On existence and bubbles
of Ramsey equilibrium
with borrowing constraints
Becker, Bosi, Le Van & Seegmuller
CNRS
CIMPA 2014, Tlemcen
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Existence and bubbles
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Ramsey conjecture
Ramsey (1928): "... equilibrium would be attained by a division into
two classes, the thrifty enjoying bliss and the improvident at the
subsistence level".
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Ramsey conjecture
Ramsey (1928): "... equilibrium would be attained by a division into
two classes, the thrifty enjoying bliss and the improvident at the
subsistence level".
In the long run, the most patient agents will hold all the capital.
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Ramsey conjecture
Ramsey (1928): "... equilibrium would be attained by a division into
two classes, the thrifty enjoying bliss and the improvident at the
subsistence level".
In the long run, the most patient agents will hold all the capital.
This conjecture was proved by Robert Becker half a century later.
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Complete markets
Individuals are allowed to borrow against future income (Le Van and
Vailakis, 2003).
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Complete markets
Individuals are allowed to borrow against future income (Le Van and
Vailakis, 2003).
The impatient agents borrow from the patient one and spend the rest
of their life to work to refund the debt.
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Complete markets
Individuals are allowed to borrow against future income (Le Van and
Vailakis, 2003).
The impatient agents borrow from the patient one and spend the rest
of their life to work to refund the debt.
Their consumption asymptotically vanishes.
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Complete markets
Individuals are allowed to borrow against future income (Le Van and
Vailakis, 2003).
The impatient agents borrow from the patient one and spend the rest
of their life to work to refund the debt.
Their consumption asymptotically vanishes.
Extension with elastic labor supply (Le Van et al., 2007).
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Incomplete markets
The existence of a steady state rests on the introduction of borrowing
constraints.
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Incomplete markets
The existence of a steady state rests on the introduction of borrowing
constraints.
These imperfections change the equilibrium properties in terms of
optimality, stationarity, monotonicity.
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Existence and bubbles
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Incomplete markets
The existence of a steady state rests on the introduction of borrowing
constraints.
These imperfections change the equilibrium properties in terms of
optimality, stationarity, monotonicity.
Optimality. The set of optimal allocations is now uninformative about
the existence of equilibrium. The Negishi’s approach no longer applies.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
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Incomplete markets
The existence of a steady state rests on the introduction of borrowing
constraints.
These imperfections change the equilibrium properties in terms of
optimality, stationarity, monotonicity.
Optimality. The set of optimal allocations is now uninformative about
the existence of equilibrium. The Negishi’s approach no longer applies.
Stationarity. At the steady state, impatient agents consume. The
steady state vanishes without borrowing constraints (Le Van and
Vailakis, 2003).
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
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Incomplete markets
The existence of a steady state rests on the introduction of borrowing
constraints.
These imperfections change the equilibrium properties in terms of
optimality, stationarity, monotonicity.
Optimality. The set of optimal allocations is now uninformative about
the existence of equilibrium. The Negishi’s approach no longer applies.
Stationarity. At the steady state, impatient agents consume. The
steady state vanishes without borrowing constraints (Le Van and
Vailakis, 2003).
Monotonicity. Borrowing constraints promote persistent cycles (Becker,
1980).
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Cycles
Under heterogenous discounting, the monotonicity property fails
(Mitra, 1979; Le Van and Vailakis, 2003; Becker, 2005; Le Van et al.,
2007).
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Cycles
Under heterogenous discounting, the monotonicity property fails
(Mitra, 1979; Le Van and Vailakis, 2003; Becker, 2005; Le Van et al.,
2007).
Borrowing constraints promote persistent cycles (Becker, 1980).
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Cycles
Under heterogenous discounting, the monotonicity property fails
(Mitra, 1979; Le Van and Vailakis, 2003; Becker, 2005; Le Van et al.,
2007).
Borrowing constraints promote persistent cycles (Becker, 1980).
Rationale for persistence: cycles of period two occur when the capital
income is decreasing in the capital stock (Becker and Foias, 1987,
1994).
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Cycles
Under heterogenous discounting, the monotonicity property fails
(Mitra, 1979; Le Van and Vailakis, 2003; Becker, 2005; Le Van et al.,
2007).
Borrowing constraints promote persistent cycles (Becker, 1980).
Rationale for persistence: cycles of period two occur when the capital
income is decreasing in the capital stock (Becker and Foias, 1987,
1994).
Ramsey conjecture still holds in the case of …nancial constraints.
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Cycles
Under heterogenous discounting, the monotonicity property fails
(Mitra, 1979; Le Van and Vailakis, 2003; Becker, 2005; Le Van et al.,
2007).
Borrowing constraints promote persistent cycles (Becker, 1980).
Rationale for persistence: cycles of period two occur when the capital
income is decreasing in the capital stock (Becker and Foias, 1987,
1994).
Ramsey conjecture still holds in the case of …nancial constraints.
But under other imperfections (distortionary taxes or market power),
a non-degenerated distribution of capital in the long run is possible.
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Existence of an equilibrium under market imperfections
Borrowing constraints.
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Existence of an equilibrium under market imperfections
Borrowing constraints.
The Negishi’s argument no longer works (FWT fails).
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
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Existence of an equilibrium under market imperfections
Borrowing constraints.
The Negishi’s argument no longer works (FWT fails).
Becker et al. (1991) prove the existence of an equilibrium with
inelastic labor supply (…xed point of a tâtonnement map).
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
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Existence of an equilibrium under market imperfections
Borrowing constraints.
The Negishi’s argument no longer works (FWT fails).
Becker et al. (1991) prove the existence of an equilibrium with
inelastic labor supply (…xed point of a tâtonnement map).
Bosi and Seegmuller (2010) give a local proof of existence with elastic
labor supply (…xed point for the policy function).
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Novelty of the model
Elastic labor supply (di¤erently from Becker et al., 1991).
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Novelty of the model
Elastic labor supply (di¤erently from Becker et al., 1991).
Global argument of existence (di¤erently from Bosi and Seegmuller,
2010).
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Existence and bubbles
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Novelty of the model
Elastic labor supply (di¤erently from Becker et al., 1991).
Global argument of existence (di¤erently from Bosi and Seegmuller,
2010).
No bubbles in a productive economy with heterogeneous agents and
imperfect markets.
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A proof of existence
The equilibrium exists in a truncated economy with uniformly
bounded allocations sets (apply a Gale and Mas-Colell (1975)
…xed-point argument as in Florenzano (1999)).
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A proof of existence
The equilibrium exists in a truncated economy with uniformly
bounded allocations sets (apply a Gale and Mas-Colell (1975)
…xed-point argument as in Florenzano (1999)).
This equilibrium is also an equilibrium of an unbounded truncated
economy.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
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A proof of existence
The equilibrium exists in a truncated economy with uniformly
bounded allocations sets (apply a Gale and Mas-Colell (1975)
…xed-point argument as in Florenzano (1999)).
This equilibrium is also an equilibrium of an unbounded truncated
economy.
Proof for an in…nite-horizon economy as a limit of a sequence of
truncated economies.
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Notation
i = 1, . . . , m, individuals.
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Notation
i = 1, . . . , m, individuals.
kit , capital.
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
Kt , aggregate capital.
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Existence and bubbles
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
Kt , aggregate capital.
Lt , aggregate labor.
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Existence and bubbles
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
Kt , aggregate capital.
Lt , aggregate labor.
δ, capital depreciation rate.
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Existence and bubbles
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
Kt , aggregate capital.
Lt , aggregate labor.
δ, capital depreciation rate.
pt , price of the good.
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Existence and bubbles
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
Kt , aggregate capital.
Lt , aggregate labor.
δ, capital depreciation rate.
pt , price of the good.
rt , return on capital.
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Existence and bubbles
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Notation
i = 1, . . . , m, individuals.
kit , capital.
lit , labor supply.
λit , leisure demand.
Kt , aggregate capital.
Lt , aggregate labor.
δ, capital depreciation rate.
pt , price of the good.
rt , return on capital.
wt , wage.
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De…nition
Given the initial capital and leisure endowments, a Walrasian
equilibrium is a sequence of prices and allocations such that
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De…nition
Given the initial capital and leisure endowments, a Walrasian
equilibrium is a sequence of prices and allocations such that
prices are positive,
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De…nition
Given the initial capital and leisure endowments, a Walrasian
equilibrium is a sequence of prices and allocations such that
prices are positive,
markets for goods, capital and labor clear,
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De…nition
Given the initial capital and leisure endowments, a Walrasian
equilibrium is a sequence of prices and allocations such that
prices are positive,
markets for goods, capital and labor clear,
production plans are optimal,
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De…nition
Given the initial capital and leisure endowments, a Walrasian
equilibrium is a sequence of prices and allocations such that
prices are positive,
markets for goods, capital and labor clear,
production plans are optimal,
consumption plans are optimal under the budget and the borrowing
constraints.
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Assumptions
Borrowing constraints: kit
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0.
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Assumptions
Borrowing constraints: kit
0.
Endowments:
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Assumptions
Borrowing constraints: kit
0.
Endowments:
positive initial aggregate endowment: ∑i ki 0 > 0,
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Assumptions
Borrowing constraints: kit
0.
Endowments:
positive initial aggregate endowment: ∑i ki 0 > 0,
one unit of leisure per period: λit = 1 lit .
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Technology:
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Technology:
CRS production function, strictly increasing and concave,
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
asymptotic properties,
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
asymptotic properties,
(∂F /∂K ) (0, m) > δ.
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
asymptotic properties,
(∂F /∂K ) (0, m) > δ.
Preferences:
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
asymptotic properties,
(∂F /∂K ) (0, m) > δ.
Preferences:
heterogeneous preferences, strictly increasing and concave,
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Existence and bubbles
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
asymptotic properties,
(∂F /∂K ) (0, m) > δ.
Preferences:
heterogeneous preferences, strictly increasing and concave,
ui (0, 0) = 0 for any i,
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Technology:
CRS production function, strictly increasing and concave,
inputs are essential: F (0, L) = F (K , 0) = 0,
asymptotic properties,
(∂F /∂K ) (0, m) > δ.
Preferences:
heterogeneous preferences, strictly increasing and concave,
ui (0, 0) = 0 for any i,
Inada conditions.
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Equilibrium of a …nite-horizon economy
Choose su¢ ciently large quantity bounds for individual capital and
consumption, and aggregate inputs.
Theorem
Under the previous Assumptions, there exists an equilibrium
m
p, r, w, ci , ki , λi
, K, L for the …nite-horizon bounded economy.
i =1
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Sketch of a long proof.
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Sketch of a long proof.
De…ne the price simplex:
4 f(p, r , w ) : p, r , w
0, p + r + w = 1g and the budget sets
CiT (p, r, w)
8
(ci , ki , λi ) 2 Xi Yi Zi :
<
pt [cit + kit +1 (1 δ) kit ] rt kit + wt (1
:
t = 0, . . . , T
and its interior
BiT (p, r, w)
8
(ci , ki , λi ) 2 Xi Yi Zi :
<
pt [cit + kit +1 (1 δ) kit ] < rt kit + wt (1
:
t = 0, . . . , T
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Existence and bubbles
λit )
9
=
λit )
9
=
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;
;
14 / 32
If w0 > 0 and rt + wt > 0, for t = 1, . . . , T , then the set BiT (p, r, w)
is nonempty and CiT (p, r, w) is the closure of BiT (p, r, w).
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If w0 > 0 and rt + wt > 0, for t = 1, . . . , T , then the set BiT (p, r, w)
is nonempty and CiT (p, r, w) is the closure of BiT (p, r, w).
We perturb the economy, providing ε units of good to any consumer
and ε units per consumer to producers.
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If w0 > 0 and rt + wt > 0, for t = 1, . . . , T , then the set BiT (p, r, w)
is nonempty and CiT (p, r, w) is the closure of BiT (p, r, w).
We perturb the economy, providing ε units of good to any consumer
and ε units per consumer to producers.
ε and kit are the same good.
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De…ne the perturbed budget sets:
CiT ε (p, r, w)
8
<
pt (cit + kit +1 )
:
(ci , ki , λi ) 2 Xi Yi Zi :
pt ε + [pt (1 δ) + rt ] (kit + ε) + wt (1
t = 0, . . . , T
λit )
BiT ε (p, r, w)
8
(ci , ki , λi ) 2 Xi Yi Zi :
<
pt (cit + kit +1 ) < pt ε + [pt (1 δ) + rt ] (kit + ε) + wt (1
:
t = 0, . . . , T
λit )
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De…ne the perturbed budget sets:
CiT ε (p, r, w)
8
<
pt (cit + kit +1 )
:
(ci , ki , λi ) 2 Xi Yi Zi :
pt ε + [pt (1 δ) + rt ] (kit + ε) + wt (1
t = 0, . . . , T
λit )
BiT ε (p, r, w)
8
(ci , ki , λi ) 2 Xi Yi Zi :
<
pt (cit + kit +1 ) < pt ε + [pt (1 δ) + rt ] (kit + ε) + wt (1
:
t = 0, . . . , T
λit )
BiT ε (p, r, w) is nonempty and CiT ε (p, r, w) = B̄iT ε (p, r, w).
Moreover the correspondence BiT ε is lower semicontinuous.
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In the spirit of Florenzano (1999), we introduce the following reaction
correspondences.
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In the spirit of Florenzano (1999), we introduce the following reaction
correspondences.
Agent i = 0 (the "additional" agent):
ϕ (p, r, w, (ch , kh , λh )m
h =1 , K, L)
80
p,~
r, w
~) 2 P :
(~
>
>
>
T
>
∑t =0 (p̃t pt )
<
(∑i [cit + kit +1 (1 δ) kit ] mε m (1 δ) ε F (Kt , Lt ))
>
>
>
+ ∑Tt=0 (r̃t rt ) (Kt mε ∑m
i =1 kit )
>
:
+ ∑Tt=0 (w̃t wt ) (Lt m + ∑m
i =1 λit ) > 0
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9
>
>
>
>
=
>
>
>
>
;
17 / 32
Agents i = 1, . . . , m (consumers-workers):
ϕi (p, r, w, (ch , kh , λh )m
h =1 , K, L)
/ CiT ε (p, r, w)
BiT ε (p, r, w) if (ci , ki , λi ) 2
BiT ε (p, r, w) \ [Pi (ci , λi ) Yi ] if (ci , ki , λi ) 2 CiT ε (p, r, w)
where Pi is the ith agent’s set of strictly preferred allocations.
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Agent i = m + 1 (the …rm):
ϕm +1 (p, r, w, (ch , kh , λh )m
h =1 , K, L)
8
~ ~
>
K,
L 2Y Z :
<
T
rt K̃t wt L̃t
∑t =0 pt F K̃t , L̃t
>
:
T
> ∑t =0 [pt F (Kt , Lt ) rt Kt wt Lt ]
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Existence and bubbles
9
>
=
>
;
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ϕi is a lower semicontinuous convex-valued correspondence for
i = 0, . . . , m + 1.
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ϕi is a lower semicontinuous convex-valued correspondence for
i = 0, . . . , m + 1.
Let v
(p, r, w, (ci , ki , λi )m
i =1 , K, L).
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ϕi is a lower semicontinuous convex-valued correspondence for
i = 0, . . . , m + 1.
(p, r, w, (ci , ki , λi )m
i =1 , K, L).
By de…nition of ϕ0 (the inequality is strict): (p, r, w) 2
/ ϕ0 (v ).
Let v
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ϕi is a lower semicontinuous convex-valued correspondence for
i = 0, . . . , m + 1.
(p, r, w, (ci , ki , λi )m
i =1 , K, L).
By de…nition of ϕ0 (the inequality is strict): (p, r, w) 2
/ ϕ0 (v ).
/ Pi (ci , λi ) Yi implies that (ci , ki , λi ) 2
/ ϕi (v) for
(ci , ki , λi ) 2
i = 1, . . . , m.
Let v
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
20 / 32
ϕi is a lower semicontinuous convex-valued correspondence for
i = 0, . . . , m + 1.
(p, r, w, (ci , ki , λi )m
i =1 , K, L).
By de…nition of ϕ0 (the inequality is strict): (p, r, w) 2
/ ϕ0 (v ).
/ Pi (ci , λi ) Yi implies that (ci , ki , λi ) 2
/ ϕi (v) for
(ci , ki , λi ) 2
i = 1, . . . , m.
By de…nition of ϕm +1 (the inequality is also strict):
/ ϕ m +1 ( v ) .
(K, L) 2
Let v
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
20 / 32
ϕi is a lower semicontinuous convex-valued correspondence for
i = 0, . . . , m + 1.
(p, r, w, (ci , ki , λi )m
i =1 , K, L).
By de…nition of ϕ0 (the inequality is strict): (p, r, w) 2
/ ϕ0 (v ).
/ Pi (ci , λi ) Yi implies that (ci , ki , λi ) 2
/ ϕi (v) for
(ci , ki , λi ) 2
i = 1, . . . , m.
By de…nition of ϕm +1 (the inequality is also strict):
/ ϕ m +1 ( v ) .
(K, L) 2
Then, for i = 0, . . . , m + 1, vi 2
/ ϕi (v ).
Let v
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
20 / 32
Apply Gale and Mas-Colell (1975) …xed-point theorem. There exists
v 2 Φ such that ϕi (v) = ? for i = 0, . . . , m + 1, that is, there exists
v 2 Φ such that the following holds.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
21 / 32
Apply Gale and Mas-Colell (1975) …xed-point theorem. There exists
v 2 Φ such that ϕi (v) = ? for i = 0, . . . , m + 1, that is, there exists
v 2 Φ such that the following holds.
Focus on "agent" i = 0. For every (p, r, w) 2 P,
T
∑ (pt
p̄t )
t =0
m
∑ [c̄it + k̄it +1
(1
δ) k̄it ]
mε
m (1
δ) ε
F (K̄t , L̄t )
i =1
!
T
+ ∑ (rt
r̄t )
t =0
m
K̄t
mε
∑ k̄it
i =1
!
m
T
+
∑ ( wt
t =0
w̄t ) L̄t
m + ∑ λ̄it
i =1
!
0
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
21 / 32
Consider i = 1, . . . , m. ci , ki , λi 2 CiT ε (p, r, w) and
h
i
BiT ε (p, r, w) \ Pi ci , λi
Yi = ? for i = 1, . . . , m. Then, for
Tε
i = 1, . . . , m, (ci , ki , λi ) 2 CiT ε (p, r, w) = B i (p, r, w) implies
T
∑ βti ui (cit , λit )
t =0
Becker, Bosi, Le Van & Seegmuller (CNRS)
T
∑ βti ui
c̄it , λ̄it
t =0
Existence and bubbles
CIMPA 2014, Tlemcen
22 / 32
Consider i = 1, . . . , m. ci , ki , λi 2 CiT ε (p, r, w) and
h
i
BiT ε (p, r, w) \ Pi ci , λi
Yi = ? for i = 1, . . . , m. Then, for
Tε
i = 1, . . . , m, (ci , ki , λi ) 2 CiT ε (p, r, w) = B i (p, r, w) implies
T
∑ βti ui (cit , λit )
t =0
T
∑ βti ui
c̄it , λ̄it
t =0
Focus on the …rm i = m + 1. For t = 0, . . . , T and for every
(K, L) 2 Y Z , we have ∑Tt=0 [p̄t F (Kt , Lt ) r̄t Kt w̄t Lt ]
∑Tt=0 [p̄t F (K̄t , L̄t ) r̄t K̄t w̄t L̄t ].
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
22 / 32
Then, we prove prices positivity and the zero pro…t:
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
23 / 32
Then, we prove prices positivity and the zero pro…t:
p̄t > 0,
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
23 / 32
Then, we prove prices positivity and the zero pro…t:
p̄t > 0,
p̄t F (K̄t , L̄t )
r̄t K̄t
Becker, Bosi, Le Van & Seegmuller (CNRS)
w̄t L̄t = 0,
Existence and bubbles
CIMPA 2014, Tlemcen
23 / 32
Then, we prove prices positivity and the zero pro…t:
p̄t > 0,
p̄t F (K̄t , L̄t ) r̄t K̄t
r̄t > 0, w̄t > 0.
Becker, Bosi, Le Van & Seegmuller (CNRS)
w̄t L̄t = 0,
Existence and bubbles
CIMPA 2014, Tlemcen
23 / 32
Finally, using prices positivity, we show the market clearing in the
perturbed economy:
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
24 / 32
Finally, using prices positivity, we show the market clearing in the
perturbed economy:
∑i [c̄it + k̄it +1
Becker, Bosi, Le Van & Seegmuller (CNRS)
(1
δ) k̄it ]
F (K̄t , L̄t )
Existence and bubbles
m (1
δ) ε
mε = 0,
CIMPA 2014, Tlemcen
24 / 32
Finally, using prices positivity, we show the market clearing in the
perturbed economy:
∑i [c̄it + k̄it +1 (1 δ) k̄it ]
K̄t mε ∑i k̄it = 0,
Becker, Bosi, Le Van & Seegmuller (CNRS)
F (K̄t , L̄t )
Existence and bubbles
m (1
δ) ε
mε = 0,
CIMPA 2014, Tlemcen
24 / 32
Finally, using prices positivity, we show the market clearing in the
perturbed economy:
∑i [c̄it + k̄it +1 (1 δ) k̄it ]
K̄t mε ∑i k̄it = 0,
L̄t ∑i 1 λ̄it = 0.
Becker, Bosi, Le Van & Seegmuller (CNRS)
F (K̄t , L̄t )
Existence and bubbles
m (1
δ) ε
mε = 0,
CIMPA 2014, Tlemcen
24 / 32
We take the limit of the perturbed economy as ε tends to zero and we
show that the limit
p, r, w, ci , ki , λi
m
i =1
, K, L
lim p (ε) , r (ε) , w (ε) , ci (ε) , ki (ε) , λi (ε)
ε !0
m
i =1
, K (ε) , L (ε)
satis…es the de…nition of Walrasian equilibrium (prices positivity,
optimal plans and market clearing).
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
25 / 32
We take the limit of the perturbed economy as ε tends to zero and we
show that the limit
p, r, w, ci , ki , λi
m
i =1
, K, L
lim p (ε) , r (ε) , w (ε) , ci (ε) , ki (ε) , λi (ε)
ε !0
m
i =1
, K (ε) , L (ε)
satis…es the de…nition of Walrasian equilibrium (prices positivity,
optimal plans and market clearing).
We prove that any equilibrium of E T is an equilibrium for the
…nite-horizon unbounded economy.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
25 / 32
We take the limit of the perturbed economy as ε tends to zero and we
show that the limit
p, r, w, ci , ki , λi
m
i =1
, K, L
lim p (ε) , r (ε) , w (ε) , ci (ε) , ki (ε) , λi (ε)
ε !0
m
i =1
, K (ε) , L (ε)
satis…es the de…nition of Walrasian equilibrium (prices positivity,
optimal plans and market clearing).
We prove that any equilibrium of E T is an equilibrium for the
…nite-horizon unbounded economy.
Simply, consider a convex combination within the bounds of the
equilibrium of the bounded economy with a candidate outside the
bounds and derive a contradiction.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
25 / 32
Equilibrium of a …nite-horizon economy
Theorem
Under the assumptions of the model, there exists an equilibrium in the
in…nite-horizon economy with endogenous labor supply and borrowing
constraints.
We denote by
m
p (T ) , r (T ) , w (T ) , ci (T ) , ki (T ) , λi (T )
, K (T ) , L (T )
i =1
an equilibrium for the truncated economy and by
m
^ ^
^
p,^
r, w
^, ^
ci , ^
ki , ^
λi
, K,
L the limit for T ! ∞ for the product
i =1
topology.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
26 / 32
Equilibrium of a …nite-horizon economy
Theorem
Under the assumptions of the model, there exists an equilibrium in the
in…nite-horizon economy with endogenous labor supply and borrowing
constraints.
We denote by
m
p (T ) , r (T ) , w (T ) , ci (T ) , ki (T ) , λi (T )
, K (T ) , L (T )
i =1
an equilibrium for the truncated economy and by
m
^ ^
^
p,^
r, w
^, ^
ci , ^
ki , ^
λi
, K,
L the limit for T ! ∞ for the product
i =1
topology.
We prove …rst that ^
ci , ^
ki , ^
λi solves the consumer’s program in the
in…nite-horizon economy.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
26 / 32
Equilibrium of a …nite-horizon economy
Theorem
Under the assumptions of the model, there exists an equilibrium in the
in…nite-horizon economy with endogenous labor supply and borrowing
constraints.
We denote by
m
p (T ) , r (T ) , w (T ) , ci (T ) , ki (T ) , λi (T )
, K (T ) , L (T )
i =1
an equilibrium for the truncated economy and by
m
^ ^
^
p,^
r, w
^, ^
ci , ^
ki , ^
λi
, K,
L the limit for T ! ∞ for the product
i =1
topology.
We prove …rst that ^
ci , ^
ki , ^
λi solves the consumer’s program in the
in…nite-horizon economy.
Then, we show the price positivity: p̂t , r̂t , ŵt > 0.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
26 / 32
Equilibrium of a …nite-horizon economy
Theorem
Under the assumptions of the model, there exists an equilibrium in the
in…nite-horizon economy with endogenous labor supply and borrowing
constraints.
We denote by
m
p (T ) , r (T ) , w (T ) , ci (T ) , ki (T ) , λi (T )
, K (T ) , L (T )
i =1
an equilibrium for the truncated economy and by
m
^ ^
^
p,^
r, w
^, ^
ci , ^
ki , ^
λi
, K,
L the limit for T ! ∞ for the product
i =1
topology.
We prove …rst that ^
ci , ^
ki , ^
λi solves the consumer’s program in the
in…nite-horizon economy.
Then, we show the price positivity: p̂t , r̂t , ŵt > 0.
The rest follows.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
26 / 32
Non-existence of bubbles
Consider the equilibrium of an in…nite-horizon economy. Take p̄t = 1
with r̄t , w̄t > 0.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
27 / 32
Non-existence of bubbles
Consider the equilibrium of an in…nite-horizon economy. Take p̄t = 1
with r̄t , w̄t > 0.
We introduce a market discount factor re‡ecting the marginal rate of
substitution between t and t + 1:
q̄t +1
max
i
Becker, Bosi, Le Van & Seegmuller (CNRS)
βi (∂ui /∂c ) c̄it +1 , λ̄it +1
=
1
(∂ui /∂c ) c̄it , λ̄it
Existence and bubbles
1
δ + r̄t +1
CIMPA 2014, Tlemcen
27 / 32
Let Q̄0 1 and Q̄t
∏ts =1 q̄s for t > 0, that is
Q̄t = ∏ts =1 (1 δ + r̄s ) 1 for t > 0.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
28 / 32
Let Q̄0 1 and Q̄t
∏ts =1 q̄s for t > 0, that is
Q̄t = ∏ts =1 (1 δ + r̄s ) 1 for t > 0.
Q̄t is the present value of a unit of capital of period t with focal date
t = 0.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
28 / 32
Let Q̄0 1 and Q̄t
∏ts =1 q̄s for t > 0, that is
Q̄t = ∏ts =1 (1 δ + r̄s ) 1 for t > 0.
Q̄t is the present value of a unit of capital of period t with focal date
t = 0.
By induction, 1 = Q̄0 = Q̄T (1
Becker, Bosi, Le Van & Seegmuller (CNRS)
δ)T + ∑Tt=1 Q̄t r̄t (1
Existence and bubbles
δ )t
1
.
CIMPA 2014, Tlemcen
28 / 32
Let Q̄0 1 and Q̄t
∏ts =1 q̄s for t > 0, that is
Q̄t = ∏ts =1 (1 δ + r̄s ) 1 for t > 0.
Q̄t is the present value of a unit of capital of period t with focal date
t = 0.
δ)T + ∑Tt=1 Q̄t r̄t (1 δ)t 1 .
At date 1, one unit of this asset will give back 1 δ unit of capital
and r̄1 unit of consumption good as its dividend.
By induction, 1 = Q̄0 = Q̄T (1
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
28 / 32
Let Q̄0 1 and Q̄t
∏ts =1 q̄s for t > 0, that is
Q̄t = ∏ts =1 (1 δ + r̄s ) 1 for t > 0.
Q̄t is the present value of a unit of capital of period t with focal date
t = 0.
δ)T + ∑Tt=1 Q̄t r̄t (1 δ)t 1 .
At date 1, one unit of this asset will give back 1 δ unit of capital
and r̄1 unit of consumption good as its dividend.
By induction, 1 = Q̄0 = Q̄T (1
At period 2, 1
capital and (1
δ unit of capital will give back (1
δ) r̄2 as its dividend.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
δ)2 unit of
CIMPA 2014, Tlemcen
28 / 32
De…nition of the Fundamental Value of capital:
∞
FV
∑ Q̄t (1
δ )t
1
r̄t
t =1
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
29 / 32
De…nition of the Fundamental Value of capital:
∞
FV
∑ Q̄t (1
δ )t
1
r̄t
t =1
The economy is said to experience a bubble if
limT !∞ Q̄T (1 δ)T > 0. Otherwise (limT !∞ Q̄T (1
there is no bubble.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
δ)T = 0),
CIMPA 2014, Tlemcen
29 / 32
Lemma
If the economy experiences a bubble, then r̄t converges to zero.
Assume that r̄t does not converge to zero. There is ρ > 0 and a
ρ for i = 1, 2, . . .
strictly increasing sequence (ti )i∞=1 such that r̄ti
For T > tn , we get
δ )T =
Q̄T (1
T
∏1
s =1
1
δ
δ + r̄s
n
∏1
1
i =1
and
0
lim sup Q̄T (1
T !∞
δ )T
lim
n !∞
1
δ
δ + r̄ti
1
1
1
δ
δ+ρ
δ
δ+ρ
n
n
=0
That is there are no bubbles.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
30 / 32
Theorem
Under the assumptions of the model (Inada included), our productive
economy experiences no bubbles.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
31 / 32
Theorem
Under the assumptions of the model (Inada included), our productive
economy experiences no bubbles.
The last (laborious) part of the proof consists in proving that r̄t does
not converge to zero.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
31 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
This equilibrium turns out to be also an equilibrium of any unbounded
economy with the same fundamentals.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
This equilibrium turns out to be also an equilibrium of any unbounded
economy with the same fundamentals.
Existence of an equilibrium in an in…nite-horizon economy as a limit of
a sequence of truncated economies.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
This equilibrium turns out to be also an equilibrium of any unbounded
economy with the same fundamentals.
Existence of an equilibrium in an in…nite-horizon economy as a limit of
a sequence of truncated economies.
Proof of non-existence of bubbles in a productive economy.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
This equilibrium turns out to be also an equilibrium of any unbounded
economy with the same fundamentals.
Existence of an equilibrium in an in…nite-horizon economy as a limit of
a sequence of truncated economies.
Proof of non-existence of bubbles in a productive economy.
We de…ne a market discount factor and the fundamental value of
capital.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
This equilibrium turns out to be also an equilibrium of any unbounded
economy with the same fundamentals.
Existence of an equilibrium in an in…nite-horizon economy as a limit of
a sequence of truncated economies.
Proof of non-existence of bubbles in a productive economy.
We de…ne a market discount factor and the fundamental value of
capital.
We show that, if the economy experiences a bubble, then the interest
rate converges to zero.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32
Conclusions
Proof of equilibrium existence in a Ramsey model with borrowing
constraints and endogenous labor supply.
We apply a …xed-point theorem by Gale-Mas-Colell to a perturbed
truncated economy.
The limit as the perturbation vanishes is an equilibrium of the
unperturbed economy.
This equilibrium turns out to be also an equilibrium of any unbounded
economy with the same fundamentals.
Existence of an equilibrium in an in…nite-horizon economy as a limit of
a sequence of truncated economies.
Proof of non-existence of bubbles in a productive economy.
We de…ne a market discount factor and the fundamental value of
capital.
We show that, if the economy experiences a bubble, then the interest
rate converges to zero.
We prove that our assumptions prevent the interest rate from going to
zero.
Becker, Bosi, Le Van & Seegmuller (CNRS)
Existence and bubbles
CIMPA 2014, Tlemcen
32 / 32