Construct the price dynamics equation pt what is the growth

Question

1. Seigniorage and Exchange Rate

Consider the following information:

(i) Money demand (notin natural logs): M_t/P_t = Y_t/(1+i)^b, where the Fisher Equation i = Π is satisfied.

(ii) Inflation (prices are in natural logs): Π = p_t+1 - p_t

(iii) Monetary rule (in natural logs): m_t = αt (with α > 0)

(iv) Output is growing over time (in natural logs): y_t = θ_t (with θ>0)

(a) Construct the price dynamics equation p_t. What is the growth rate of P_t? Explain your result.

(b) Given your answer in (a), find the money growth rate α that maximizes Seigniorage, and it corresponding maximum Seigniorage in period t = 9

(c) Suppose that the government only wants to collect $8 in Seigniorage in period t=9. If θ=0.5, find the money growth rate(s) that achieve(s) this objective.

Use the above information with two exceptions: (i) income is constant (θ=0), and (ii) the monetary rule is now:

       {(2  if s=t,t+1,t+2,...,t+T-1)}

m_s =

       {(4) if s=t+T,t+T+1,...)}

(d) If the PPP and UIP equations are satisfied with p_t* = i_t* = 0, find the exchange rate dynamics equation and plot -with precision- its time path.