V =lnc 1 +βlnc 2 (1) subject to the following feasibility constraints: c 1 +x y 1 (2)

Transcription

1 Problemset, Econ502 Question : Cake-eating problem Crusoe s lifetime welfare is given by: V =lnc +βlnc 2 () subject to the following feasibility constraints: c +x y (2) c 2 ( δ)x (3) wherexisthepieceofcakeputasideforperiod2y istheinitial endowment, i.e. the cake. Substitute constraint 3 into 2 to get: c + c 2 δ y (4) The Lagrangean for this problem can then be written as: ( L=lnc +βlnc 2 +λ y c c ) 2 δ Takingfirstorderconditionswithrespecttoc c 2 yields: (5) c =λ (6) β = λ c 2 δ Combining these two equations, we have: (7) c 2 c =β( δ) (8) Fromconstraints23wecanrewritec c 2 intermsofxget: x= β +β y (9) Question 2: Quantities Prices Each agent s budget constraints are: The market clearing conditions are: c +x y (0) c 2 y 2 +(+r)x ()

2 c y c 2 y 2 (2) Thenwecefineacompetitive equilibriumforthiseconomy asasetof allocations{c,c 2,x}apricersuchthat: a) The allocations maximize agents utility given their budget constraints givenpricer; b)atthisprice,allmarketsclear. Nowcombine0toget: c + c 2 +r y + y 2 +r So that the Lagrangean is: L= [ c σ +βc σ ] 2 +λ (y + y 2 σ +r c c ) 2 +r Solvingforthefirstorderconditionswithrespecttoc c 2 wehave: (3) (4) c σ =λβc σ 2 = λ +r Combine these two equations to get: (5) ( c2 ) σ =β(+r) (6) c Imposing market clearing we obtain the equilibrium expression for the interest rate: ( ) σ y2 =β(+r) (7) y Note that all agents are identical. Now suppose that y 2 = ϕy thus ϕ σ /β=(+r). Ifϕincreasesthentheinterestratemustalsoincrease,orput differently, the relative price of the good in period to the good in period 2 must increase. This is because the marginal utility of consumption in period hasincreased: ifyouknowyouwillgetmoreofthegoodinperiod2youshould askformoreinexchangeofthegoodfromperiod. Finally, the equilibrium expressions for c c 2 are given by the market clearing conditions x can be computed by substituting c or c 2 in one of the budget constraints. Question 3 Endogenous labour-leisure (i) Period budget constraint is: c +x w n +r k +π (8) 2

3 Period 2 budget constraint is: The Lagrangean is thus: c 2 w 2 n 2 +r 2 k 2 +π 2 +( δ)k 2 (9) L=ln(c n )+ln(c 2 n 2 )+λ [w n +r k +π c k 2 +( δ)k ] (20) (ii) First order conditions: +λ 2 [w 2 n 2 +r 2 k 2 +π 2 +( δ)k 2 c 2 ] (2) (c ): (c 2 ): (n ): (n 2 ): c n λ =0 (22) β c 2 n 2 λ 2 =0 (23) c n +λ w =0 (24) β c 2 n 2 +λ 2 w 2 =0 (25) (k 2 ): λ +λ 2 (r 2 + δ)=0 (26) Thefirstorderconditionswithrespecttoλ λ 2 givesyouback8 9 with the constraint binding. Notefrom26that: Combine22,2326toget: λ λ 2 =r 2 + δ (27) (c n ) = β[r 2+ δ] (28) c 2 n 2 Dothesamewiththefirstorderconditionsofn n 2 get: w (c n ) = β[r 2+ δ] w 2 (c 2 n 2 ) (29) Equation 28 gives the trade-off of forgoing a unit of consumption today for a unit of consumption tomorrow while equation 29 gives the trade-off between working today against working tomorrow. (iii) 3

4 Note that the problem of the firm is static. The objective function of the firmisthus: π t =maxk tl t w t l t r t k t t=,2 (30) Solvingforthefirstorderconditionsofthefirmwegetforeachperiod: ( ) lt (k t ): r t = (3) k t (l t ): w t =( ) ( kt l t ) (32) Now,from2829wehavethatw =w 2. Moreover,using2224we getw =w 2 =. Isolate the capital-labor ratio in 32 obtain, using the result for w above: ( ) k l = (33) The capital-labor ratio is a function of parameters only is thus stationary. (iv) A competitive equilibrium for this economy is a set of allocations for households{c,c 2,k 2,n,n 2 },asetofallocationsforfirms{k,k 2,l,l 2 }asetof prices{w,w 2,r,r 2 }suchthat: a) Given prices budget constraints, households maximize their utility; b) Given prices technology, firms maximize profits; c) Markets clear. Thereare3marketssoweshouldhave3marketclearingconditionsineach period: c +x y (34) c 2 ( δ)k 2 y 2 (35) n t =l t fort=,2 (36) k firms t =k hh t fort=,2 (37) Wenowhaveallequationsneededtosolvefortheequilibriumvaluesofthe endogenousvariables. Rememberthatk isagivenforhouseholds. Nowsolving forl using32fromperiodweget: 4

5 [ ] l = k (38) frommarketclearingweknowthatn =l. Usingthisresultin3 we have: thus: ( ) l r = =l k = k [ ] k k (39) [ r = Itiseasytoseethatoutputinperiodis: ] (40) y =k l = [ ] k (4) Letusproceedinthesamemannertosolvefortheendogeneousvariablesof period 2: Asbeforeuse32tosolveforlaborinperiod2: [ ] k2 l 2 = (42) Frommarket clearingwe know thatl 2 =n 2. Notehoweverthatthis time k 2 isnotagivenweneedtosolveforit. Use3toget: [ r 2 = ] Knowingthatδ=using3435wehave: (43) c 2 =y 2 =k 2l 2 = c =y k 2 (44) [ ] k2 (45) Plugtheexpressionsforr 2,c c 2 in28tosolvefork 2. Youshouldget after some algebraic manipulations: k 2 = β +β [y n ] (46) Wenowhaveeverythingweneedtosolvefortherestofthechoicevariables. Togetc c 2,plugtheresultfork 2 in3435. Finally,y 2 =k 2 l 2. 5

6 (v) Notethatk 2 isafunctionofk thatperiod2variablescanbewritten intermsofk. Itiseasytoseethatthethetimepathsfortheeconomywitha higherk shouldlieabovethosefortheeconomywithalowerk. 6