Part 5
1. Recursive formulation of competitive equilibrium. Complete markets.
This material is a version of Chapter 12 (12.7 and 12.10) in Ljungqvist-Sargent.
We start from a recursive formulation of the planner’s problem, and then we
decentralize the solution. This is a generalization of the procedure we adopted in
the discussion of the RBC model.
The stochastic productivity shocks are governed by a Markov chain [z ∈ S,
π(z 0 |z)] (a discrete version of the AR(1) process).
1.1. Recursive formulation of the planner’s problem
(
)
X
v(K, z) = max 0 u(C, 1 − N) + β
π(z 0 |z)v(K 0 , z 0 ) ,
C,N,K
z0
subject to
zF (K, N) = C + K 0 − (1 − δ)K.
F (K, N) is homogeneous of degree one.
The state variables are K, z. Hence, the optimal decision rules are denoted as
C = Ωc (K, z),
N = Ωn (K, z),
K 0 = Ωk (K, z).
The first-order conditions for the planner’s problem are
Ul (K, z) = Uc (K, z)zFn (K, z)
X
π(z 0 |z)Uc (K 0 , z 0 ) [z 0 Fk (K 0 , z 0 ) + 1 − δ] ,
Uc (K, z) = β
z0
where the Envelope Theorem was used in the Euler equation and
Uc (K, z) ≡ u1 (Ωc (K, z), 1 − Ωn (K, z)),
Ul (K, z) ≡ u2 (Ωc (K, z), 1 − Ωn (K, z)),
Fk (K, z) ≡ F1 (K, Ωn (K, z)),
Fn (K, z) ≡ F2 (K, Ωn (K, z)).
1.2. Recursive formulation of competitive equilibrium
We decentralize the economy using Arrow securities, which are one period-assets
which have state-contingent payoffs.
There are three types of agents:
Households, who consume, save and supply labor,
firms of type I, which produce output from hiring labor and renting the existing
capital stock, and
firms of type II, which rent the capital to firms of type I and borrow from
households to finance next-period capital stock.
The prices as functions of the aggregate state K, z are defined as r(K, z), w(K, z)
and Q(z 0 |K, z). The first is the rental price of capital, the second is the wage rate,
and the third is the current price of a claim to one unit of consumption in next
period–when the current state is K, z and the future realization z 0 . The latter
price represents a state-contingent 1/R.
The household’s problem
We define a as total claims to consumption goods held by the household at
the beginning of the period. The household’s problem is to choose the amounts of
assets a(K 0 , z 0 ) for each future state K 0 , z 0 . Note that the aggregate capital stock
K 0 is known at time t.
J(a, K, z) =
max
c,n,a(K 0 ,z 0 )
subject to
c+
X
z0
(
u(c, 1 − n) + β
X
z0
)
π(z 0 |z)J(a(K 0 , z 0 ), K 0 , z 0 ) ,
Q(z 0 |K, z)a(K 0 , z 0 ) ≤ w(K, z)n + a.
The household enters the period with a and receives labor income. This total
can be spent either on consumption or purchasing contingent claims on nextperiod consumption goods at the prices Q(z 0 |K, z). Next period, of course, only
one of the z 0 s will realize, and the household will receive then a(K 0 , z 0 ).
The household’s decisions rules are denoted as
c = σ c (a, K, z),
n = σ n (a, K, z),
a(K 0 , z 0 ) = σ a (a, K, z; z 0 ).
2
The first-order conditions for this problem are
ul (a, K, z) = uc (a, K, z)w(K, z),
uc (a, K, z)Q(z 0 |K, z) = βπ(z 0 |z)uc (σ a (a, K, z; z 0 ), K 0 , z 0 ),
where
uc (a, K, z) ≡ u1 (σ c (a, K, z), 1 − σ n (a, K, z)),
ul (a, K, z) ≡ u2 (σ c (a, K, z), 1 − σ n (a, K, z)).
z 0 ∈ S,
Firm of type I
This firm maximizes profits from production of output using rented capital
and labor. Hence, the problem of this firm is static.
©
ª
max zF (kI , nI ) − r(K, z)kI − w(K, z)nI .
kI ,nI
The first-order conditions are
r(K, z) = zF1 (kI , nI ),
w(K, z) = zF2 (kI , nI ).
Because F (·) is homogeneous of degree one, profits of this firm are zero.
Firms of type II
This firm specializes in investment. It intermediates between households and
firms of type I by issuing state-contingent assets to the households, investing in
capital and renting it to firms of type I.
We formulate the optimization of this firm as a two-period problem, where the
firm issues securities each period to finance all the capital stock for next period.
(Following production, the firm is left with no funds for investment because the
proceeds from selling the depreciated capital and the current rental income is used
to pay the return on the assets issued last period).
0
The state-contingent rental revenue for next-period capital will be r(K 0 , z 0 )kII
0
and the liquidation value will be (1 − δ)kII . Because the contingent price of funds
to be repaid next period is Q(z 0 |K, z), the present value of future streams are
computed using these prices.
This firm’s problem is
(
)
X
0
0
max
Q(z 0 |K, z) [r(K 0 , z 0 ) + 1 − δ] kII − kII .
0
kII
z0
3
Explanation:
Net profits are the difference between the present value of future income stream
and current investment.
Why summation over z 0 instead of expected value? For each z 0 , the firm can sell
today contingent assets for the total of [r(K 0 , z 0 ) + 1 − δ] kII for each one of the z 0
realizations. Hence, the firm can sell assets for all possible future z 0 realizations.
(Next period, or course, only one state will realize).
The first-order condition here is simply
X
1=
Q(z 0 |K, z) [r(K 0 , z 0 ) + 1 − δ] .
z0
0
Note that this condition does not determine kII (and it implies zero profits).
The future capital stock is determined by the saving decision. In other words,
this first-order condition represents an horizontal supply of assets (demand for
funds) at the price Q(z 0 |K, z). The quantity traded will depend on the demand
for state-contingent assets.
1.3. Competitive equilibrium conditions
Here the equilibrium conditions are formulated.
1. Households and firms take the aggregate K as given, but equilibrium in the
rental market of capital implies that
K = kI = kII .
2. Labor market equilibrium requires
N = nI = σ n (a, K, z)).
3. The state-contingent assets issued by type II firms should equal the statecontingent assets demanded by households.
(a) The state-contingent assets supplied equal [r(K 0 , z 0 ) + 1 − δ] K 0 . This is
the amount of assets firm of type II can issue contingent on the specific realization
z 0 . It includes the contingent rental income plus the principal after-depreciation
value.
(b) The state-contingent assets demanded is, as stated before, σa (a, K, z; z 0 ).
Hence, equilibrium for the z 0 contingent asset requires
σ a (a, K, z; z 0 ) = [r(K 0 , z 0 ) + 1 − δ] K 0 .
4
4. The goods market clears, i.e.,
σ c ([r(K, z) + 1 − δ] K, K, z)+K 0 −(1 − δ) K = zF (K, σn ([r(K, z) + 1 − δ] K, K, z)).
Note that this equation can be written as
K 0 = G(K, z) ≡ zF (K, σ n ([r(K, z) + 1 − δ] K, K, z))+(1 − δ) K−σ c ([r(K, z) + 1 − δ] K, K, z),
which describes the evolution of aggregate capital.
Formal definition of the equilibrium
A recursive competitive equilibrium with Arrow securities is a set of
price functions r(K, z), w(K, z) and Q(z 0 |K, z),
and household decision rules σ c (a(K, z), K, z), σ n (a(K, z), K, z), σ a (a(K, z), K, z; z 0 )
such that:
• σ c (a(K, z), K, z), σ n (a(K, z), K, z), and σ a (a(K, z), K, z; z 0 ) solve the household’s problem given r(K, z), w(K, z), Q(z 0 |K, z).
• r(K, z) = zF1 (K, σ n ([r(K, z) + 1 − δ] K, K, z)),
and w(K, z) = zF2 (K, σ n ([r(K, z) + 1 − δ] K, K, z)), (these are the conditions for solving the problem of firm of type I),
X
•
Q(z 0 |K, z) [r(K 0 , z 0 ) + 1 − δ] = 1, (the condition for firm of type II),
z0
• the goods and the labor markets clear, i.e., K 0 = zF (K, σn ([r(K, z) + 1 − δ] K, K, z))+
(1 − δ) K − σ c ([r(K, z) + 1 − δ] K, K, z) (or K 0 = G(K, z), as mentioned
above).
1.4. Using the planner’s problem to solve for the competitive equilibrium
Exercise: Show that the solution of the planner’s problem satisfies the first-order
conditions of all units.
5
2. Borrowing Constraints I
2.1. Self insurance with a single constant-return asset
This discussion is from Ljungqvist-Sargent, Chapter 16.
Here we turn to a setup where there is only one asset with a fixed (“risk-free”)
return. Hence, households cannot purchase contingent claims for the entire range
of realizations of uncertainty in the future. Only in the obvious case there is only
one possible realization, i.e., the case of certainty, this asset is an Arrow security.
We start from the deterministic case.
2.1.1. Nonstochastic environment
The household faces a certain sequence of income flows {yt }∞
t=0 . The only asset
available is a one-period debt instrument b with a net interest rate r, where (1 +
r)β = 1. The problem of a household is to maximize
∞
P
β t u(ct ),
0 < β < 1,
t=0
subject to the sequence of budget constraints
ct + bt =
1
bt+1 + yt .
1+r
The function u is strictly increasing, twice differentiable and it satisfies the Inada
conditions. Here, bt is the debt at the beginning of period t. The expenses appear
on the left-hand side of the budget constraint, and the sources for this expenses
appear on the right-hand side. The Inada conditions imply that ct > 0. This is
important for the implications of the borrowing constraint.
The “natural borrowing constraint”
How large can bt be? The “natural borrowing constraint” corresponds to the
household’s maximal ability to pay, i.e., that level of bt that can be honored if all
income flows are devoted to debt repayment, and hence ct = 0 for all t.
¶j
∞ µ
X
1
yt+j ≡ b̄t .
bt ≤
1+r
j=0
In other words, this is the maximal debt that can be repaid at time t by selling
the rights to all future income.
6
With an initial b0 < b̄0 , the present value of consumption should satisfy the
constraint
or
¶t
∞ µ
X
1
ct ≤ b̄0 − b0 .
1+r
t=0
¶t
¶t
∞ µ
∞ µ
X
X
1
1
ct ≤
yt − b0 ,
1
+
r
1
+
r
t=0
t=0
With an equality, this is the intertemporal budget constraint that can be obtained
by solving forward the sequence of periodical budget constraints
ct + bt =
1
bt+1 + yt
1+r
and using the transversality condition
µ
¶t
1
lim
bt = 0.
t→∞ 1 + r
Hence, the imposition of a “natural borrowing constraint” implies the imposition
of the transversality condition (also referred to as the “no Ponzi games” condition).
Formulation of the household’s problem with a borrowing constraint
“Ad-hoc” constraint: b̃ < b̄. If b̃ = b̄, the transversality condition suffices.
½
µ
¶
³
´¾
∞
P
1
t
Max β u(ct ) + λt
bt+1 + yt − bt − ct + μt b̃t+1 − bt+1
,
1+r
t=0
The first order conditions are
uc (ct ) − λt = 0,
λt ≥ 0,
λt
1
− βλt+1 − μt = 0,
1+r
μt ≥ 0,
t = 0, 1, 2...
Because of the Inada condition, uc (ct ) = λt > 0. Substituting,
uc (ct )
1
− βuc (ct+1 ) − μt = 0,
1+r
7
or
uc (ct ) = uc (ct+1 ) + (1 + r)μt .
Hence,
uc (ct ) ≥ uc (ct+1 ).
If the household is borrowing constrained at time t, current consumption has to
be lower than the desired level for consumption smoothing, i.e.,
uc (ct ) > uc (ct+1 ) ⇒ ct < ct+1 .
If the borrowing constraint never binds, uc (ct ) = uc (ct+1 ) for all t. Then, we
get the “permanent income hypothesis” we discussed earlier, where the optimal
policy is to set consumption equal to “permanent income”.
Assuming for simplicity that b0 = 0, permanent income is defined as ȳ such
that
¶t
¶t
∞ µ
∞ µ
X
X
1
1
1+r
ȳ.
yt =
ȳ =
1
+
r
1
+
r
r
t=0
t=0
This level of income is a weighted average of income flows with weights equal to
the corresponding discount factors. Setting
ct = c̄ = ȳ for all t
satisfies both the first-order conditions with equality–i.e., consumption smoothing–
and the intertemporal budget constraint, which implies satisfying the natural borrowing constraint.
Specific ad-hoc borrowing constraint: no borrowing allowed – Skip.
Just for showing that the consumption profile in non-decreasing. Only intuitive
discussion.
Assume now that the household is not allowed to borrow, i.e., bt ≤ 0, and
therefore for consumption smoothing it has to have nonnegative assets. Here, we
define
at ≡ −bt + yt .
Hence, at is the total amount of resources the household could spend in period t,
i.e., it includes current income. The borrowing constraint takes here the form
at ≥ yt .
8
The first-order conditions are, similarly as in the case of the ad-hoc constraint
above,
uc (ct ) ≥ uc (ct+1 ),
but now equality holds when ct < at . Then, ct = ct+1 holds. When the household
is borrowing constrained,
ct < ct+1 ,
and hence
ct = at ⇒ at+1 = yt+1 .
In the latter case, the household does not have any assets at the end of period t.
From the above it follows that
ct ≤ ct+1 .
That is, the consumption profile cannot decline. This asymmetry follows from the
asymmetric restriction on assets. They have to be nonnegative
It can never occur that
ct > ct+1 ,
because then the household is better off by saving and achieving consumption
smoothing.
The borrowing constraint will typically apply when the income profile is increasing. Higher future incomes requires borrowing for consumption smoothing,
but this is not allowed here.
2.1.2. Stochastic environment
We do not impose here a borrowing constraint. We’ll see that the solution satisfies
the “natural borrowing constraint”.
The household’s problem is here to maximize
E0
∞
P
β t u(ct ),
0 < β < 1,
t=0
subject to the sequence of budget constraints
at+1 = (1 + r)(at − ct ) + yt+1 .
As previously, there is only one asset available, a, which yields the net rate of
return r, with (1+r)β = 1. The function u is strictly increasing, twice differentiable
9
and it satisfies the Inada conditions. Income yt is identically and independently
distributed (i.i.d.) with mean μy .
The Bellman equation associated with this problem is
V (a) = max {u(c) + βEV [(1 + r)(a − c) + y 0 ]} ,
c
and the corresponding Euler equation is
uc (c) = β(1 + r)EVa [(1 + r)(a − c) + y 0 ] ,
which, since Va (a) = uc (c), (which we showed earlier own by differentiating the
Bellman equation), can be expressed as
uc (c) = Euc (c0 ).
Marginal utility is a martingale. A stochastic sequence {xt } that satisfies E [x0 |J] =
x is defined as a martingale adapted to the information set J. In this case, J = x.
This case is also denoted a “random walk,” or alternatively, a process with a “unit
root.”
Quadratic preferences
We investigate first the quadratic utility function
1
u(c) = − (γ − c)2 ,
2
0 < γ < ∞.
Marginal utility is here γ − c.
Note two things about this utility function:
1. It is not strictly increasing as assumed for the general case. Beyond the
satiation level γ, utility decreases with consumption.
2. The Inada condition uc (c) → ∞ as c → 0 is not satisfied. Here uc (c) → γ
as c → 0. Hence, consumption is not constrained to be nonnegative.
The Euler equation takes here the form
c = Ec0 ,
i.e.,
c0 = c + ε0 ,
10
where ε0 has zero mean (it should be unpredicted given all the information available currently). Correspondingly, consumption is a martingale, and hence it is
unbounded under the current preferences.
Robert Hall (1978) tested this equation, which has the strong econometric
implication that when regressing c0 on any number of current variables, once c is
included in the regression, none of the others can have any explanatory power.
The resulting process for assets has the form
a0 = a + ξ 0 ,
where ξ is white noise, i.e., i.i.d. with mean zero. Correspondingly, assets (or
debt) are also a martingale. Showing this, the exact form of ξ, and its link with
ε are left as an exercise.
Note that here
(
µ
¶t )
n
o
1
at = 0,
E0 lim at = a0 ⇒ E0 lim
t→∞
t→∞ 1 + r
for any finite starting assets a0 . Hence, the transversality condition holds in expected value.
Solving forward the sequence of budget constraints yields
(∞ µ
(∞ µ
¶t )
¶t )
X
X
1
1
E0
ct = E0
yt + a0 − y0 .
1+r
1+r
t=0
t=0
That is, the “natural borrowing constraint” is satisfied in expected values.
To solve the model we can use the same dynamic programming procedure
applied to the Euler equation. In this case, however, the equation is linear and
hence there is no need for approximation.
Conjecture that c = A + Ba. Substituting the conjecture: c0 = A + Ba0 into
the Euler equation c = Ec0 gives
c = E (A + Ba0 ) ,
c = A + B (1 + r) (a − c) + BE (y 0 ) .
c=
A + Bμy + B (1 + r) a
.
B (1 + r) + 1
The rest of the solution is left for the homework.
11