Reference no: EM131755970

Question - Optimal Monetary Policy in the Basic Framework

In this problem, we review the derivation of the Friedman rule.

Suppose firms only require labor, n_t, to produce output y_t, i.e. y_t = f(n_t) = n_t. Given firms earn zero profits (in this setup), this tells us: P_ty_t = W_tn_t, where P_t is the price of the consumption good and W_t is the nominal wage. Note, real wages, w_t, are defined such that w_t = W_t/P_t. Assume the representative consumer maximizes period-t utility function, u(c_t, 1 - n_t), by optimally choosing consumption, leisure, and assets.

The consumer faces budget constraint:

P_tc_t + (1/1+i_t ) B_t + M_t + S_ta_t = W_tn_t + M_t-1 + B_t-1 + (S_t + D_t)a_t-1 + τ_t,

where τ_t is a government policy at time t to be defined below, it is the nominal interest rate at time t, S_t is the price of the real asset at time t, and the remaining quantities, B_t, M_t, D_t, 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: P_tc_t = M_t.

Recall, also that if we define the real rate as r_t, 1 + r_t = ((S_t+1+D_t+1)/P_t+1)/(S_t/P_t). It also may be useful to define: P_t^b = 1/1+i_t.

Government money growth, g_t, follows rule: M_t = M_t-1 + τ_t, where τ_t = M_t-1g_t. Notice, this implies: M_t = M_t-1(1 + g_t)

We also have resource constraint: c_t = n_t.

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 w_t = 1∀t. We have the cash-in-advance constraint c_t = M_t/P_t ⇒ (M_t/M_t-1)/(P_t/P_t-1) = c_t/c_t-1 ⇒ 1+g_t/1+π_t = c_t/c_t-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?