Question $2$: The Zero Lower Bound and Expectations

A central bank operates under a ZLB constraint where nominal rates cannot drop below $0\%.$ The relationship between inflation expectations $({}_e),$ real interest rates $(r),$ and output is given by:

$r=i-{}_e$;$Y=120-5r$

$($a$)$ If the central bank commits to keeping nominal rates at $0\%,$ calculate the output level when inflation expectations are $1\%,2\%,$ and $-1\%.$

$($b$)$ Using the above equations, demonstrate how a deflationary spiral could occur if inflation expectations continuously decline. Solve iteratively for $Y$ and ${}_e$ starting at ${}_e=0\%$ and assuming a decline of $1\%$ per iteration in expectations.

Homework $6$

ECON $600$

Submission Date: $9$ November $2024$ Monday

$($c$)$ Graph the output path over time and identify any points of equilibrium. Discuss the effectiveness of unconventional monetary policy tools such as quant$$tative easing in breaking this cycle.