Lecture2

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.
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
+
p2 p1
y2
.
Therefore,
p1 p2
=
y2 βy1
.
Connecting the equilibrium relative price in the consumption functions yields
the equilibrium consumption allocations:
c1 = y1,
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.
• 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
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
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
– Representative firm’s optimization problem:
max
k
P R = p2[f (k) + (1 − δ)k] − qk
FONC:
∂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 → −∞.
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
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
• 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
consumer’s optimization
⇒
p1 p2
=
I M RS
relative prices:
1/c1 β/c2
=
p1 p2
(an E Euler equation for c2 and use the resulting expression to
substitute out c2 from the budget constraint. This yields the period-1 con-
(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 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]
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
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