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EconS 500:Macroeconomic Theory I
Mark J. GibsonWashington State University
Fall 2013
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Standard utility functions
Consider the utility function
t=0 tu(ct),
where 0 < < 1 is the discount factorStandard assumptions on the period utility function u():
It is twice continuously differentiableIt is strictly increasing: u0(c) > 0 for all c > 0It is strictly concave: u00(c) < 0 for all c > 0It satisfies the Inada conditions:
limc!0
u0(c) =
limc! u
0(c) = 0
These assumptions guarantee a unique interior solution toa consumers problem
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Common period utility functions in macroPower/isoelastic/CRRA utility:
u(c) =
(c11
1 if > 0 and 6= 1log c if = 1
Note that, by lHpitals rule,
lim!1
c1 11 = log c
The coefficient of relative risk aversion is
cu00(c)
u0(c)=
The intertemporal elasticity of substitution is
log(ct+1/ct) log(u0(ct+1)/u0(ct))
=1
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Competitive equilibrium
In a competitive equilibrium, all consumers/householdsand firms take prices as givenA competitive equilibrium is a list of equilibrium objectsthat satisfy certain conditionsA competitive equilibrium typically requires that
consumers/households maximize utility subject to budgetconstraintsfirms maximize profits subject to technology constraintsmarkets clearthe governments budget constraints are satisfied
We will consider three types of competitive equililbria inthis course:
ArrowDebreuSequential marketsRecursive
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Pure-exchange model with infinitely lived consumers
Assumptions of the model:
Time is indexed by t = 0, 1, ...In each period there is a single nonstorable consumptiongoodThere are I consumers indexed by i = 1, 2, ..., IConsumer i has the sequence of endowments of theconsumption good fitgConsumer i has standard utility function
t=0 tiui(cit)
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ArrowDebreu equilibrium
In an ArrowDebreu equilibrium, all market transactionsoccur in period 0Market participants buy and sell futures contracts. Forexample, in this model pt is the price of a contract thatdelivers one unit of the consumption good in period tEach consumer has a single lifetime budget constraint
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Defining an ArrowDebreu equilibrium
An ArrowDebreu equilibrium is fcit, ptg such that:Consumer i chooses fcitg to solve
maxt=0 tiui(cit)s.t. t=0 ptcit =
t=0 pt
it
cit 0Markets clear in each period:
Ii=1 cit =Ii=1
it
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Notes on the definitionWhen do we use hats?
A hat on on object indicates that it is an equilibrium objectParameters do not get hatsChoice variables of consumers do not get hats. Forinstance, consumer i can choose any value for cit, but c
it is
the particular value the consumer chooses when prices areat their equilibrium values
In this course, budget constraints and market-clearingconditions will always hold with equality, but with certainnonstandard utility functions they might not. In suchcases we can write the budget constraint as
t=0 ptcit t=0 pt
it
and the market-clearing conditions as
Ii=1 cit Ii=1
it, = if pt > 0
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Characterizing the solution to a consumers problem
The Lagrangian for consumer i is
Li =t=0 tiui(cit) + it=0 ptit
t=0 ptc
it
and i 0 is the Lagrange multiplier on consumer isbudget constraintThe KuhnTucker first-order conditions are
Licit
= tiu0i(c
it) ipt 0, = if cit > 0
Lii
=t=0 ptit t=0 ptc
it 0, = if i > 0
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Characterizing the solution to a consumers problem
If the solution is interior, as when the Inada conditionshold, then the first-order conditions hold with equality:
tiu0i(c
it) =
ipt
t=0 ptcit =t=0 pt
it
We can eliminate the Lagrange multipliers by writing
u0i(cit)
iu0i(c
it+1)
=pt
pt+1
This is known as the intertemporal condition or Euler equation
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First definition of Pareto efficiency
An allocation fcitg is Pareto efficient if it is feasible,
Ii=1 cit Ii=1
it
and cit 0, and if there does not exist an alternative feasibleallocation fcitg such that
t=0 tiui(cit) t=0
tiui(c
it)
for all i, with strict inequality for at least one i
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Second definition of Pareto efficiency
An allocation fcitg is Pareto efficient if there exist welfare weightsfig, i 0 for all i, with strict inequality for at least one i, suchthat fcitg solves the social planners problem:
maxfcitgIi=1 i
t=0
tiui(c
it)
s.t. Ii=1 cit Ii=1
it
cit 0
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Characterizing the solution to the social plannersproblem
The Lagrangian is
L =Ii=1 it=0
tiui(c
it) +t=0 t
Ii=1 it
Ii=1 c
it
The first-order conditions for an interior solution are
Lcit
= itiu0i(c
it) t = 0
Lt
=Ii=1 it Ii=1 c
it = 0
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Welfare theorems
The first and second welfare theorems are the mostimportant theorems in all of economicsThe First Welfare Theorem says that, under certainconditions, a competitive equilibrium allocation is ParetoefficientThe Second Welfare Theorem says that, under certainconditions, a Pareto-efficient allocation can be supportedas a competitive equilibrium allocation if there areappropriate transfersThere are many different proofs of the welfare theoremswith varying levels of generality. We will start by provingthe theorems for this simple economy using calculus
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ArrowDebreu equilibrium with transfersAn ArrowDebreu equilibrium with transfers is fcit, pt, Tig suchthat:
Consumer i chooses fcitg to solvemaxt=0 tiui(cit)
s.t. t=0 ptcit =t=0 pt
it + T
i
cit 0Markets clear in each period:
Ii=1 cit =Ii=1
it
The governments budget constraint holds:
Ii=1 Ti = 0
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First Welfare Theorem
If fcit, ptg is an ArrowDebreu equilibrium, then fcitg is aPareto-efficient allocation.
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Proof of the First Welfare TheoremTo show that fcitg is a Pareto-efficient allocation, we can showthat it is feasible and that there exist welfare weights fig andLagrange multipliers ftg such that
itiu0i(c
it) = t.
Since fcit, ptg is an ArrowDebreu equilibrium, fcitg is feasibleand there exist Lagrange multipliers fig such that
tiu0i(c
it) =
ipt.
Thus we set
t = pt
i = 1/i.
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Second Welfare Theorem
If fcitg is a Pareto-efficient allocation, then there exist pricesfptg and transfers fTig such that fcit, pt, Tig is anArrowDebreu equilibrium with transfers.
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Proof of the Second Welfare TheoremWe want to show that fcitg is feasible and that there exist pricesfptg, transfers fTig, and Lagrange multipliers fig such that
tiu0i(c
it) =
ipt
t=0 ptcit =t=0 pt
it + T
i.
Since fcitg is a Pareto-efficient allocation, it is feasible and thereexist welfare weights fig and Lagrange multipliers ftg suchthat
itiu0i(c
it) = t.
Thus we set
pt = tTi =t=0 ptcit
t=0 pt
it
i= 1/i.
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Sequential markets equilibrium
In a sequential markets equilibrium, market transactionsoccur in each period, so there is a sequence of marketsEach consumer has a budget constraint for each periodTo allow for intertemporal trade (borrowing and lending),there is a one-period bond that earns simple interest. Ifbit+1 is the amount of the bond purchased by consumer i inperiod t, then the consumer receives amount (1+ rt+1)bit+1in period t+ 1, where rt+1 is the interest rateWe typically assume that bi0 = 0 for all i, but we couldalternatively pick nonzero amounts for initial bonds aslong as i bi0 = 0
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Normalization
If consumers choices of goods are homogenous of degreezero in pricesmeaning that if market prices double thenthe quantities chosen remain the samethen we cannormalize the price of one good each time markets areopen without loss of generalityIn an ArrowDebreu equilibrium, markets are open once,so we typically normalize p0 = 1. Then the period-0consumption good is the numraireIn a sequential markets equilibrium, markets are openevery period, so we typically normalize pt = 1 for all t.Then the period-t consumption good is the numraire inperiod tNote that, if there is more than one good in a period, onlythe price of one of the goods can be normalized
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Ruling out Ponzi schemes
An example of a Ponzi scheme is when someone borrowsfunds and repays the lender with interest by borrowingeven more funds. This can continue until no one is willingto lend to the borrower anymoreIf this continues forever, the persons debt tends towardinfinityFor a sequential markets equilibrium to exist, we need torule out this kind of Ponzi schemeThis requires that we add a no-Ponzi condition to theconsumers problem:
bit B,where B is sufficiently large that the condition rules outPonzi schemes but does not otherwise bind in equilibrium
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Defining a sequential markets equilibrium
A sequential markets equilibrium is fcit, bit, rtg such that:Consumer i chooses fcit, bitg to solve
maxt=0 tiui(cit)s.t. cit + b
it+1 =
it + (1+ rt)b
it
cit 0, bit B, bi0 = 0Markets clear in each period:
Ii=1 cit =Ii=1
it
Ii=1 bit = 0
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Characterizing the solution to a consumers problemThe Lagrangian for consumer i is
Li =t=0 tiui(cit) +t=0
it
it + (1+ rt)b
it cit bit+1
The first-order conditions are
Licit
= tiu0i(c
it) it = 0
Libit+1
= it + it+1(1+ rt+1) = 0
Liit
= it + (1+ rt)bit cit bit+1 = 0
We can eliminate the Lagrange multipliers by combiningthe first two conditions:
u0i(cit)
iu0i(c
it+1)
= 1+ rt+1
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Euler equation
The Euler equation is
u0i(cit)
iu0i(c
it+1)
= 1+ rt+1
The left side is the marginal rate of intertemporalsubstitution and the right side is the gross real interest rate
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Equivalence of ArrowDebreu and sequential marketsequilibria
Though the market structures of ArrowDebreu andsequential markets equilibria are very different, there is anequivalence between themWe will specify two propositions that establish theequivalence
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Establishing the equivalence
Both equilibria satisfy the feasibility constraintsWe can show that both equilibria satisfy the samefirst-order conditionsWe can show that the budget constraints are equivalent
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Equivalence of first-order conditions
In an ArrowDebreu equilibrium, we obtained the Eulerequation
u0i(cit)
iu0i(c
it+1)
=pt
pt+1
In a sequential markets equilibrium, we obtained the Eulerequation
u0i(cit)
iu0i(c
it+1)
= 1+ rt+1
This implies a relationship between prices in anArrowDebreu equilibrium and interest rates in asequential markets equilibrium:
ptpt+1
= 1+ rt+1
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Equivalence of budget constraints: from SME to ADEMultiply the period-0 budget constraint by p0 andmultiply the period-t+ 1 budget constraint by
pt+1 =pt
1+ rt+1
Then we have
p0ci0 + p0bi1 = p0
i0
p1ci1 + p1bi2 = p1
i1 + p0b
i1
p2ci2 + p2bi3 = p2
i1 + p1b
i2
...
After summing the constraints and canceling terms, wehave
t=0 ptcit + limt! ptbit+1 =t=0 ptit
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Equivalence of budget constraints: from SME to ADE
The no-Ponzi condition and the market-clearing conditionfor bonds imply that bit+1 is bounded from above andbelow, so
limt! ptb
it+1 = 0
This is known as a transversality condition. It says that, inthe limit, the present value of a consumers assets goes tozeroThus we have an ArrowDebreu budget constraint:
t=0 ptcit =t=0 pt
it
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Equivalence of budget constraints: from ADE to SME
To go from ADE to SME, we need to calculate eachconsumers bond holdings (or net asset position)There are two equivalent approachesWith each approach, we multiply the period-0 sequentialmarkets budget constraint by p0 and multiply theperiod-t+ 1 sequential markets budget constraint by
pt+1 =pt
1+ rt+1
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Equivalence of budget constraints: from ADE to SME
With the first approach, we sum the constraints
p0ci0 + p0bi1 = p0
i0
p1ci1 + p1bi2 = p1
i1 + p0b
i1
p2ci2 + p2bi3 = p2
i2 + p1b
i2
...
pt1cit1 + pt1bit = pt1
it1 + pt2b
it1
to obtaint1s=0 pscis + pt1bit =
t1s=0 ps
is
Thusbit =
1pt1
t1s=0 ps(
is cis)
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Equivalence of budget constraints: from ADE to SME
With the second approach, we sum the constraints
ptcit + ptbit+1 = pt
it + pt1b
it
pt+1cit+1 + pt+1bit+2 = pt+1
it+1 + ptb
it+1
...
to obtain
s=t pscis + lims! psbis+1 =s=t psis + pt1bit
Thusbit =
1pt1
s=t ps(c
is is)
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Equivalence of budget constraints: from ADE to SME
Using the above calculations of bond holdings, we obtain
t1s=0 pscis + pt1bit =t1s=0 ps
is
s=t+1 pscis =s=t+1 ps
is + ptb
it+1
If we take the ADE budget constraint and subtract thesetwo equations, we obtain
ptcit pt1bit = ptit ptbit+1We rearrange and use pt1/pt = 1+ rt to obtain
cit + bit+1 =
it + (1+ rt)b
it
This is the sequential markets budget constraint
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Proposition: from ADE to SME
If fcit, ptg is an ArrowDebreu equilibrium, then fcit, bit, rtg is asequential markets equilibrium, where
rt+1 =pt
pt+1 1
bit =1
pt1 s=t ps(c
is is)
(The value of r0 does not matter because initial bond holdingsare zero.)
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Proposition: from SME to ADE
If fcit, bit, rtg is a sequential markets equilibrium, then fcit, ptg isan ArrowDebreu equilibrium, where
p0 2 (0,)pt+1 =
pt1+ rt+1
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Discount bonds
An alternative to using one-period bonds with simpleinterest is to use one-period discount bondsWith discount bonds, a person who lends amount qtbit+1 inperiod t receives amount bit+1 in period t+ 1They are called discount bonds because, typically, qt < 1With discount bonds, the consumers problem is
maxt=0 tiui(cit)s.t. cit + qtb
it+1 =
it + b
it
cit 0, bit B, bi0 = 0Either way is equivalent
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Overlapping generations (OG) models
Infinitely lived consumers (ILC) models are good forstudying issues that do not depend on consumers lifecycles (for example, growth or business cycles)OG models are good for studying issues where consumerslife cycles matter (for example, saving for retirement or theeffects of public pensions)In OG models, time lasts forever but consumers have finitelifetimesIf each generation lives for n periods, then there are noverlapping generations in each periodIn period 0, there are n 1 initial generations that wereborn before period 0
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Other differences between ILC and OG models
The welfare theorems do not hold for OG modelsThere is only inside money in ILC models, but there can beoutside money in OG models
Inside money is in zero net supply (borrowing must equallending). For example, in our pure-exchange ILC model,we had i bit = 0Outside money is created by a government or by nature andis not in zero net supply. Examples are fiat money andgold. As a result, in OG models, the sum over individualsbond holdings need not equal zero
In OG models, consumers do not necessarily discount thefuture
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Notation and utility functions in OG modelsI let subscripts denote the current time period and letsuperscripts denote the time period in which a generationwas bornFor example, ctt+1 is consumption in period t+ 1 of thegeneration born in period tSuppose that consumers live for two periods and u(ctt, c
tt+1)
is the lifetime utility function of an individual born inperiod tStandard properties of u(, ) are as follows:
It is twice continuously differentiable in each argumentIt is strictly increasing in each argument: uj(c1, c2) > 0It is strictly concave in each argument: ujj(c1, c2) < 0The Inada conditions hold:
limcj!0
uj(c1, c2) =
limcj!
uj(c1, c2) = 0
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Common utility functions for two-period lives
CES utility:
u(c1, c2) =
( c1 + (1 )c2
1/if < 1 and 6= 0
c1 c12 if = 0
Note that
lim!0
c1 + (1 )c2
1/= c1 c
12
The elasticity of intertemporal substitution is
log(c2/c1) log(u1(c1, c2)/u2(c1, c2))
=1
1
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A pure-exchange model with two overlappinggenerations
Assumptions of the model:
There is a single nonstorable consumption good in eachperiodThe consumer born in period t = 0, 1, ... lives for twoperiods, has goods endowments (tt,
tt+1), and has
standard utility function u(ctt, ctt+1)
There is an initial old consumer with goods endowment10 and standard utility function u
1(c10 )The initial old consumer is also endowed with amount mof fiat money, where m 2 (,)
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ArrowDebreu equilibriumAn ArrowDebreu equilibrium is fct1t , ctt, ptg such that:
The initial old consumer chooses c10 to solve
max u1(c10 )
s.t. p0c10 = p010 +m
c10 0The consumer born in period t = 0, 1, . . . chooses ctt andctt+1 to solve
max u(ctt, ctt+1)
s.t. ptctt + pt+1ctt+1 = pt
tt + pt+1
tt+1
ctt 0, ctt+1 0Markets clear in each period:
ct1t + ctt =
t1t +
tt
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Characterizing the solution to the initial olds problem
The budget constraint is
p0c10 = p010 +m
This is one equation in one unknown, so we do not need totake first-order conditions
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Characterizing the solution to every other generationsproblem
The Lagrangian for the consumer born in period t = 0, 1, ...is
Lt = u(ctt, ctt+1) + t(pttt + pt+1tt+1 ptctt pt+1ctt+1)The first-order conditions are
Ltctt
= u1(ctt, ctt+1) tpt = 0
Ltctt+1
= u2(ctt, ctt+1) tpt+1 = 0
Ltt
= pttt + pt+1tt+1 ptctt pt+1ctt+1 = 0
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Characterizing the solution to every other generationsproblem
After eliminating the Lagrange multiplier, we have
u1(ctt, ctt+1)
u2(ctt, ctt+1)
=pt
pt+1ptctt + pt+1c
tt+1 = pt
tt + pt+1
tt+1
This is a system of 2 equations in 2 unknowns
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The role of fiat money
What is the equilibrium allocation if m = 0? Autarky (notrade)Why?
The initial olds budget constraint implies that c10 = 10
The market-clearing condition in period 0 then implies thatc00 =
00
The initial youngs budget constraint then implies thatc01 =
01
And so on
Fiat money allows there to be trade between the youngand old generations
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How many equilibria are there?
If m = 0, then the unique equilibrium is autarkyIf m 6= 0, then there is a continuum of equilibria, one foreach p0 2 (0,)When there is a nonzero amount of fiat money, thiseconomy is not homogeneous of degree zero in prices andwe cannot normalize p0 = 1 without loss of generality
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Sequential markets equilibriumA sequential markets equilibrium is fct1t , ctt, btt+1, rt+1g and p0such that:
The initial old consumer chooses c10 to solve
max u1(c10 )
s.t. c10 = 10 +m/p0
c10 0The consumer born in period t = 0, 1, . . . chooses ctt, c
tt+1,
and btt+1 to solve
max u(ctt, ctt+1)
s.t. ctt + btt+1 =
tt
ctt+1 = tt+1 + (1+ rt+1)b
tt+1
ctt 0, ctt+1 0
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Sequential markets equilibrium
Markets clear in each period:
ct1t + ctt =
t1t +
tt
b01 = m/p0bt+1t+2 = (1+ rt+1)b
tt+1
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Characterizing the solution to the initial olds problem
The budget constraint is
c10 = 10 +m/p0
This is one equation in one unknown, so we do not need totake first-order conditions
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Characterizing the solution to every other generationsproblem
The Lagrangian for the consumer born in period t is
Lt = u(ctt, ctt+1) + tt(tt ctt btt+1)+ tt+1(
tt+1 + (1+ rt+1)b
tt+1 ctt+1)
The first-order conditions are
Ltctt
= u1(ctt, ctt+1) tt = 0
Ltctt+1
= u2(ctt, ctt+1) tt+1 = 0
Ltbtt+1
= tt + tt+1(1+ rt+1) = 0
and the two budget constraints
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Characterizing the solution to every other generationsproblem
After eliminating the Lagrange multipliers, we have
u1(ctt, ctt+1)
u2(ctt, ctt+1)
= 1+ rt+1
ctt + btt+1 =
tt
ctt+1 = tt+1 + (1+ rt+1)b
tt+1
This is a system of 3 equations in 3 unknowns
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Stationary equilibriumA stationary equilibrium (or steady state) is a sequentialmarkets equilibrium in which the equilibrium objects areconstant over timeHere a stationary equilibrium consists of cy, co, b, r, and p0such that
u1(cy, co)u2(cy, co)
= 1+ r
cy + b = ttco = t1t + (1+ r)b
co + cy = t1t +tt
b = m/p0b = (1+ r)b
Clearly a stationary equilibrium only exists if t1t = oand tt = y for all t
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Golden rule allocation
A golden rule allocation is a feasible stationary allocationthat maximizes the utility of future generationsHere the golden rule allocation is cy and co such that theysolve
max u(cy, co)s.t. cy + co = y +o
cy 0, co 0Exercise: Show that there are two stationary equilibriahere and that only one satisfies the golden rule
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Excess demand functions and offer curves
An alternative way of characterizing an ArrowDebreuequilibrium is to use excess demand functionsLet the excess demand functions of a consumer born inperiod t = 0, 1, . . . be y(pt, pt+1) = ctt(pt, pt+1)tt whenyoung and z(pt, pt+1) = ctt+1(pt, pt+1)tt+1 when oldLet the excess demand function of the initial old consumerbe z1(p0, m) = m/p0Then an equilibrium price sequence fptg satisfies
z1(p0, m) + y(p0, p1) = 0z(pt, pt+1) + y(pt+1, pt+2) = 0
An offer curve shows the relationship between y and z, withy on the horizontal axis and z on the vertical axis
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Adding population dynamics to the basic model
Suppose now that nt1 is the measure of old people inperiod t and nt is the measure of young people in period t.All people in a generation are identicalFor our purposes, a measure is the length of an interval onthe real lineFor example, since there is measure n1 of initial oldconsumers, there is an initial old consumer at each pointon the interval [0, n1]
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ArrowDebreu equilibriumAn ArrowDebreu equilibrium is fct1t , ctt, ptg such that:
Each initial old consumer chooses c10 to solve
max u1(c10 )
s.t. p0c10 = p010 +m/n
1
c10 0Each consumer born in period t = 0, 1, . . . chooses ctt andctt+1 to solve
max u(ctt, ctt+1)
s.t. ptctt + pt+1ctt+1 = pt
tt + pt+1
tt+1
ctt 0, ctt+1 0Markets clear in each period:
nt1ct1t + ntctt = n
t1t1t + nttt
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Sequential markets equilibriumA sequential markets equilibrium is fct1t , ctt, btt+1, rt+1g and p0such that:
Each initial old consumer chooses c10 to solve
max u1(c10 )
s.t. c10 = 10 +m/(p0n
1)
c10 0Each consumer born in period t = 0, 1, . . . chooses ctt, c
tt+1,
and btt+1 to solve
max u(ctt, ctt+1)
s.t. ctt + btt+1 =
tt
ctt+1 = tt+1 + (1+ rt+1)b
tt+1
ctt 0, ctt+1 0
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Sequential markets equilibrium
Markets clear in each period:
nt1ct1t + ntctt = n
t1t1t + nttt
n0b01 = m/p0nt+1bt+1t+2 = n
t(1+ rt+1)btt+1
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Stationary equilibrium
Even with population change, we can have a stationaryequilibrium if
t1t = ott = y
nt = (1+ g)nt1
The stationary resource constraint, for example, is then
co + (1+ g)cy = o + (1+ g)y
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Adding money supply dynamics to the basic model
Suppose now that the initial old generation receives amount m0of fiat money and the old consumer alive in period t = 1, 2, . . .receives amount mt mt1 of fiat money
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ArrowDebreu equilibriumAn ArrowDebreu equilibrium is fct1t , ctt, ptg such that:
The initial old consumer chooses c10 to solve
max u1(c10 )
s.t. p0c10 = p010 +m0
c10 0The consumer born in period t = 0, 1, . . . chooses ctt andctt+1 to solve
max u(ctt, ctt+1)
s.t. ptctt + pt+1ctt+1 = pt
tt + pt+1
tt+1 +mt+1 mt
ctt 0, ctt+1 0Markets clear in each period:
ct1t + ctt =
t1t +
tt
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Sequential markets equilibriumA sequential markets equilibrium is fct1t , ctt, btt+1, rt+1, ptg suchthat:
The initial old consumer chooses c10 to solve
max u1(c10 )
s.t. c10 = 10 +m0/p0
c10 0The consumer born in period t = 0, 1, . . . chooses ctt, c
tt+1,
and btt+1 to solve
max u(ctt, ctt+1)
s.t. ctt + btt+1 =
tt
ctt+1 = tt+1 + (1+ rt+1)b
tt+1 + (mt+1 mt)/pt+1
ctt 0, ctt+1 0
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Sequential markets equilibrium
Markets clear in each period:
ct1t + ctt =
t1t +
tt
btt+1 = mt/pt
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Models with production
So far we have studied pure-exchange models to learn thebasic conceptsNext we will study models with productionWe will start with simple neoclassical growth models, onewith an infinitely lived consumer and one withoverlapping generations
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Standard production functions
Let capital and labor be the factors of production and letthe production function be f (k, l)Standard properties of f (, ) are:
It is homogeneous of degree oneIt is twice continuously differentiable in each argumentIt is strictly increasing in each argument: fj(k, l) > 0It is strictly concave in each argument: fjj(k, l) < 0The Inada conditions hold:
limk!0
f1(k, l) = liml!0
f2(k, l) =
limk!
f1(k, l) = liml!
f2(k, l) = 0
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Common production functions in macroCES production function:
f (k, l) =
(k + (1 )l)1/ if < 1 and 6= 0kl1 if = 0
Note that
lim!0
(k + (1 )l)1/ = kl1
The elasticity of substitution between the two factors is
log(l/k) log(f1(k, l)/f2(k, l))
=1
1 The CobbDouglas production function, f (k, l) = kl1,is used the most because the income shares for capital andlabor are constant, which is roughly consistent withaggregate data
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Neoclassical growth modelAssumptions of the model:
There is a single good in each period that can be used forconsumption and investmentThere is a representative consumer who is endowed with lunits of labor in each period and k0 units of initial capitaland has a standard lifetime utility function
t=0 tu(ct)There is a representative firm with standard productionfunction f (kt, lt)The resource constraint in period t is
ct + xt = f (kt, lt)
The law of motion of capital is
kt+1 = (1 )kt + xt,where 0 < 1 is the depreciation rate
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The representative firms problem
A firm maximizes profits each period subject to thetechnology constraintIn period t, the representative firm chooses kt, lt to solve
max ptf (kt, lt) rtkt wltThe profit-maximization conditions are
rt = ptf1(kt, lt)
wt = ptf2(kt, lt)
Each factor is paid the value of its marginal productWhen defining an equilibrium, you can either write downthe firms maximization problem or write down theprofit-maximization conditions. I usually prefer to writedown the profit-maximization conditions
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Zero profits
There are zero profits in equilibriumTo see why, we use Eulers theorem for homogeneousfunctions: If g(, ) is continuously differentiable andhomogeneous of degree n, then
g1(x, y)x+ g2(x, y)y = ng(x, y)
Since f (, ) is homogeneous of degree one, profits arepi = pf (k, l) rkwl= pf (k, l) pf1(k, l)k pf2(k, l)l= 0
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Investment
I usually assume that the consumer makes the investmentdecisions. As an exercise, you can show that it isequivalent if the firm makes the investment decisionsInvestment is puttyputty if it can be converted into theconsumption good. In this case, xt (1 )ktInvestment is puttyclay if it cannot be converted into theconsumption good. In this case, xt 0We will assume that investment is puttyputty unlessspecified otherwise
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ArrowDebreu equilibriumAn ArrowDebreu equilibrium is fct, xt, kt, lt, pt, rt, wtg such that:
The consumer chooses fct, xt, ktg to solvemaxt=0 tu(ct)
s.t. t=0 pt(ct + xt) =t=0(rtkt + wt l)
kt+1 = (1 )kt + xtct 0, xt (1 )kt, kt 0, k0 = k0
The profit-maximization conditions hold:
rt = ptf1(kt, lt)
wt = ptf2(kt, lt)
Markets clear:
ct + xt = f (kt, lt)
lt = l
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Characterizing the solution to the consumers problem
The Lagrangian is
L =t=0 tu(ct)+t=0(rtkt +wt l)
t=0 pt(ct + xt)
+t=0 t((1 )kt + xt kt+1)
The first-order conditions are
Lct
= tu0(ct) pt = 0Lxt
= pt + t = 0Lkt+1
= t + rt+1 + t+1(1 ) = 0
and the budget constraint, the law of motion of capital,and the initial condition on capital
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Euler equationBy combining the first-order conditions and eliminatingthe Lagrange multipliers, we obtain the Euler equation
u0(ct)u0(ct+1)
=rt+1pt+1
+ 1
By substituting in the profit-maximization condition forcapital, we obtain the Euler equation entirely in terms ofquantities:
u0(ct)u0(ct+1)
= f1(kt+1, lt+1) + 1
By substituting in the market-clearing conditions and thelaw of motion of capital, we obtain a second-orderdifference equation in fktg:
u0(f (kt, l) + (1 )kt kt+1)u0(f (kt+1, l) + (1 )kt+1 kt+2) = f1(kt+1, l) + 1
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Transversality conditionWe obtained a second-order difference equation as anecessary condition for a solution to the consumersproblemIs it sufficient for a solution? NoFor a unique solution, a second-order difference equationrequires two additional conditions on fktgWe only have one: k0 = k0We need to specify a second condition, which is known asthe transversality condition:
limt! ptkt+1 = 0
or, in terms of fktg only,limt!
tu0(f (kt, l) + (1 )kt kt+1)kt+1 = 0StokeyLucasPrescott provides a proof, which you willwork through on a problem set
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Necessary and sufficient conditionsThe necessary and sufficient conditions for fct, xt, kt, lt, pt, rt, wtgto be an ArrowDebreu equilibrium are:
t=0 pt(ct + xt) =t=0(rtkt +wt l)
kt+1 = (1 )kt + xtk0 = k0
ptpt+1
=u0(ct)
u0(ct+1)=
rt+1pt+1
+ 1 rt = ptf1(kt, lt)wt = ptf2(kt, lt)
ct + xt = f (kt, lt)lt = l
limt! ptkt+1 = 0
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Simplifying the definition of equilibrium
We can simplify the definition of equilibrium by making thefollowing substitutions:
Let lt = l and eliminate the labor market clearing conditionLet xt = kt+1 (1 )kt and eliminate the law of motion ofcapital
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ArrowDebreu equilibrium againAn ArrowDebreu equilibrium is fct, kt, pt, rt, wtg such that:
The consumer chooses fct, ktg to solvemaxt=0 tu(ct)
s.t. t=0 pt(ct + kt+1 (1 )kt) =t=0(rtkt + wt l)
ct 0, kt 0, k0 = k0The profit-maximization conditions hold:
rt = ptf1(kt, l)
wt = ptf2(kt, l)
Markets clear:
ct + kt+1 (1 )kt = f (kt, l)
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Sequential markets equilibriumA sequential markets equilibrium is fct, kt, bt, rkt , rbt , wtg such that:
The consumer chooses fct, kt, btg to solvemaxt=0 tu(ct)
s.t. ct + kt+1 (1 )kt + bt+1 = rkt kt + wt l+ (1+ rbt )btct 0, kt 0, k0 = k0, bt B, b0 = 0
The profit-maximization conditions hold:
rkt = f1(kt, l)
wt = f2(kt, l)
Markets clear in each period:
ct + kt+1 (1 )kt = f (kt, l)bt = 0
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Characterizing the solution to the consumers problem
The Lagrangian is
L =t=0 tu(ct) +t=0 t(r
kt kt +wt l+ (1+ r
bt )bt
ct kt+1 + (1 )kt bt+1)The first-order conditions are
Lct
= tu0(ct) t = 0Lkt+1
= t + t+1(rkt+1 + 1 ) = 0Lbt+1
= t + t+1(1+ rbt+1) = 0
and the budget constraints
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Euler equations
After eliminating the Lagrange multipliers, we have twoEuler equations
u0(ct)u0(ct+1)
= rkt+1 + 1 u0(ct)
u0(ct+1)= 1+ rbt+1
Notice that, using the profit-maximization condition forcapital, we obtain the same Euler equation as in anArrowDebreu equilibrium:
u0(ct)u0(ct+1)
= f1(kt+1, l) + 1
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No arbitrage
A fundamental concept in economics and finance is thatthere can be no arbitrage in equilibrium (that is, pricescannot be such that a trader could make positive profitswith zero risk)The first-order conditions imply that, for an interiorsolution, the no-arbitrage condition on capital and bonds is
rkt+1 = rbt+1
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A note on bonds
With only a representative consumer, there is no one totrade with, which is why bt = 0 for all tIn this case, we can simplify the definition of a sequentialmarkets equilibrium by leaving out bondsIf there are at least two different agents that can borrowand lend, then we must include bonds for markets to becomplete
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Sequential markets equilibrium againA sequential markets equilibrium is fct, kt, rt, wtg such that:
The consumer chooses fct, ktg to solvemaxt=0 tu(ct)
s.t. ct + kt+1 (1 )kt = rtkt + wt lct 0, kt 0, k0 = k0
The profit-maximization conditions hold:
rt = f1(kt, l)
wt = f2(kt, l)
Markets clear in each period:
ct + kt+1 (1 )kt = f (kt, l)
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Steady state
A steady state for this model is c, k, r, and w such that, ifk0 = k, then the sequential markets equilibrium is ct = c,kt = k, rt = r, and wt = w for all tIf k0 6= k, then the equilibrium capital stock will convergeto the steady state over time
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The recursive language
Consider the following maximization problem insequential form:
v(x0) = maxxt+12(xt)
t=0 tr(xt, xt+1)
We can equivalently state this problem recursively as
v(x) = maxx02(x)
r(x, x0) + v(x0)
This is known as Bellmans equation, which is a functionalequation (meaning that the solution is a function)Recursive notation:
There are no time subscriptsA prime denotes the value of a variable next period
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Recursive terminology
Consider the recursive dynamic program
v(x) = maxx02(x)
r(x, x0) + v(x0)
v(x) is the value functionx is the state variablex0 is the control variable(x) is the feasible set of values of x0 given xr(x, x0) is the period return function is the discount factorx0(x) is the decision rule or policy function
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Optimal growth as a sequential problem
The social planner chooses fct, ktg to solvev(k0) = maxt=0 tu(ct)
s.t. ct + kt+1 (1 )kt = f (kt, l)ct 0, kt 0, k0 = k0
Equivalently, the social planner chooses fktg to solvev(k0) = max
kt+12(kt)t=0 tu(f (kt, l) + (1 )kt kt+1),
where(kt) = [0, f (kt, l) + (1 )kt]
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First-order conditions for the sequential problem
The Lagrangian is
L =t=0 tu(ct) +t=0 t
f (kt, l) ct kt+1 + (1 )kt
The first-order conditions are
Lct
= tu1(ct) t = 0Lkt+1
= t + t+1(f1(kt+1, l) + 1 ) = 0
and the resource constraintWe obtain the Euler equation
u0(ct)u0(ct+1)
= f1(kt+1, l) + 1
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Optimal growth as a recursive dynamic program
The social planner chooses c(k) and k0(k) to solve
v(k) = max
u(c) + v(k0)
s.t. c+ k0 (1 )k = f (k, l)c 0, k0 0
Equivalently, the social planner chooses k0 to solve
v(k) = maxk02(k)
u(f (k, l) + (1 )k k0) + v(k0) ,
where(k) = [0, f (k, l) + (1 )k]
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First-order conditions for the recursive problem
The Lagrangian is
L = u(c) + v(k0) + f (k, l) c k0 + (1 )kThe first-order conditions are
Lc= u1(c) = 0
Lk0
= v1(k0) = 0
and the resource constraintThis is not useful yet because we do not know v()
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Envelope conditionThe envelope condition requires that a small change in thestate variable changes each side of Bellmans equationequallyIn this case, we take the partial derivative with respect to kof each side of the functional equation and equate them:
v1(k) = f1(k, l) + 1
If we move the envelope condition forward one period, wehave
v1(k0) = 0f1(k0, l) + 1
We can plug this back into the first-order conditions to getthe Euler equation:
u1(c)u1(c0)
= f1(k0, l) + 1
This is equivalent to the Euler equation from the sequentialproblem
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Can we solve for the value function?There are only a few dynamic programs that have analyticsolutionsMost require numeric solutionsThe optimal growth problem only has an analytic solutionwhen
f (k, l) = kl1
u(c) = log c = 1
To solve for the value function, we use guess and verifyWe guess that
v(k) = a+ b log k,
where a and b are undetermined coefficientsWe verify that the guess is correct and then solve for a andb
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Recursive competitive equilibrium
We can reinterpret the model as having measure one ofidentical consumersIn a recursive competitive equilibrium, we distinguishbetween the state of the individual and the state of theeconomyHere let k be the capital stock of an individual and let K bethe aggregate capital stockThen consumers decisions are functions of (k, K), whileprices and aggregates are functions of K onlyThere must be consistency between individuals decisionsand aggregates
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Recursive competitive equilibrium
A recursive competitive equilibrium is v(k, K), c(k, K), k0(k, K),C(K), K0(K), r(K), and w(K) such that:
The consumer chooses c(k, K) and k0(k, K) to solve
v(k, K) = max
u(c) + v(k0, K0(K))
s.t. c+ k0 (1 )k = r(K)k+w(K)lc 0, k0 0
The profit-maximization conditions hold:
r(K) = f1(K, l)w(K) = f2(K, l)
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Recursive competitive equilibrium
The aggregate consistency conditions hold:
C(K) = c(K, K)K0(K) = k0(K, K)
Markets clear in each period:
C(K) + K0(K) (1 )K = f (K, l)
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Neoclassical growth model with overlappinggenerations
Assumptions of the model:
There is a single good in each period that can be used forconsumption and investmentThere are two generations in each periodThe initial old consumer has standard utility functionu1(c10 ) and is endowed with k
10 units of initial capital
and amount m of fiat moneyThe consumer born in period t = 1, 2, . . . has standardutility function u(ctt, c
tt+1) and is endowed with l units of
labor when young and none when oldThere is a representative firm with standard productionfunction f (, )The resource constraint in period t is
ct1t + ctt + k
tt+1 (1 )kt1t = f (kt1t , l)
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ArrowDebreu equilibrium
An ArrowDebreu equilibrium is fct1t , ctt, kt1t , pt, rt, wtg such that:The initial old consumer chooses c10 and k
10 to solve
max u1(c10 )
s.t. p0c10 = r0k10 + (1 )p0k10 +m
c10 0, k10 = k10The consumer born in period t = 1, 2, . . . chooses ctt, c
tt+1,
and ktt+1 to solve
max u(ctt, ctt+1)
s.t. pt(ctt + ktt+1) + pt+1c
tt+1 = wt l+ rt+1k
tt+1 + (1 )pt+1ktt+1
ctt 0, ctt+1 0, ktt+1 0
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ArrowDebreu equilibrium
The profit-maximization conditions hold:
rt = ptf1(kt1t , l)
wt = ptf2(kt1t , l)
Markets clear:
ct1t + ctt + k
tt+1 (1 )kt1t = f (kt1t , l)
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Characterizing the solutions to consumers problems
The solution to the initial olds problem is characterized by
p0c10 = r0k10 + (1 )p0k10 +mk10 = k
10
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Characterizing the solutions to consumers problems
The Lagrangian of the consumer born in t = 0, 1, . . . is
Lt = u(ctt, ctt+1) + t(wt l+ rt+1ktt+1 + (1 )pt+1ktt+1 pt(ctt + ktt+1) pt+1ctt+1)
The first-order conditions are
Ltctt
= u1(ctt, ctt+1) tpt = 0
Ltctt+1
= u2(ctt, ctt+1) tpt+1 = 0
Ltktt+1
= tpt + tpt+1
rt+1pt+1
+ 1 = 0
and the budget constraint
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Characterizing the solutions to consumers problems
The solution to the problem of the consumer born int = 1, 2, . . . is characterized by
ptpt+1
=u1(ctt, c
tt+1)
u2(ctt, ctt+1)
u1(ctt, ctt+1)
u2(ctt, ctt+1)
=rt+1pt+1
+ 1
pt(ctt + ktt+1) + pt+1c
tt+1 = wt l+ rt+1k
tt+1 + (1 )pt+1ktt+1
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Sequential markets equilibriumA sequential markets equilibrium is fct1t , ctt, kt1t , btt+1, rkt , rbt+1, wtgand p0 such that:
The initial old consumer chooses c10 and k10 to solve
max u(c10 )
s.t. c10 = (rk0 + 1 )k10 +m/p0
c10 0, k10 = k10The consumer born in period t = 1, 2, . . . chooses ctt, c
tt+1,
ktt+1, and btt+1 to solve
max u(ctt, ctt+1)
s.t. ctt + ktt+1 + b
tt+1 = wt l
ctt+1 = (rkt+1 + 1 )ktt+1 + (1+ rbt+1)btt+1
ctt 0, ctt+1 0, ktt+1 0
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Sequential markets equilibrium
The profit-maximization conditions hold:
rt = f1(kt1t , l)
wt = f2(kt1t , l)
Markets clear:
ct1t + ctt + k
tt+1 (1 )kt1t = f (kt1t , l)
b01 = m/p0bt+1t+2 = (1+ rt+1)b
tt+1
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Characterizing the solutions to consumers problems
The solution to the initial olds problem is characterized by
c10 = rk0k10 + (1 )k10 +m/p0
k10 = k10
The Lagrangian of the consumer born in t = 0, 1, . . . is
Lt = u(ctt, ctt+1) + tt(wt l ctt ktt+1 btt+1)+ tt+1((r
kt+1 + 1 )ktt+1 + (1+ rbt+1)btt+1)
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Characterizing the solutions to consumers problems
The first-order conditions are
Ltctt
= u1(ctt, ctt+1) tt = 0
Ltctt+1
= u2(ctt, ctt+1) tt+1 = 0
Ltktt+1
= tt + tt+1
rkt+1 + 1 = 0
Ltbtt+1
= tt + tt+1
1+ rbt+1= 0
and the budget constraints
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Characterizing the solutions to consumers problems
The solution to the problem of the consumer born int = 1, 2, . . . is characterized by
u1(ctt, ctt+1)
u2(ctt, ctt+1)
= rkt+1 + 1
u1(ctt, ctt+1)
u2(ctt, ctt+1)
= 1+ rbt+1
ctt + ktt+1 + b
tt+1 = wt l
ctt+1 = (rkt+1 + 1 )ktt+1 + (1+ rbt+1)btt+1
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Equilibrium path
We can plug the budget constraints and profit-maximizationconditions into the Euler equation to obtain
u1(f2(kt1t , l)l ktt+1 btt+1, (f1(ktt+1, l) + 1 )(ktt+1 + btt+1))u2(f2(kt1t , l)l ktt+1 btt+1, (f1(ktt+1, l) + 1 )(ktt+1 + btt+1))
= f1(ktt+1, l) + 1 If m = 0, then btt+1 = 0 for all t and we have a first-orderdifference equation in capital:
u1(f2(kt1t , l)l ktt+1, (f1(ktt+1, l) + 1 )ktt+1)u2(f2(kt1t , l)l ktt+1, (f1(ktt+1, l) + 1 )ktt+1)
= f1(ktt+1, l)+ 1