1. Transversality conditions, ``no-Ponzi'' conditions, and ``intertemporal budget constraints''2. Intertemporal budget constraints3. No-Ponzi conditions4. Something completely different5. Real Business Cycles6. But Don't We Know Prices are Sticky?7. 8. The Firm9. Government10. InterpretationEco504, Part II Spring 2004 C. Sims1. Transversality conditions, “no-Ponzi” conditions, and“intertemporal budget constraints”Borrowing Constraints: When agents are modeled as able to raise resourcesby issuing securities or borrowing, there must be some constraint that prevents theirraising arbitrarily large resources by issuing arbitrarily large amounts of securities.Otherwise they will consume, or issue dividends (in the case of firms) in arbitrarilylarge, and infeasible, amounts. Our examples have used simple deterministic boundson borrowing: Bt≥ −¯B or Ct≤ Wt.2. Intertemporal budget constraints• Relate the borrowing constraint to ability to pay.• A complete-markets notion.• Firm: market net worth remains non-negative• Consumer: with a market stochastic discount factor Φt,Bt≤ Et∞Xs=1βsΦt+sΦt(Yt+s− Ct+s) . (1)for any feasible consumption plan. That is, this constraint defines the set offeasible consumption plans.• Corresponds to a period budget constraint and no-Ponzi condition of the formCt≤ Yt− β−1Φ−1tBt−1+ Bt(2)limT →∞EtβTΦt+TBt+T ≤ 0 . (3)• Verifiable at t because there are markets at t in which the Φtvalues corre-sponding to every future contingency are quoted and b ecause all probabilitiesare known. If (as usual in such a model) Ct< 0 is impossible but Ctarbitrarilyclose to 0 is possible, then we just check, by lo oking at the market value oftraded assets, thatBt≤ Et∞Xs=1βsΦt+sΦtYt+s3. No-Ponzi conditions• Hybrids of the ITBC and B ≤¯B.• Apply to incomplete-markets situations.• No uniquely accurate or reasonable way to set them up.c2004 by Christopher A. Sims. This document may be reproduced for educational and researchpurp ose s, s o long as the copies contain this notice and are retained for personal use or distributedfree.2• With single-period budget constraint Bt= Ct− Yt+ Rt−1Bt−1, solve forwardto obtainBt=TXs=1sYv=1R−1t+s−1(Yt+s− Ct+s) +TYs=1R−1t+s−1BT +1. (4)• If the last term in this expression goes to zero as T → ∞, get something like(1).• Equivalent to what is usually called the no-Ponzi condition, which is the re-quirement that the last term in (4) go to zero as T → ∞.• Important differences from (1):– the condition is often written (as here) with no expectation operator infront of it;∗ With an E in front, it seems to imply that running a risk that some-thing will happen that makes payment impossible is OK, so long asthat is offset by a sufficient probability of having more than enoughresources to pay. This implicitly assumes that B is not a risk-freebond.∗ Without an E in front, it is extremely restrictive, often implying anagent can’t take out any loans at a risk-free rate, because it requiresthat even in extremely improbable worst-case scenarios the agentmust be able to pay back the debt for sure.– the discounting is done using some existing market return (here Rt), notthe ideal complete-markets stochastic discount factor.4. Something completely differentTransversality. Borrowing constraints, no-Ponzi conditions, and intertemporal bud-get constraints are all inequalities, bounding debt or net worth from below. They areperceived by agents as set externally. The TVC is a condition for optimization, in mosteconomic models playing the role of ruling out solutions in which wealth grows rapidlyforever but is never used to provide consumption or dividends.5. Real Business CyclesWhat is the real business cycle theory or school?• It might seem obvious: an approach that attempts to explain business fluctua-tions as efficient responses of producers and consumers to random variation inthe technological environment. And this characterization is partially correct.• But there are papers that are by RBC economists and in the RBC style thatexplore sticky prices (Chari, K ehoe and McGrattan, e.g.) and that explorethe implications of financial frictions (several papers by Christiano and Eichen-baum, e.g.). So what else characterizes the RBC style?– stochastic general equilibrium modeling ;3– much more readiness to devote resources to computation of solutions tononlinear GE models, and to simplifying models so that such solutions arepossible;– willingness to leave prices out of the model, particularly the overall pricelevel, and to ignore implications of the model for price behavior;– adherence to “calibration” rather than “estimation and testing” as thecriterion for assessing a model’s fit;All these criteria have become fuzzier over time, with some of the RBC char-acteristics becoming common outside the school (like stochastic GE modelingand, sadly, calibration) and some outside characteristics (like price stickinessand statistical assessment of fit) showing up at least occasionally in RBC work.6. But Don’t We Know Prices are Sticky?Transactions prices, measured directly, might be far from the true spot prices oftheory, and thus display a lot of stickiness whose real effect is small.Consumer’s objective:maxCs,Ls,Bs,YsE"∞Xt=0βtU(Ct, 1 − Lt)#Consumer’s constraints:λ: Ct+BtPt+ τt≤YtPt+ πt+Rt−1Bt−1Ptµ: Yt≤ wt(Lt− (1 − δ)Lt−1) + (1 − δ)Yt−1Unusual variables: Y : wage bill; w: wage on new contracts; δ: rate of contractdissolution; τ: taxes.7.FOC’s:∂C: D1Ut= λt∂L: D2Ut= µtwt− β(1 − δ)Et[µt+1wt+1]∂Y :λtPt= µt− β(1 − δ)Etµt+1∂B:λtPt= βEtRtλt+1Pt+14Wage=MUL/MUC:D2UtD1Ut=wtPt− β(1 − δ)Etµt+1Pt+1µtPtwt+1Pt+11 − β(1 − δ)Etµt+1µt(5)Forward-looking:wt= Et"∞Xs=0βs(1 − δ)sD2Ut+sµt#µt= Et"∞Xs=0βs(1 − δ)sD1Ut+sPt+s#(6)8. The Firmobjective:maxLs,Ys,xsE"∞Xt=0βtΦtxt#constraints:ζ: xt≤ Atf(Lt) −YtPtν: Yt≥ wt(Lt− (1 − δ)Lt−1) + (1 − δ)Yt−1FOC’s:∂x: Φt= ζt∂Y :ζtPt= νt− β(1 − δ)Etνt+1∂L: ζtAtf0(Lt) =νtwt− β(1 − δ)Et[νt+1wt+1]Wage=MPL:Atf0(Lt) =wtPt− β(1 − δ)Etνt+1Pt+1νtPtwt+1Pt+11 − β(1 − δ)Etνt+1νt(7)5Forward looking:νt= Et"∞Xs=0βs(1 − δ)sζt+sPt+s#= Et"∞Xs=0βs(1 − δ)sD1Ut+sPt+s#(8)9. GovernmentBudget Constraint:BtPt+ τt= Rt−1Bt−1Ptbehavior: The