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Lecture 14: Time and Uncertainty

Advanced Microeconomics I

Yosuke YASUDA

Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp

November 18, 2014

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Interpretation of General Equilibrium Model

The simplest general equilibrium models take no explicit account of time or uncertainty; a one-period model where all allocation is

(i) at a single date ⇒ timeless?

(ii) in a given certain environment ⇒ no uncertainty?

Make the markets for goods (i) over time or (ii) under uncertainty look just like those in the general equilibrium model.

All we have to do is re-interpret the commodity space, Rn. Then, the same formal results as before will follow:

(i) Establish an intertemporal equilibrium and intertemporally efficient allocation.

(ii) Establish an equilibrium for goods across uncertain events and an efficient allocation of risk bearing.

2 / 14

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Incorporate Time and Uncertainty

To incorporate Time Let xkt denote the amount of good k consumed at date t. If there are two goods k = 1, 2, and two dates t = 1, 2, then,

A consumption bundle (x11, x12, x21, x22) is a vector of four numbers. That is, there are four distinct goods.

⇒ x12 is the amount of good k = 1 consumed at date t = 2.

Ex k = {apple, orange} and t = {today, tomorrow}. To incorporate Uncertainty Let xks denote the amount of good k consumed at state of nature (/the world) s.

⇒ x12 is the amount of good k = 1 consumed at state s = 2.

Ex k = {umbrellas, sunscreen} and s = {sunny, rainy}.

⇒ Incorporate both via contingent commodities.

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Intertemporal Preferences

We assume that the consumer has preferences over streams of consumption over time. The following simplifications are common:

Additively Separable Form

U (c1, ..., cT) = XT

t=1

ut(ct)

This form allows different utility function in different period, ut. The next form assumes the same utility function in each period, but with (exponential) discounting by a discount factor δt.

Time-Stationary Form

U (c1, ..., cT) = XT

t=1

δtu(ct)

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Overlapping Generations Model (1)

Consider an economy with the following with the following structure called overlapping generations (OLG) model.

Each period, n agents are born; each lives for two periods. At any time after the first period, 2n agents are alive: n young agents and n old agents.

Each agent has an endowment of 2 units of consumption when she is born, and

Indifferent between consumption when she is young and old. Theorem 1

There is a competitive equilibrium in this OLG model wherept is non-decreasing int and every agent consumes her endowment when she is young, which is not Pareto efficient.

⇒ Is this violating the first welfare theorem??

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Overlapping Generations Model (2)

Proof.

Suppose that each member of generation t + 1 transfers one unit of its endowment to generation t. Now generation 1 is better off since it receives 3 unit of consumption in its lifetime. None of the other generations are worse off.

⇒ Pareto improvement on the original equilibrium!

Q Why does the first welfare theorem fail?

A There are an infinite number of goods:

The value of both the aggregate consumption stream and the aggregate endowment become infinite.

The contradiction in the last step of the proof of the first welfare theorem no longer holds.

⇒ One should be very careful in extrapolating results of models

with finite horizons to models with infinite horizons. 6 / 14

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Uncertainty in Edgeworth Box (1)

Consider an economy consisting of two consumers and one good, wheat. The endowments of the agents in wheat depend on the weather: Agent 1 (/2) has an endowment of

w11 (/w12) if the weather is nice ⇒ good 1. w12 (/w22) if the weather is bad ⇒ good 2.

Suppose that consumer i has preferences over the contingent consumption plans that satisfy expected utility hypothesis:

Ui(xi1, xi2) = π1ui(xi1) + π2ui(xi2)

where π12) is the objective probability of nice (bad) weather. Assume further that consumers are risk averse, i.e., ui is concave.

Fg Draw the Edgeworth Box. What if an agent is risk neutral?

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Uncertainty in Edgeworth Box (2)

Analyze the competitive equilibrium in this exchange economy.

Rm The marginal rate of substitution is constant along the diagonal, i.e., xi1 = xi2, since

∂Ui/∂xi2

∂Ui/∂xi1 = π2

π1

dui/∂xi2 dui/∂xi1 =

π2

π1

.

When there is no macroeconomic risk, i.e., w11+ w21 = w12+ w22: The equilibrium must be an allocation on the diagonal in which agents’ consumption is independent of the weather. Both consumers are perfectly insured: xi1 = xi2.

The price ratio is equal to the ratio of probabilities: pp1

2 =

π1 π2.

When there is macroeconomic risk, i.e., w11+ w21 > w12+ w22:

p1 p2 <

π1

π2 must hold and insurance is imperfect for both agents.

8 / 14

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Contingent Commodities

Each (contingent) commodity is characterized by

1 what it is (its description)

2 where it is available (its location)

3 when it is available (its date)

4 under what condition it is deliverable (the state of the world) Definition 2

A contingent commodity xkts is a promise of delivery of a particular good or service k at a particular date t if an uncertain event s actually occurs.

The price is not necessary the price of a definite consumption. Instead, the price of a contingent commodity, a specific good deliverable if a specified event occurs.

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Arrow-Debreu Equilibrium (1)

In principle, time/date can be incorporated in the state of nature. Consider an exchange economy with I agents and K goods:

Distinguish two dates: date 0 (ex ante), date 1 (ex post). There are S mutually exclusive state of nature.

At date 0, the future is uncertain. At date 1, each agent observes the realized state and consumes accordingly. Let xiks be the quantity of good k ∈ K consumed in state s ∈ S by agent i ∈ I.

Let wksi be the initial endowment of good k ∈ K for agent i ∈ I in state s ∈ S.

If a market for each good k and every state s is created at date 0, we have a system of (complete) Arrow-Debreu markets.

10 / 14

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Arrow-Debreu Equilibrium (2)

Let pks be the system of prices for k ∈ K and s ∈ S. Each consumer i solves the following problem:

max Ui(xi) =X

s∈S

πsui(xis) s.t. X

s∈S

X

k∈K

pksxiksX

s∈S

X

k∈K

pkswiks

Definition 3

An Arrow-Debreu equilibrium is a system of prices p ∈ RKS+ and an allocation (x∗1, ..., xI) such that

1 Optimization x∗i is a solution to the above maximization problem for p for any i = 1, ..., I.

2 Feasibility Pi∈Ix∗i =Pi∈Iwi.

Rm The existence of equilibrium and the fundamental theorems holds under the assumptions similar to the standard model.

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Arrow-Debreu Equilibrium (3)

Definition 4

An allocation (x∗1, ..., xI) is called

1 ex ante Pareto efficient if there exists no feasible allocation x such that

Ui(xi) ≥ Ui(x∗i) for all i ∈ I, and Uj(xj) > Uj(xj) for some j ∈ I.

2 ex post Pareto efficient if there exists no state s and no feasible allocation xs such that

ui(xis) ≥ ui(x∗is ) for all i ∈ I, and uj(xjs) > uj(x∗js ) for some j ∈ I. Theorem 5

An ex ante Pareto efficient allocation is always ex post Pareto

efficient, but the converse is not true. 12 / 14

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Arrow-Debreu Securities

Rm The previous theorem implies that re-opening the (spot) markets after the realization of the state s would not lead to any trade since the allocation is ex post Pareto efficient as well. Definition 6

An Arrow-Debreu security is a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state of nature occurs and pays zero in all the other states. Suppose there exist no market for contingent commodities, but Arrow-Debreu securities for all state s ∈ S are exchanged at t = 0.

Each agent can reallocate her resources among different states of nature by using the securities markets.

If the expectations of spot market prices at t = 1 are all correct, the Arrow-Debreu equilibrium allocation is achieved. Given this perfect foresight, we can organize the economy with S + K markets rather than S × K contingent markets.

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Arrow-Debreu Model: Remarks

It is remarkable how much additional mileage we are able to get from a model that appears entirely static and deterministic simply by reinterpreting its variables! The below is important remarks on Arrow-Debreu model saying it assumes that

1 There is perfect monitoring in the sense that it is not possible for a firm or consumer to claim that he can supply more units of a basic good in state s at date t than he actually can supply. Thus, bankruptcy is assumed away.

2 There is perfect information in the sense that all agents are informed of the state when it occurs at each date. If only some were informed of the state, they might have an incentive to lie about which state actually did occur.

3 All contracts are perfectly enforced.

Clearly, each of these assumptions is strong and rules out important economic settings.

14 / 14

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