Lecture 1: Foundations of Neoclassical Growth Acemoglu 2009, … · In macroeconomics, it is...
Transcript of Lecture 1: Foundations of Neoclassical Growth Acemoglu 2009, … · In macroeconomics, it is...
Lecture 1: Foundations of Neoclassical Growth(Acemoglu 2009, Chapter 5)
Adapted from Fabrizio Zilibotti’s notes
Kjetil Storesletten
August 24, 2018
Kjetil Storesletten () Lecture 1 August 24, 2018 1 / 53
Solow ModelFocus on capital accumulation, constant saving rate:
k (t) = s · f (k (t))− (n+ δ) · k (t)
output
k(t)
f(k*)
k*
f(k(t))
sf(k*)sf(k(t))
consumption
investment
0
(δ+n)k(t)
Figure: Investment and consumption in the steady-state equilibrium withpopulation growth.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 2 / 53
Foundations of Neoclassical Growth
More satisfactory to specify the preference orderings ofindividuals and derive their decisions from these preferences
Enables better understanding of the factorsaffecting savings decisions
Enables to discuss the “optimality”of equilibria
Allows researchers to ask whether the decentralized equilibriaof growth models can be “improved upon”
Notion of improvement: Pareto optimality
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 3 / 53
Preliminaries I
Consider an economy consisting of a unit measureof infinitely-lived households
E.g., the set of households H could be representedby the unit interval [0, 1]
This representation as a continuum emphasizes that eachhousehold is infinitesimal and has no effect on aggregates
Can alternatively think of H as a countable set of the formH = {1, 2, ...,M} with M → ∞, without any loss of generalityAdvantage of unit measure: averages and aggregates are the same
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 4 / 53
Example
Suppose all agents have the same income, yj = y
Clearly, average income is y
With a countable set of agents, total income is
M
∑j=0yj = My
While with a unit measure, y is both average and aggregate:∫ 1
0y (j) dj = y
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 5 / 53
Preliminaries II
How to model households’behavior in a world that goes on for ever(t → ∞)?
1 “Infinitely lived”or consisting of overlapping generationswith full altruism linking generations
2 overlapping generations
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Preliminaries III
We make standard assumptions on preferencesso that they can be represented by utility functions
In particular, each household i has an instantaneous utility function
ui (ci (t))
ui : R+→ R is increasing and concaveand ci (t) is the consumption vector of household i .
Sometimes also referred to as the “felicity function”
Two major assumptions in writing an instantaneous utility function1 consumption externalities are ruled out2 overall utility is time separable
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Infinite Horizon
Suppose households discount the future “exponentially”(or “proportionally” in discrete time)
Thus household preferences at time t = 0 are
Ui =∞
∑t=0
βti ui (ci (t))
where βi ∈ (0, 1) is the discount factor of household i
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 8 / 53
Time Consistency
Time separability and exponential discountingensure “time-consistent”behavior.
A solution {x (t)}Tt=0 (possibly with T = ∞) is time consistent if:
I whenever {x (t)}Tt=0 is an optimal solution starting at time t = 0,
then
{x (t)}Tt=t ′ is an optimal solution to the continuationdynamic optimization problem starting from time t = t ′ ∈ [0,T ].
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 9 / 53
Uncertainty
So far, we have ignored uncertainty:the sequence {ci (t)}∞
t=0 is known with certainty.
With uncertainty, preferences of household i at time t = 0can be represented by the following (von Neumann-Morgenstern)expected utility function:
Ei0
∞
∑t=0
βti ui (ci (t))
Ei0 is the expectation operator with respect to the
information set available to household i at time t = 0.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 10 / 53
Heterogeneity
So far we indexed individual utility function, ui (·),and the discount factor, βi , by “i”
Households could also differ according to their income processes.E.g., effective labor endowments of {ei (t)}∞
t=0,labor income of {ei (t)w (t)}∞
t=0
But at this level of generality, the problem is not tractable
In macroeconomics, it is standard to assume theexistence of a representative household.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 11 / 53
The Representative Household I
An economy admits a representative householdif the preference (demand) side can be representedas if a single household made the aggregate consumptionand saving decisions subject to a single budget constraint
This description of a representative household is purely positive
There is a stronger notion of “normative” representative household:if we can use the utility function of the representative householdfor welfare comparisons
Simplest case that will lead to the existence of a representativehousehold: suppose each household is identical
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The Representative Household II
I.e., same β, same sequence {e (t)}∞t=0 and same
u (ci (t))
where u : R+→ R is increasing and concave and ci (t) is theconsumption of household i .
The demand side can be represented as the solution to
U repr = max∞
∑t=0
βtu (c (t)) ,
β ∈ (0, 1) is the common discount factor andc (t) the consumption level of the representative household
Admits a representative household rather trivially
Representative household’s preferences, U repr ,can be used for both positive and normative analysis.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 13 / 53
The Representative Household III
If instead households are not identical, one would like to model thecollection of their behaviors as if the demand side were generated bythe optimization decision of a single (representative) household:
Problems:1 The representative household will have positive,but not always a normative meaning
2 Models with heterogeneity often do not lead to behavior that can berepresented as if it were generated by a representative household
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The Representative Household IVDebreu-Mantel-Sonnenschein theorem ["anything goes"]:any vector of equilibrium prices, p, and any associatedvector of (excess) demand functions, x (p) , consistentwith a budget constraint, can arise from theoptimizing behavior of a set H of households.Problem: the fact that aggregate demand functionscome from optimizing behavior of households putsno restrictions on the form of these demands
I E.g., x (p) does not necessarily possess a negative-semi-definiteJacobian or satisfy the weak axiom of revealed preference(requirements of demands generated by individual households)
Hence without imposing further structure, it is impossible toderive x (p) from the optimizing behavior of a single householdThe source of problem is income effects:
I special (but reasonably realistic) preference functions,and restrictions on distribution of income and/or insurance marketsrule out arbitrary aggregate demand functions
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The Representative Household V
Theorem (Gorman’s Aggregation Theorem)Consider an economy with a set H of households.Suppose that the preferences of household i ∈ H can berepresented by an indirect utility function of the form
v i(p, y i
)= ai (p) + b (p) y i ,
then these preferences can be aggregated and represented bythose of a representative household, with indirect utility
v (p, y) =∫i∈H
ai (p) di + b (p) y ,
where y ≡∫i∈H y
idi is aggregate income.
Note: as usual, any indirect utility function v i(p, y i
)must be homogeneous of degree 0 in p and y .
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 16 / 53
The Representative Household VI
The Gorman form implies linear Engel Curves(i.e., the derivative of an individual’s demandfor a commodity with respect to his income is constant)
The proof follows from Roy’s identity
x ij(p, y i
)= −
∂v i(p,y i)∂pj
∂v i (p,y i )∂y i
= −∂ai (p)
∂pj+ ∂b(p)
∂pjy i
b (p)
→∂x ij(p, y i
)∂y i
=∂b (p)
∂pj
1b (p)
indep. of y i
where x ij(p, y i
)is the Marshallian demand of good j
for a consumer with income y i
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 17 / 53
The Representative Household VII
Quasi-linear structure limits the extent of income effectsand enables the aggregation of individual behavior
Note that (under no uncertainty) all we require is the existenceof a monotonic transformation of the indirect utility functionthat takes the Gorman form
Contains some commonly-used preferences in macroeconomics
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Example: CES Preferences I
A very common class of preferences: constant elasticity ofsubstitution (CES) preferences or Dixit-Stiglitz preferences
Suppose each household denoted by i ∈ H has total income y i
and preferences defined over j = 1, ...,N goods
U i(x i1, ..., x
iN
)=
[N
∑j=1
(x ij) σ−1
σ
] σσ−1
,
where σ ∈ (0,∞) is the elasticity of substitution,σ = d ln
(xjxi
)/d ln
(UxiUxj
)
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 19 / 53
Example: CES II
Each consumer faces a vector of prices p= (p1, ..., pN )
The indirect utility function is given by [please show that!]
v i(p,y i
)=
y i[∑Nj=1 p
1−σj
] 11−σ
,
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 20 / 53
Example: CES Preferences III
Satisfies the Gorman form(and is also homogeneous of degree 0 in p and y)
Therefore, this economy admits a representative householdwith indirect utility:
v (p,y) =y[
∑Nj=1 p
1−σj
] 11−σ
y is aggregate income given by y ≡∫i∈H y
idi
The utility function leading to this indirect utility function is
U (x1, ..., xN ) =
[N
∑j=1(xj )
σ−1σ
] σσ−1
Preferences closely related to CES preferences will be keyin ensuring balanced growth in neoclassical growth models
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 21 / 53
The Representative Household VIII
Converse to Gorman Theorem:I unless we put some restrictions on the distribution of income, Gormanpreferences are not only suffi cient to admit a representative household,but also necessary
I I.e., if the indirect utility functions of the households do not take theGorman form, there will exist some distribution of income such thataggregate behavior cannot be represented as if it resulted fromrepresentative household
Gorman preferences also imply the existence ofa normative representative household
Recall an allocation is Pareto optimal if no household can be madestrictly better off without some other household being made worse off
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The Representative Household IX
Theorem (Existence of a Normative Representative Household)Consider an economy a set H of households.Suppose that the preferences of each household i ∈ Htake the Gorman form, v i
(p, y i
)= ai (p) + b (p) y i
1 Then any allocation that maximizesthe utility of the representative household,v (p, y) = ∑i∈H a
i (p) + b (p) y , with y ≡ ∑i∈H yi ,
is Pareto optimal2 Moreover, if ai (p) = ai for all p and all i ∈ H,then any Pareto optimal allocation maximizesthe utility of the representative household
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 23 / 53
Infinite Planning Horizon I
Most growth and macro models assume that individualshave an infinite planning horizon
Two reasonable microfoundations for this assumption:1 “Poisson death model”or the perpetual youth model2 Intergenerational altruism
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Poisson Death Model I
Individuals are finitely-lived, but not aware of when they will die1 Strong simplifying assumption:likelihood of survival to the next age in reality is not a constant
2 But a good starting point, tractable and implies expected lifespanof 1/ν < ∞ periods, can be used to get a sensible value of ν
Suppose each individual has a standard instantaneousutility function u (c), and a “true”or “pure”discount factor β
Normalize u (0) = 0 to be the utility of death
Consider an individual who plans to havea consumption sequence {c (t)}∞
t=0 (conditional on living).
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 25 / 53
Poisson Death Model IIIndividual would have an expected utility at time t = 0 given by
U (0) = u (c (0)) + β (1− ν) u (c (1)) + βνu (0)
+β2(1− ν)2 u (c (2)) + β
2(1− ν) νu (0) + ...
=∞
∑t=0
(β (1− ν)
)tu (c (t))
=∞
∑t=0
βtu (c (t)) , (1)
The second line collects terms and uses u (0) = 0The third line defines β ≡ β (1− ν)as the “effective discount factor.”
Isomorphic to model of infinitely-lived individuals,up to reinterpretation of β.Note that ∑∞
t=0 βtu (c (t)) is the expected utility;probabilities have been substituted away.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 26 / 53
Intergenerational Altruism I
Imagine an individual who lives for one periodand has a single offspring (who will also live for a single periodand has a single offspring etc.)
Individuals not only derives utility from consumptionbut also from the bequest they leave to their offspring
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 27 / 53
Intergenerational Altruism II
“Pure altruism:” each agent cares aboutthe utility of his offspring (with some discount factor).
Let discount factor between generations be β
Assume offspring will have an income of w without the bequest.
Then the utility of the individual can be written as
u (c (t)) + βV (b (t) + w) ,
V (·): continuation value, the utility that the offspring will obtainfrom receiving a bequest of b (t) (plus having his own w)
Value of the individual at time t can in turn be written as
V (b (t − 1) + w) = maxc (t)+b(t)≤b(t−1)+w
{u (c (t)) + βV (b (t) + w)}
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 28 / 53
Intergenerational Altruism II
V (b (0) + w) = maxc (1)+b(1)≤b(0)+w
{u (c (1)) + βV (b (1) + w)}
V (b (1) + w) = maxc (2)+b(2)≤b(1)+w
{u (c (2)) + βV (b (2) + w)}
V (b (2) + w) = maxc (3)+b(3)≤b(2)+w
{u (c (3)) + βV (b (3) + w)}...
Hence, by repeated substitution one obtains
V (b (0) + w) = maxc (1),c (2),c (3),...,b(T )
u (c (1)) + βu (c (2)) + β2u (c (3))
+...+ βT (V (b (T ) + w))
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 29 / 53
Intergenerational Altruism III
Canonical form of a dynamic programming representationof an infinite-horizon maximization problem.
If limT→∞ βT (V (b (T ) + w)) = 0, this dynamicprogramming representation is equivalent to
maxc (0),c (1),c (2),...
∞
∑s=0
βsu (ct+s )
Equivalent to infinite planning horizon.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 30 / 53
Problem Formulation
Discrete time infinite-horizon economy and suppose that the economyadmits a representative household
Once again ignoring uncertainty, the representative household has thet = 0 objective function
∞
∑t=0
βtu (c (t)) ,
with a discount factor of β ∈ (0, 1)In continuous time, this utility function of the representativehousehold becomes ∫ ∞
0exp (−ρt) u (c (t)) dt
where ρ > 0 is now the discount rate of the individuals.
The book discusses the equivalence in mathematical terms
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 31 / 53
Welfare Theorems (Preference Side)
Establish the connection between Pareto optimaand competitive equilibria
Assume a finite number of consumers (H is finite)and a countable (possibly infinite) number of commodities
Let commodities be indexed by j and let
x i≡{x ij}∞
j=0be household i’s consumption bundle,
and ωi≡{
ωij
}∞
j=0be household i’s endowment bundle
Each household in H has a well defined preference ordering overconsumption bundles that can be represented by a utility function
Let u ≡{ui}i∈H be the set of utility functions
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 32 / 53
Welfare Theorems (Production Side)
Production side: finite number of firms represented by FEach firm f ∈ F is characterized by production set Y f , specifyinglevels of output firm f can produce from specified levels of inputs
E.g.: if there are only labor and a final good, Y f includes pairs(−l , y) such that with labor input l the firm can produce at most y
Assume that the technology features constant returns to scale
Ownership structure of firms:if firms make profits, they should be distributed to some agents
θif is the share of profits of firm f that will accrue to household i
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 33 / 53
Welfare Theorems (Allocation)
An economy E is described by E ≡ (H,F ,u,ω,Y,X, θ),where X denotes the choice set and Y the aggregate production set.An allocation is (x, y) such that x and y are feasible, that is,x ∈ X, y ∈ Y, and ∑i∈H x
ij ≤ ∑i∈H ωi
j +∑f ∈F yfj for all j ∈N
A price system is a sequence p≡{pj}∞j=0, such that pj ≥ 0 for all j
We can set one of these prices to unity (choice of numeraire)
We define the inner product p · x ≡ ∑∞j=0 pjxj
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 34 / 53
Welfare Theorems (Competitive Equilibrium)
Definition A competitive equilibrium for the economyE ≡ (H,F ,u,ω,Y,X, θ) is given by an allocation(x∗ =
{x i∗}i∈H , y
∗ ={y f ∗}f ∈F
)and a price system p∗
such that
1 The allocation (x∗, y∗) is feasible, i.e., x i∗ ∈ X i for alli ∈ H, y f ∗ ∈ Y f for all f ∈ F and
∑i∈H
x i∗j ≤ ∑i∈H
ωij + ∑
f ∈Fy f ∗j for all j ∈N.
2 For every firm f ∈ F , y f ∗ maximizes profits3 For every consumer i ∈ H, x i∗ maximizes utility4 All markets clear: ∑i∈H x
i∗j = ∑i∈H ωi
j +∑f ∈F yf ∗j for
all j ∈N.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 35 / 53
Welfare Theorems (Pareto Optimality)
Definition A feasible allocation (x, y) for economyE ≡ (H,F ,u,ω,Y,X, θ) is Pareto optimalif there exists no other feasible allocation (x, y) such that
ui(x i)≥ ui
(x i)for all i ∈ H
with at least one strict inequality
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 36 / 53
First Welfare Theorem
Local nonsatiation of consumer preferences: for any bundle of goodsthere is always another bundle of goods arbitrarily close that ispreferred to it.
Formally, ∀x ∈ X and ∀ε > 0, ∃x′∈ X s.t. ‖x′ − x‖ ≤ ε and x′ ispreferred to x. If there are no "bads", it is equivalent to monotonicity.
Definition Household i ∈ H is locally non-satiated if at each x i ,ui(x i)
is strictly increasing in at least one of its arguments at x i
Theorem (First Welfare Theorem I)Suppose that (x∗, y∗, p∗) is a competitive equilibrium.Assume that all households are locally non-satiated.Then (x∗, y∗) is Pareto optimal
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 37 / 53
Second Welfare Theorem (Preliminaries) I
Converse of the First Welfare Theorem: Any Pareto optimumallocation can be decentralized as a competitive equilibrium
But requires assumptions such as the convexity of consumptionand production sets and preferences and additional requirements
Contains an “existence of equilibrium argument,”whichruns into problems in the presence of non-convexities
Under convex preferences and technology,it is easy to prove the Second Welfare Theoremwhen the number of commodity is finite
The extension to ∞ commodities is more complexbut also important for us. It requires a finite set ofcommodities to be "salient" in preferences and technology
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 38 / 53
Second Welfare Theorem (Preliminaries) II
Recall that the consumption set of each individual i ∈ H is X i
A typical element of X i is x i =(x i1, x
i2, ...
), where x it can be
interpreted as the vector of consumption of individual i at time t
Similarly, a typical element of the production setof firm f ∈ F , Y f , is y f =
(y f1 , y
f2 , ...
)Let us define the "truncations"x i [T ] =
(x i0, x
i1, x
i2, ..., x
iT , 0, 0, ...
),
y f [T ] =(y f0 , y
f1 , y
f2 , ..., y
fT , 0, 0, ...
)
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 39 / 53
Second Welfare Theorem I
Consider a Pareto optimal allocation (x∗∗, y∗∗)in an economy described by ω,
{Y f}f ∈F ,
{X i}i∈H, and
{ui (·)
}i∈H.
Suppose all production and consumption sets are convex,and all agents have continuous, convex preferencessatisfying local non-satiation
Suppose also 0 ∈ X i for each i ∈ H
Suppose ∃ T such that for any x , x ′ ∈ X i with ui (x) > ui (x ′),ui (x [T ]) > ui (x ′ [T ]) for all T ≥ T and for all i ∈ H,and that ∃ T such that if y ∈ Y f ,then y [T ] ∈ Y f for all T ≥ T and for all f ∈ F .
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 40 / 53
Second Welfare Theorem II
Then there exist p∗∗ and (ω∗∗, θ∗∗) such that in economyE ≡ (H,F ,u,ω∗∗,Y,X, θ∗∗),
1 ω∗∗ satisfies ω = ∑i∈H ωi∗∗;2 (profits are maximized) for all f ∈ F ,
p∗∗ · y f ∗∗ ≥ p∗∗ · y for all y ∈ Y f ;
3 (utilities are maximized) for all i ∈ H,
if x i ∈ X i involves ui(x i)> ui
(x i∗∗
), then p∗∗ · x i ≥ p∗∗ · w i∗∗,
where w i∗∗ ≡ ωi∗∗ +∑f ∈F θi∗∗f y f ∗∗.
Moreover, if p∗∗ ·w∗∗ > 0 [i.e., p∗∗ · w i∗∗ > 0 for each i ∈ H],then economy E has a competitive equilibrium (x∗∗, y∗∗,p∗∗).
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 41 / 53
Second Welfare Theorem (Comments) I
Remark 1:I If we have a finite commodity space, say with K commodities, then thehypothesis that there exists T with the properties defined above wouldbe satisfied automatically, by taking T = T = K .
I In dynamic economies, its role is changes in allocations at very far inthe future should not have a large effect.
Remark 2:I The conditions are stronger than for the First W.Th.i.e., stronger conditions are required for anyPareto optimal allocation to be decentralizable
I The decentralization of a Pareto optimumis subject in general to reallocation of endowments.In reality this may be diffi cult (or costly) to achieve
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 42 / 53
Second Welfare Theorem (Comments) II
Under the conditions of the SWT a competitive equilibrium must exist
The SWT motivates many macroeconomists to look for the set ofPareto optimal allocations instead of explicitly characterizingcompetitive equilibria
Real power of the Theorem in dynamic macro models comes when wecombine it with models that admit a representative household
Enables us to characterize the optimal growth allocation thatmaximizes the utility of the representative household and assert thatthis will correspond to a competitive equilibrium
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 43 / 53
Sequential Trading and Arrow-Debreu Equilibrium
GE models assume that all commodities are tradedat a given point in time– once and for all.
Arrow-Debreu Equilibrium (AD):all households trade at t = 0 and purchase or sell irrevocableclaims to commodities indexed by date and state of nature.
In models of economic growth, it is realistic to modeltrade as taking place at different points in time
Sequential Trade Equilibrium (ST):separate markets open at each t, and households trade labor,capital and consumption goods in each such market.
With complete markets and time-consistent preferences,AD equilibrium and ST equilibrium are equivalent.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 44 / 53
A Simple Example
Consider a simple dynamic exchange economy, t ∈ {0, 1}.There are 2 goods at each date, j ∈ {A,B}.Denote the consumption of good j by household i at time t as x ij ,t .
Goods are perishable, so that they are indeed consumed at time t.
Each household i ∈ H has a vector of endowment(ωiA,t ,ω
iB ,t
)at time t, and (time-consistent) preferences
U =1
∑t=0
βti ui (x iA,t , x iB ,t) .
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 45 / 53
Arrow-Debreu EquilibriumAn AD equilibrium is given by two vectors
x∗ = (xA,0, xB ,0, xA,1, xB ,1)
p∗ = (pA,0, pB ,0, pA,1, pB ,1)
where xj ,t ={x ij ,t}i∈H
is a vector of households’consumptions
and p∗A,0 = 1 (numeraire)Each agent i ∈ H trades only at t = 0 subject to a budget constraint
x iA,0 + pB ,0xiB ,0 + pA,1x
iA,1 + pB ,1x
iB ,1
≤ ωiA,0 + pB ,0ω
iB ,0 + pA,1ω
iA,1 + pB ,1ω
iB ,1
Moreover, market clearing requires
∑i∈H
x ij ,t = ∑i∈H
ωij ,t for j ∈ {A,B} and t ∈ {0, 1}
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 46 / 53
Sequential Trading Equilibrium IEquilibrium with sequential trading:
I Markets for goods dated t open at time t.I There is a bond b in zero net supply that can be traded at t = 0(with T + 1 periods, there would be T one-period bonds, b1 traded att = 0, b2 traded at t = 1, etc.)
I b pays one unit of one of the numeraire good (A) at time one(with T + 1 periods, bt pays one unit of A at time t).
Let q denote the price of the bond expressed in units of Aat time zero (with T periods, there are T such prices, qt).Thus, a household can purchase a unit of bond at time 0 by payingq units of good A and will receive one unit of good A at time 1Since each bond is in zero net supply, market clearing requires
∑i∈H
bi = 0
With T periods
∑i∈H
bit = 0 for each t = 0, 1, ...,T .
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 47 / 53
Sequential Trading Equilibrium II
At time one, each individual uses his endowment plus (or minus)the proceeds from the bonds to buy commodities at date 1
With ST, it is convenient (and legitimate) to choose aseparate numeraire for each date, p∗∗A,0 = p
∗∗A,1 = 1
Therefore, the budget constraint of household i ∈ Hgiven equilibrium price vector (p∗∗, q∗∗):
x iA,0 + p∗∗B ,0x
iB ,0 ≤ ωi
A,0 + p∗∗B ,0ω
iB ,0 − q∗∗bi ,
x iA,1 + p∗∗B ,1x
iB ,1 ≤ ωi
A,1 + p∗∗B ,1ω
iB ,1 + b
i .
(recall: bi can be negative - an agent who borrows)
Let a ST equilibrium be denoted by (p∗∗,q∗∗, x∗∗,b∗∗)
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 48 / 53
Equivalence TheoremTheorem (Sequential Trading) If (p∗, x∗) is an AD equilibrium,
then there exists a ST equilibrium (p∗∗,q∗∗, x∗∗,b∗∗),such that x∗= x∗∗,p∗∗j ,t = p
∗j ,t/p
∗A,t for all j ∈ {A,B, ...} and t and
q∗∗t = p∗A,t/p∗A,t−1 for all t > 0.
Conversely,if (p∗∗,q∗∗, x∗∗,b∗∗) is a ST equilibrium,then there exists an AD equilibrium (p∗, x∗)with x∗= x∗∗,p∗j ,t = p
∗∗j ,tp∗A,t for all j and t, and p
∗A,t = q
∗∗t · q∗∗t−1 · ... · q∗∗1
for all t > 0.
In growth theory we focus on economies with STand assume that there exist assets to transfer resources across dates.These assets might be riskless bonds in zero net supply,or even the capital stock.The interest rates, rt = 1/qt , is the key relative pricein dynamic macroeconomic models.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 49 / 53
Optimal Growth in Discrete Time I
Economy characterized by an aggregate production function,and a representative household.
Optimal growth problem in discrete time with no uncertainty,no population growth and no technological progress:
max{c (t),k (t)}∞
t=0
∞
∑t=0
βtu (c (t))
subject to
k (t + 1) = f (k (t)) + (1− δ) k (t)− c (t) ,
k (t) ≥ 0 and given k (0) = k0 > 0.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 50 / 53
Optimal Growth in Discrete Time II
The optimal growth model can be solved in a number of differentways (e.g., infinite dimensional Lagrangian), but the most convenientis by dynamic programming
V (k) = maxc ,k ′
{u (c) + βV
(k ′)}
s.t. k ′ = f (k) + (1− δ) k − c
Note:I We can either solve it the optimal growth modeland appeal to the Second Welfare Theorem...
I ... or directly solve for competitive equilibria
The two problems will be formally identical
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 51 / 53
Optimal Growth in Continuous Time
The formulation of the optimal growth problemin continuous time is very similar:
max[c (t),k (t)]∞t=0
∫ ∞
0exp (−ρt) u (c (t)) dt (2)
subject tok (t) = f (k (t))− c (t)− δk (t) ,
k (t) ≥ 0 and given k (0) = k0 > 0.Most convenient way of characterizing the solution tothis problem is via optimal control theory.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 52 / 53
Conclusions
The models we study in this course are examplesof more general dynamic general equilibrium models.
First and the Second Welfare Theorems are essential.
The most general class of dynamic general equilibrium models are nottractable enough to derive sharp results about economic growth.
Need simplifying assumptions, the most important one beingthe representative household assumption.
Sequential trade is the natural assumption in dynamic economies.However, we can exploit equivalence results with Arrow-Debreuequilibrium and invoke standard results in GE theory.
Kjetil Storesletten (University of Oslo) Lecture 1 August 24, 2018 53 / 53