Problem set 4
1. Theory
Consider the following social planner's problem {or Pareto problem).1
ii
vt^e |o:i] vt
given
Assume the following; functional forms and law of motion for Lechnol-
yt = ztf(kt,ht) = ^h\-°,     Vt:o € (0tl) ^t-i-i = PZt + (1 ~P}& + Mm-i»      Vl,p€ [0,1|,J= 1 where {w}^ is a white noise process.
In steady statn. where the mine of the shock is unity (2 = 1), compute the following endogenous parameters ("the Latin letters'1) as functions of the structural parameters, i.e. technology and preference paramc-ters (uthe Greek letters"):
1 As we covered in class, our interest In the soda] planners problem is based an L be fact that tbe solution to tbe social planners problem is the competitive equilibrium allocation. Thai is, there exists a set of prices such that tbe optimum solution can be decentralized as a competitive equilibrium with a price system that has an inner product representation. The social planner's problem is much easier to soLve since we get rid of the prices and the individuals' budget constraint.
(a) the investment/capital ration
(b) the capital/Lahor ratio3
(c) the capital/output ratio
(d) the factor prices
(c) capitals and labors sbarra of national income
(f) the investment/output ratio
[g) the consumption/output ratio
Hint: use L ho Law of motion for capltaJ.
Hint: use Lhe intertemporal opLimality condition, the E-uLer equation.
2, Computations
Consider a model economy where the social planner chooses an infinite sequence of consumption and next period's capital stock {Q.fci-riU^o in order to
DO
max    K(V &ulet)
suhject to
fco > 0. given
The model economy is exposed to an iid stochastic shock in each period,
7(€r = [4.95,5.05| with associated probabilities
*i =Pr{7e =7i) = -5 T3 = Pr{7e =72J = &
Assume the following functional forms
ft = 7(F(fc(, 1) = Ti*?i     Vt,Q € (0,1)
Compute the value function and the decision rules to this deterministic problem using Bellman's method of successive iterations, also called value function iterations.
t. Reformulate this problem as a dynamic programming problem, i.e. write up the Bellman equation. What are the control variahlc{s) and the (endogenous and exogenous) state variables)?
ii. Define the parameters, compute the steady state value of the capital stock fc% discrctizc the state space by constructing a grid on the capital stock with g values k € X = \k\ < k2 < -   < kg] with k\ > 0 in the neighborhood of the steady state.
For these computations, set a = 35*0 = .98,5 = .025,a = 2, and 7 = 5.
in. Construct consumption and welfare matrices
Compute the two (g x g) dimensional consumption matrix C\ and C2, conditional on the value of the stochastic shock, with the value of consumption for all the (g x g) values of k and Next compute two {g x g)-dimcnsionaI matrices with the utility of consumption for all the {g x g) values of k and fc*.
iv. Define the initial value function and compute the Hrst-period value function
Define an initial (g x l)-dimcnsional vector (of zeros) for the initial value function tTo- Compute the (g x 1) dimensional vector of one-period value functions:
The operator T maps hounded continuous functions into hounded continuous functions (a contraction mapping)
Tiv = on (M) = m«{tfiY)}. T2v = oa(*,7) = ff«{03 + 0m(*'.Y)}.
v. Continue by iterating on Bellman's equation until convergence whereby we have computed a close approximated true value function by solving Bellman's equation
Set v = Tv and continue iterations
Ttv  =   ^{yi+fl(l.(*1h(^fYi)]' + ir»h(*'.YJ)]'))}.
T2v = 5g{u» + fl(i.(*i[w(ft',Yi)]' + n[w(*',Y»)]/))}
vi. Compute the approximated true decision rules, fr' = g{k) and c - Aka - kf, hased on the approximated true value function by solving Bellmans equation.
i. Maximize the finaL approximated value functions and find the index row number, j% where the maximized value for each k for cadi of the two functions.
ii. Compute the decision rules for capital from the index, j, k? = k{j). Compute the decision rule for consumption rcsidually
vii. Compare the numeric approximation with the true value function. Set 7j = 72, a = 1 (Le. ufa) = In 4) and 6 = 1. Save the value functions alter four distinct number of iterations and compare these with the true value function for this prohlem which you found in Prohlem L
3, Measurement
Compute annua! real interest rate for the period 1979 to 2006 using annualized 3M NIBOR rates from Norgcs Bank
http://wvu•norges-bank.no/Pages/Article____41851.aspx
and CPI series from Statistics Norway
http://statbank.ssb.n0//stat1stifckbanken/def ault.fr .asp?PLanguage=!
3. Measurements
Historical growth and business cycles statistics
L Goto http://wvu.ssb.no/histstat/
and then choose "Bmttonasjonalprodukt.cttcranvendcisc. Faste 2000-priser. 1865-2006^
http://vuy.ssb.no/histstat/aarbok/ht-0901-355.hxal Download the data ii. Plot G.\P on a log-scale
in. Compute the average annual GNP growth rate h/w 1865 and 2005.
iv. Compute the shares of private consumption, government consumption, gross investment and net export. Plot these together and comment.