Suggested Solutions to Homework #4 Econ 511b (Part I), Spring 2004

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1 Suggested Solutions to Homewok #4 Econ 5b (Pat I), Sping Conside a neoclassical gowth model with valued leisue. The (epesentative) consume values steams of consumption and leisue accoding to P t=0 βt U (c t,l t ),wheel t is hous of leisue in peiod t. The felicity function U is stictly concave and stictly inceasing in both of its aguments. Output is poduced accoding to y t = F (k t,n t ),wheen t = L l t is hous of labo supply in peiod t (F is the total numbe of hous available fo eithe leisue o wok in peiod t) and F exhibits constant etuns to scale. Output in peiod t canbeusedfoeitheconsumptionc t o investment i t and capital accumulates accoding to k t+ =( δ) k t + i t. (a) Display Bellman s equation fo the poblem faced by the social planne in this economy. Identify clealy the state and contol (o choice) vaiable(s). The ecusive fomulation fo the planning poblem is o v k = max nc,l,k 0oU c, l + βv c + k 0 = F (k, L l)+( δ)k v k ³ =max {l,k 0 } U F (k, L l)+( δ)k k 0, l + βv Fom the way we wite it, we can see that the state vaiable is k, andcontol vaiables ae l, k 0. (b) Deive the fist-ode and envelope conditions fo the planning poblem. The F.O.C. is ª l : U c, l F2 (k, L l) =U 2 c, l o nk 0 ³ 0 0 : U c, l = βv k Fom Envelope Theoem, we have v 0 k = U c, l F (k, L l)+( δ) Iteate fowad fo one peiod, it becomes ³ 0 ³ v 0 k = U c 0, l 0 ³F (k 0,L l 0 )+( δ) Plug it into F.O.C., we get the final optimality conditions: ª lt : U ct, l t F2 (k t,l l t )=U 2 ct, l t ª + : U ct, l t = βu ct+, l t+ F (k t+,l l t+ )+( δ)

2 (c) Use the conditions fom pat (b) to detemine the economy s steady state. Show how the steady state depends on pimitives and compae you esults to those fo a gowth model without valued leisue. In steady state, the optimality condition becomes nl o : U ³c, l F 2 (k,l l )=U 2 ³c, l n k o : F (k,l l )= ( δ) β whee c = F (k,l l ) δk. We can see that k and l depends on both β,δ, poduction techonology F (k t,n t ), and utility function U (c t,l t ). In a gowth model without valued leisue, the steady state is detemined by the equation F (k,l)= ( δ) β which does not depend on the utility function U (c t,l t ). Now let s compae two models. Fist, in the model with leisue choice we add an additional equation which states that the maginal ate of substitution between consumption and leisue must equal to the maginal ate of tansfomation. Second, the equation about k is the same, except that the level of steady-state leisue is diffeent. Thid, fo the equation F (k,l l )= ( δ), dueto β the diffeence in the steady-state leisue level, the steady-state capital stock is also diffeent. Fo example, if F 2 > 0 asinthecaseofcobb-douglaspoduction function, the capital stock in the model with leisue will be lowe than that without leisue choice (since L l <L). (d) Let F (k, n) =k α n ( α) and U (c, l) =λ log (c) +( λ)log(l), whee 0 < α< and 0 <λ<. Solve explicitly fo the steady state in tems of paametes. With F (k, n) =k α n ( α) and U (c, l) =λ log (c)+()log(l), the steady-state condition becomes n l o : n k o : α Ã λ ³ k α L l ( α) ( α) δk Ã! k α L l = ( δ) β k L l! α = l which leads to n = Ã λ λ α α l = L n k = µ β ( δ) α α δ β ( δ) α n c = k α (n ) ( α) δk! + L 2

3 2. (a) Caefully define a ecusive competitive equilibium fo the neoclassical gowth model with valued leisue. (Hint: You need two functions to descibe the behavio of the aggegate economy.) A ecusive competitive equilibium fo the neoclassical gowth model with valued leisue is a set of functions: pice function : k,w k policy function : k 0 = g k,l = gl value function : v aggegate state : k 0 = G k, l = l(k) such that: () Given k 0 = G k,k 0 = g k,l = gl and v solves consume s poblem: v = max U (c, l)+βv 0, k {c,l,k 0 } (2) Pice is competitively detemined: c + k 0 = k k +( δ) k + w k (L l) k 0 = G k k = F k, L l(k) w k = F 2 k, L l(k) (3) Consistency: G k = g k l(k) = g l (b) Find the (functional) fist-ode conditions of a typical consume who takes as given the economy s aggegate laws of motion. Solve a typical consume s poblem v = max {l,k 0 } U k 0 = G k We get the F.O.C. as ³ k k +( δ) k + w k (L l) k 0,l {l} : U (c, l) w k = U 2 (c, l) nk 0o : U (c, l) =βv ³k 0, k 0 + βv 0, k 3

4 Use the envelope condition We get the optimality condition: v = U (c, l) k +( δ) {l t } : U (c t,l t ) w k t = U2 (c t,l t ) {k t+ } : U (c t,l t )=βu (c t+,l t+ ) k t+ +( δ) Coespondingly, the functional F.O.C. is {l t } : U ct,g l, k t w = U2 ct,g l, k t {k t+ } : U ct,g l, k t = βu ct+,g l +, k t+ G +( δ) whee c t = k t +( δ) k t + w k t L gl, k t gk, k t, k = F k, L l = F k, L gl,andw k = F2 k, L l = F2 k, L gl. (c) Impose equilibium conditions on the fist-ode conditions fom pat (b) and veify that the esulting equations ae identical to the fistode conditions associated with the planning poblem fo this economy. Impose the equilibium conditions k t = k t,l t = l t,wehave {l t } : U ct, l t F2,L l t = U2 ct, l t {k t+ } : U ct, l t = βu ct+, l t+ F,L l t +( δ) whee c t = F,L l t +( δ) k t + F 2,L l t L lt + = F k t,l l t +( δ) k t+ which ae identical to the fist-ode conditions associated with the planning poblem fo this economy. 3. Conside a competitive equilibium one-secto gowth model without valued leisue in which consumes own capital and labo and ent thei sevices to fims. Thee is no uncetainty and the felicity function of a typical consume has constant elasticity of intetempoal substitution σ. (a) Show that the decision ule of a typical consume takes the fom: k 0 = µ k + λ k k whee the functions µ and λ satisfy the following pai of functional equations (i.e., these two equations must hold fo all values of k): µ k w + ³ 0 = w k k k k k λ k 4

5 and ³ β λ k = k k σ, whee k 0 = µ k + λ k k. In these equations, k is the individual s holdings of capital, k is aggegate capital, k is the ental ate of capital plus one minus the depeciation ate, and w k is the wage ate. (Hint: Obtain the Eule equation fo a typical consume, guess that the consume s decision ule takes the conjectued fom, and then find estictions that the coefficients µ k and λ k must satisfy in ode fo the Eule equation to hold fo all values of k and k.) A ecusive competitive equilibium fo this economy is a set of functions: pice function : k,w k policy function : k 0 = g value function : v tansition function : k 0 = G k such that: () k 0 = g and v solves consume s poblem: v c σ ³ 0 = max σ + βv k 0, k {c,k 0 } (2) Pice is competitively detemined: c + k 0 = k k + w k k 0 = G k k = F k, + δ w k = F 2 k, (3) Consistency: G k = g() By taking F.O.C. and using envelope condition, we can get the Eule equation as βu 0 (c t+ ) k u 0 t+ = (c t ) µ σ ct+ β k t+ = c t c t+ = β k t+ σ c t k t+ + + w k t+ +2 = β k σ t+ k t + w k t + 5

6 Since this is a second-ode linea diffeence equation in k t,weconjectuethe solution fom as k 0 = µ k + λ k k Plug back into Eule equation, we get LHS = k t+ + + w k t+ +2 = k t+ µ + λ + w + µ k t+ + λ + µ + λ = k t+ + λ + µ w + + and RHS = β k t+ σ k t + w k t µ + λ = β k t+ σ k t + w k t TomakeLHSandRHSequatefoeveyk t and k t+, we must have () : + + λ = β σ + k t (2) : µ k t w + + = β k σ t+ w k t i.e. ³ () β λ k σ = k k (2) µ k w h + = β i w k k σ µ k w ³ + λ k = k k w k k µ k w + ³ 0 = w k k k k k λ k This finishes the poof. (b) Suppose now that thee ae two (types of) consumes in the economy who diffe only in thei initial capital holdings. Each consume epesents half of the economy s population. Use the esult fom pat (a) to ague that a edistibution of capital (holding aggegate capital constant) acoss the two consumes at time 0 has no effect on equilibium inteest ates and wages. 6

7 To pove the esult, we need to see what is the diffeence between a epesentative agent economy and heteogeneous agent economy. So we fist clealy define a ecusive competitive equilibium fo heteogeneous agent economy. A ecusive competitive equilibium fo the economy with two types of agents is a set of functions: pice function : k,w k policy function : k 0 i = g i ki, k, k 2 value function : v i ki, k, k 2 tansition function : k 0 i = G i k, k 2 whee k = (k 2 + k 2 ),suchthat: 0 () Given G k, k 2 and G2 k, k 2,k i = g i ki, k, k 2 and vi ki, k, k 2 solve consume s poblem: ³ v i ki, k, k 2 = max {c i,ki} u 0 i (c i )+βv i ki, 0 k 0, k 0 2 (2) Pice is competitively detemined: c i + k 0 i = k k i + w k k 0 = G k, k 2 k 0 2 = G 2 k, k 2 k = F k, + δ w k = F 2 k, (3) Consistency: G i k, k 2 = gi (k i, k, k 2 ) Notice the diffeence between the definition of ecusive competitive equilibium we met befoe: in heteogeneous agent economy, nomally the whole distibution of state vaiable counts. Now we stat to pove the esult. Due to the special pefeence elation hee, fist we assume that only aveage capital level mattes fo the evolution of the economy, i.e. k 0 = G k. Then the esult fom pat (a) tells us that in ecusive competitive equilibium we will have k 0 i = µ k + λ k k i Theefoe, we have k 0 = 2 ³ k 0 + k 0 2 = µ k + λ k k 7

8 Theefoe, indeed the evolution of aveage capital stock does not depend on the distibution of capital. In othe wods, this veifies ou guess that k 0 = G k. Since equilibium inteest ates and wages depend only on the aggegate aveage capital stock (emembe that k = F k, + δ and w k = F2 k, ), a edistibution of capital (holding aggegate capital constant) acoss the two consumes at time 0 has no effect on equilibium inteest ates and wages. 8