Considerahouseholdwithu(c) = ln(c) thatfacesadeterministiclife-cycle problem. Forthefirst T1 = 40periodsitreceivesearningsy = 0.5and forthenext T2 = 20periodsitdoesnotreceiveanything. Thehousehold...

Considerahouseholdwithu(c) = ln(c) thatfacesadeterministiclife-cycle problem. Forthefirst T1 = 40periodsitreceivesearningsy = 0.5and forthenext T2 = 20periodsitdoesnotreceiveanything. Thehousehold discountsfutureatrate β = 0.9andcansaveorborrowataninterestrate r = 0.1.
1. Solvethehousehold’sproblemnumerically. Thatmeanscomputethe matricesforvaluefunctionsandpolicyfunctionsforconsumptionand wealthforallperiods. Usetheequidistantgridforwealthwithsize N = 401,smallestelement-2andlargestelement6. 2. Inonefigureplotconsumptionpolicyfunctionsforwealthlevelsabove -1whenthehouseholdis5,10,35,40and45yearsold.3 3 Lineswillnotbesmooth,that’sfine. 3. Simulatehousehold’sbehavioriftheinitialwealthiszero. Plot(separately)consumptionandwealthpaths.4 4 Thepathforconsumptionmaynotbe smooth,forthewealthitshouldbe. 4. Simulatehousehold’sbehavioriftheinitialwealthis1. Plotconsumptionandwealthpaths. 5. Supposenow T2 = 25,sothatthehouseholdliveslongerbuttheworkingperiodisthesame. Solvethemodelandsimulatehousehold’sbehavioriftheinitialwealthiszero. Plotconsumptionandwealthpaths. 6. Supposenowu(c) = c1−γ−1 1−γ with γ = 0.5. T2 isattheinitialvalue of20. Solvethemodelandsimulatehousehold’sbehavioriftheinitial wealthiszero. Plotconsumptionandwealthpaths. Compareplotswith thosefrom(3). Whatarethedifferencesandwhatisthereasonforthem? 7. Supposenowr = 1/9. Solvethemodelunderbothutilityspecifications andsimulatestartingfromwealthofzero. Compareconsumptionand wealthpathsacrossthetwospecifications. Whatarethedifferencesand why?