Introduction to Dynamic modeling, Summaries of Macroeconomics

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Lecture 5

Preliminaries: Building Dynamic Models

Econ 602 Spring 2020

Ibn Haldun University

April 13, 2020

A Simple Dynamic Economy General Principles

General Principles for Specifying a Model

A model typically has at most 3 types of entities that take decisions. (^1) Households: Preferences over commodities,endowment of commodities. (^2) Firms: Production technology available (^3) Government: Policy instruments controlled. Information set of decision makers Equilibrium concept: How agents perceive their power to affect market prices. Competitive equilibrium: All agents in the model are price takers.

Assumptions about the utility function

The above utility function satisfies the following assumptions. (^1) Time separability: Total utility from a consumption allocation ci^ equals the discounted sum of period utilities. Period utility at time t only depends on the consumption in that period, not on consumption in other periods. (^2) Time discounting: β < 1 implies that agents are impatient. β is the subjective time discount factor. The subjective discount rate ρ is defined as β = (^1) +^1 ρ. Often intimately related to the interest rate in the economy. (^3) Other standard properties of the utility function is that it is continuous, twice continuously differentiable, strictly increasing (U′(c) > 0), strictly concave (U′′(c) < 0) and satisfies the Inada conditions. limc→ 0 U′(c) = +∞ limc→+∞ U′(c) = 0

Endowment Streams

Agents have deterministic endowment streams ei = (eit )∞ t= 0 of the consumption goods given by

e t^1 =

2 if t is even 0 if t is odd

e t^2 =

0 if t is even 2 if t is odd

There is no risk in this model and both agents know their endowment pattern perfectly in advance. All information is public i.e. all agents know everything. At period 0, two agents meet at a central market place and trade all commodities i.e. trade consumption of all future dates. Let pt denote the price in period 0 of one unit of consumption good to be delivered in period t. Both agents behave competitively, they take the sequence of prices (pt )∞ t= 0 as given and beyond their controls and they make decisions.

Arrow-Debreu Equilibrium

Note that the budget constraint can be written as ∞

t= 0

pt (eti − cit ) ≥ 0 (6)

The quantity eti − cti is the net trade of consumption of agent i for period t which may be negative or positive. In equilibrium, prices are right in the sense that they induce agents to choose consumption so that total consumption equals total endowment in each period.

Definition of Equilibrium

Definition A competitive (Arrow-Debreu) equilibrium, are prices { pˆt }∞ t= 0 and allocations ({cti }∞ t= 0 )i =1,2 such that (^1) Given { pˆt } t∞= 0 , for i=1,2 { cˆti }∞ t= 0 solves the optimization problem expressed in 2. (^2) Goods market clearing condition holds. i.e. there is no free disposal of goods e t^1 + e t^2 = cˆ^1 t + cˆ t^2 (7)

Solving for the Equilibrium

Using the optimality condition expressed in Equation (12) together with the budget constraint we can solve for the optimal sequence of consumption bundles of household i as a function of the infinite sequence of prices and endowments in the economy. i.e.

cti = cti ({pt }∞ t= 0 ) Plugging this into the market clearing condition cti ({pt }∞ t= 0 ) + cti ({pt } t∞= 0 ) = e t^1 + e^2 t ∀t

This is a system of infinite equations in an infinite number of unknowns ({pt }∞ t= 0 ). Hard to solve. Remedy: Reduce the number of equations and unknowns to a smaller number.

Solving for the Equilibrium

For our simple economy it is easy to achieve a solution. Sum Equation (12) across agents to obtain pt+ 1 (c t^1 + 1 + c t^2 + 1 ) = β pt (c t^1 + c t^2 ) Using the goods market clearing condition, we find that

pt+ 1 e t^1 + 1 + e t^2 + 1 = β pt e^1 t + e t^2 and hence

pt+ 1 = β pt therefore equilibrium prices are of the form

pt = β t^ po Without loss of generality, we can set po = 1, i.e. make consumption in period 0 the numeraire.

Solving for the Equilibrium

The two agents differ only along one dimension: agent 1 is rich first. This is an advantage given that prices are falling over time. The right hand side of the budget equation becomes ∞

t= 0

p ˆt e t^1 = 2

∞

t= 0

β^2 t^ =

1 − β^2 For agent 2 it becomes ∞

t= 0

p ˆt e t^2 = 2 β

∞

t= 0

β^2 t^ =

2 β 1 − β^2 The equilibrium allocation is then given by

c ˆ t^1 = cˆ o^1 = ( 1 − β )

1 − β^2

1 + β

c ˆ t^2 = cˆ o^2 = ( 1 − β )

2 β 1 − β^2

2 β 1 + β

Solving for the Equilibrium

These allocations obviously satisfy c ˆ^1 t + cˆ t^2 = 2 = eˆ^1 t + eˆ t^2 ∀t The mere fact that agent 1 is rich at first makes him consume more in each period. Note that there is substantial trade going on In each even period, the first agent delivers 2 − (^1) +^2 β to the second agent. In all odd periods, the second agent delivers 2 − (^12) + ββ to the first agent. Note that this trade is mutually beneficial for both agents

Pareto Optimality Welfare Theorems

For this economy the competitive equilibrium is socially optimal. First let’s define what socially optimal means. Notion of optimality: Pareto efficiency Loosely speaking an allocation is Pareto efficient if it is feasible and if there is no other feasible allocation that makes no household worse off and at least one household better off. The precise definition of a feasible allocation is

Definition An allocation (c^1 t , c t^2 )∞ t= 0 is feasible if and only if (^1) cti ≥ 0 ∀t, for i=1,2. (^2) c t^1 + c t^2 = e t^1 + e t^2 ∀t

Feasibility requires that consumption is nonnegative and satisfies the resource constraint for all period t=0,1...

Pareto Optimality and Welfare Theorems

A more proper definition of Pareto efficiency would then be

Definition An allocation (c^1 t , c t^2 )∞ t= 0 is Pareto efficient if it is feasible and if there is no other feasible allocation ( c˜ t^1 , c˜ t^2 )∞ t= 0 such that u( c˜i^ ) ≥ u(ci^ )for both i=1,2. u( c˜i^ ) > u(ci^ ) for at least one i=1,2.

Important note: Pareto efficiency has nothing to do with fairness in any sense. An allocation in which agent 1 consumes everything in every period and agent 2 starves is Pareto efficient. We can only make agent 2 better off by making agent 1 worse off.

Negishi’s Method to Compute Equilibria

In the example above, it was straightforward to compute the competitive equilibrium by hand. This is usually not the case for the dynamic general equilibrium models. Now we describe a method to compute equilibria for economies in which the welfare theorem holds. The main idea is to compute the Pareto optimal allocations by solving an appropriate social planner’s problem. The social planner problem is a simple optimization problem and does not involve any prices. If the first welfare theorem holds, then all competitive equilibrium allocations are Pareto optimal. Hence by solving for the Pareto optimal allocation we will then have solved for the competitive equilibrium. Equilibrium prices that support these economies will be derived from the appropriate Lagrange multipliers.

Negishi’s Method to Compute Equilibria

Benevolent social planner Objective: To maximize the overall welfare in the economy. The weighted sum of utilities of all agents Constraint: Does not take prices into account. Only feasibility of the allocations matter. Consider the following social planner’s problem.

max {c^1 t ,c t^2 }∞ t= 0

α u(c^1 ) + ( 1 − α )u(c^2 ) (13)

= max {c t^1 ,c^2 t } t∞= 0

∞

t= 0

β t^ [ α ln(c t^1 ) + ( 1 − α )ln(c t^2 )]

s.t. cit ≥ 0 ∀t, for i=1,2. c t^1 + c t^2 = e t^1 + e t^2 = 2 ∀t for a Pareto weight of α ∈ [0, 1].