1. Endogenous Growth with Human Capital
Consider the following endogenous growth model with both physical capital (k (t))
and human capital (h (t)) in continuous time. The representative household solves the
problem
max
Z
1
exp (
t)
c (t)1
1
_
_ g 0
h(t)
fc(t);k(t);
s:t:
k_ (t) + h_ (t) = zk (t) h (t)1
c (t) 0:
dt
k (t)
h (t)
c (t) ;
Both k (0) and h (0) are given at time 0.
(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming problem.
(b) Derive the Euler Equation for this problem.
(c) Assume that at t = 0, the marginal product of k (0) and h (0) are equal to each
other. Derive the restriction on k (0) and h (0).
(d) Show that under the restriction in Part (c), the model features endogenous growth
with a constant growth rate over time.
(e) Under the same conditions that apply in Part (c) and Part (d), solve the optimal
consumption at t (c (t)) as a function of k (t).
2. Capital and Labor Taxation with Capital Utilization
Consider a one-sector real business cycle model. The household is endowed with 1 unit
of time in every period. The preference of the representative household is given by:
E0
1
X
t
[ log (ct ) + (1
) log (1
nt )] ;
t=0
where ct and nt are private consumption and labor hours, respectively, and
2 (0; 1).
The production technology is characterized by a Cobb-Douglas production function
with endogenous capital utilization:
yt = F (kt ; nt ; zt ; ht )
zt (ht kt ) n1t
:
Here the continuous variable ht 2 [0; 1] measures the “utilization level”of capital stock
in t, and it is a choice variable in every period. The aggregate resource constraint
reads:
ct + kt+1 + G = yt + (1
(ht )) kt ;
where G is an exogenously-given constant government expenditure in every t. Unlike the standard RBC model, the depreciation rate of the capital is endogenously
determined from a function:
h!
(ht ) = 0 + 1 t ;
!
where 0 0; 1 0 and ! > 1. Thus, the depreciation rate of capital in period t is
an increasing and convex function of the utilization level ht .
The TFP zt is stochastic and evolves according to:
log (zt+1 ) = log (zt ) +
t+1 ;
where f t+1 g1
t=0 is an independent and identically distributed sequence of shocks drawn
from a normal distribution N (0; 2 ) and 0
< 1.
At the beginning of t = 0, k0 > 0 is given.
(a) Write the Bellman equation of the social planning problem of maximizing the
welfare of the household.
(b) Derive the FOC of choosing ht in Part (a). From the FOC in ht , derive the optimal
ht as a function of (zt ; kt ; nt ) (assuming an interior solution of ht ).
(c) Suppose that the average behavior of the real-world data can be captured by the
social planner solution. Assume that you have observed (ht ; kt ; nt ; yt ) for a long
period of time. Design a procedure to calibrate the production parameter ( ) and
TFP shock process ( ).
(d) Consider a sequential-trading environment with a …scal authority. The household
owns the capital, and supply capital and labor to the …rm in each period. The
representative …rm maximizes one period pro…t by solving the problem
n
o
1
zt ht ktd
ndt
(ht ) ktd rt ktd wt ndt ;
max
fktd ;ndt ;ht g
h!
where (ht ) = 0 + 1 !t . Notice that we have assumed that the …rm pays the
depreciation (ht ) ktd after production.
The household’s budget constraint in a competitive equilibrium is
ct + kt+1 = 1
where
tively.
k
t
2 [0; 1] and
n
t
k
t
rt kt + (1
n
t ) wt nt
+ kt ;
2 [0; 1] are capital and labor income tax rates, respec-
The tax and spending policies are subject to a balanced-budget requirement, i.e.,
the tax revenue needs to equal to G in every period. Suppose that kt is determined
through an exogenous …scal rule: kt = k (Kt ; zt ), where Kt is the aggregate capital stock in period t. nt is endogenously determined to balance the budget in t.
Carefully de…ne a recursive competitive equilibrium for this economy. Be sure to
write down the household’s dynamic programming problem, the …rm’s FOC, the
government problem, and the equations which determine the pricing functions.
You do NOT need to solve the pricing functions explicitly.
(e) Suppose that at time 0 the …scal authority is given a one-time opportunity to
k
n
deviate from the …scal rule
and choose
to maximize the welfare of
0; 0
the representative household. The economy is otherwise the same as that in Part
(d). In particular, the government still balances its budget for every t 0, and
all future tax rates (i.e., for t
1) will again follow the …scal rule in part (e).
Without doing any actual calculation, discuss the optimal choice of k0 ; n0 and
give economic intuition.
3. Inequality and Growth in a Neoclassical Model
Consider an in…nite-horizon neoclassical growth model with a continuum of households
of two types, i = 1 and 2. Each household type represents 1 unit of the economy’s
population, and shares the same time-separable preference with discount factor 2
(0; 1):
1
X
t
U ci =
log cit :
t=0
Every type is endowed with one unit of labor (n = 1) in every t and supply it inelastically. There is a continuum of representative …rm, with the production function
Yt = zKt Nt1
;
2 (0; 1);
where z is the exogenous technology, and Kt and Nt are the aggregate capital and
labor used by the …rm, respectively.
The resource constraint at time t is
Ct + Kt+1 = Yt + (1
) Kt ;
where Ct = c1t + c2t is the aggregate consumption, and 2 (0; 1] is the capital depreciation rate. At the beginning of t = 0, K0 > 0 is given.
(a) Consider the social planner’s problem. The social planner maximizes a social
welfare function
1
U c1 + 2 U c2 ;
or equivalently,
1
X
t
1
log c1t +
2
log c2t
;
t=0
i
where
is a positive weight with 1 + 2 = 1. The social planner chooses
1
2
fcit gi=1 ; Kt+1
subject to the resource constraint. Write the Bellman equation
t=0
of the social planning problem and derive the Euler equation. Argue that the
optimal solution of aggregate capital (Kt+1 )1
t=1 is identical to the representativeagent Ramsey growth model with the same K0 . Find the steady state of the
2
economy fci gi=1 ; C ; K .
(b) Suppose that the average behavior of the real-world data can be captured by the
2
social planner solution. Assume that you have observed fcit gi=1 ; Ct ; Kt ; Yt for
a long period of time. From your solution in Part (a), design a procedure to
2
calibrate i i=1 .
(c) Let the capital depreciate fully in every t, i.e., = 1. Find the policy function of
optimal capital accumulation in the planner’s problem in closed form.
(d) Consider a competitive equilibrium setup with a …scal authority in discrete time.
Like in the standard model, the households own the capital and labor and rent
them to the …rm in each period. The competitive …rm makes decisions of renting
capital and labor to maximize one-period pro…t.
The …scal authority chooses income tax rate ( t ) and transfer (T Rt ) for each t,
subject to a balanced-budget requirement
T Rt =
t Yt :
is determined through an exogenous constitution rule t = (Kt1 ; Kt2 ), where
Kti is the average capital stock for type i in period t. Like in a representativeagent model, notice that Kti is equal to kti in equilibrium in every t, but a (small)
household in type i may not take this into account in making its own decision on
1
i
.
kt+1
t=0
t
The budget constraint of the household i is
i
cit + kt+1
= (1
) kti + (1
i
t ) yt
1
+ T Rt ;
2
where yti is the total pre-tax income of the household i. The initial capital (k01 ; k02 )
is given, with 0 < k01 < k02 and k01 + k02 = K0 .
Carefully de…ne a recursive competitive equilibrium for this economy. Be sure to
write down households’dynamic programming problem and functional forms of
pricing functions explicitly.
4. Overlapping Generations with Growth in Population and Money
Consider a pure-exchange economy with two overlapping generations in each period.
There is a single nonstorable consumption good in each period. Each consumer is
endowed with ! y units of the good when young and ! o units when old. Each initial
old consumer has standard utility function v(c0 1 ). Each consumer born in period
t = 0; 1; : : : has standard utility function u(ctt ; ctt+1 ). Let N t denote the measure of
consumers born in period t; it evolves according to
N t = nN t 1 .
Let Mt denote the aggregate supply of …at money in period t; it evolves according to
Mt = gMt 1 .
Newly printed money is equally distributed among the old consumers in each period,
so each old consumer in period t is endowed with amount (Mt Mt 1 )=N t 1 of …at
money.
(a) De…ne an Arrow–Debreu equilibrium.
(b) De…ne a sequential markets equilibrium.
(c) Characterize the solution to each consumer’s problem in a sequential markets
equilibrium. (Be sure to eliminate any Lagrange multipliers.)
(d) Consider a stationary equilibrium. Specify the 5 equations that the objects
fcy ; co ; b; rg must satisfy. Brie‡y argue that there are two stationary equilibria.
(e) Characterize the golden rule allocation (that is, set up the problem of maximizing
the utility of some future generation subject to the stationary resource constraint
and characterize the solution). Which stationary equilibrium satis…es the golden
rule? If the supply of money is not zero in that stationary equilibrium, what is
the value of g?
5. Cash-Credit Goods Economy and the Friedman Rule
Consider a standard cash-credit goods economy. Good 1 must be purchased with cash,
while good 2 must be purchased with credit. Bonds are not state-contingent and earn
gross return R(st ). At the beginning of each period, assets are traded in a centralized
securities market. Then money is used to purchase good 1. Finally, labor is paid in
cash. The representative consumer has expected utility function
X1 X
t
(st ) log c1 (st ) + log c2 (st ) l(st )2 =2 .
t
t=0
s
The resource constraint is
c1 (st ) + c2 (st ) + g(st ) = z(st )l(st ).
The government purchases goods g(st ), taxes labor income at rate (st ), and …nances
changes in money supply M (st ) through open-market operations.
(a) De…ne a sequential markets equilibrium.
(b) Characterize the solution to the consumer’s problem. (You need not eliminate
the Lagrange multipliers.)
(c) Derive the implementability constraint.
(d) Set up the Ramsey problem and characterize the solution.
(e) Prove that the Friedman rule (R(st ) = 1) is optimal.