In the OLG model, agents live for two periods. they work and divide their labour income between consumption and

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Dwayne Jordan
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1 1 The Overlapping Generatins Mdel (OLG) In the OLG mdel, agents live fr tw perids. When ung the wrk and divide their labur incme between cnsumptin and savings. When ld the cnsume their savings. As the name f the mdel implies, generatins verlap s in each perid there are ung agents saving and ld agents cnsuming. This mdel allws us t cnsider tw issues, the cnsequences f finite time hrizns and the cnsequences f difference between agents. We begin with the agent's cnsumptin saving decisin. Cnsumers brn in generatin t get labur incme w t and cnsume c t when ung and c t = (1+r t+1 )(w t -c t ) when ld. Cnsumers maximize U(c t,c t ). Fr example, the might maximize lg(c t ) + (1/(1+d))lg(c t ) r slightl mre generall u(c t ) + (1/(1+d))u(c t ) fr sme functin u. We assume that the chse c t ratinall s 1) u'(c t ) - ((1+r t+1 )/(1+d))u'(c t ) = 0 Fr a fixed wage w t, different interest rates r t+1 can cause a pattern f cnsumptin like that shwn in figure ne r like that shwn in figure tw. Where x = (c (w t,r t+1 ),c (w t,r t+1 )). An increase in r t+1 must cause an increase in c t but ma cause c t t increase r t decrease.

2 In the simplest tpe f OLG mdel there are n lng lived assets. This means that there is n wa t save s 1+r t+1 = 0 (see Samuelsn 1958 r Cass and Yaari 1968). In this case peple starve when the are ld.

2 2 In the simplest tpe f OLG mdel there are n lng lived assets. This means that there is n wa t save s 1+r t+1 = 0 (see Samuelsn 1958 r Cass and Yaari 1968). In this case peple starve when the are ld. This illustrates ne unusual feature f OLG mdels; the market utcme ma be ver bad. In fact the market utcme ma be Paret inferir t sme ther feasible utcme. The grim wrld described abve can clearl be imprved if the ung give sme fractin f their incme t the ld each perid. Everne can be made better ff b such a scial securit sstem. Indeed, the ppulatin grws b a factr 1+n each generatin, everne can get 1+n units f cnsumptin gd when ld fr each unit f cnsumptin gd given awa when ung. The ptimal stead state is achieved when each f the ung cntributes x units t the scheme and each f the ld get x(1+n) units and 2) u'(w t -x) - ((1+n)/(1+d))u'(x(1+n)) = 0.

3 3 This is what a scial planner wh cared nl abut the stead state level f happiness wuld require peple t d. Such a desirable utcme can be achieved withut a scial securit sstem. Let's sa that there is ne unit f intrinsicall wrthless asset, sa pieces f paper with pictures printed n them (mne). If these cannt be cunterfeited, the ma be valuable in equilibrium in an OLG mdel. Each generatin f ung ma bu the mne f the ld because the knw that the new ung will bu the mne frm them when the are ld. If p t is the amunt f mne required t bu ne unit f gds at time t then the return t hlding mne is p t /p t+1, fr each unit f cnsumptin gd given up at t ne gets p t /p t+1 units f gd when ld. In ther wrds 1+r t+1 = p t /p t+1. In the case f w t fixed at w, ttal prductin in time t is N t w where N t is the number f ung brn at t. Since the gd can nt be stred, ttal cnsumptin equals ttal prductin s 2) c t-1 N t-1 + c t N t = wn t This means that the value f the mne 1/p t is given b 3) 1/p t = N t-1 c t-1 = N t (w - c t ) A little algebra tells us that 4) c t = w - (1/(1+n))c t-1. This and the first rder cnditin f the ung 5) u'(c t ) - p t /((1+d)p t+1 )u'(c t ) = 0 mean that we can calculate the equilibrium p t+1 frm p t. This

4 4 calculatin can be dne graphicall where c t is calcualted frm c t-1 b drawing the hrizntal line t the graph f 4) the line thrugh (w,0) with slpe -1+n. Then c t = 1/(N t p t+1 can be calculated using equatin 5 b drawing the vertical line t the ffer curve, the graph f equatin 5. The ptimal stead state defined b 2 can alwas be an equilibrium f the verlapping generatins ecnm with mne. The value f the mne at t has t equal xn t fr x as defined in equatin 2. This means that mne has t becme mre and mre valuable as the price f the gd declines b a factr 1/(1+n) each perid. This is an equilibrium n matter what the shape f the ffer curve. The ptimal stead state is nt the nl pssible equilibrium f the verlapping generatins ecnm with mne. Figure 3 shws an inflatinar equilibrium in which the stck f mne becmes less an less valuable cmpared t wn t. This implies that thse unfrtunate enugh t be brn at a high t starve when the are ld which is ver sad.

5 5

6 6 In cntrast if the ffer curve is as in figure 4 a deterministic ccle is pssible in which e.g. even numbered generatins cnsume a large amunt bth when ung and when ld, but dd numbered generatins cnsume little when ung and when ld which is less sad but nt fair.

7 7 As figure 5 shws the ptimal stead state can be a stable equilibrium s the ecnm ends up there even if it started smewhere with p different frm the ptimal p but nt t far frm it. These figures als illustrate anther dd feature f verlapping generatins ecnmies, the equilibrium utcme is nt unique. Equatins 4 and 5 allw us t calculate p 2, p 3 &c fr man different values f p 1. This means that knwledge f the tastes and the technlg des nt enable us t sa if the market utcme will be Paret ptimal r ver ver bad. In the case f the ffer curve in figure 5 there are als equilibria in which smething trul strange happens. In perid 3,

8 the Walrasian auctineer flips a cin. If it cmes up heads (testa s t) then p 3 is high s the ld have lw cnsumptin c 2 is lw and c 3 is high and vice versa if the cin cmes up tails (crce s c). When 2 nd generatin agents are ung, the knw this will happen and the chse cnsumptin under uncertaint. The in each case (testa r crce) there is a series f prices which makes markets clear. Thus the equilibrium can be stchastic even if tastes and technlg are nt. In figure 6 etc. is et cetera, that is I claim that I culd shw the cntinuatin f the testa equilibrium (if I culd draw better). 8

CAUSAL INFERENCE. Technical Track Session I. Phillippe Leite. The World Bank

CAUSAL INFERENCE. Technical Track Session I. Phillippe Leite. The World Bank CAUSAL INFERENCE Technical Track Sessin I Phillippe Leite The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Phillippe Leite fr the purpse f this wrkshp Plicy questins are causal