Solutions Problem Set 2 Macro II (14.452)web.mit.edu/14.452/www/pdf/ps2_ Solutions Problem Set 2...

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Transcript of Solutions Problem Set 2 Macro II (14.452)web.mit.edu/14.452/www/pdf/ps2_ Solutions Problem Set 2...

  • Solutions Problem Set 2 Macro II (14.452)

    Francisco A. Gallego

    04/22

    We encourage you to work together, as long as you write your own solutions.

    1 Intertemporal Labor Supply Consider the following problem. The consumer problem is:

    Max {Ct},{Nt}

    E0

    Ã t=TX t=o

    βt {χ1C 1t + χ2N 2t } !

    s.t.

    At+1 = R(At +NtWt − Ct)

    Where C is consumption, N is labor supply, A0 is initial wealth, R = 1+ r, and the greek letters are parameters. Only W is stochastic 1). Derive and interpret the first-order conditions for Ct and Nt.

    What are the intertemporal and intratemporal optimal conditions There are at least two (equivalent ways of setting up and solving this prob-

    lem). I will implement both of them

    1. Lagrangian method I:

    In this case the Lagrangian is:

    Max {Ct},{Nt},{At+1}

    £ = E0

    Ã t=TX t=o

    βt {χ1C 1t + χ2N 2t − λt(At+1 −R(At +NtWt − Ct)} !

    s.t. A0

    F.O.C.

    {Ct} : ∂£

    ∂Ct = 0⇔ χ1 1C 1

    −1 t = λtR (1)

    1

  • {Nt} : ∂£

    ∂Nt = 0⇔ −χ2 2N 2−1t = λtWtR

    The interpretation is more or less natural. The first tells you that the marginal utility of Ct has to be equal to the marginal utility of wealth (do not worry too much about R because it is an implication of the fact that C in this case is realized at the beginning of the period t—so the value of saving one unit of consumptions is 1 times R). The second tells you that marginal disutility of work has to be equal to the marginal utility of wealth times how much consumption you can get from one our of work (the wage rate). Why no expectation? This is important, at time t, the only source of uncertainty is realized (Wt) and, roughly speaking, λt is a function of expected future wages (see below).

    Combining both equations we get the intratemporal condition:

    −χ2 2N 2−1t =Wχ1 1C 1 −1

    t

    To get the intertemporal condition, we need another FOC:

    ∂£

    ∂At+1 = 0⇔ βtλt = βt+1λt+1R

    Using the FOC for consumption, we get our intertemporal condition:

    βtχ1 1C 1−1

    t = β t+1RE(χ1 1C

    1−1 t+1 )

    1 = βRE

    "µ Ct+1 Ct

    ¶ 1−1

    #

    2. Lagrangian method II:

    Notice that you can iterate the constraint and write the intertemporal budget constraint.

    TX t=0

    R−tCt = A0 + TX t=0

    R−tNtWt

    Two comments: (1) Why no expectation? This has to hold exactly. (2) what AT+1?

    Max {Ct},{Nt}

    £ = E0

    Ã t=TX t=o

    βt {χ1C 1t + χ2N 2t }− λ( TX t=0

    R−tCt −A0 − TX t=0

    R−tNtWt)

    !

    F.O.C.

    2

  • {Ct} : ∂£

    ∂Ct = 0⇔ βtχ1 1C 1

    −1 t = λR

    −t (2)

    {Nt} : ∂£

    ∂Nt = 0⇔ −βtχ2 2N 2−1t = λWR−t (3)

    If you combine both equations, you get the same intratemporal condition as before.

    To get the intertemporal condition write the FOC for t+ 1 taking expec- tations at t.

    {Ct+1} : ∂£

    ∂Ct+1 = 0⇔ Et

    h βt+1χ1 1C

    1 −1

    t+1 − λR−t−1 i = 0

    Using the FOC for Ct:

    Et

    h βt+1χ1 1C

    1−1 t+1 − χ1βt 1C 1

    −1 t R

    −t−1 i = 0

    and you get the same as before:

    1 = βRE

    "µ Ct+1 Ct

    ¶ 1−1

    #

    What is more interesting about this method is that λ can be interpreted as the sufficient statistic you need to solve the problem

    Notice that (1) and (2) imply that:

    λtR = λ (βR) −t

    How can you interpret this condition?

    Moreover, we can plug (1) and (3) into the budget constraint and we will get a decreasing implicit function of λ as a function of {Wi} with i ≥ t and, of course, some constant terms (among them R). Thus, the only thing you need to compute λ is to know something about {Wi}. Therefore, you will keep your λ constant in the future if the realizations of {Wi} were more or less what you expected in the past.

    2). What is the link between Ct and Nt in this model? What assumption(s) is (are) producing this result The only link between both variables is λ (or λt). This implies the intratem-

    poral condition and the negative correlation between both variables ifW is kept constant. And, the lack of correlation if W increases without affecting λ. Assumption: (1) both terms are additively separable in the utility function

    and (2) perfect capital markets (what may happen if we introduce frictions as At > 0)

    3

  • 3). How can you analyze changes to wages that do not affect the expected wealth of the consumers? (Hint: what is λ—the Lagrange multiplier—in this model?). Let’s define the wealth-constant elasticity of labor supply as η = ∂ lnNt∂ lnWt when wealth is constant. What is η in this model? Is it positive or negative? Why? Given our previous discussion, these are changes that do not affect λ, there-

    fore we can take logs in (3) and get:

    log( −χ2 2 λ (βR)

    −t ) + ( 2 − 1) log(Nt) = logWt

    Then

    η = log(Nt)

    logWt =

    1

    ( 2 − 1) Notice that η > 0. There are several ways of understanding this. Here is

    one: First, notice that obviously χ

    1 > 0, χ2 < 0, 1 < 1. Why? People like C,

    dislike N, and marginal utility of consumption is decreasing in general. Therefore if the instantaneous utility function is to be concave (and, therefore, the solution we just proposed has sense), it has to be true that:

    ∂2U

    ∂N2 = χ2 2 ( 2 − 1)N 2

    −2 t < 0⇒ ( 2 − 1) > 0

    4). Take the FOCs and discuss the effect of the following situations on {Ct} and {Nt} .

    1. • Cross-sectional differences associated with permanent dif- ferences in human capital (assume W varies in the cross- section) Higher human capital will probably imply higher W . Therefore, λ ↓, C ↑. What about N ? Assume all other parameters do not vary in the cross-section and take two individuals i and j

    log( Nj Ni ) =

    1

    ( 2 − 1)

    µ log

    Wj Wi

    + log λj λi

    ¶ So effect is ambiguous. One of them is richer, but at the same time the alternative cost of leisure is higher.

    • Life-cycle changes in wages (i.e. if Mincerian equations are correct, wages follow an inverted-U behavior over the life- cycle) Notice that these evolutionary changes were predictable at the be- ginning of the life-cycle and, therefore, as W changes, λ is constant. Thus C is also constant—see (2). But given that η > 0 and λ is constant, N moves together with W .

    4

  • • Unexpected temporary shocks to wages If change is temporary, it shouldn’t affect λ by much, then C is constant, and N moves together with W .

    • Unexpected permanent shocks to wages In this case λ should be affected. Therefore, C moves in the same direction as W and the effect on N is unclear.

    2 Log-Linear RBC Consider the following problem. Consumers maximize

    maxEt

    ⎡⎣ ∞X j=0

    bjU( eCt+j , Nt+j) ⎤⎦

    Subject to eKt+1 = Rt eKt +fWtNt − eCt U( eCt, Nt) = log³ eCt´+ φ log(1−Nt)

    while firms solven Nt, eKto = argmax

    N,K ZtF (AtN, eK)− (Rt + δ − 1) eK −fWtN

    F (AtN, eK) = Y = Z eK1−α(AtN)α and we assume

    At+1 = γAt.

    This is the standard RBC model. trending variables are those with a hat. 1) De-trend the model Define V = VA .

    1. Utility function:

    U(Ct, Nt) = log (CtAt)+φ log(1−Nt) = log (At)| {z } Does not matter

    +log (Ct)+φ log(1−Nt)

    2. Consumers B.C.Kt+1At+1 = RtKtAt +WtAtNt − CtAt

    Kt+1At+1 = RtKtAt +WtAtNt − CtAt

    ⇔ Kt+1 At+1 At| {z } γ

    = RtKt +WtNt − Ct

    ⇔ γKt+1 = RtKt +WtNt − Ct

    5

  • 3. Profits

    {Nt,Kt} = argmax N,K

    Zt (KtAt) 1−α (AtN)

    α − (Rt + δ − 1)AtKt −AtWtNt

    ⇔ {Nt,Kt} = argmax N,K

    Zt (Kt) 1−α (N)α − (Rt + δ − 1)Kt −WtNt

    2)Solve for the F.O.C of the consumer problem, either using a La- grangian or a Bellman Equation. You should find two final equations, the Euler Equation and the labor supply.

    1. Bellman equation

    V (K) = Max C,N

    {log (C) + φ log(1−N) + bE [V (K0)]}

    s.t.

    γK0 = RK +WN − C

    FOC:

    ∂V

    ∂C = 1

    C + bE

    ∙ ∂ [V (K0)]

    ∂K0 ∂K0

    ∂C

    ¸ = 1

    C − b

    γ E

    ∙ ∂ [V (K0)]

    ∂K0

    ¸ = 0 (4)

    ∂V

    ∂N = − φ

    1−N +bE ∙ ∂ [V (K0)]

    ∂K0 ∂K0

    ∂N

    ¸ = − φ

    1−N + Wb

    γ E

    ∙ ∂ [V (K0)]

    ∂K0

    ¸ = 0

    (5)

    Envelopment theorem:

    ∂V

    ∂K = bE

    ∙ ∂ [V (K0)]

    ∂K0 ∂K0

    ∂C

    ¸ =

    Rb

    γ E

    ∙ ∂ [V (K0)]

    ∂K0

    ¸ (6)

    (a) Euler Equation:

    Using (15) and (4), we get: ∂V

    ∂K =

    R

    C

    Then, we get: 1

    C =

    b

    γ E

    ∙ R0

    C 0

    ¸ = 0

    (a) Labor supply (5) and (4) imply:

    1−N = Cφ W

    (7)

    6

  • 2. Lagrangean:

    Max {Ct},{Nt}

    £ = E

    Ã t=TX t=o

    bt {log (C) + φ log(1−N)− λt(γKt+1 − (RtKt +WtNt − Ct))} !

    FOC

    {Ct} : 1

    C = λt (8)

    {Ct} : − φ

    1−N = λtWt (9)