ECON4310: Macroeconomics
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ECON4310: Macroeconomics
Espen Henriksen
September 10, 2008
ECON4310: Macroeconomics
Introduction Partial and general equilibrium 2/31
General and partial equilibrium
General equilibrium:
1 Decision makers (indviduals, firms, ...) take prices as given
and decide on quantities
2 Markets clear, i.e. “for given decisions over quantities what
must prices be for supply to equal demand.”
Partial equilibrium:
Decision makers (indviduals, firms, ...) take prices as given
and decide on quantities
State variables:
In general equilibrium, the state is fully described by the
quantities – i.e. prices are never state variables
In partial equilibrium, prices might be state variables
ECON4310: Macroeconomics
Introduction Indirect utility 3/31
Indirect utility and profit functions
The logic of the indirect utility function or a reduced form
profit function generated by an optimizing firm underlies
dynamic programming.
This static problem ignores dynamics – savings, spending on
durable goods, uncertainty about the future
These are all important when individuals make decisions – and
material we will study in this class
We will start by reviewing the indirect utility function
ECON4310: Macroeconomics
Introduction Indirect utility 4/31
Indirect utility function
v (p, m) = max u (x)
subject to
px = m
where p is a vector of prices, x is a vector of consumption goods,
and m is income.
The first order condition is given by
uj (x)
pj
= λ for all j
where λ is the shadow price of the budget constraint and uj (x) is
the marginal utility from good j.
ECON4310: Macroeconomics
Introduction Indirect utility 5/31
v (p, m) is the maximized utility from the current state (p, m).
Someone in this state can be predicted to attain this utility
One does not need to know what that person will do with his
income; it’s enough to know what his income is and what the
prices he will face.
This powerful logic underlies dynamic programming.
ECON4310: Macroeconomics
Introduction Indirect utility 6/31
Indirect profit function
π (w, p, k) = max
l
pf (k, l) − wl
A labor demand function depends on (w, p, k)
As with v (p, m), π (w, p, k) summarizes the value of the firm given
factor prices w, the product price p and the stock of capital k
Not specified where k, p, w, m come from. In general equilibrium:
k control (decision) variable and endogenous state variable
p and w are never state variables in general equilibrium –
prices clear markets.
m might follow some endogenous or exogenous process
ECON4310: Macroeconomics
Introduction Overview 7/31
Building the toolbox
A B
C D
No uncertainty Uncertainty
No decisions
Decisions
A : Solow growth model
B : Value iteration on a Markov process
C : Ramsey growth model
D : Stochastic neoclassical growth model
ECON4310: Macroeconomics
Introduction Overview 8/31
First time
Product approach to output: yt = ¯zf (kt, ¯nt) .
Income approach to output: yt = rtkt + wt¯nt.
Expenditure approach to output: yt = ct + xt.
Law of motion for capital: kt+1 = (1 − δ)kt + xt.
Behavioral assumption:
it = σyt
ECON4310: Macroeconomics
Introduction Overview 9/31
This time
max
{ct,xt}∞
t=0
∞
t=0
βt
u (ct, 1 − ¯nt)
subject to
Product approach to output: yt = ¯zf (kt, ¯nt) .
Income approach to output: yt = rtkt + wt¯nt.
Expenditure approach to output: yt = ct + xt.
Law of motion for capital: kt+1 = (1 − δ)kt + xt.
ECON4310: Macroeconomics
Introduction Overview 10/31
Next time
max
{ct,xt}∞
t=0
E0
∞
t=0
βt
u (ct, 1 − ¯nt)
subject to
Product approach to output: yt = ztf (kt, ¯nt) .
Income approach to output: yt = rtkt + wt¯nt.
Expenditure approach to output: yt = ct + xt.
Law of motion for capital: kt+1 = (1 − δ)kt + xt.
Total factor productivity: zt = ρzt−1 + εt, ε ∼ N(0, σ2
),
estimated from {zt}T
t=0 ln zt = ln yt − α ln kt − (1 − α) ln nt.
ECON4310: Macroeconomics
Introduction Overview 11/31
The time after that
max
{ct,xt}∞
t=0
E0
∞
t=0
βt
u (ct, 1 − nt)
subject to
Product approach to output: yt = ztf (kt, nt) .
Income approach to output: yt = rtkt + wtnt.
Expenditure approach to output: yt = ct + xt.
Law of motion for capital: kt+1 = (1 − δ)kt + xt.
Total factor productivity: zt = ρzt−1 + εt, ε ∼ N(0, σ2
),
estimated from {zt}T
t=0 ln zt = ln yt − α ln kt − (1 − α) ln nt.
ECON4310: Macroeconomics
Ramsey model 12/31
Class objectives today
Solve the Ramsey-Cass-Koopmans model
(decisions, but no uncertainty)
Developed by Ramsey (1928), Cass (1965), Koopmans (1965).
Avoids all market imperfections and all issues raised by
heterogenous households and links between generations.
More specifically, we will solve
The sequential problem, abstracting from labor-leisure choice
and uncertainty.
First in finite time, then with infinite horizon.
Give a recursive formulation of the same problem
ECON4310: Macroeconomics
Ramsey model 13/31
The Ramsey optimal savings problem
Mathematical strategy:
Maximizing a function (total utility) of an infinity of variables
(consumption and capital at each date) subject to
technological constraints.
Set up the problem in continuous time (we’ll solve it in
discrete time) and characterized the utility-maximizing
dynamics.
Under suitable assumptions, capital stock should converge
monotonically to the level that, if sustained, maximizes
consumption per unit of time.
Key: Solve for a sequence of allocations.
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Ramsey model 14/31
Finite time, social planner’s problem
max
{ct,xt}T
t=0
T
t=0
βt
u (ct, 1 − ¯nt)
subject to
Product approach to output: yt = ¯zf (kt, ¯nt) .
Expenditure approach to output: yt = ct + xt.
Law of motion for capital: kt+1 = (1 − δ)kt + xt.
with
k0 > 0, kt+1 ≥ 0, ct > 0.
ECON4310: Macroeconomics
Ramsey model 15/31
Simplifying
max
{ct,xt}T
t=0
T
t=0
βt
u (ct)
subject to
ct + kt+1 ≤ f (kt) + (1 − δ)kt
with
k0 > 0, kt+1 ≥ 0, ct > 0.
ECON4310: Macroeconomics
Ramsey model Finite time 16/31
Lagrangian / Kuhn-Tucker
Solve:
L =
T
t=0
βt
[u (ct) − λt [ct + kt+1 − f (kt) − (1 − δ) kt] + µtkt+1]
First-order conditions are:
∂L
∂ct
= βt
[u′
(ct) − λt] = 0, t = 0, . . . , T
∂L
∂kt+1
= −βt
λt + βt
µt + βt+1
λt+1 [1 + f′
(kt+1) − δ] = 0, t = 0, . . . , T − 1
∂L
∂kT +1
= −βT
λT + βT
µT = 0, t = T
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Ramsey model Finite time 17/31
Simplifying
First-order conditions are:
∂L
∂ct
= u′
(ct) − λt = 0, t = 0, . . . , T
∂L
∂kt+1
= −λt + µt + βλt+1 1 + f′
(kt+1) − δ = 0, t = 0, . . . , T − 1
∂L
∂kT+1
= −λT + µT = 0, t = T
with complementary slackness conditions
µtkt+1 = 0
λt [ct + kt+1 − f (kt) − (1 − δ) kt] = 0
λt, kt+1, µt ≥ 0
ECON4310: Macroeconomics
Ramsey model Finite time 18/31
u′
(ct) > 0 ∀t
⇒ λt > 0 ∀t
⇒ µT > 0 t = T
⇒ kT+1 = 0 t = T
k0 > 0, t = 0
⇒ kt+1 > 0, t = 0, . . . , T − 1
⇒ µt = 0, t = 0, . . . , T − 1
⇒ −λt + βλt+1 1 + f′
(kt+1) − δ = 0, t = 0, . . . , T − 1
⇒ −u′
(ct) + βu′
(ct+1) 1 + f′
(kt+1) − δ = 0, t = 0, . . . , T − 1
⇒ β
u′ (ct+1)
u′ (ct)
1 + f′
(kt+1) − δ = 1, t = 0, . . . , T − 1
ECON4310: Macroeconomics
Ramsey model Finite time 19/31
Identified system of equations
T + 2 unknown variables
{kt}T+1
t=0
T + 2 equations
k0 given
β
u′ (kt+2 − f (kt+1) − (1 − δ) kt+1)
u′ (kt+1 − f (kt) − (1 − δ) kt)
1 + f′
(kt+1) − δ = 1
kT+1 = 0
Since the number of unknowns is equal to the number of
equations, the difference equation system will
typically have a solution (existence),
and under appropriate assumptions on primitives,
there will be only one such solution (uniqueness).
ECON4310: Macroeconomics
Ramsey model Finite time 20/31
Conditions
A brief look at the conditions under which there is only one
solution to the first-order conditions – or, alternatively, under
which the first-order conditions are sufficient.
u is concave.
Since the sum of concave functions is concave it follows that
U = T
t=0 u(ct) is concave in the vector {ct}T
t=0.
f is concave.
From the definitions of a convex set and a concave function it
follows that the constraint set is convex in {ct, kt+1}T
t=0.
So, concavity of the functions u and f makes the overall
objective concave and the choice set convex, and thus the
first-order conditions are sufficient.
ECON4310: Macroeconomics
Ramsey model Finite time 21/31
The Euler Equation
βRt+1
u′(ct+1)
u′(ct)
= 1
where for the neoclassical growth model
Rt+1 = 1 + f′
(kt+1) − δ.
u′
(ct)
marginal disutility of saving one more unit
= β
discount factor between two periods rate of ret
ECON4310: Macroeconomics
Ramsey model Finite time 22/31
Determinants of saving
1 The the concavity of utility / consumption “smoothing”:
if the utility function is strictly concave, the individual prefers
a smooth consumption stream.
2 Discounting / Impatience:
via β, we see that a low β (a low discount factor, or a high
discount rate) will tend to be associated with low ct+1’s and
high ct’s.
3 The return to savings: f′(kt+1) − δ clearly also affects
behavior, but its effect on consumption cannot be signed
unless we make more specific assumptions. Moreover, kt+1 is
endogenous, so when f′ nontrivially depends on it, we cannot
vary the return independently.
ECON4310: Macroeconomics
Ramsey model Infinite horizon 23/31
Infinite horizon
Models with an infinite time horizon demand more advanced
mathematical tools.
Consumers in our models are now choosing infinite sequences.
These are no longer elements of Euclidean space Rn, which
was used for our finite-horizon case.
A basic question is when solutions to a given problem exist.
Suppose we are seeking to maximize a function
Several issues arise.
How do we define continuity in this setup?
What is an open set?
What does compactness mean?
We will not answer these questions here.
ECON4310: Macroeconomics
Ramsey model Infinite horizon 24/31
Maximization of utility under an infinite horizon
Will mostly involve the same mathematical techniques as in
the finite-horizon case.
In particular, (Kuhn-Tucker) first-order conditions (Euler
equation): barring corner constraints; choosing a path,
defined for any k0, such that the marginal effect of any choice
variable on utility is zero.
In finite time horizon, the final zero capital condition was key
to determining the optimal path of capital: it provided us with
a terminal condition for a difference equation system.
In the infinite time case, there is no such final T: the
economy will continue forever. Therefore, the difference
equation that characterizes the first-order condition may have
an infinite number of solutions.
ECON4310: Macroeconomics
Ramsey model Infinite horizon 25/31
Transversality condition
Need some other way of pinning down the households’ choice.
Turns out that the missing condition is analogous to the
requirement that the capita stock be zero at T + 1
The missing condition is called the transversality condition –
typically, a necessary condition for an optimum.
lim
t→∞
βt
λtkt+1 = 0.
It expresses the following idea:
It cannot be optimal for the household to choose a capital
sequence such that, in present-value utility terms, the shadow
value of kt remains positive as t goes to infinity. This could
not be optimal because it would represent saving too much: a
reduction in saving would still be feasible and would increase
utility.
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Ramsey model Recursive formulation 26/31
Recursive formulation
Alternative mathematical strategy:
Seek the optimal savings/investment function directly and
then to use this function to compute the optimal sequence of
investments from any initial capital stock.
This way of looking at the problem – decide on the immediate
action to take as a function of the current situation – is called
a recursive formulation.
It exploits the observation that a decision problem of the same
structure recurs every period.
ECON4310: Macroeconomics
Ramsey model Recursive formulation 27/31
Recursive formulation
v(ks) ≡ max
{ct,kt+1}∞
t=s
∞
t=s
βt−s
u (ct)
subject to
ct + kt+1 ≤ f (kt) + (1 − δ) kt ∀t ≥ s
v(kt) = max
kt+1
{u (ct) + βv (kt+1)}
Solution characterized by decision rules/functions
kt+1 = g(kt)
ct = f (kt) + (1 − δ) − g(kt)
ECON4310: Macroeconomics
Ramsey model Recursive formulation 28/31
A few concepts
1 A functional equation is an equation in terms of
independent variables, and also unknown functions, which are
to be solved for.
Usually the term functional equation is reserved for equations
that are not in some simple sense reducible to algebraic
equations.
Simple examples. Which functions satisfy the following
equations?
f(xy) = f(x) + f(y) ?
f(x + y) = f(x)f(y) ?
ECON4310: Macroeconomics
Ramsey model Recursive formulation 29/31
A few concepts cont’d
2 State variable(s) – the set of variables that are sufficient to
summarize all information (past and present) needed to solve
the forward looking optimization problem. We distinguish
between endogenous and exogenous state variables.
3 Control variable(s) – the variable(s) that is/are chosen.
4 Stationarity – essential for recursive programming is that all
relations can be expressed without an indication of time. In a
given state the optimal choice of the agent will be the same
regardless of “when” he optimizes.
5 Structural parameters (“the greeks”) and endogenous
variables (“the latins”) – essentially, what is ”the structure of
the world”, i.e. independent of choices, and what is
determined by peoples choices.
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Ramsey model Uncovering the value function 30/31
Contraction mapping
In general, v(·) is unknown, but the Bellman equation can be used
to find it.
In most of the cases we will deal with, the Bellman equation
satisfies a contraction mapping theorem, which implies that
1 There is a unique function v(·) which satisfies the Bellman
equation.
2 If we begin with any initial function v0(k) and define
vi+1(k) = max
c,k′
[u(c) + vi(k′
)]
subject to
c + k′
= f(k) + (1 − d)k
for i = 0, 1, 2, . . . then limi→∞ vi+1(k) = v(k).
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Ramsey model Uncovering the value function 31/31
How to find the true value function
The above two implications give us two alternative means of
uncovering the value function.
1 Implication 1 above, if we are fortunate enough to correctly
guess the value function v(·); then we can simply plug v(kt+1)
into the right side and then verify that v(kt) solves the
Bellman equation. This procedure only works in a few cases.
2 Implication 2 above is useful for doing numerical work. One
approach is to find an approximation to the value function
and iterate.