtes or to the same commodty at dfferent dates makes no dfference. Ths result holds generally. It may appear that ths approach s n trouble when there s uncertanty because t requres the agents to decde how much they wsh to consume at all future dates. But the approach s easly extended to cover the case of uncertanty by defnng a commodty to be ndexed by date and state of the world (e.g. an umbrella tomorrow, f t rans ). The mcro course wll handle these ssues n full glory. Alternatvely, there could be a sequence of markets. At each date, agents can buy and sell one perod bonds. Gvng up one unt of consumpton today buys a bond that promses (1+r) unts of consumpton tomorrow. Note the close relatonshp between the Arrow-Debreu prce p and the nterest rate r. If we defne p = 1/(1+r) the agents budget constrants and the descrpton of the equlbrum s the same n both arrangements. Ths s also a general result: the two setups can used nterchangeably and yeld the same allocaton, f markets are complete, whch essentally means that for each possble state of the world at each date, there exsts an asset that pays precsely n that state/date. 2 Adoptng the sequence of markets approach, we can wrte the household problem as max u( c1) + βu( c2) s.t. b = e 1 c1 ; c 2 = b ( 1+ r) + e2 In the frst perod, the household saves e 1 c 1 unts of account, for whch he buys b bonds, whch cost 1 unt of account a pece. In the second perod, the household receves the prncpal and nterest on the bonds purchased and uses t together wth the endowment to buy c 2. 2 We wll not go nto the detals of what complete markets mean. Suffce t to note that n the models consdered here markets are almost always complete. 6
Here, I have taken the lberty of normalzng all prces to one! Why can I do that? I can normalze p 1 and p 2 because I can choose the unts of account n both perods. In other words, the prce p 1 n ths economy s meanngless. It says: you need to gve up p 1 date 1 unts of account to buy one unt of c 1. Smlarly for p 2. Note that I can use dfferent unts of account at dfferent dates. Ths would not be the case f there was a way to carry unts of account from one perod to the next (as n the case where the unt of account s a commodty lke money). In ths economy bonds allow me to transfer unts of account from perod to perod, but the bonds have a real rate of return whch s endogenous. Ther nomnal return smply adjusts to get the same equlbrum real return no matter how I choose p 2 or p 1. And I can set the prce of a bond to 1 by choosng unts for bonds. If you don t beleve any of ths, smply set up the model wth prces at every date that may dffer from 1. You wll fnd that all prces drop out and the equlbrum s the same no matter how you choose them. The two perod budget constrants can then be combned nto a present value budget constrant: e 1 + e2 /( 1+ r) = c1 + c2 /(1 + r). The frst-order condtons are u ( c ) = λ, β u '( c2 ) = λ /(1 + r ) ' 1 Combnng them yelds u ( c ) = β(1 + r) u'( ) ' 1 c2 whch s known as an Euler equaton. It descrbes the ntertemporal tradeoff faced by the household: gvng up one unt of consumpton today costs u ( c ). Next perod, the household gans (1+r) unts of ' 1 consumpton, but these are dscounted at rate β. The Euler equaton states that a small reallocaton between consumpton today and tomorrow along the budget lne must leave utlty unchanged. That s, contemplate gvng up 1 2 ε (1 r dc = ε at date 1. The utlty cost s u c ) ε. Tomorrow, ths allows to consume an addtonal dc = + ) leadng to a utlty gan of β u ( c 2 ) ε (1 + r). Settng both equal yelds the Euler equaton. The same condton would hold wth more than two perods. Good 2 e 1 (1+r) ( 1 c 2 c 1 e 1 Good 1 7
One mplcaton of ths model s the Permanent Income Hypothess. A household s optmal consumpton path only depends on total wealth W, not on the ndvdual endowments separately (hs savngs do!). That s, the tmng of ncome over the lfe-cycle should not affect consumpton n any way. Ths predcton fals emprcally (Carroll and Summers 1991). Another mplcaton s Rcardan Equvalence: any polcy that only changes the tmng of lump-sum tax payments over the lfe-cycle (but leaves the present value unchanged) should have no effect on consumpton. We wll talk about ths n detal later on. 3. An Example Wth Trade So far there has never been trade n equlbrum because all agents were dentcal. Now we gve up ths assumpton and assume nstead that there are N agents who receve endowment e 1 when young, but nothng when old, and N agents who receve e 2 when old but nothng when young. Just to be pedantc, we wll go through all the steps agan. We frst need to solve the problems for all agents. Now we have two types of agents: households who receve early endowments and those who receve late endowments. 3.1 Households A household wth early endowment solves I max u( c1 ) + βu( c2 ) s.t. I I I e1 c1 + p c2 =. A household wth late endowments solves II max u( c1 ) + βu( c2 ) s.t. II II II p e2 c1 + p c2 =. We could now wrte out separate frst-order condtons for each household type, but t s easer to wrte a generc problem for household type s as s max u( c1 ) + βu( c2) s.t. s s s s W c1 + p c2 =. where wealth levels are W I = e 1 and W II = p e2. Assumng log utlty, we know that the decson rules are s s 1 β s s 2 β (4) c = W /(1 + ) and p c = W β /(1 + ), ( 1 s c2 s A soluton to the household problem of type s s then a par c, ) that satsfes (4). 8
3.2 Market clearng Aggregate demand for good s now D ( p,...) II N s s s (,, s = I h = c 1 p e1 e2 = I I 1 = N c ( p, e, e I 2 ) + N c II ) ( p, e II 1, e II 2 ) Smlarly, aggregate supply s here, then reduces to (5) I II S ( p,...) = N e + N e. Market clearng, n the specal case consdered I II N e = N c + N c. 3.3 Compettve Equlbrum A CE s an allocaton ( c s ; = 1,2; s = I, II) and a prce p that satsfy: 2 optmalty condtons for each household type (4 equatons) 2 market clearng condtons We have 5 varables and 6 equatons, one of whch s redundant by Walras law. In the log utlty case: e 1 = e1 /( 1+ β) + p e2 /(1 + β) = W /(1 + β) e = W β /(1 + β) (1/ ), 2 p where W = e1 + p e2. Ths s not surprsng: If every household spends the same fracton of ts h h endowment on good 1 ( c 1 = W /(1 + β)), then aggregate spendng on good 1 s that same fracton of the aggregate endowment, N W. Takng ratos yelds the market clearng prce: p = βe 1 / e 2. Ths makes sense: The prce for good 2 s hgher f there s more demand for t (β ) or less supply of t. Equlbrum consumpton levels are then c1 I = e1 /(1 + β) and 2 = e1 β /( 1+ β) ( e2 / βe1 ) = e2 /(1 + β) c I c1 II = p e2 /( 1+ β) = e1 β /(1 + β) and c2 II = e2 β /(1 + β). Ths fortunately adds up to the endowments as t should. Note the extremely odd outcome: household I receves fracton 1/(1+β) of both goods, regardless of hs relatve endowment (the beauty of log utlty ). At ths pont t s useful to revew how ths analyss fts nto the general setup presented earler. 9
1. The descrpton of the economy s our startng pont: 2 N households wth log utlty and a partcular endowment pattern. 2. We then solved the problems of all agents, whch n ths case means: the problems of two types of households. Snce we had already done that more generally before, we smply wrote down the polcy functons (4). Both budget constrants are redundant n ths case, not because of Walras law, but because they are bult nto the decson rules. 3. We next stated the market clearng condtons (5). 4. We then defned CE and characterzed t. A techncal detal: We usually talk about N as the number of households. Strctly speakng, to make ths model work, we need nfntely many households, so that each one s small and acts as a prce taker. For practcal purposes, we may smply assume a large, fnte N. A common alternatve s to assume that there s a contnuum of households of measure N. Ths s convenent because t allows us to normalze ths measure to N = 1. We then have a sngle representatve household of each type. Ths s what we wll assume n the future. 10
4. Addng Producton The next step s to add producton. The economy agan lasts for two perods and s populated by N = 1 dentcal households. The only fnancal market s the bond market wth nterest rate r as descrbed earler. In a bt more detal, the prmtves of the economy are: Preferences: u c ) + βu( ) ( 1 c2 Endowment: e 1 receved at date 1 Technology: Storng k at date 1 yelds f(k) at date 2. It s common to mpose Inada condtons on f. Ths means f '(0) =, f '( ) = 0, f ' > 0, f < 0. As we wll see qute often, Inada condtons rule out corner solutons (k = 0). Markets: Households consume and produce (store) usng technology f. In addton to the goods markets at both dates, there s a bond market at date 1, where households can buy or sell one perod bonds wth nterest rate r. The nterest s, of course, to be determned n equlbrum. 4.1 The household problem The household maxmzes u c ) + βu( ) subject to the budget constrants ( 1 c2 e 1 = c 1 + k + b and c 2 = b (1 + r) + f ( k), takng the endowment e 1 and the nterest rate r as gven. The present value budget constrant s: [ c f ( k)]/(1 + r = e1 c1 k 2 ) Lagrangean: Γ = u c ) + βu( c ) + λ ( e c k [ c f ( k)]/(1 + )) ( 1 2 1 1 2 r FOC: u ( c ) = λ β u '( c2 ) = λ /(1 + r ) f '( k) = 1+ r ' 1 The soluton to the household problem now conssts of (functons) (c 1, c 2, λ, k, b) that satsfy the 3 FOCs and the 2 budget constrants. The only added tem s the FOC for k whch smply requres that the two assets the household has access to should have the same rate of return. 4.2 Equlbrum A compettve equlbrum s defned as a lst of varables (λ, c 1, c 2, b, k, r) that satsfy the household optmalty condtons and market clearng. There are 3 market clearng condtons: one for bonds and two for goods. Let s frst turn to bond market clearng. The market clearng condton s b = 0. More 11
h precsely, ndexng each household by h = 1,, N the market clearng condton s b = 0. But we know that all b h are the same so that N b = 0. Of course, we know that there s no trade n equlbrum because all households are dentcal, but we stll need the market clearng condton to fgure out the prce (r). The budget constrants then mply goods market clearng at both dates: e 1 = c 1 + k (endowments can be eaten or stored) and c = f ( ). Wthout trade, households are essentally operatng under autarky, 2 k although they all could trade, f they only wanted to. Ths means we have 6 ndependent equatons (5 from the household and b = 0) that solve for the 6 unknowns. Note that Walras law n ths case makes both goods market clearng condtons redundant. 4.3 A log-utlty example To obtan a closed form soluton, assume u ( c) = ln( c) and then a lst (λ, c 1, c 2, k, r) whch satsfes θ f ( k) = k. A compettve equlbrum s 1/ c = λ, ( 1+ r ) / c = λ, θk = 1+ r, e c + k 1 β 2 Smple algebra allows to solve for k: θ 1 θ 1 = 1, k = c2 θ β( 1+ r) k = 1/( e1 k) θ 1 θ βθk k = βθ k = 1/( e k) / 1 e 1 k = k /( θβ) k = e1 /[ 1/( θβ) + 1] = e1 θβ /(1 + θβ) The comparatve statcs results are ntutve: k rses (r falls) as households become more patent (hgher β) or have larger perod 1 endowments. 4.4 Frms as separate agents So far the household has been a consumer-producer. More commonly, t s assumed that households consume and supply factors of producton (captal and labor) to frms. In many models both approaches lead to dentcal outcomes. Addng frms n the prevous model nvolves the followng modfcatons. Assume there s a contnuum of frms of measure one, whch s modeled as a sngle representatve frm. The frm rents captal (k F ) from the sngle representatve household at rental prce q to maxmze current perod profts: F = F π ( k ) = f ( k ) q k. The FOC s f ( k ) q. Note the trck: f the frm does not own any assets, ts problem s statc. Convnce yourself that Inada condtons on f ensure an nteror soluton. A soluton to the frm s problem s a par (π, k F ) that satsfes the frm s FOC and the defnton of profts. 12 F F
We also need to specfy what fracton of captal deprecates n producton. In general, rentng k to the frm results n output of f(k) + (1 δ) k. There s a unt measure of households. Each owns the same fracton κ of the representatve frm and s not allowed to trade ths ownershp (alternatvely one could ntroduce a stock market n whch shares of the frm are traded). Therefore, each household receves the same fracton κ of aggregate profts, whch t takes as gven. In partcular, the household s k does not affect the amount of profts t receves. How much captal to rent to frms and how many shares to own are two separate decsons. The household budget constrants are therefore c 1 = e 1 k and k ( 1+ r) + κ π = c2, whch mples a present value budget constrant of e c = c κ π) /(1 + ). Note that household 1 1 ( 2 r problem s somorphc to one wth bonds and second perod endowment κ π. Snce all households are dentcal and the mass of frms equals that of households, κ = 1. The FOCs are unchanged. A soluton to the household problem s a vector (c 1, c 2, k) that satsfes 2 FOCs and one budget constrant. Exercse: Modfy the model to add an equty market n whch households can buy and sell shares of the frm at date 1. Derve an equaton that characterzes the value of the frm. A compettve equlbrum s a lst of prces (r, q) and quanttes (c 1, c 2, k, k F, π) such that each agent s FOCs and constrants are satsfed and markets clear. Whch markets? There s a rental market for captal whch clears f k F = k. Goods market clearng at date 1 requres c 1 = e 1 k. In perod 2, the frm supples f ( k F ) and households demand c 2. In addton, households eat the left-over captal. Therefore: f ( k) + (1 δ) k = c2. We therefore have 7 varables that satsfy the followng condtons the frm s FOC and the defnton of π (2 equatons) the household optmalty condtons (3 equatons) 3 market clearng condtons How many of these equatons are ndependent? Clearly perod 1 goods market clearng s the same as the perod 1 budget constrant. Imposng captal market clearng on the perod 2 budget constrant yelds perod 2 goods market clearng. So both goods market clearng condtons are redundant. One equaton s stll mssng, whch s essentally an accountng dentty that relates the rental prce of captal pad by the frm (q) to the rental rate receved by the households (r). The household receves (1 δ) k n left-over captal and rental payments of q k. Therefore ( 1+ r ) k = (1 δ) k + q k or r = q δ. Common specal cases are no deprecaton (δ = 0) or full deprecaton (δ = 1). Sometmes deprecaton s wrapped nto the producton functon. I.e., the producton functon becomes g ( k) = f ( k) + (1 δ) k, whch changes the defnton of q: q = g ( k) = f ( k) + 1 δ. Otherwse ths s 13
equvalent to havng no deprecaton, except that g does not obey Inada condtons. For our purposes, the default assumpton s the frst one: nvestng k yelds (1+r) k = (1+q δ) k ncome next perod. Note: Not all models assume that consumpton goods can be converted one-for-one nto captal, so that the relatve prce of captal (the purchase prce, not q) does not equal one. We wll look at such models much later (see the secton on mult-sector models). 5. Readng CM ch. 1 14