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# MIT 14 06 - Arrow-Debreu Markets and Consumption Smooth-ing

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Chapter 4Applic atio ns4.1 A rrow-Debreu M arkets and Consump tion Smooth-ing4.1.1 The Intertemporal Budget• For any given sequence {Rt}∞t=0, pic k an arbitrar y q0> 0 and define qtrecursively b yqt=q0(1 + R0)(1 + R1)...(1 + Rt).qtrepresents the price of period−t consumption relative to period−0 consumption.• Multiplying the period-t budget by qtandaddingupoverallt, we get∞Xt=0qt· cjt≤ q0· xj089George-Marios Angeletoswherexj0≡ (1 + R0)a0+ hj0,hj0≡∞Xt=0qtq0[wtljt− Tjt].The abo ve represents the intertemporal budget constraint. (1+R0)aj0is the household’sfinancial we alth as of period 0.Tjtis a lump-sum tax obligation, which may depend onthe identit y of household but not on its choices. hj0is the present value of labor incomeas of period 0 net of taxes; we ofte n call hj0the hous ehold ’s human wealth as of period0. The sum xj0≡ (1 + R0)aj0+ hj0represents the household’s eﬀective wealth.• Note that the sequence of per-period budgets and the in tertemporal budget constraintare equivalent.We can then write household’s consumption problem as follow smax∞Xt=0βtU(cjt,zjt)s.t.∞Xt=0qt· cjt≤ q0· xj04.1.2 Arrow-Debreu versus Radner• We no w introduce uncertainty...• Let q(st) be the period-0 price of a unite of the consumable in period t and event stand w(st) the period-t wage rate in terms of period-t consumables for a given event st.q(st)w(st) is then the period-t and event-stwage rate in terms of period-0 consunam-90Lecture Notesbles. We can then write househo ld’s consumption proble m as follo wsmaxXtXstβtπ(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 household’s financial we alth as of period 0.Tj(st) is a lump-sum taxobligation, which may depend on the identity of household but not on its choices. hj0is the present value of labor income as of period 0 net of taxes; w e often call hj0thehousehold’s human wealth as of period 0. The sum xj0≡ (1 + R0)aj0+ hj0represents thehousehold’s eﬀective wealth.4.1.3 The Consumption Problem with C EI S• Suppose for a momen t that preferences are separable bet ween consumption and leisureand are homothetic with respect to consumption:U(c, z)=u(c)+v(z).u(c)=c1−1/θ1 − 1/θ• Letting µ be the Lagrange multiplier for the intertemporal budget constrain t, the F O Csimp lyβtπ(st)u0¡cj(st)¢= µq(st)91George-Marios Angeletosfor all t ≥ 0. Ev aluating this at t =0, we infer µ = u0(cj0). It follo ws 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 relativ e to period 0 equals the marginalrate of in tertemporal substitution between 0 and t.• Solving qt/q0= βtπ(st)£cj(st)/cj0¤−1/θfor cj(st) givescj(st)=cj0£βtπ(st)¤θ·q(st)q0¸−θ.It follows that the present value of consumption is giv e n byXtXstq(st)cj(st)=q−θ0cj0∞Xt=0£βtπ(st)¤θq(st)1−θSubstituting in to the resource constraint, and solving for c0, we 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 0.4.1.4 Intertemporal Consumption Smoothing, with No Uncertain t y• Consider for a momen t the case that there is no uncertain ty, so that cj(st)=cjtandq(st)=qtfor all st.92Lecture Notes• Then, the riskles s bond and the Arrow securities satisfy the following arbitrage condi-tionqt=q0(1 + R0)(1 + R1)...(1 + Rt).Alternatively,qt= q0h1+eR0,ti−twhereeR0,trepresents the “a verage” interest rate bet ween 0 and t. Next, note that m0is decreasing (incre as ing) 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 decreasing (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 Suppose preferences are seperable between consumption and leisure andhomothetic in consumption (CEIS). T hen, the optimal consump tion 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.mtis a decreasing (increa sing) function of qτfor an y τ ≥ t if and only θ>1 (θ<1).That is, the marginal propensity to save out of eﬀective wealth is increasing (decreasing) infuture interest rates if and only if the elasticity of intertemporal substitution is higher (lower)93George-Marios Angeletosthan unit. Moreover, for given prices, the optimal consumption path is inde pendent of thetimining of either lab or income or taxes.• Ob viously, a similar result holds with uncertain ty, as long as there are complete Arro w -Debreu mark ets.• Note that any expected c h ang e in income has no eﬀect o n consumption as long asit does not aﬀect the present value of labor income. Also, if there is an innovation(unexpected ch a n ge) in income, consumption will incr ease today and for ever by anamount proportional to the innovation in the annuity value of labor income.• To see this more clearly, suppose that the in terest rate is constan t and equal to thediscount rate: Rt= R =1/β − 1 for all t. Then, the marginal propensity to consumeism =1− βθ(1 + R)1−θ=1− β,the consumption rule in period 0 becomescj0= m ·£(1 + R)a0+ hj0¤and the Euler condition reduces tocjt= cj0Therefore, the consumer choose a totally flat consumption path, no matter what is thetime variation in labor income. And any unexpected change in consumption leads toa parallel shift in the path of consumption b y an amount equal to the annuit y valueof the change in labor income. This is the manifestation of intertemporal consumptionsmoothing.94Lecture Notes• More generally, if the interest rate is higher (lower) than the discount rate, the pathof consumption is smooth but has a positive (negative) trend. To see this, note thatthe Euler co ndition islog ct+1≈ θ[β(1 + R)]θ+logct.4.1.5 I ncomplete Markets and Self-Insurance• The abov e analysis has assumed no uncertain ty, or that mark ets are complete. Ex-tending the model to introduce idiosy nc ratic uncertaint y in labor income would imp lyan Euler condition of the formu0(cjt)=β(1 + R)Etu0(cjt+1)Note that, because of the conv exity of u0, as long as Vart[cjt+1] > 0, we have Etu0(cjt+1) >u0(Etcjt+1) and thereforeEtcjt+1cjt> [β(1 + R)]θThis

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