14.06 Lecture N otesInterm ediate M acroeconom icsGeorge-Marios AngeletosMIT Depar tment of EconomicsSpring 2004Chapter 4App l icatio ns4.1 Con sum ption Smoothing4.1.1 The Intertemporal Budget• For any given sequence of interest rates {Rt}∞t=0, pick an arbitrary q0> 0 and defineqtrecursively byqt=qt−11+Rt,that is,qt=q0(1 + R1)...(1 + Rt).qtrepresents the price of period−t consumption relative to period−0 consumption.• The buget in period t is given byct+ at+1≤ (1 + Rt)at+ ytwhere atdenotesassetsandytdenotes labor income.89George-Marios Angeletos• Multiplying the period-t budget by qtandaddingupoverallt, we getTXt=0[qtct+ qtat+1] ≤TXt=0[qt(1 + Rt)at+ qtyt] .Using the fact that qt(1 + Rt)=qt−1, we haveTXt=0qt(1 + Rt)at= q0(1 + R0)a0+TXt=1qt−1atso that the abo ve reduces toTXt=0qtct+ qTaT +1≤ q0(1 + R0)a0+TXt=1qtyt• A s suming either that the agen t dies at finite time without leaving any bequests, inwhich case aT +1=0, or that the time is infinite, in which case w e impose qTaT +1→ 0as T →∞, we conclude that the intertemporal budget constraint is giv en byTXt=0qtct≤ q0(1 + R0)a0+TXt=1qtyt,where T<∞ (finite horizon) or T = ∞ (infinite horiozon). The in terpretation issimple: The presen t value of the consumption the agen t enjo ys from period 0 and oncan not exceed the value of initial assets the agent has in period 0 plus the presen tvalue of the labor income the agent receiv es from period 0 and on.• We can rewrite the intertem poral budget asTXt=0qtct≤ q0x0wherex0≡ (1 + R0)a0+ h0,h0≡∞Xt=0qtq0yt.90Lecture Notes(1 + R0)a0is the hou se h old ’s financial wealth as of period 0.h0isthepresentvalueoflabor income as of period 0; we often call h0the househo ld’s human we alth as of period0. The sum x0≡ (1 + R0)a0+ h0represents the household’s effective wealth.• Note that the sequence of per-period budgets and the in tertemporal budget constrain tare equivalen t. We can then write the household’s consumption problem as follo w smaxTXt=0βtU(ct)s.t.TXt=0qtct≤ q0x0• N ote that the abo ve is lik e a “static” consumption problem: Inter pret ctas differentconsumption goods and qtas the price of these goods. This observation relates to thecontext of Arrow -Debreu markets that we discuss later.4.1.2 C onsump tion Smoothing• T he Lagrangian for the household’s problem isL =TXt=0βtU(ct)+λ"q0x0−TXt=0qtct#where λ is the shado w cost of resources for the consumer (that is, the Lagrange mul-tiplier for the in t ertemporal budget constrain t).• The FOCs giveU0(c0)=λq0for period 0 and similarlyβtU0(ct)=λqtfor any period t.91George-Marios Angeletos• Suppose for a momen t that the interest rate equals the discoun t rate in all periods:Rt= ρ ≡ 1/β − 1.Equivale ntly,qt= βtq0TheFOCsthenreducetoU0(ct)=λq0for all t, and thereforect= c.for all t. That is, the level of consump tio n is the same in all periods.• But ho w is the value of c determ ined? From the in tertemporal budget, using qt= βtq0and ct= c, we inferq0x0=TXt=0qtct=11 − βq0cand thereforec =(1− β)x0=(1− β)[(1+R0)a0+ h0]That is, the household consum es a fraction of his initial effective w e alth . This fractionis give n by 1 − β.4.2 Arrow-Debreu Markets4.2.1 Arrow-Debreu v ersus Radner• We no w in troduce uncertainty...92Lecture Notes• Let q(st) be the period-0 price of a unite of the consumable in period t and ev ent stand w( st) the period-t wage rate in terms of period-t consu mab les for a given even t st.q(st)w(st) is then the period-t and ev e nt-stwage rate in terms of period-0 consunam-bles. We can then write household’s consum p tion pro ble m as followsmaxXtXstβtπ(st)U¡cj(st),zj(st)¢s.t.XtXstq(st) · cj(st) ≤ q0· xj0wherexj0≡ (1 + R0)a0+ hj0,hj0≡∞Xt=0q(st)w(st)q0[lj(st) − Tj(st)].(1 + R0)aj0is the hous e hold’s financial wealth as of period 0.Tj(st) is a lump-sum taxobligation, which ma y depend on the identit y of household but not on its c hoices. hj0is the present value of labor income as of period 0 net of taxes; we often call hj0thehousehold’s human wealth as of period 0. The sum xj0≡ (1 + R0)aj0+ hj0represents thehousehold’s effective wealth.4.2.2 The Consumption Problem with CEIS• Suppose for a mom ent that preferences are separable between consumption and leisureand are homothetic with respect to consumption:U(c, z)=u(c)+v(z).u(c)=c1−1/θ1 − 1/θ93George-Marios Angeletos• L ettin g µ be the Lagrange multiplier for the in tertemporal budget constraint, the F O Csimp lyβtπ(st)u0¡cj(st)¢= µq(st)for all t ≥ 0. Ev aluating this at t =0, we infer µ = u0(cj0). It follows thatq(st)q0=βtπ(st)u0(cj(st))u0(cj0)= βtπ(st)µcj(st)cj0¶−1/θ.That is, the price of a consumable in period t relative to period 0 equals the marginalrate of in tertemporal substitution between 0 and t.• S o lv ing qt/q0= βtπ(st)£cj(st)/cj0¤−1/θfor cj(st) giv escj(st)=cj0£βtπ(st)¤θ·q(st)q0¸−θ.It follo ws that the prese nt value of consum pt ion is giv e n byXtXstq(st)cj(st)=q−θ0cj0∞Xt=0£βtπ(st)¤θq(st)1−θSubstituting into the resource constraint, and solving for c0, w e concludecj0= m0· xj0wherem0≡1P∞t=0£βtπ(st)¤θ[q(st)/q0]1−θ.Consumption is thus linear in effective wealth. m0represent the MPC out of effectivewealth as of period 0.94Lecture Notes4.2.3 Intertemporal Consumption Smoothing, with No Uncertain t y• C onsider for a moment the case that there is no uncertainty, so that cj(st)=cjtandq(st)=qtfor all st.• Then, the riskless bond and the Arrow securities s atisf y the follo win g arbitrage condi-tionqt=q0(1 + R0)(1 + R1)...(1 + Rt).Alternatively,qt= q0h1+eR0,ti−twhereeR0,trepresents the “a verage” interest rate between 0 and t. Next, note that m0is decreas ing (increasing ) in qtif and only if θ>1 (θ<1). It follows that the marginalpropensity to save in period 0, which is simply 1 − m0, is decreas ing (increasing) ineR0,t, for any t ≥ 0, if and only if θ>1 (θ<1).• A similar result applies for all t ≥ 0. We concludeProposition 22 Su ppose preferences are seperable between consum ption and leisure andhomothetic in consumption (CEIS). Then, the optimal consumption is linear in contempo-raneous effective wealth:cjt= mt· xjtwherexjt≡ (1 + Rt)ajt+ hjt,hjt≡∞Xτ=tqtqt[wτljτ− Tjτ],mt≡1P∞τ=tβθ(τ −t)(qτ/qt)1−θ.95George-Marios Angeletosmtis a decreasing (increasing) function of qτfor any τ ≥ t if and only θ>1
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