Grasping Reality on TypePad, by Brad DeLong

Suppose that we have a system in which future unemployment u_t+1 is a function of variables V (which include past, present, and expected future unemployment and inflation rates), of the stimulative impact of monetary policy m, and of a shock:

(1) u_t+1=ƒ(V_t) + m_t + ε^u_t+1 ;

in which inflation is a function of variables V (which include past, present, and expected future unemployment and inflation rates) and of a shock:

(2) π_t+1 = j(V_t) + ε^π_t+1 ;

and in which there is an objective function h depending on present and future values of unemployment and inflation (but not directly on monetary or fiscal policy variables):

(3) h(π_t, π_t+1, …; u_t, u_t+1, …)

At each point in time the monetary authority will choose an optimal m'_t in order to maximize the expected value of that objective (3) subject to (1) and (2), and the unemployment rate will consequently be:

(4) u_t+1=ƒ(V_t) + m'_t + ε^u_t+1

Now consider what happens if we replace (1) with (1a), with the difference between (1) and (1a) being that fiscal policy g also affects unemployment:

(1a) u_t+1=ƒ(V_t) + m_t + g_t + ε^u_t+1

Then as long as m'_{t is the optimal choice of m for the system (1), (2), and (3):}

(5) m"^*_t = m'_t − g_t

is the optimal choice of m’ for the system (1a), (2), and (3).

Why? In system (1a), (2), and (3), define:

(6) n_t = m_t + g_t

Then (1a) becomes:

(7) u_t+1=ƒ(V_t) + n_t + ε^u_t+1

By hypothesis, the objective h is maximized for the system of (7), (2), and (3) by choosing:

(8) n'_t =m'_t

And so the objective is maximized for the system of (1’), (2), and (3) by choosing:

(9) m"_t =m'_t - g_t

You take your optimal policy if fiscal policy were neutral--if g_t=0--and you then subtract the stimulative value of the actual g_t from it.

The important point here is that m and g cannot enter into the objective function h directly, but only indirectly through their effects on inflation and unemployment.

For this reason this argument breaks down at the zero nominal lower bound. At the zero lower bound the central bank does not care only about inflation and unemployment. It cares as well about the magnitude of the non-standard monetary policy measures it must take in order to achieve its net monetary policy impetus value m.