Chapter 1: General Equilibrium with Uncertainty

1. Introduction

Consider an economy with 2 consumers and 1 consumption good. There are 2 periods, t = 0 (today) and t = 1 (tomorrow). The agents do not know the state of the world at t = 1. To simplify, we will assume that there are two possible states (alternatives), e = e₁, e₂. The probabilities of each state are: p(e₁) = π, p(e₂) = 1 − π.

ππππ 1111−−−− ππππ

e₁ e₂ t = 0

t = 1

w₁¹

w₁²

w₂¹

w₂² Agent 1 Agent 2

x₁¹

x₁²

x₂¹

x₂² Agent 1 Agent 2

w₁¹

w₁²

w₂¹

w₂²

x₁¹

x₁²

x₂¹

x₂² +

+

+

+

= w¹

= w²

=

=

ππππ

1111−−−− ππππ

We consider that there is a unique good in the economy that may be consumed only at t = 1 (that is, there is no consumption at t = 0.) Let us use the following notation:

• x^s_i is the amount of the good that agent i consumes in state e_s.

• w^s_i are the initial endowments that agent i has in state es.

• w^s = w^s₁+ w₂^s, s = 1, 2.

• The utility functions of the agents are

u_i(x¹_i, x²_i) = πu_i(x¹_i) + (1 − π)u_i(x²_i) i = 1, 2 i.e. agents maximize expected utility.

1

We may represent this situation using Edgeworth’s box

x¹₁ x²₁

x¹₂ x²₂

x¹₁ x²₁

(w¹₁+ w¹₂, w²₁+ w²₂)

2. Pareto Efficiency

Let us compute the Pareto efficient allocations. These are solutions of the following optimization problem

max πu₁(x¹₁) + (1 − π)u₁(x²₁) subject to πu₂(x¹₂) + (1 − π)u₂(x²₂) ≥ ¯u

x¹₁+ x¹₂ = w¹ x²₁+ x²₂ = w² The Lagrangian is

L = πu₁(x¹₁)+(1−π)u₁(x²₁)+λ ¯u − πu₂(x¹₂) − (1 − π)u₂(x²₂)
+µ₁ w¹− x¹₁− x¹₂
+µ₂ w²− x²₁− x²₂
The first order conditions are

∂L

∂x¹₁ : πu⁰₁(x¹₁) = µ₁

∂L

∂x¹₂ : λπu⁰₂(x¹₂) = µ₁

∂L

∂x²₁ : (1 − π)u⁰₁(x²₁) = µ₂

∂L

∂x²₂ : λ(1 − π)u⁰₂(x²₂) = µ₂

πu⁰₁(x¹₁) = λπu⁰₂(x¹₂) (1 − π)u⁰₁(x²₁) = λ(1 − π)u⁰₂(x²₂)

Observe that the marginal rate of substitution for agent i = 1, 2 is

πu⁰_i(x¹_i) (1 − π)u⁰_i(x²_i)

Simplifying the above equations, we see that

u⁰₁(x¹₁) = λu⁰₂(x¹₂) (2.1)

u⁰₁(x²₁) = λu⁰₂(x²₂) (2.2)

and dividing the above equations, we obtain that

(2.3) u⁰₁(x¹₁)

u⁰₁(x²₁) = u⁰₂(x¹₂) u⁰₂(x²₂)

These equations, together with the feasibility conditions

x¹₁+ x¹₂ = w¹ x²₁+ x²₂ = w²

determine the Pareto efficient allocations.

Example 1. Suppose that

u₁(x) =√

x, w₁¹ = 0, w₁² = 1 u₂(x) =√

x, w₂¹ = 1 = w₂² π = 1/4

w₁¹= 0

w₁²= 1

w₂¹= 1

w₂²= 1

π = 1 / 4

π = 1 / 4 π = 1 / 4 π = 1 / 4 1111−−−− π π π π

Initial endowments Agent 1 Agent 2

u₁(x) = x u₂(x) = x The conditions for Pareto optimality

u⁰₁(x¹₁)

u⁰₁(x²₁) = u⁰₂(x¹₂) u⁰₂(x²₂) become

x²₁ x¹₁ = x²₂

x¹₂ which together with the feasibility condition

x¹₁+ x¹₂ = 1 x²₁+ x²₂ = 2 determine the Pareto efficient allocations. Taking

x = x¹₁ y = x²₁ we have

x¹₂ = 1 − x¹₁ = 1 − x x²₂ = 2 − x²₁ = 2 − y and substituting in the first order condition, we obtain

y

x = 2 − y 1 − x

that is x = y/2 and the Pareto efficient allocations are of the form x¹₁ = y/2 x¹₂ = 1 − y/2

x²₁ = y x²₂ = 2 − y 0 ≤ y ≤ 2

In the Edgeworth box, we see that the Pareto efficient allocations determine the straight line x¹₁ = x²₁

2

(1,2)

x¹₁ x²₁

2.1. One risk neutral agent and one risk averse agent. Suppose now that

• agent 1 is risk averse: u⁰⁰₁ < 0. (Equivalently, u⁰₁ is strictly decreasing)

• agent 2 is risk neutral: u⁰₂ is constant.

In this case,

u⁰₂(x¹₂) u⁰₂(x²₂) = 1 So, the first order Pareto optimality condition 2.3 is

u⁰₁(x¹₁) u⁰₁(x²₁) = 1 that is,

u⁰₁(x¹₁) = u⁰₁(x²₁)

and since u⁰₁ is strictly decreasing, the first order Pareto optimality condition becomes x¹₁ = x²₁

That is, agent 1 insures himself completely.

Example 2. In example 1, suppose that agent 2 is risk neutral. That is, u₁(x) =√

x, w₁¹ = 0, w₁² = 1 u₂(x) = x, w₂¹ = 1 = w₂²

π = 1/4

w₁¹= 0

w₁²= 1

w₂¹= 1

w₂²= 1

π = 1 / 4

π = 1 / 4 π = 1 / 4 π = 1 / 4 1111−−−− π π π π

Agent 1 Agent 2

u₁(x) = x u₂(x) = x Initial endowments

Thus, the Pareto efficient allocations are

(1,2)

x¹₁ x²₁

1

1

3. Exchange Economies

Now, we introduce prices that allow to contract (at t = 0) the delivery of 1 unit of the good at each state.

ππππ 1111−−−− ππππ

p₁ p₂

t= 0 t= 1

The model is the as follows.

• At t = 0 the ‘Walrasian auctioneer’ announces prices p₁, p₂.

• For each state of nature s = 1, 2, the price p_s allows to purchase one unit of the good to be delivered if state e_s occurs at t = 1.

• That is, agent i = 1, 2 can purchase the bundle (x¹_i, x²_i) at t = 0 for the price p1x¹_i + p2x²_i. This bundle allows agent i to consume x¹_i units of the good if state e1 occurs and x²_i units of the good if state e2 happens.

• The bundle (x¹_i, x²_i) is a contingent consumption plan.

• These prices are paid at t = 0.

• The income of agent i = 1, 2 at t = 0 is

p₁w¹_i + p₂w_i²

• Thus, at t = 0 agent i = 1, 2 chooses the bundle that maximizes his utility subject to his budget constraint,

max πu_i(x¹_i) + (1 − π)u_i(x²_i) (3.1)

subject to p₁x¹_i + p₂x²_i = p₁w_i¹+ p₂w_i²

• The solution to the above optimization problem determines the demand of each agent: x¹_i(p), x²_i(p).

• The equilibrium prices (p₁, p₂) are those that clear the market, state by state x¹₁(p) + x¹₂(p) = w¹₁+ w¹₂

(3.2)

x²₁(p) + x²₂(p) = w²₁+ w²₂ Observation 3. The first order condition of problem 3.1 is

MRS = p₁ p₂ that is,

πu⁰_i(x¹_i)

(1 − π)u⁰_i(x²_i) = p₁

p₂ i = 1, 2

In particular, the marginal rates of substitution for both agents coincide. Hence, the competitive equilibria are Pareto efficient.

Example 4. Let us compute the competitive equilibrium in example 1. The economy is described by

u₁(x) =√

x, w₁¹ = 0, w₁² = 1 u₂(x) =√

x, w₂¹ = 1 = w₂² π = 1/4

w₁¹= 0

w₁²= 1

w₂¹= 1

w₂²= 1

π = 1 / 4

π = 1 / 4 π = 1 / 4 π = 1 / 4 1111−−−− π π π π

Initial endowments Agent 1 Agent 2

u₁(x) = x u₂(x) = x

Let p = (p₁, p₂) be the equilibrium prices. The condition for Pareto efficiency is x²₁ = 2x¹₁

On the other hand, the first order conditions for utility maximization, πu⁰_i(x¹_i)

(1 − π)u⁰_i(x²_i) = p₁

p₂ i = 1, 2 for agent 1, imply that

p₁ p₂ = 1

3 s

x²₁ x¹₁ =

√2 3

(In the second equality, we have used the Pareto efficiency condition, x²₁ = 2x¹₁). We see that the equilibrium prices are

p₁ =√

2, p₂ = 3 The budget restriction for agent 1 is

p₁x¹₁ + p₂x²₁ = p₂

and, since the equilibrium allocations are Pareto efficient, we have that p₂ = p₁x¹₁+ 2p₂x¹₁ = (p₁+ 2p₂)x¹₁ so,

x¹₁ = p₂

p₁+ 2p₂ = 3

√2 + 6, x²₁ = 6

√2 + 6 and the demand of agent 2 is

x¹₂ = 1 − 3

√2 + 6, x²₂ = 2 − 6

√2 + 6

Observation 5. Coming back to the general model, if, in addition one of the agents, for example agent 2, is risk neutral, then

u⁰₂(x¹₂) u⁰₂(x²₂) = 1 so, the equilibrium prices verify that

π

1 − π = p₁ p₂

And if, in addition, agent 1 es strictly risk averse, the above observation implies that in the competitive equilibrium

x¹₁ = x²₁ Example 6. In example 1, suppose that

u₁(x) =√

x, w₁¹ = 0, w₁² = 1 u₂(x) = x, w₂¹ = 1 = w₂²

π = 1/4 That is, agent 2 is risk neutral.

w₁¹= 0

w₁²= 1

w₂¹= 1

w₂²= 1

π = 1 / 4

π = 1 / 4 π = 1 / 4 π = 1 / 4 1111−−−− π π π π

Agent 1 Agent 2

u₁(x) = x u₂(x) = x Initial endowments

Then, the equilibrium prices are

p₁ = 1

4, p₂ = 3 4 the demand of agent 1 verifies that

x¹₁ = x²₁ = x and the budget restriction

p₁x¹₁+ p₂x²₁ = p₁w₁¹+ p₂w₁²

is 1

4x + 3 4x = 3

4 so the equilibrium allocation is

x¹₁ = x²₁ = 3

4 For agent 1

and

x¹₂ = 1

4, x²₂ = 5

4 For agent 2

3.2. Interpretation. One way to interpret 6 is in terms of exchange economies. In state e₂ agent 1 transfers the amount

w₁²− x²₁ = 1 − 3 4 = 1

4 (= x²₂− w₂²) to agent 2. In exchange, agent 1 receives

x¹₁− w¹₁ = 3

4 (= w¹₂− x¹₂) from agent 2, in state e₁.

W₁¹= 0

W₁²= 1

W₂¹= 1

W₂²= 1

ππππ

1111−−−− ππππ

Initial endowments Agent 1 Agent 2

Consumption Agent 1 Agent 2

x₁¹= 3 / 4

x₁²= 3 / 4

x₂¹= 1 / 4

x₂²= 5 / 4 3 / 4

1 / 4

4. Insurance markets

Another possible interpretation of example 6 is in terms of insurance. Agent 1 has initially a wealth w₁⁰ = 1 in t = 0. And with probability

π = 1 4

may suffer a loss of D units of wealth. That is, he faces the following situation

(1) In state e₁, which happens with probability π = 1/4, agent 1 loses D = 1 units, so w₁¹ = w⁰₁− D = 0.

(2) In state e₂, which happens with probability π = 3/4, agent 1 does not suffer any loss, so w²₁ = w₁⁰ = 1.

Graphically,

w₁⁰ - D

ππππ

1111−−−− ππππ

Agent 1

w₁⁰

w₂⁰

ππππ

1111−−−− ππππ

Agent 2

w₂⁰

w₂⁰ w₁⁰

w₁⁰ - D -q

αααα + α + α + α + α ππππ

1111−−−− ππππ

αααα

w₂⁰ + q

αααα

αααα ππππ

1111−−−− ππππ

αααα

w₂⁰ + q

αααα

αααα

Suppose now that agent 2, who is risk neutral, represents an insurance company. Initially, the insurance company owns one unit w⁰₂ = 1 of the good at t = 0 and it is not subject to risk. That is, in both states e₁ and e₂ keeps its wealth w¹₂ = 1 = w²₂.

Agent 2 offers to agent 1 the possibility of insuring α units of wealth at the price q, per each insured unit. From the point of view of agent 1, the situation is as follows

• At t = 0 agent 1 buys α units of the insurance and pays the amount qα to agent 2. He keeps w⁰₁− qα = 1 − qα.

• If the state e₁ happens (with probability π = 1/4) agent 1 suffers a loss of D = 1 units of his wealth. Therefore agent 2 pays to agent 1 the amount insured α. The remaining wealth of agent 1 is x¹₁ = w⁰₁− qα − D + α = α − 1.

• If the state e₂ occurs (with probability π = 3/4) gente 1 does not suffer any loss. Therefore, he does not receive any compensation from agent 2. The remaining wealth of agent 1 is x²₁ = w₁⁰− qα = 1 − qα.

Suppose also that the insurance market is perfectly competitive. Therefore, the expected benefit for insurance companies is 0. From the point of view of the insurance company (agent 2), if agent 1 purchases α units of the insurance, the situation is the following,

• At t = 0 it receives qα units from agent 1 and starts with w⁰₂+ qα = 1 + qα.

• If state e₁ occurs agent 1 loses D = 1 units and agent 2 pays the amount α insured. The remaining wealth of agent 2 is w₂¹ = w⁰₂ + qα − α = 1 + qα − α.

• If state e₂ occurs, agent 1 does not suffer any loss and agent 2 does not have to pay any compensation. The remaining wealth of agent 2 is w²₂ = w⁰₂+ qα = 1 + qα.

The condition of expected 0 zero profit is

w₂⁰ = π(w₂⁰+ qα − α) + (1 − π)(w₂⁰+ qα) = w₂⁰+ qα − πα that is, the competitive price of each insured unit is

q = π This is the actuarially fair price.

Given the price

q = π

agent 1 chooses the amount α of insurance that maximizes his expected utility, πu1(w₁⁰− qα − D + α) + (1 − π)u1(w₁⁰− qα)

The first order condition is

π(1 − q)u⁰₁(w₁⁰− qα − D + α) = (1 − π)qu⁰₁(w⁰₁− qα) and since q = π this condition implies

u⁰₁(w₁⁰− qα − D + α) = u⁰₁(w₁⁰− qα) But, since agent 1 is risk averse, we have that u⁰₁ is decreasing. Hence,

w₁⁰− qα − D + α = w₁⁰− qα and we see that

α = D = 1

that is, agent 1 insures himself completely. The consumption of the agent is

ππππ 1111−−−− ππππ

ππππ D

w₁⁰–

ππππ D

ππππ D

Note that EV = w₁⁰ − πD is the expected consumption of the agent when he buys no insurance.

That is, when there is perfect competition among the insurance companies, the expected utility of the agent is u(EV).

On the other hand, the amount paid by to the insurance company by the agent is p^e= πD. Letting I^e be the amount of insurance bought by the agent, we see that the insurance policy in the competitive equilibrium is (p^e, I^e) = (πD, D).

Plugging in the values

α = D = 1, q = π = 1 4 we see that at t = 0, agent 2 charges the amount

1 4

to agent 1, in exchange for giving him full insurance. In either state the wealth of agent 1 is 1 − 1

4 = 3

4 = EV Agent 2 has

1 4

1 + 1

4− 1 = 1 4 and in state e₂ does not pay anything to agent 1 and has

1 + 1 4 = 5

4

5. Perfectly Discriminating Monopoly The model is the as follows.

• Agent 2 is a monopolist. He knows the utility function of agent 1.

• At t = 0, agent 2 proposes the feasible ‘contract’ (x¹₁, x²₁) to agent 1. That is, he proposes agent 1 the previous contingent consumption plan.

• In above contract, agent 2 consumes the rest of the resources (x¹₂, x²₂) = (w¹₁+ w₂¹, w₁²+ w₂²) − (x¹₁, x²₁)

• If, at t = 0, agent 1 accepts the contract, then consumption at t = 1 is the one described by the above contract.

• If, at t = 0, agent 1 rejects the contract, then at t = 1 each agent consumes his initial endowments (w_i¹, w_i²).

The monopolist (agent 2) chooses a contract (x¹_i, x²_i) such that he maximizes his utility and keeps agent 1 at his reservation utility

max πu2(x¹₂) + (1 − π)u2(x²₂)

subject to πu₁(x¹₁) + (1 − π)u₁(x²₁) ≥ πu₁(w¹₁) + (1 − π)u₁(w₁²)

This is coincides exactly with the problem of computing the Pareto efficiency allocations keeping the utility of agent 1 equal to ¯u = πu₁(w₁¹) + (1 − π)u₁(w₁²). The condition of Pareto optimality is again

u⁰₁(x¹₁)

u⁰₁(x²₁) = u⁰₂(x¹₂) u⁰₂(x²₂)

Example 7. Suppose in example 1, agente 2 is a perfectly discriminating monopoly u₁(x) =√

x, w₁¹ = 0, w₁² = 1 u₂(x) =√

x, w₂¹ = 1 = w₂² π = 1/4

w₁¹= 0

w₁²= 1

w₂¹= 1

w₂²= 1

π = 1 / 4

π = 1 / 4 π = 1 / 4 π = 1 / 4 1111−−−− π π π π

Initial endowments Agent 1 Agent 2

u₁(x) = x u₂(x) = x

Then, agent 1 is kept at his reservation utility

¯ u1 = 1

4

√ 0 + 3

4

√ 1 = 3

4

On the other hand, this utility is attained at an allocation of the form

x¹₁ = x²₁ 2

That is,

3 4 = 1

4 q

x¹₁+3 4

q

x²₁ = 1 4

q x¹₁+3

4 q

2x¹₁

That is,

3 = q

x¹₁

1 + 3√ 2

and we obtain that

x¹₁ = 9 1 + 3√

2
²

(1,2)

x¹₁ x²₁

1

1

p₁x¹₂+ p₂x²₂= c

Competitive allocation

Competitive Economy: General Case

inititial endowments

Agent 2 is a perfectly discriminating monopoly

5.1. One risk neutral agent and one risk averse agent. Suppose now that agent 2 is risk neutral. Assume this agent is an insurance company that behaves as a monopolist and agent 1 purchases insurance from agent 2. We define the certainty equivalent CE as the consumption level that leaves the agent indifferent between accepting the gamble and consuming CE. That is,

u₁(CE) = πu₁(w₁⁰− πD) + (1 − π)u₁(w⁰₁) And the risk premium is

RP = EV − CE The amount paid by agent 1 to the monopolist is

w⁰₁− CE = EV +p^e− CE = p^e+ RP

Therefore, the insurance policy that the agent 1 purchases from the monopolist is (p^m, I^m) = (p^e+ RP, D)

Example 8. Suppose that in example 1, agent 2 is a perfectly discriminating monopoly, who in addition is risk neutral.

u₁(x) =√

x, w₁¹ = 0, w₁² = 1 u₂(x) = x, w₂¹ = 1 = w₂²

π = 1/4

w₁¹= 0

w₁²= 1

w₂¹= 1

w₂²= 1

π = 1 / 4

π = 1 / 4 π = 1 / 4 π = 1 / 4 1111−−−− π π π π

Agent 1 Agent 2

u₁(x) = x u₂(x) = x Initial endowments

Pareto efficiency requires that

x¹₁ = x²₁

The reservation utility of agent 1 is

3 4 = 1

4 q

x¹₁+3 4

q x¹₁ =

q x¹₁

and we obtain that

x¹₁ = CE = 9 16

and

x²₁ = 7

16 x²₂ = 23 16

(1,2)

x¹₁ x²₁

1

1

p₁x¹₂+ p₂x²₂= c p₁=

ππππ

ππππ

p₁ x¹₁ + p₂ x²₁= c

Actuarially fair insurance

Competitive Economy: Agent 2 is risk neutral

inititial endowments

Agent 2 is a perfectly discriminating monopoly

EV CE

6. Labor Contracts Example 8 may be interpreted in the following way.

• In example 8 we may consider that agent 2 (the principal) has 1 unit of the good and hires agent 1 to do some work. In state e₁, the worker is not efficient and produces 0 units of the good. In state e₂, the worker is efficient and produces 1 unit of the good.

• The principal offers a wage w to the worker in a ‘take it or leave it’ offer. The worker accepts if his reservation utility u^R₁ = 3/4 is below the utility from accepting the contract.

• The principal pays to agent 1 the minimum amount that leaves the agent indifferent between accepting and his reservation utility, u^R₁ = 3/4. The utility function of agent 1 on monetary amounts is √

x. His utility will be equal to u^R₁ = 3/4 if the salary is w = 3

4

2

= 9 16 Agent 1 will accept this salary from the principal.

• On the other hand, agent 2, receives in each state the amount initial endowment + value of production − salary

so, in state e1 he receives

1 − 9 16 = 7

16 and in state e₂ he receives

1 + 1 − 9 16 = 23

16