derive the optimal value of loss functiona

Question

Derive the Optimal Value of Loss Function

A speculative attack and the consequent currency crisis may not be due to excessive money-growth or other misaligned fundamentals, but to self-fulfilling panics under fixed exchange rates." To explain this, consider a small open economy where the policy maker wants to minimize a quadratic loss function given by:

L = ½ (βΠ²_t + x²_t)

Where, 0<β<1

Subject to the budget constraint:

Rb_t = x_t + Θ (Π_t - Π^e_t)

Where 0< Θ<1

and, Π_t = sctual rate of exchange rate devaluation (equal to the inflation rate)

x_t = ow of net tax revenue (policy variable)

b_t = interest rate (equal to the given world interest rate)

R = interest rate (equal to the given world interest rate)

Π^e_t = exogenously given expected rate of devaluation (inflation)

Assume that Purchasing Power Parity holds so that rate of inflation and exchange rate devaluation coincides.

(a) What is the interpretation of the term Θ (Π_t -  Π^e_t )?

(b) If the policy maker does not pre-commit to 'not to devalue', calculate the optimal value of and the loss function.

(c) Derive the optimal value of and the loss function when the policy maker has pre-committed 'not to devalue'.

(d) Show that the loss is higher when the policy maker has pre-committed not to devalue.

(e) What is the condition under which the policy maker finds that it is optimal to devalue even if he faces an additional exogenous cost of devaluation, C > 0.

(f) Show from the conditions you had derived earlier that devaluation is the equilibrium outcome when the expectations of devaluation are sufficiently high.