Econ 4310 Fall 2008 Chen
Problem Set 4 Due on October 14th/16th, 2008
1. Theory: Intertemporal Labor Supply
Consider the following problem
T
(1 ~ ht)
max     E^ßU log(ct) +
{d,ht,At }f=0    t=0 [
s.t.
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At+i   =   R (At + wtht - ct) , Ao given
where ct is consumption, ht is labor supply, Ao is initial wealth, R = 1 + r is the gross rate of return for saving. Here only wt, the wage rate, is stochastic.
(a) Derive and interpret the first order conditions for ct, ht and At+i. What are the intertemporal and intratemporal optimality conditions?
(b) Judging from the first order conditions, which variables link ct and ht in our model? What is the economic interpretation of this variable?
(c) Define the Frisch elasticity of labor supply as
That is, Frisch elasticity of labor supply capture the elasticity of hours worked to wage rate, given the marginal utility of consumption unchanged. In other words, Frisch elasticity measures the substitution effects of a change in wage rate on labor supply. (Why is that? Find out what marginal utility of consumption in the first order condition is equal to and you will get the answer.) Compute r in this model.
(d) Use the Focs and discuss the effect of the following situations on ct, and ht
d ln ht
I u' (c) constant
d ln wt
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i. Unexpected temporary shocks to wages ii. Unexpected permanent shocks to wages
2. Computation
Consider the following version of the real business cycle model. The economy is populated by a large number of identical agents with total measure one. Each agents has a time endowment of 1 per period. Agents rank life time consumption and leisure according to
EY,Plu (ct,ht), P 2 (0,1)
t=0
There is a large number of identical firms with aggregate technology given by
yt = ezt F (kt,ht)
where zt is a discrete state Markov process with transaction probability matrix II and stationary distribution P. The function F is assumed to be strictly increasing in each of its arguments, strictly concave, constant returns to scale and it also satisfies the INADA conditions. The technology shock has an unconditional mean of zero and it follows a two—state Markov process; the state space of this Markov chain is Z = {—"} . Denote the transition matrix II for this Markov process as
I
where the ) element TTij = prob {zt+i = Zj p zt = Zj}. The law of motion for the aggregate capital stock is given by
kt+i = (1 - 5) kt + it
where 5 is the depreciation rate of capital. The resource constraint is given by ct + it = yt
(a) Write the Bellman's equation for the social planner's problem. State clearly what are the endogenous state variables, the exogenous state variables and the control variables in this problem
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(b) Now assume that
u (ct,ht)   =   log(ct) - ipht F (kt,ht)   = k?h\-a
Compute the deterministic steady-state for the social planner's problem. A satisfactory answer should consist of three equations that can be used to solve for steady state value of k,c,h.
(c) Calibrate this model economy, so that its competitive allocation matches the long run observations
• c/y = 0.85 . k/y = 3.0,
• labor share of income is equal to 70%,
• and the fraction of leisure in total nonsleeping hours is equal
to 80%.
In particular you need to choose values for a, 8, (3, tp to match the above targets.
(d) We now would like to solve the Bellman's equation in part (a) by the method of value function iterations. To simply the problem, in this exercise we shut down the fluctuation of technology shock and let zt = 0 for all t. (Note without aggregate uncertainty, the model setup is very close to that in our term paper without labor supply restriction except the utility specification).
i. Without aggregate uncertainty, what are the state variables and control variables now?
ii. Discretize the state space into S = {ki, k2,.., k^g}. Use ki = 0.85k and kkg = 1.15k, where k is the steady state capital you calculated in part (b). Discretize the choice set of h into grid space {hi, hi, .., hhg}, use hi = 0.8h and hhg = 1.2h. Use kg = 101 and hg = 51.
iii. Construct consumption and welfare matrix Compute the (hg X kg X kg) dimension consumption matrix C, where element C (i,j,n) denote the value of consumption when h = hi,k! = kj, and k = kn. Next compute the ( hg X kg X kg) dimensional matrix for the current period with element U (i,j,n) denoting the utility value when h ^= hi, k ^= kj, and k ^= kn.
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iv. Defined the initial guess for value function and compute the value function by value function iteration
Define an initial (kg X 1) dimensional vector (of zeros) for the initial guess of value function v$. Compute the (kg X 1) dimensional vector of one-period value functions: The operator T maps bounded continuous functions into bounded continuous functions (contracting mapping)
whereby we have computed a closely approximated true value function by solving Bellman equation
(Hint: when you write the code, you may define obj fun (:,:, n) = U (:,:, n) + /Jlv^ for each k = kn, where the superscript T on v denote the transpose of vector, 1 denotes a column vector of lwith length hg. Then use the max operator (twice) to find the optimal k0 and h in a two-dimensional grid space. In other words, for each k = kn we are now searching on a two dimensional grid space defined on h and k0 to find the maximum value). v. Update the guess for the value function by setting vo = Tv and continue by iterating on Bellman equation until the distance between Tv and vo is very small.
(e) Plot the decision rules for k0, h and c.
(Tv) (k) = max
k',h
(Tv) (k) = max | U + /31 [v0 (k'
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