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1-35.86. Yaari M. E. (1965). Uncertain lifetime, life insurance, and the theory of the consumer,Review of Economic Studies. 32. pp. 137–150.104Appendix 1Variablesoutputconsumption of differentiated good, Dixit-Stiglitz aggregatorC ( C s , C b ) aggregate(savers’ and borrowers’ consumption)price indexPfirm profitПsbh (h ,h )total hours worked (by savers, by borrowers)Wreal wageinflation ratennatural interest raterinominal interest ratesbT ( T , T ) lump-sum taxes (taxes on savers, taxes on borrowers)YsBs ( b )Bg ( b g )GGU ( g U )bonds issued by savers (real)GP ( g P )Dproductive government expenditure (log-deviation)government bonds (real)total government expenditure (Dixit-Stiglitz aggregator)utility-enhancing government expenditure (log-deviation)debt limitA1.1 Derivation of the model equations1.1.1 Derivation of private and public demandAs the steps of derivation are the same for private and public demand, here thederivation of the public demand on a differentiated good i is presented.

Government demandis obtained analogously. Consumer of type i solves the following minimization problem:1min ci ( j )  pt ( j )cti ( j )dj ,t0(A1.1)1 ≥ (∫0 ()(−1)/1)/(−1). / −1 1 i  −1/ i,L =  pt ( j )c ( j )dj +   Ct −   ct ( j )dj 00(A1.2)1/ −1 1 i  −1/ Ltiii−1/ = 0.=p(j)−c(j)c(j)dj ttt tcti ( j )0(A1.3)1ititit1051Taking into account that Ct1/ =  cti ( j ) −1/ dj ]1/ −1 , we obtain:0−i −1/t tpt ( j ) =  c p ( j) → c ( j) = C  t i  , t 1/tit( j )C( −1)/−1 pt ( j )iiCt =   i  Ctdj  0   t   /( −1)it /( −1)− 1   1 1−=  i    pt ( j )dj  t   0Cti .(A1.4)1/( −1)1Therefore,  =   pt1− ( j )dj 0= Pt .itSubstituting this expression in the demand function we obtain:− p ( j) c ( j) = C  t . Pt itit(A1.5)By aggregating by two types of consumers we get:− p ( j) ct ( j ) = Ct  t . Pt (A1.6)Using the same procedure, government demand can be obtained:− p ( j) gt ( j ) = Gt  t . Pt (A1.7)1.1.2 Utility maximization problemE0   t (i ) U i ((Ct (i)) − ti ( ht (i)) + ti (GtU )  with i = s or b ,(A1.8)t =01s.t.

B (i) + Wt Pht t (i ) +  Пt (i ) = (1 + it −1 ) Bt −1 (i ) + PCt t (i ) + PTt t (i ) ,iti(A1.9)0(1 + rt )Bt (i ) Dt (i ) .Pt(A1.10)1L = E0   (i)[U (Ct (i)) −  (ht (i)) +  (G ) + 1t (i)( B (i) + Wt Pt ht (i) +  Пt (i) −i0tt =0iititUtiti0106(A1.11)−(1 + it −1 ) Bt −1 (i ) − PCt t (i ) − PTt t (i )) +  2 t (i )( Dt − (1 + rt )Bt (i ))] .PtLti= 0  U ci (Ct (i)) = 1t (i) Pt ,Ct (i)(A1.12)Lti= 0  ci (ht (i)) = 1t (i) PWt t,ht (i)(A1.13)Lti1 + rt= 0  1t (i) −  (i) Et 1,t +1 (i)(1 + it ) − 2t (i)= 0,Bt (i)Pt(A1.14)and the slackness condition:2 ≥ 0, ( ) ≥ (1 + ) ()and  2t (i )( Dt − (1 + rt )Bt (i ))=0.PtSavers' consumption and labor supplyFor savers debt constraint is not binding, thus, the Lagrangian multiplier of thisconstraint 2t ( s) = 0 from the slackness condition. Tanking this into account and substitutingthe definition of 1t ( s ) from the first order condition into the last one, we get the Eulerequation for the saver:( )(U cs Cts =  (1 + it ) Et [U cs Cts+1) PPt].(A1.15)t +1Combining (A1.12) and (A1.13) labor supply function can be obtained:Wt = hs (hts )U cs (Cts ).(A1.16)Borrowers' consumption and labor supplyIn the case of borrowers debt limit constraint is binding.

Borrowers’ consumption canbe defined by budget constraint where real value of debt is expressed through the exogenousdebt limit:1D1C = − Dt −1 + t + Wt b htb +  Пtb − Tt b .1 + rtPt0btLabor supply of a borrower is obtained from the utility maximization problem:107(A1.17)Wt = hb (htb ).U cb (Ctb )(A1.18)1.1.3 FirmsThe demand on a good j consists of private and public demand of this good.Thus, market clearing condition for good j can be written as follows:yt ( j ) = ct ( j ) + gt ( j ) .(A1.19)Taking into account that:− p ( j) ct ( j ) = Ct  t , Pt (A1.20)− p ( j) gt ( j ) = Gt  t , Pt (A1.21)output of good j can be written as follows: p ( j) yt ( j ) = (Ct + Gt )  t Pt −(A1.22)Taking into account market clearing condition for the goods market Yt = Ct + Gt it canbe written as follows:− p ( j) yt ( j ) = Yt  t . Pt (A1.23)Profit maximization problem and price settingThere is a continuum of firms measure one, where a fraction  can set their prices at allperiods, while 1 −  of firms set their prices one period in advance.Firms maximize the infinite sum of profits using  t =  s 1st − (1 −  s ) 2bt as a discountfactor, where  1st and  2st are Lagrangian multipliers from utility maximization problem:Et t=0t  pt ( j) yt ( j) −Wt Pht t ( j ) ,(A1.24)− p jt s.t.

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