Question - Optimal Monetary Policy in the Basic Framework
In this problem, we review the derivation of the Friedman rule.
Suppose firms only require labor, nt, to produce output yt, i.e. yt = f(nt) = nt. Given firms earn zero profits (in this setup), this tells us: Ptyt = Wtnt, where Pt is the price of the consumption good and Wt is the nominal wage. Note, real wages, wt, are defined such that wt = Wt/Pt. Assume the representative consumer maximizes period-t utility function, u(ct, 1 - nt), by optimally choosing consumption, leisure, and assets.
The consumer faces budget constraint:
Ptct + (1/1+it ) Bt + Mt + Stat = Wtnt + Mt-1 + Bt-1 + (St + Dt)at-1 + τt,
where τt is a government policy at time t to be defined below, it is the nominal interest rate at time t, St is the price of the real asset at time t, and the remaining quantities, Bt, Mt, Dt, and at, are nominal debt, money, dividends, and real assets at time t, respectively.
In each period, the consumer also faces cash-in-advance constraint: Ptct = Mt.
Recall, also that if we define the real rate as rt, 1 + rt = ((St+1+Dt+1)/Pt+1)/(St/Pt). It also may be useful to define: Ptb = 1/1+it.
Government money growth, gt, follows rule: Mt = Mt-1 + τt, where τt = Mt-1gt. Notice, this implies: Mt = Mt-1(1 + gt)
We also have resource constraint: ct = nt.
3(a): Write down the sequential Lagrangian (notice this is an infinite-horizon problem and also do not forget the cash in-advance constraint) and derive the consumption-leisure optimality condition.
Note: You will want to follow the same first three steps as in question 2, i.e. write out the sequential Lagrangian, write out the first order conditions, and show clearly how you rearrange these first order conditions and which first order conditions are plugged into one another.
3(b): Also derive the consumption-savings optimality condition.
3(c): Let wt = 1∀t. We have the cash-in-advance constraint ct = Mt/Pt ⇒ (Mt/Mt-1)/(Pt/Pt-1) = ct/ct-1 ⇒ 1+gt/1+πt = ct/ct-1.
Rewrite the consumption-savings and consumption-leisure optimality conditions to finish our description of an equilibrium.
Don't forget to plug in using the resource constraint and our assumption on wages.
These equations should not be written in steady state.
3(d): Notice with the assumptions from part (c), in steady state,
the consumption-savings optimality condition is: u2(c¯,1-c¯)/u1(c¯,1-c¯) = 1+i/1+2i, and the consumption-leisure optimality condition is: 1 = β(1 + r). Rearranging, we have: 1 + r = 1/β.
Recall, the Fisher equation in steady state is: (1 + π)(1 + r) = (1 + i).
Hence, (1 + π) = β(1 + i) is the Fisher equation after plugging in the consumption-leisure optimality condition.
Also, notice that 1 + π = 1 + g in steady state.
Thus, (1 + g) = β(1 + i).
Rearranging, i = (1+g/β) - 1.
Thus, from u2(c¯,1-c¯)/u1(c¯,1-c¯). = 1+i/1+2i, we have that: u2(c¯,1-c¯)/u1(c¯,1-c¯) = 1+g/2+2g-β.
(d)(i): What value of g delivers u2(c¯,1-c¯)/u1(c¯,1-c¯) = 1 (which can be shown is the solution of the social planner who maximizes the consumer's utility)?
(d)(ii): What value of i is consistent with such a value of g?