上财罗大庆高宏课件 (2)

+
p2 p1
y2
.
Therefore,
p1 p2
=
y2 βy1
.
Connecting the equilibrium relative price in the consumption functions yields
the equilibrium consumption allocations:
c1 = y1,
function:
c2
=
1
β +
β
p1y1 p2
+
y2
.
– Market Clearing: c1 = y1, and c2 = y2.
Clearing of the goods market in period 1 requires c1 = y1. This implies:
y1
=
1
1 +
β
y1
k=x c1 + x = y1
c2 = y2 + (1 − δ)k + f (k)
– Representative consumer’s optimization problem:
max
c1,c2,x
u(c1, c2) = log c1 + β log c2
s.t. p1c1 + p2c2 = p1(y1 − x) + p2y2 + qx + P R
sumption function:
c2
=
β
·
p1 p2
·
c1
p1y1
+
p2y2
−
p1c1
−
p2β
p1 p2
c1
=
0
⇒
c1
=
1
1 +
β
y1
+
p2 p1
y2
.
2
Plugging this result in the Euler equation yields the period-2 consumption
– The central planner’s problem is then: max log c1 + β log c2 s.t. c1 = y1 and c2 = y2.
– The solution is obviously to set c1 = y1 and c2 = y2.
3
– The equilibrium relative price is then calculated using the equality of relative
c2 = y2 + (1 − δ)k + f (k).
That is
max U = log(y − k) + β log(kα)
k
FONC:
∂U ∂k
=
−
y
1 −
k
+
β
αkα−1 kα
= 0.
So we have
y
1 −k
=
αβ k
⇒
k
=
αβ 1 + αβ
y.
Plugging the solution for k in the planner’s resource constraints yields
• Additional notation: – q: price of capital – x: amount of capital sold to firms – k: physical capital used by firms in production – δ: capital depreciation rate – f (k): production function
u(c1, c2) = log c1 + β log c2 f (k) = kα
y1 = y, y2 = 0, and δ = 1. The central planner’s problem:
max
c1,c2,k
s.t.
u(c1, c2) = log c1 + β log c2 c1 + k = y1
p1c1 + p2c2 ≤ p1y1 + p2y2.
Now we have a simple 2-period model: max u(c1, c2) s.t. p1c1 + p2c2 ≤ p1y1 + p2y2 1
Solving the model:
• A Competitive Equilibrium is a set of price (p1, p2) and an allocation (c1, c2) such that
consumer’s optimization
⇒
p1 p2
=
I M RS
k→0
k→∞
• A competitive equilibrium with production is a price system (p1, p2, q) and an allocation (c1, c2, x, k) such that
1. the representative consumer maximizes utility subject to his budget constraint, given prices; 2. the representative firm maximizes profits, given prices and technology; 3. prices clear all markets, i.e.
relative prices:
1/c1 β/c2
=
p1 p2
(an Euler Equation)
We can solve the Euler equation for c2 and use the resulting expression to
substitute out c2 from the budget constraint. This yields the period-1 con-
c2 = y2.
Let’s normalize p2 = 1. Then the competitive equilibrium prices and allocations are:
(p1, p2) =
y2 βy1
,
1
and (c1, c2) = (y1, y2).
• Solving for a competitive equilibrium (c.e.) — central planning problem A central planner maximizes the welfare of all agents subject to the economy’s resource constraint. Since all agents are identical in the current model, we can restrict our attention to the welfare of a representative agent.
II. Two-Period Economy Models
(I) A Deterministic Endowment Economy Assumptions:
• Environment – no uncertainty (deterministic) – no money (real model) – 2 periods – a large number of identical consumers – exchange economy – nonstorable endowment – competitive markets
The First Order Necessary Conditions (FONC) are:
∂L ∂c1
=
1 c1
−
λp1
=
0
∂L ∂c2
=
β c2
−
λp2
=
0
∂L ∂λ
=
p1y1 + p2y2 − p1c1 − p2c2 = 0
Eliminating λ we get intertemporal marginal rate of substitution IMRS =
5
– The central planner’s problem must also be modified to take into account production:
max u(c1, c2) s.t. c1 + k ≤ y1 c2 ≤ y2 + (1 − δ)k + f (k)
• Example: Solving for a c.e. with production — central planning problem. Consider an economy where
– The representative consumer’s problem:
max u(c1, c2) = log c1 + β log c2 s.t. p1c1 + p2c2 = p1y1 + p2y2 The Lagrangian is:
L = log c1 + β log c2 + λ[p1y1 + p2y2 − p1c1 − p2c2]
• Representative consumer has preferences represented by the utility function u(c1, c2). – ct is period t consumption, t = 1, 2 – u(·) is concave
• Representative consumer receives endowment stream (y1, y2). • Representative consumer’s budget constraint is
price with the IMRS (Euler equation), evaluated at the equilibrium consump-
tion allocation:
p1 p2
=
1/c1 β/c2
=
y2 βy1
.
– Again, normalizing p2 = 1, we get the c.e.:
• Assumptions about f (k):
4
– f (0) = 0 – f > 0 and f < 0 (production function is increasing and concave) – Inada conditions:
lim f (k) = ∞ and lim f (k) = 0.
c1
=y−k
=
1 1 + αβ
y
c2 = kα =
αβ 1 + αβ
y
α
.
Finally, we find equilibrium prices:
6
– Just as in the endowment economy, the consumer’s optimization problem
generates an Euler equation stating that relative price equals the IMRS:
(p1,
p2)
=
(
y2 βy1
,
1)
and
(c1, c2) = (y1, y2).
(II) A Deterministic Production Economy
• Environment: – no uncertainty – no money (real model) – 2 periods – a large number of identical consumers and firms – number of consumers = number of firms – production economy (production occurs in period 2) – consumers own firms and capital stock (by holding shares)
– the consumer maximizes utility subject to his budget constraint, given prices; – prices clear market: c1 = y1, and c2 = y2. • Solving for a competitive equilibrium (c.e.)
– Representative firm’s optimization problem:
max
k
P R = p2[f (k) + (1 − δ)k] − qk
FONC:
Hale Waihona Puke ∂P R ∂k=p2[f
(k)
+
1
−
δ]
−
q
=
0.
– Note that in equilibrium, we must have q = p1. Otherwise, if q > p1, x → ∞; if q < p1, x → −∞.