How to Monetary policy

Question 1

Commitment rule

Answer 1

1. Start with some intuition: Under commitment, the government introduce the monetary rule, π_t. Then the privates observe this and the theata, which is the natural level of output. Then the privates form expectations upon that information. This make the policy ex post optimal
   
   2. Write the optimal monetary policy π_t, write it wtih and withouut psi.
   3. Take the expectation to the optimal policy, which is based om the agents information of theta. In comitment, the optimal rule would be equal to the expected vaulue, because the privates trust the CB
   4. Government minimezes expected loss choosing policy parameters, given the calculated equlibrium inflation and output.
      a. Insert the values in the loss function given in the assignment, where we then take the expected value to all of it, based on the agents information of theta. Some values go out, because the government has no incentive to repsond to shocks that are observed by the agents (not able to stabilize, Lucas critique).
      b. Take FOC to the unobserved shocks, and plug them into the calculated equlibrium inflation and output. Denote them with a c for commitment, and then you have the governments optimal rule under commitment.

Question 2

Discretion rule

Answer 2

1. Start with some intuition: Under discretion, the government does not introduce the monetary rule, π_t, it is decided after the households made their expectations. The privates can only observe theta, which is the natural level of output.
2. Do not take the expectations yet.
3. To show the inflation bias, write the loss function without the curly things, only as pi. The difference is that we do not take expectation to the loss function in the beginning, because the household have not made them yet.

Government minimizes loss choosing policy parameters, given the calculated equilibrium inflation and output, as before.
a. Insert the values in the loss function given in the assignment, without the curly things, due to no optimal policy from CB.
b. Take FOC to pi, Denote with D for discretion, and now for the expected inflation take the expectations to pi. Then isolate expected pi.
c. plug them into the calculated equlibrium inflation and output. Denote with D for discretion.

Question 3

How to do the conservative CB.

Answer 3

1. Assume monetary policy is delegated to an independent central banker, lambda CB < lambda, which make him conservative. He trade off lower inflation against lower pass-through of supply shocks.
2. Find equilibrium inflation an output growth under CB
   - Substitute lamba CB in on labda´s place
3. Find the optimal rule
   - Plug in the equilibrium values in the loss function with expectations.
   - Take expectations, everything with unexpected shocks goes out.
   - Take FOC wrt. lambda CB and isolate.

done

Question 4

What can we take privates expectations to.

Answer 4

E[?|theta], they only have information about theta. Everything that is unexpected desapiers.