Exercise 2
Consider a government with preferences represented by the loss function
$L = \pi^2 + au^2$,
where $\pi$ is the inflation rate, $u$ is the unemployment rate, and $a$ is a positive constant in-
dicating the strength of the government's preference for low and stable unemployment
relative to low and stable inflation.
The expectations-augmented Phillips curve linking inflation and unemployment is:
$u = u_n - b(\pi - \pi^e)$,
where $\pi^e$ denotes inflation expectations, $u_n$ is the natural rate of unemployment, and $b$
is a positive constant. Expectations of inflation are assumed to be rational. Assume that
monetary policy is able to control the inflation rate $\pi$ directly.
Suppose initially that the government chooses a monetary policy without being restricted
by any past commitments (the government acts with discretion). This means inflation ex-
pectations $\pi^e$ are taken as given when monetary policy is chosen.
(a) (1 pt) Derive the first-order condition for the inflation rate $\pi$ that minimises the loss
function subject to the Phillips curve for given $\pi^e$.
As expectations are formed rationally, everyone anticipates the actions of the central
bank determined in part (a).
(b) (2 pts) Find the equilibrium unemployment and inflation rates in terms of the pa-
rameters $a$, $b$, and $u_n$.
(c) (2 pts) Suppose political pressure forces the government to shift its focus away from
inflation towards unemployment. Interpreting this as an increase in $a$ in the loss
function, what are the effects on the equilibrium values of $u$ and $\pi$? Explain your
findings with reference to the policy ineffectiveness proposition.
The government announces it will follow a rule that strictly targets inflation $\pi$.
(d) (2 pts) Assuming that everyone believes the rule will be followed, find the optimal
rule for the inflation rate that minimises the loss function. What is the equilibrium
unemployment rate when the government follows the rule?
(e) (3 pts) Once everyone chooses inflation expectations consistent with your answer to
part (d), suppose the government is now free to choose inflation taking $\pi^e$ as given as
it did in part (a). Solve for the resulting values of $u$ and $\pi$. By comparing your answer
to part (d), explain why the optimal rule for inflation is time inconsistent.
