Question 1

(a). Write down the problem of an agent that maximizes ex-ante utility in autarky. Find the conditions that characterise the allocation in autarky. Explain how the allocation changes with β.

Autarky is a situation where no trading takes place between agents. Each agent needs to provide for his own needs in an autarky, ie he independently chooses the amount of I that he wants to invest in the long run technology. The issue of liquidity insurance arises here.

Every agent wants to maximise his ex-ante utility but the problem is that at time t=0 he does not know about his type whether he wants to consume early at t=1 or late at t=2 resulting in asymmetric information. Hence, there is a risk that more than is optimal may be invested.

The conditions that characterise the allocation in autarky are bounded by the constraints of C₁ and C₂. If agent decides to consume early, he will get savings (1-I) and liquidated investment (É­I).

C₁ = 1 – I + É­I = 1 – I (1-É­)

If agent decides to consume late, he will obtain savings (1-I) and returns from investment (RI).

C₂ = 1 – I + RI = 1 + I(R-1)

Agent will choose his consumer profile (C₁, C₂) that will maximise his ex-ante utility U based on the above constraints.

However, the allocation is not efficient in autarky as shown in the next part of the question.

Max U(C₁,C₂) = u(C₁) + βu(C₂) = [1 – I + É­I]+ [1 – I + RI]= 2+ É­I + RI

We set up the lagrangian method to explain the allocation changes in β where the constraint in the below equation is the maximum utility.

L = πu(C₁) + (1-π)βu(C₂) + λ[2+ É­I + βRI]

 = π + λÉ­I = 0

 = (1- π)β + λRI = 0

 = 2+ É­I + βRI = 0

Home

Complementary Slackness Condition: λ^*[2+ É­I + βRI] = 0

If values were given for the variables, we could even have solved and get the value of β. If a value close to zero is obtained for β, it means agent is impatient anda value close to one indicates that agent is patient.

This argument is further supported by the marginal rate of substitution concept where  = R. If β=0, no returns obtained as the agent wants to consume immediately. If β=1, returns will result for the patient agent. Hence, it shows that the discount factor β will not change the basic results of the model.

(b) Write down the conditions that characterise the Pareto-optimal allocation. Show that autarky is not efficient. Explain how the allocation changes with β.

The conditions that characterise the allocation in autarky are bounded by the constraints of C₁ and C₂.

π₁C₁ = 1 – I   => C₁ = 

(1-π)C2= RI   => C₂ = 

The constraints can be combined in a single one.

π₁C_{1 +} (1-π)= 1

The key result is that allocation is inefficient in autarky as shown below:

Recall in autarky: C₁ = 1 – I + É­I = 1 – I (1-É­)

C₂ = 1 – I + RI = 1 + I(R-1)