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Lecture 4: Infinite-horizon dynamics

- Introduction
- Dynamic models with finite time horizon
- Infinite horizon: dynastic models
- Infinite horizon: overlapping generations models of exchange economies
- Infinite horizon: overlapping generations models with production
- Remarks

Aim of lecture 4

- Explicitating dynamic behavior of consumers and producers
- Highlighting main issues related to models with infinite horizon models
- Highlighting main differences between dynastic and OLG models of infinite horizon

Introduction

- Recapitulating lectures 1-3
- Competitive equilibrium can be represented as welfare optimum with welfare weights that are such that budgets hold (Negishi)
- This facilitates direct link to welfare analysis, among other things the welfare gains from (tax) reforms
- Other formats, such as open economy format, full format, and CGE format are useful for the application of general equilibrium theory to specific issues
- Within CGE format, macro-economic mechanisms, including savings rules can be included by closure rules, but care should be taken that the closure rule does not become the dominant mechanism in the model
- Within all formats, direct and indirect taxes can be introduced

Introduction (continued)

- Models discussed so far had no explicit time subscript
- However, commodity classification can distinguish between different dates of delivery
- Consumer and producer decisions can then be given a dynamic interpretation, but time remains implicit
- Now: time dependence is made explicit and specific dynamics are specified
- First: consider models with finite time horizon

Dynamic models with finite time horizon

- Dynamics
- Number and composition of commodities: new products entering the economy via endogenous or exogenous processes
- Number and composition of agents
- Social classes
- Age cohorts
- Occupational groups
- Some of these changes can be represented through migration (lecture 1), some are changes in the characteristics of the agents themselves
- Entry and exit of producers: are all producers potentially active in all periods or not?

Dynamics in producer decisions

- Goods produced in t+1 cannot serve as inputs for production in t
- Production process must be separable over time and can be represented as:
- Where is the net supply, is the stock of capital goods used as inputs, and stands for the stocks of capital goods that will made available as inputs for the production in time t+1
- Capital goods are goods that can be transferred through time. They can be produced, increase at exogenous rates or remain fixed over time

Dynamics in producer decisions (continued)

- Optimization problem of producer:
- As in lecture 3, additional restriction and variable is added to recover price; in this case price of capital
- Storage is interpreted as a productive activity, transforming goods at time t to goods at time t+1

Dynamics in producer decisions (continued)

- The objective of the firm is to maximize the sum of its profits over the finite time horizon.
- This does not rule out negative receipts at specific points in time
- It enables firms are to borrow to invest in expanding production capacity in later periods
- Note: there is no interest rate in the model: the ratio represents discounting as well as relative scarcity of each of the commodities in the model

Dynamics in consumer decisions

- Consumer maximizes intertemporal utility function subject to intertemporal budget
- Separability of utility over time not obvious, but is often imposed (“no regret”)
- Common specification of intertemporal utility function assumes that the discount factor is constant over time for consumer i. More general specifications can be used that maintain the “no regret” property (viz. chapter 2, section 2.2.5)

Dynamics in consumer decisions (continued)

- Intertemporal budget:
- This can be rewritten to explicitate net savings and net accumulation of worth :

Dynamics in consumer decisions (continued)

- Prices in budgets are discounted at commodity-specific rates, just as in producer case
- Differences between prices in different time periods cannot be interpreted as interest rate nor as rate of discount by individuals (viz prices in different countries and exchange rate)

Dynamic implications of market demand equilibrium conditions

- In additional to intertemporal budgets and intertemporal profits, at each time t, commodity balances must hold
- This implies that at each time t, there are no net aggregate savings or lendings: savings and lendings act as transfers between agents
- This also implies that there can be no accumulation of aggregate net worth over time, and no debt inherited by the economy as a whole:
- Prices adjust to restore equilibrium between demand and supply at every time t: the “rate of interest” that restores balance between savings and lendings is implicitly accounted for by these commodity-specific prices

Finite horizon dynamics in CGE

- CGE model for T-period equilibrium has exactly the same structure as CGE model of Chapter 3, except for commodity classification
- Dynamics with respect to production enter by specifying investment functions for replacement of existing capital goods and new capital goods. Disaggregation of capital into different vintages is possible
- Investment can also be set exogenously as part of public consumption, in which case some closure rule is needed
- Consumer dynamics enter by specifying savings functions that may or may not be derived from intertemporal optimization by consumers. If investment is set exogenously, consumer savings may also be assumed to be the adjusting variable, and no behavioral equation is then specified
- Population dynamics can be made endogenous by specifying functional relations between economic variables and population growth, e.g. by basing migration decisions on wage differentials

Difficulties in going from finite to infinite horizon

- Number of commodities goes to infinity
- Number of prices goes to infinity
- Number of agents goes to infinity (OLG)
- In production economies: value of production and hence of consumption can become infinite

Infinite horizon: dynastic model

- intertemporal welfare program for given welfare weights:

Dynastic model (continued)

- Finite and constant number of dynasties over the whole horizon of the model
- Infinite horizon may cause solution of program to be unbounded (production can become infinite)
- Therefore, impose additional constraints to ensure boundedness of dynastic utility function (CD1)
- A stronger assumption is to assume CD1 with the additional requirement of equal discount rate for all consumers.
- Then, the dynastic utility function can be written as:
- Note: if discount rates differ between consumers, then eventually, only the most patient one will consume

Dynastic model (continued)

- With constant and identical discount rate for all consumers, difficulties associated with infinite horizon can be avoided:
- determine expenditure as difference between full expenditure and consumer surplus, using the extended (homogeneous of degree one) utility function with as in lecture 1.
- And using the derived intertemporal utility function, the dynastic model can be reformulated as a dynamic program:

Dynastic models: multiplicity and indeterminacy

- If the shadowprices on capital use are unique, then they are a continuous function of welfare weights and there may be several regular equilibria, but their number is finite
- when period-specific distortions are introduced, then the number of equations and unknowns becomes infinite and it may happen that there is a continuum of equilibria
- This is referred to as indeterminacy
- Geanakoplos and Polemarchakis (1991) suggest this gives room for policy intervention (e.g. through institution with infinite horizon budget)
- Indeterminacy makes it impossible to determine the value of stocks without having to impose some arbitrary valuation at time T

Dynastic models: dynamics and steady states

- Steady state: using the optimal values of and as initial values will reproduce the same and a new , proportional to the previous one by the same factor .
- In steady state of dynastic models, welfare weights are not such that budgets hold, but such that aggregate capital stocks and prices of capital remain constant (upto scaling)
- Convergence to steady state:
- global asymptotic convergence not natural property of trajectory
- Single man-made capital good that is also consumed (Ramsey model): unique steady state and convergence
- multicommodity case: convergence depends on discount rate, curvature of utility and transformation function

Dynastic models: real business cycles

- Research in RBC concentrates on reproducing moments of observed business fluctuations in macro-economic variables
- using stochastic dynastic models with agents having rational expectations
- models are simple: one representative dynastic consumer, one producer
- stochastic element is introduced in transformation function

Dynastic models: rbc models (continued)

- RBC literature uses stochastic version of dynastic model:
- For given and the distribution of the scalar error term , and where E[] denote the expectation operator.
- Usually, is specified according to the autocorrelated process

, where follows some stochastic distribution that is independent across time periods and is a non-negative autocorrelation term.

Dynastic models: rbc models (continued)

- Under appropriate assumptions on the way the disturbance enters the transformation function, it is possible to represent the program as a single-period dynamic programming formulation:
- The basic problem is that is not known, and only an approximation of it can be obtained from a finite data set
- Working with first-order conditions directly is not possible since the number of equations would be infinite

Dynastic models: rbc models (continued)

- Solving problem for finite horizon, finite number of states
- Solving problem for finite horizon, continuum of states
- Sampling from the continuum of states, approximation of the value function in the neighborhood of the steady state, or of the distribution
- Solving problem for infinite horizon, finite number of states
- Approximate value function, deriving policy functions from infinite number of first order conditions, approximate model itself
- Solving the model for infinite horizon and infinite number of states
- Approximate the model itself or the Euler equations (first order conditions). But: convergence of Euler equations is not guaranteed (Judd, 1998).

Dynastic models (concluding)

- Dynastic models are natural counterpart of welfare program
- Strength of dynastic models
- Possibility to reformulate program in dynamic programming format
- Efficiency of the solution
- Weaknesses of dynastic models
- Generations alive at time t do not possess the resources available at that time
- Assumptions on discount rate unrealistic

Full format representation of OLG model

- Infinite number of consumers and budgets
- Infinite number of goods and prices

OLG models: excess demand representation

- OLG model is started in year t=1, with consumer born in t=0 coming in with a claim M0 and with commodity endowment . His consumption is a given parameter in his optimization problem:
- Claim M0 is a financial claim not backed by commodities that can be sold to new generations Walras’ law will not hold over any finite number of periods

OLG models: claims

- Different types of claims:
- is equal to , where is a given vector expressed in terms of commodities (a real claim); this is the claim used in Chapter 8
- is a fixed amount (a nominal claim)
- is enforced by a government through a direct tax levied on the generation born in t=1, which in turn will have a claim on the next generation

OLG models: equilibrium with real claims

- The allocation supported by the price vectors , which are bounded for any finite t, is an OLG pure exchange equilibrium if consumers solve:
- and all markets clear:

OLG models: efficiency and indeterminacy

- Inefficiency of OLG models stems from the possibility to shift debt to the future indefinitely
- Efficiency can be ensured by ensuring that incomes converge to zero at infinity, since then, shifting the burden to the future is not possible . In exchange economy, assuming prices go to zero in infinity establishes zero income at infinity
- indeterminacy:
- infinite number of prices, so equilibrium problem is in infinite dimensional space
- existence proof depends on arbitrary set consumption in period T
- in one-commodity economy with two-period lived consumers, there is no indeterminacy

OLG models: dynamics and steady-state

- As in the case of dynastic models, path followed through time may diverge, converge, cycle, or be chaotic
- Dynamics of pure exchange model are described by time invariant excess demand function where is the net demand by the old, while is the net demand by the young
- A steady state is a price vector and a discount factor such that

which by homogeneity of is equivalent to

OLG models: steady state characterization

- Rewriting of condition yields:
- This implies that steady state can be of two types
- if . The value of the claim is zero real steady state
- if , there is no restriction on and there may be savings monetary steady state.
- If , the steady state describes a path with prices converging to zero, hence, steady state is efficient
- In monetary steady state, convergence to a steady state is problematic

OLG models with production

- Introduction of production allows savings to be specified by buying and selling capital stock
- Discussion on inefficiency can be extended to production case
- The same holds for indeterminacy
- In OLG with production, there are no nominal claims, so every steady state is a real steady state
- if , prices go to zero and the steady state is efficient
- if , then prices rise over time, and the steady state is inefficient
- if , then . Efficiency cannot be assured

OLG models (concluding)

- OLG models are natural counterpart of competitive equilibria
- Strength of OLG models
- Welfare of future generations does not depend on benevolence of single planner as in dynastic models
- Increases in population size and shifts in composition of population can be easily accommodated
- Generations that live in the present have full control over present resources
- Weaknesses
- No possibility to represent marriages and no explicit relation between ancestors and children
- Efficiency needs to be ensured explicitly by additional assumptions

Discussion of main theorems

- Proposition 8.2 (existence of dynastic Negishi equilibrium)
- Proving (8.8) is a standard convex program, using assumed boundedness of production (PD1:3), and utility (CD1:5). Convexity is shown and closedness follows from Lucas/Stokey
- Since it is a standard convex program, the value function is bounded and continuous
- Then, perturbation theorem and envelope theorem can be used to further characterize value function
- For Negishi update, budget deficits need to be calculated.
- Using boundedness of value function and assumed strict concavity of Wi, budget deficits can be expressed as function of finite number of parameters
- Construct fixed point mapping as in Proposition 3.1

Discussion of main theorems (continued)

- Proposition 8.5 (existence of equilibrium of pure exchange OLG model with claims)
- Prove equilibrium in truncated model, along lines of proposition 3.3
- Extending T to infinity
- Since prices cannot be normalized over infinite time horizon, reformulate model with consumer-specific normalizations
- From (1) it follows that there is at least one optimal value for consumption in period T by consumer born in T
- As T increases, the number of optimal values decreases, and with that, the number of optimal trajectories decreases
- To prove that there exists an optimal solution at infinity, define price vectors that are the optimal prices for t
T. This implies construction of a sequence of prices - The set of all possible trajectories and corresponding price sequences is compact (Tychonoff theorem: the infinite-dimensional product of compact sets is compact)
- Therefore, the set of all possible trajectories forms a nonincreasing sequence of non-empty sets and the set is nonempty at infinity
- Recover prices from the consumer-specific prices
- Show that consumer demand is continuous in infinite time using A.6.2.
- Therefore, consumer demand converges as as the set of all possible trajectories converges for T to infinity

Discussion of main theorems (continued)

- Proposition 8.7 (existence of equilibrium of OLG model with production
- As in 8.5, first prove equilibrium in truncated model
- Then, extend T to infinity
- By same reasoning as in 8.5, there is at least one optimal value for consumption and one optimal value of capital stock in period T, and the set of all possible trajectories forms a nonincreasing sequence of non-empty sets and the set is nonempty at infinity
- Recover prices from the consumer-specific prices. Here, an additional problem is that welfare may become unbounded. Boundedness of the value of initial capital stock is proved by contradiction: if it were unbounded, the income of consumer 0 would be infinite, and since his utility function is non-satiated by assumption, his demand would be infinite. This cannot be an equilibrium and therefore, the value of initial capital stocks must be bounded
- Show that consumer demand is continuous in infinite time using A.6.2. as in 8.5
- Therefore, consumer demand converges as as the set of all possible trajectories converges for T to infinity

Remarks

- Four important differences between dynastic and OLG models
- in the dynastic model, generations do not overlap
- each generation is altruistic
- in the OLG model, generations have to buy capital but receive endowments freely, whereas in the dynastic model, consumers receive capital and possibly endowments as an explicit gift
- in the dynastic model, the individual consumer can borrow from every other agent in the economy; in the OLG model, the young cannot borrow from the old, since they cannot repay them later

Remarks (continued)

- It is possible to rewrite a dynastic model of infinite horizon as an infinite number of consumers who live for one period only and care for the welfare of their immediate descendants:
- If dynasties no longer care for their descendants, they are no longer linked, the wealth constraint needs to be split and welfare weights given to each dynasty.
- If this occurs an infinite number of times, we are in the OLG framework
- The inclusion of altruism in OLG models introduces a dynastic feature, since it links generations.

Remarks (continued)

- In e.g. applications of dynamic models to environmental issues, one would like to impose a social discount rate that is lower than that of individuals
- In OLG models, the interpretation as a dynastic model shows that welfare weights of the individuals living at time t are determined endogenously. Hence, the discount factor is also endogenous.
- In dynastic models, every dynasty has an explicit discount rate, and imposing a low discount rate, equal across dynasties, is possible in principle, but the value of such an exercise for policy purposes is questionable

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