Lecture 7:

Competitive Equilibrium

Patrick Macnamara
ECON80301: Advanced Macroeconomics
University of Manchester

Fall 2024
Today’s Lecture

Two main topics today:

• Competitive equilibrium
• Solving for the transition between two steady states of the NGM

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Competitive Equilibrium

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Competitive Equilibrium, 1/2
• We are interested in the allocations that actually arise when firms and
consumers interact. They may not necessarily coincide with the Social
Planner’s Problem (SPP).
• Multiple competitive equilibrium (CE) concepts:
• Arrow Debreu Equilibrium (ADE)
• Sequential Market Equilibrium (SME)
• Recursive Competitive Equilibrium (RCE)
• Welfare theorems apply in the case of the NGM

SPP ⇔ ADE ⇔ SME ⇔ RCE

• In practice, solving SPP much easier than solving for CE.
• Solve SPP, then use resulting allocation to back out prices which support the
CE.
• In the heterogeneous agent models coming later in this course, we will be
breaking the link between SPP and CE. Thus, we’ll have no choice but to solve
for the CE directly.
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Competitive Equilibrium, 2/2

I will discuss these competitive equilibrium concepts in the context of two models:
1. The deterministic NGM
• No risk
• I’ll discuss the SPP, ADE, SME and RCE concepts
• This will help set the stage for the transition between steady states
(coming later in this lecture).
2. A basic endowment economy with risk
• This model will feature complete financial markets.
• I will discuss the SPP, ADE and SME concepts
• This will help set the stage for next week when we start discussing incomplete
markets models with heterogeneous agents.

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Competitive Equilibrium
No Risk

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Deterministic NGM: Social Planner’s Problem

• We’ve already seen this. Let F (kt , nt ) be the production function and define
f (kt ) = F (kt , 1) + (1 − δ)kt .
• In sequential form:
∞
β t u (f (kt ) − kt+1 ) 0 ≤ kt+1 ≤ f (kt )
X
max∞ subject to
{kt+1 }t=0
t=0 k0 given
• In recursive form:

V (k) = max {u (f (k) − k ′ ) + βV (k ′ )}

k ′ ∈[0,f (k)]

• The solution is efficient, by construction.

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Arrow Debreu Equilibrium (ADE)

Arrow-Debreu market structure:

• Single market that takes place at time 0.
• At date 0, goods for all future dates are traded.
• After market closes, agents just carry out trades
agreed upon in period 0.
• All contracts perfectly enforceable.
Assumptions:
• Households own capital stock, supply labor and capital to a representative firm.
• Labor and capital markets are perfectly competitive.

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ADE: Prices

For each period, there are 3 goods that can be traded:

• Final output good, yt
pt is the price of the period-t final output good, quoted in period 0. Period 0
output is the numeraire, thus p0 = 1.
• Labor services, nt
wt is the real wage in period t, quoted in period 0, in terms of the period-t
output good.
pt wt is the price in terms of the period-0 output good.
• Capital services, kt
rt is the rental price of capital, quoted in terms of the period-t output good.
pt rt is the price in terms of the period-0 output good.

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ADE: Firms

Given prices {pt , wt , rt }∞

t=0 , representative firm solves

∞
X
π = max∞ pt (yt − rt kt − wt nt ) (1)
{kt ,nt }t=0
t=0

subject to

yt = F (kt , nt ) t ≥ 0
kt , nt ≥ 0

Note, there’s nothing dynamic about the firm’s problem.

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ADE: Households
Given prices {pt , wt , rt }∞
t=0 , representative household solves
∞
β t u(ct )
X
max (2)
{ct ,it ,xt+1 ,kt ,nt }∞
t=0 t=0

subject to
∞
X ∞
X
pt (ct + it ) ≤ pt (rt kt + wt nt ) + π
t=0 t=0
xt+1 = (1 − δ)xt + it ∀t ≥ 0
0 ≤ nt ≤ 1 ∀t ≥ 0
0 ≤ kt ≤ xt ∀t ≥ 0
ct , xt+1 ≥ 0 ∀t ≥ 0
x0 given

We are carefully distinguishing between the capital stock xt and capital services kt
supplied to firms.
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ADE: Definition of Equilibrium
An Arrow Debreu Equilibrium (ADE) consists of
(a) prices {pt , wt , rt }∞ t=0 ,
(b) allocations for the firm {ktd , ntd , yt }∞ t=0 , and
(c) allocations for the household {ct , it , xt+1 , kts , nts }∞
t=0
(d) initial value of capital, x0
such that
(1) Given prices {pt , wt , rt }∞ d d ∞
t=0 , the firm’s allocation {kt , nt , yt }t=0 solves the firm’s
problem in (1).
(2) Given x0 and prices {pt , wt , rt }∞ t=0 , the household’s allocation
s s ∞
{ct , it , xt+1 , kt , nt }t=0 solves the household’s problem in (2).
(3) Markets clear
yt = ct + it (Goods Market)
ntd = nts (Labor Market)
ktd = kts (Capital Services Market)
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Sequential Markets Equilibrium (SME)

• ADE is perhaps a bit unrealistic in that the market is only open at date 0.
• However, in a sequential market equilibrium (SME), markets are open every
period and the resulting allocation is exactly the same.

• SM market structure:
• Markets open every period.
• As in Arrow-Debreu, households own the capital stock, supply labor and capital
to a representative firm. Fixing the Output Good’s Price at One:
• In every period, the output good is the numeraire. This removes any nominal distortions and
allows focus on real variables like wages,
capital rental rates, and output

Only prices we have now are real wage (wt ) and real rental rate of capital (rt ).
numeraier Definition
A unit of exchange or product that serves as a benchmark for comparing the value of similar products or financial instruments

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 SME: Firms

Taking as given (wt , rt ), representative firm solves

max {F (kt , nt ) − wt nt − rt kt } (3)

kt ,nt

Firm’s problem is static, since households own the capital stock.

the firm does not decide on the accumulation or maintenance of
capital over time. The firm’s problem reduces to maximizing profits
given the rental rate of capital and wages each period, without needing
to consider how today’s choices will impact its future production capabilities. Type te

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SME: Households

Given prices {wt , rt }∞

t=0 , representative household solves

∞
β t u(ct )
X
max ∞ (4)
{ct ,kt+1 }t=0
t=0

subject to

ct + kt+1 − (1 − δ)kt = wt + rt kt ∀t
ct , kt+1 ≥ 0 ∀t
k0 given

Note, we are taking some shortcuts here. Implicitly, we are using nts = 1 and that
firm profits are zero.

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SME: Definition of Equilibrium
A sequential markets equilibrium (SME) consists of
(a) prices {wt , rt }∞
t=0 ,
(b) allocations for the firm, {ntd , ktd }∞
t=0 ,
(c) allocations for the household, {ct , kt+1s
}∞
t=0 , and
s
(d) initial value of capital, k0 = k0
such that
(1) For each t ≥ 0, given prices (wt , rt ), the firm’s allocation (ntd , ktd ) solves the
firm’s problem in (3).
(2) Given k0 and prices {wt , rt }∞ s ∞
t=0 , the household’s allocation {ct , kt+1 }t=0 solves
the household’s problem in (4).
(3) Markets clear for all t ≥ 0:
ntd = 1 (Labor Market)
ktd = kts (Capital Market)
F (ktd , ntd ) s
= ct + kt+1 − (1 − δ)kts (Goods Market)
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Recursive Competitive Equilibrium (RCE)

RCE Assumptions:
• The starting point for the RCE is typically the sequential formulation.
• To determine the state variables, we need to think about what minimal
information the household needs to solve its dynamic problem.
• Therefore, we will utilize the “Big K, little k” trick:
• k (little k) is the representative household’s stock of capital.
• K (big K) is the aggregate stock of capital.
• Of course, k = K , but since we’re solving for the competitive equilibrium, we
don’t want households to incorporate how their decisions affect K ′ tomorrow
and thus affect prices tomorrow.

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RCE: Firms

• Taking as given (w , r ), firms solve

max {F (k, n) − wn − rk}

k,n

• Firm’s FOC:

w = Fn (k, n)
r = Fk (k, n)

• Firm’s problem is so simple that we can use the FOCs to define the prices as a
function of the aggregate state:

w (K ) = Fn (K , 1)
r (K ) = Fk (K , 1)

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RCE: Households

• Household’s problem:

V (k, K ) = max′ {u(c) + βV (k ′ , K ′ )} (5)

c,k

subject to

c + k ′ = w (K ) + (1 − δ + r (K ))k
K ′ = Λ(K )

where K ′ = Λ(K ) is the law of motion for aggregate capital. Households need
to know this in order to forecast future prices.

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RCE: Definition of Equilibrium
A recursive competitive equilibrium (RCE) is defined as
(a) the value function V (k, K ),
(b) policy functions c(k, K ), g(k, K ),
(c) pricing functions w (K ), r (K ), and
(d) aggregate law of motion Λ(K )
such that
(1) Given w (K ), r (K ), Λ(K ), the value function V (k, K ) solves the household’s
Bellman equation in (5) and c(k, K ) and g(k, K ) are the associated policy
functions.
(2) The pricing functions are w (K ) = Fn (K , 1), r (K ) = Fk (K , 1)
(3) Markets clear
c(K , K ) + g(K , K ) = F (K , 1) + (1 − δ)K
(4) The law of motion Λ(K ) is consistent with household choices:
Λ(K ) = g(K , K )

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Competitive Equilibrium
With Risk

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Competitive Equilibrium in Model with Risk

• Next week, we will start studying Bewley models

(e.g., Bewley (1986), Huggett (1993), Aiyagari (1994)).
These are heterogeneous agent models with idiosyncratic uncertainty and
incomplete financial markets.

• To set the stage for these models, let’s discuss the ADE and SME equilibrium
concepts in a model with risk. This model will feature complete financial
markets.
• Welfare theorems will apply and risk will be perfectly shared across households.
Type text here
• Distribution of income across households will be irrelevant.

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Risk Model: Representation of Risk

• Let st ∈ S be a particular realization of an event at date t, where

S = {η1 , . . . , ηN } is the set of possible events that can happen in period t.
• S is finite and same for all periods.
• WLOG, let’s suppose ηj = j.

• Let s t = (s0 , s1 , . . . , st ) denote an event history, containing all event

realizations up to period t.
• Let S t denote the set of possible event histories of length t.
• Let πt (s t ) denote the probability of a event history s t .
Assume πt (s t ) > 0 for all s t .

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Risk Model: Households

• Two agents, indexed by i = 1, 2.

• Agent i has an endowment, eti (s t ), which depends on s t .
• Example:
( (
2 if st = 1 0 if st = 1
et1 (s t ) = et2 (s t ) =
0 if st = 2 2 if st = 2

• Each household has the same preferences:

∞ X
β t πt (s t )u(cti (s t ))
X

t=0 s t ∈S t

where consumption of household i, cti (s t ), is indexed not only by time but also
the realized event history.

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Risk Model: Arrow-Debreu Market Structure

• All trade takes place at period 0, before any risk has been realized (including s0 ).
• Prices are indexed by event histories, in addition to time.
pt (s t ) is the price of one unit of consumption, quoted in period 0, delivered in
period t iff event history s t has been realized.
• This is the key insight of Arrow-Debreu.
Consumption at the same date, but for different event histories, can be thought
of as different commodities.

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Risk Model: Definition of ADE

An Arrow-Debreu equilibrium (ADE) consists of prices {p̂t (s t )} and

allocations {ĉt1 (s t )}, {ĉt2 (s t )} such that
1. Given {p̂t (s t )}, for i = 1, 2, {ĉti (s t )} solves
∞ X
β t πt (s t )u(cti (s t ))
X
max
i
{ct (s t )} t=0 s t ∈S t

subject to
∞ X ∞ X
p̂t (s t )cti (s t ) ≤ p̂t (s t )eti (s t )
X X

t=0 s t ∈S t t=0 s t ∈S t

2. Markets clear

ĉt1 (s t ) + ĉt2 (s t ) = et1 (s t ) + et2 (s t ) ∀t ≥ 0, ∀s t ∈ S t

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Risk Model: Sequential Markets Structure

• Trade takes place sequentially in each period and event-history.

• We need to introduce Arrow securities, contingent claims that pay out only
for a particular realization of st+1 tomorrow.
• Let qt (s t , st+1 ) be the price in period t, given history s t , of a contract that pays
out one unit of consumption in period t + 1 iff tomorrow’s event is st+1 .
• Let at+1
i (s t , st+1 ) denote the quantities of these securities bought (or sold) at
period t by agent i.
• Period-t budget constraint of agent i, given event history s t :

cti (s t ) + qt (s t , st+1 )at+1

i
(s t , st+1 ) ≤ eti (s t ) + ati (s t )
X

st+1 ∈S

We assume a0i (s0 ) = 0 for all s0 ∈ S.

By purchasing a portfolio of Arrow securities, households can insure themselves
against unfavorable states or benefit from favorable ones, allowing for optimal
consumption in each state.

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Risk Model: Definition of SME
A sequential markets equilibrium (SME) consists of prices {q̂t (s t , st+1 )} and
allocations {ĉti (s t ), {ât+1
i
(s t , st+1 )}} for i = 1, 2 such that
(1) Given {q̂t (s t , st+1 )}, {ĉti (s t ), {ât+1
i
(s t , st+1 )}}, for i = 1, 2, solves
∞ X
β t πt (s t )u(cti (s t ))
X
max
t=0 s t ∈S t

subject to

cti (s t ) + q̂t (s t , st+1 )at+1

i
(s t , st+1 ) ≤ eti (s t ) + ati (s t )
X

st+1 ∈S
i
at+1 (s t , st+1 ) ≥ −Āi (Natural borrowing limit)

(2) Markets clear:

ĉt1 (s t ) + ĉt2 (s t ) = et1 (s t ) + et2 (s t ) ∀t, ∀s t ∈ S t

1
ât+1 (s t , st+1 ) + ât+1
2
(s t , st+1 ) = 0 ∀t, ∀s t ∈ S t , ∀st+1 ∈ S
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Risk Model: Social Planner’s Problem

• The social planner maximizes the weighted sum of the utility of the two agents,
subject to the allocation being feasible.
∞ X h i
β t πt (s t ) αu(ct1 (s t )) + (1 − α)u(ct2 (s t ))
X
max2
{ct1 (s t )},{ct (s t )} t=0 s t ∈S t

subject to

ct1 (s t ) + ct2 (s t ) = et1 (s t ) + et2 (s t ) ∀t, ∀s t ∈ S t

where α is a Pareto weight.

• Solution to this problem gives us Pareto efficient allocations.
• Competitive equilibrium allocation corresponds to one particular value of α.

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Risk Model: Negishi’s Method

It’s easier to solve the SPP and then use Negishi’s method to solve for the prices
that support the efficient allocation, rather than solving for the CE directly.
Negishi’s method:
Step 1 Solve the social planner’s problem, to obtain efficient allocations (indexed by
the Pareto weight α).
Step 2 Compute transfers, indexed by α, necessary to make the efficient allocation
affordable. Use the Lagrange multipliers from the SPP as prices.1
Step 3 Find the Pareto weight, α̂, that makes the transfer functions equal to zero.
Step 4 The efficient allocation corresponding to α̂ is the equilibrium allocation. The
prices are the SPP Lagrange multipliers.

1
The price p0 (s0 ) is normally normalized to 1 for some state s0 , but the planner’s problem may
not be set up to guarantee this. It doesn’t matter, but you can divide the Lagrange multiplier in
the planner’s problem by some constant to ensure that p0 (s0 ) = 1.
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Risk Model: Solving the SPP, 1/2
• FOC for the social planner’s problem:

λt (s t ) = β t πt (s t )αu ′ (ct1 (s t ))
λt (s t ) = β t πt (s t )(1 − α)u ′ (ct2 (s t ))

where λt (s t ) = Lagrange multiplier on the resource constraint. SPP FOC

• This implies

u ′ (ct1 (s t )) 1−α
′ 2 t
=
u (ct (s )) α

The ratio of marginal utilities between the two agents is constant over time and
across all states of the world.
• If we make the usual assumptions about u(c), then ct1 (s t )/ct2 (s t ) is constant
over time and across all states.2
2
The usual assumptions are u(c) is increasing, twice continuously differentiable, a strictly
concave function, and satisfies the Inada conditions.
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Risk Model: Solving the SPP, 2/2
• Let’s assume u(c) = ln c. Then:

ct2 (s t ) 1−α pareto weight

1 t
=
ct (s ) α

Define et (s t ) = et1 (s t ) + et2 (s t ) as the aggregate endowment. Use the resource

constraint, ct1 (s t ) + ct2 (s t ) = et (s t ), to get:
resource constraint

ct1 (s t ) = αet (s t ) ct2 (s t ) = (1 − α)et (s t )

• Endowment risk is perfectly shared. The only endowment risk that affects
consumption is aggregate risk. Consumption doesn’t depend on the specific
history s t .
• Since u ′ (c) = 1/c, Lagrange multiplier is then

λt (s t ) = β t πt (s t )αu ′ (ct1 (s t )) = β t πt (s t )/et (s t )

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Risk Model: Solving for the CE, 1/3

• FOC for the social planner’s problem:

λt (s t ) = αβ t πt (s t )u ′ (ct1 (s t )) = (1 − α)β t πt (s t )u ′ (ct2 (s t ))

• FOC for Arrow-Debreu:

µi pt (s t ) = β t πt (s t )u ′ (cti (s t ))

where µi is the Lagrange multiplier on household i’s BC. ADE FOC

• If we set pt (s t ) = λt (s t ), µ1 = 1/α, µ2 = 1/(1 − α), the FOC are identical.

• Notice that pt (s t ) = λt (s t ) = β t πt (s t )/et (s t ). Prices only depend on et (s t ).
Distribution of income is irrelevant.
• We can then solve for the α that coincides with the competitive equilibrium.

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Risk Model: Solving for the CE, 2/3
• Recall the household’s budget constraint in Arrow-Debreu:
∞ X h i
pt (s t ) cti (s t ) − eti (s t ) = 0
X

t=0 s t ∈S t

• Set pt (s t ) = λt (s t ) and compute transfers required to make the Pareto efficient

allocation affordable for household i:
∞ X
β t πt (s t ) h i
cti (s t ; α) − eti (s t )
X
ti (α) =
t=0 s t ∈S t et (s t )
| {z }
pt (s t )
Fixing the Output Good’s Price at One:

• Solve for that α̂ such that ti (α̂) = 0 for i = 1, 2. Using ct1 (s t ; α) = αet (s t ):
∞ X
et1 (s t )
β t πt (s t )
X
α̂ = (1 − β)
t=0 s t ∈S t et (s t )

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Risk Model: Solving for the CE, 3/3

• FOC for SME problem:

γti (s t ) = β t πt (s t )u ′ (cti (s t ))
qt (s t , st+1 )γti (s t ) = γt+1
i
(s t+1 )

where γti (s t ) is the Lagrange multiplier on household i’s BC. Note that the
natural borrowing limit won’t be binding. SME FOC

• Notice that γt (s ) = µi pt (s ) at the efficient solution. The second FOC then

i t t

implies lagrangian multiplier

proportional to price, one i have solution to AD, i have ??
pt+1 (s t+1 ) Fixing the Output Good’s Price at One:
qt (s t , st+1 ) =
pt (s t )
Using the equilibrium allocation we can back out sequential market prices.

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Risk Model: Summary

• Key results come from the assumption of complete markets.

• These equilibrium concepts tell us what exactly is needed in order to get full risk
sharing: a full set of Arrow securities.
• Don’t need to solve for CE directly. We can solve the SPP, then back out prices
which support the efficient allocation from the SPP.

• Next week, we will consider models with incomplete markets.

• Consumers will only have access to one non-contingent security.
• There will be imperfect risk-sharing and now heterogeneity will matter for prices.
• Resulting equilibrium will not be efficient. We must solve for the CE directly
(this is harder).

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Transition Between Steady States

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Transition Between Steady States

• Last week, we introduced aggregate shocks into the NGM. However, there is an
intermediate case we can consider: the transition between two steady states of
the deterministic NGM.
• It is sometimes easier to solve for a transition path than a model with aggregate
shocks.
• Example: suppose there is an unexpected and permanent increase in A at date
t0 in the NGM, where F (k) = Ak α .
What is the path of capital along the transition?
We will consider two cases:
(1) Transition between two steady states of the social planner’s problem
(this is easy)
(2) Transition between two steady states of the competitive equilibrium
(this is harder)

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Example: Deterministic NGM

k*2

Capital, k t

k*1 impossible path

actual path

t0
Time, t

Initial steady state (k1∗ ) and final steady state (k2∗ ) are easy to determine. But how
do we connect the two steady states?
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Example: Deterministic NGM (SPP)

new g(k)

k'=g(k) k*2

old g(k)

k*1

old policy
new policy
next period capital
k*1 k*2
k

Easy to use new policy function to trace out path for capital. Notice there will be no
overshooting as we converge.
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Transition Between Steady States (SPP)

• Given initial k1∗ , use new policy function g(k) to compute the path for kt along
the transition to the new steady state:

kt = k1∗ (increase in A at date t)

kt+1 = g(kt ) (use new policy function)
kt+2 = g(kt+1 )
..
.

• This is easy because this is the social planner’s version of the NGM (there are
no prices). The solution is efficient.
• It’s much harder to solve for the competitive equilibrium in which agents
forecast the future evolution of prices.

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Transition Between Steady States (CE), 1/2

Solving for the transition path in the competitive equilibrium (CE) case is harder.
Step 1 Given a shock in period 0, suppose that at some date T in the future, the
economy has converged to the new steady state. Set T sufficiently large.3
Step 2 Solve for the initial and final steady state levels of capital (K0 and KT ).4
Step 3 Start with a guess for the transition path (K0 , K1 , . . . , KT ) where K0 is our
initial steady state and KT is our final steady state.
Step 4 Given guess (K0 , K1 , . . . , KT ), solve for the household’s optimal
(k0 , k1 , . . . , kT −1 , kT ), imposing that k0 = K0 and kT = KT .

3
But do not set it too large either, as that will make convergence more difficult, at least in the
NGM. T = 20 should be sufficient in the NGM.
4
Often, we have to solve for these. In the NGM, we can compute these directly using either
1/β = 1 − δ + r (kss ) or 1/β = 1 − δ + F ′ (kss ).
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Transition Between Steady States (CE), 2/2

Step 5 If kt is sufficiently close to Kt for t = 0, . . . , T , stop.

Otherwise, return to Step 4 with a new guess.
Construct the new guess, Ktn+1 , as a convex combination of kt and the old
guess Ktn :
Ktn+1 = ωkt + (1 − ω)Ktn t = 0, . . . , T
Since convergence is not guaranteed here, we need to use a small ω
(e.g., ω = 0.01).

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Transition: Solving for k0 , k1 , . . . , kT , 1/3
Given a guess for the path of aggregate capital, (K0 , K1 , . . . , KT ), how do we solve
for the household’s optimal (k0 , k1 , . . . , kT )?
• Given aggregate capital K , we can compute prices:

w (K ) = Fn (K , 1) = (1 − α)AK α
r (K ) = Fk (K , 1) = αAK α−1

• Use Euler Equation and impose k0 = K0 and kT = KT :

u ′ (ct ) = βu ′ (ct+1 ) [1 − δ + r (Kt+1 )] t = 0, . . . , T − 2

k0 = K0
kT = KT

where ct = w (Kt ) + (1 − δ + r (Kt ))kt − kt+1

Solve the system of T + 1 equations for k0 , . . . , kT .

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Transition: Solving for k0 , k1 , . . . , kT , 2/3
• When u(c) = ln c, the T + 1 equations can be written as a linear system of
equations.
• Re-write Euler Equation:

u ′ (ct ) = βu ′ (ct+1 ) [1 − δ + r (Kt+1 )]

1 β [1 − δ + r (Kt+1 )]
=
ct ct+1
1 βRt+1
= (6)
wt + Rt kt − kt+1 wt+1 + Rt+1 kt+1 − kt+2

where wt ≡ w (Kt ) and Rt ≡ 1 − δ + r (Kt ).

• Re-write (6) as a second-order difference equation:

βRt Rt+1 kt − (1 + β)Rt+1 kt+1 − kt+2 = wt+1 − βRt+1 wt

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Transition: Solving for k0 , k1 , . . . , kT , 3/3

• To summarize, we need to solve the following system of T + 1 equations:

βRt Rt+1 kt − (1 + β)Rt+1 kt+1 − kt+2 = wt+1 − βRt+1 wt

for t = 0, . . . , T − 2
k0 = K0
kT = KT

• Write these equations as a linear system of equations

Ax = d

and then solve for x = [k0 , k1 , . . . , kT ]′ using x = A−1 d (very easy in Matlab).

46 / 48
Example: NGM Transition Path (CE)
iter
iter
iter
iter====141,
101,
111,
121,
131,
11,
21,
31,
41,
51,
61,
71,
81,
91,
1, err
err
err
err====8.552722e+01
1.866792e+00
5.358780e-01
3.252580e-01
2.230087e-01
1.590315e-01
1.164959e-01
8.704375e-02
6.603566e-02
5.070573e-02
3.931682e-02
3.073275e-02
2.418582e-02
1.914325e-02
1.522709e-02
400
5.5
12
7
6
True
6.5
10
350 K (guess)
5.5 k (decision)
5
8
6
300
5
5.5
4.5
6
250
4.5
4
5
200
4
4.5
2
4
150
3.5
0
4
3.5
100
3.5
-2
3
3
50
-4
3

2.5
-6
0
-20 -15 -10 -5 0 5 10 15 20

47 / 48
What’s Next?

• Homework 7 available now on Blackboard.

48 / 48
Risk Model: Solving the SPP
• Set up Lagrangian:
utility
∞ X
X
β t πt (s t ) αu(ct1 (s t )) + (1 − α)u(ct2 (s t ))
 
L=
t=0 s t ∈S t
X∞ X
λt (s t ) et1 (s t ) + et2 (s t ) − ct1 (s t ) − ct2 (s t )
 
+
t=0 s t ∈S t
resource constraint

• FOC:
∂L
= β t πt (s t )αu ′ (ct1 (s t )) − λt (s t ) = 0
∂ct1 (s t )
∂L
= β t πt (s t )(1 − α)u ′ (ct1 (s t )) − λt (s t ) = 0
∂ct2 (s t )

• This implies:
λt (s t ) = β t πt (s t )αu ′ (ct1 (s t ))
λt (s t ) = β t πt (s t )(1 − α)u ′ (ct2 (s t )) Back

1/3
Risk Model: Solving the ADE

• Set up Lagrangian for i = 1, 2:

∞ X
β t πt (s t )u(cti (s t ))
X
L=
t=0 s t ∈S t
 
∞ X ∞ X
pt (s t )eti (s t ) − pt (s t )cti (s t )
X X
µi 
t=0 s t ∈S t t=0 s t ∈S t
budget constraint
• FOC:
probability
∂L t t ′ i t t
= β π t (s )u (ct (s )) − µi pt (s ) = 0
∂cti (s t )
for particular event history

• Therefore:
µi pt (s t ) = β t πt (s t )u ′ (cti (s t )) Back

2/3
Risk Model: Solving the SME
• Set up Lagrangian for i = 1, 2:
∞ X
X
L= β t πt (s t )u(cti (s t ))+
t=0 s t ∈S t
 
XX X
γti (s t ) eti (s t ) + ati (s t ) − cti (s t ) − qt (s t , st+1 )at+1
i
(s t , st+1 )
t=0 s t ∈S t st+1 ∈S

• FOC:
∂L
= β t πt (s t )u ′ (cti (s t )) − γti (s t ) = 0
∂cti (s t )
∂L
i = −qt (s t , st+1 )γti (s t ) + γt+1 i
(s t+1 ) = 0
∂at+1 (s t , st+1 )

• Thus:
γti (s t ) = β t πt (s t )u ′ (cti (s t ))
qt (s t , st+1 )γti (s t ) = γti (s t+1 ) Back

3/3