Question $3$: Inflation Credibility and Supply Shocks

A small open economy is experiencing a negative supply shock due to rising global oil prices. The short$-$run aggregate supply $($AS$)$ curve is defined as:

${}_t={}_e+(Y_t-Y^P)+P_{\text{o}\text{i}\text{l}}$

Where, ${}_t=$ current inflation rate; ${}_e=$ expected inflation; $Y_t=$ Actual output; $Y^p=$ Potential Output; $=$ output gap sensitivity $(0\mathrm{.}3)$;

$u=$ oil price sensitivity $(0\mathrm{.}4)$; $P_{\text{o}\text{i}\text{l}}=$ percentage change in global oil prices.

Assume the central bank is targeting an inflation rate of $3\%.$

$($a$)$ Current Inflation: Oil prices increase by $20\%,$ expected inflation is $4\%,$ and actual output is $5\%$ below potential. Calculate the current inflation rate.

$($b$)$ Output Adjustment: The central bank implements a contractionary policy to bring inflation back to the $3\%$ target. Determine the output adjustment needed to achieve this target if $P_{\text{o}\text{i}\text{l}}$ remains constant and re re stays at $4\%.$

$($c$)$ Credibility Dynamics: If the public perceives the central bank to be ineffective and expected inflation rises to $6\%,$ recalculate the inflation rate assuming output remains at its adjusted level. Discuss the implications for the central bank's credibility and suggest a mathematical strategy for minimizing welfare loss, given:

$L=({}_t-{}^{**})2+(Y_t-Y^{}{)}^2$; where $=0\mathrm{.}8$