Life-Cycle ModelMotivation: We now combine the production struc-ture of the Solow growth model with optimizing con-sumers. This will give us some of the same insightsof growth theory with the ability to conduct policyanalysis as it is done in microeconomics.1Model Ingredients:1. Utility: U(cyt,cot+1)=α log cyt+(1− α)logcot+12. Demographic Structure: agents live two periodsand die. They produce children that live for two peri-ods and die. There are as many young agents N as oldagents N each time period. There are infinitely-manyperiods.22. Endowments: each young agent has one unit ofwork time, whereas each old agent cannot work. Atthe beginning of time (t = 1) there are some initialold agents who own all the physical capital .4. Technology:Ct+ It= Yt= F (Kt,Lt)=AKβtL1−βtKt+1= Kt(1 − δ)+ItIntuition:Agents enjoy consuming both when young and whenold. Given that they have no labor income when old,how can they consume in old age?Answer: Retirement saving. There is no governmentforcing the young to provide social security and thereare no altruistic links across generations.3How do Ktand ktevolve?cyt= αwtand c0t+1=(1−α)wt1/(1+rt+1)at+1= wt− ct=(1− α)wtKt+1= Nat+1= N(1−α)wt= N(1−α)(1−β)AKβtN−βkt+1=(1− α)(1 − β)Akβt45Summary:We just worked out that the savings of young agentsdetermines how the capital-labor ratio evolves overtime. Once we know this, we can easily determinehow many other variables evolve over time. The logicfollows closely the logic for how output, wages andinterest rates evolve in the Solow model.6How do Wages Move?Capital Ktand labor Lt= N are known. Inputs aresupplied inelastically at this date.Supply curve of labor is inelastic, whereas the demandcurve is the marginal product of labor.Implication: wage is where supply and demand crosswt= MPL =(1− β)Akβt7Summary:1. Capital-Labor Ratio: kt+1=(1− α)(1 − β)Akβt2. Output-Labor Ratio: yt= F (kt, 1) = Akβt3. Wage: wt= MPL =(1− β)Akβt4. Investment: it= kt+1− kt(1 − δ)8Investment Paths are Tricky:it= kt+1− kt(1 − δ)=[kt+1− kt]+δktGraphically: Gap in law of motion + depreciated cap-ital termUpshot: It is possible that over time itmay at firstincrease and then decrease to a steady state level!9Main Properties:1. One steady state with a positive capital-labor ratio.2. Convergence to this steady state.3. Factor prices are marginal products10A One-Time Immigration:Israel in late 1980’s experienced a large immigrationof Soviet Jews. There was a relatively short windowfor this immigration. We will view this as a one-time,permanent increase in the population from within theLife-Cycle model.Implications from Life-Cycle model??111. Capital-labor ratio: falls initially but increases subsequentlyback to steady state.2. Output: output growth should be temporalily high after theimmigration.3. Investment: should be an investment boom.4. Wage: workers should be grumbling as wages should betemporarily depressed.5. Interest Rate and Return to Capital: should be temporarilyhigh after the immigration.12Welfare Analysis:In economics it is standard to use the Pareto criterion.We say that a feasible allocation is Pareto efficient ifthere is no other feasible allocation that makes someagent strictly better off and makes no other agentworse off. We will ask whether or not equilibria withinthe Life-Cycle model are Pareto efficient.13PROPOSITION: Consider a generalized version of thelife-cycle model where the technology can change overtime and where agents have a well defined marginalrate of substitution.If the allocation produced by the model has 1+rt>1+ for all time periods t ≥ 1 for some number >0,then the allocation produced by any such model isPareto efficient.14Strategy: Argue any Pareto improvement is infeasible.Step 1: Improve utility for the old at t.GiveΔ> 0Step 2: Compensation to young:Δ × MRS(cyt,cot+1)=Δ× (1 + rt+1)Step 3: (Snowball Effect) Δ×(1+rt+1)×MRS(cyt+1,cot+2)1516Conclusion:The initial “gift” of Δ > 0 is not consistent with fu-ture generations maintaining utility level. Young needto be compensated. Compensation grows with eachgeneration. It eventually becomes larger than GDP!Thus, we conclude this potential Pareto improvementleads to a violation of resource feasibility.17Conclusion Continued:Our analysis of the Life-Cycle model concludes that,at least under the conditions of the Proposition, thecompetitive market structure leads to an allocationthat cannot be improved according to the Pareto cri-teria. If we later use this model to analyze policychoices, then we will end up concluding that thereis no clear-cut role for government policy to achievePareto improvements for the people who live in themodel.18Conclusion Continued:If one wants a theoretical structure that allows a gov-ernment to achieve Pareto improvements, then it wouldneed some model elements beyond those consideredso far. The only ”problem” highlighted within themodel is the overaccumulation of physical capital. Somepossibilities: allow public goods and various uninsuredshocks.19Connection to Golden Rule:Interest Rate Condition: 1 + rt> 1Simple Model: If there is no technological change thenbeing below the Golden Rule is equivalent to a positivereal interest rate each period!20Marginal Conditions and Data Implications:1. −MRS(ct,ct+1)=αu (ct)(1−α)u (ct+1)=1+rt+12. 1 + AF1(kt+1, 1) − δ =1+rt+1Emphasize: 1 ties consumptiongrowthtotheinter-est rate, whereas 2 ties the real interest rate to thecapital-labor ratio and technology level. 1 also saysthat future consumption level should be predictablefrom the current level and from the interest