Marco Del Negro, Teoria y Politica Monetaria, Spring 2000

Problem set 4
Solutions

a) If you substitute the government budget constraint:

$$M^s_{t+1}+B^s_{t+1}=M^s_t+B^s_t(1+R_t)+p_tg_t-p_t\tau_t$$

into the agent's budget constraint:

$$M_{t+1}+B_{t+1}+p_tc_t=M_t+B_t(1+R_t)+p_ty_t-p_t\tau_t$$

(you know that in equilibrium the constraint holds with equality), after imposing the equilibrium conditions:

$$M^s_t=M_t\mbox{, }B^s_t=B_t$$

you obtain the condition:

c_t=y_t-g_t

Given that both y_t and g_t are constant, this implies:

c=y-g

The first order condition with respect to money are unchanged. The equilibrium condition for real money balances is therefore:

$$(1+\mu-\beta)u_c(y-g,m^*)=\beta u_m(y-g,m^*)$$

and this is the condition determining m^*.
b) The case in which taxes are distortive is more complicated. Now consumption depends in fact on the choice of the government between financing spending with taxes or with seignorage.
If you substitute the government budget constraint:

$$M^s_{t+1}+B^s_{t+1}=M^s_t+B^s_t(1+R_t)+p_tg_t-p_t\tau_t$$

into the agent's budget constraint:

$$M_{t+1}+B_{t+1}+p_tc_t=M_t+B_t(1+R_t)+p_ty^*_t-p_t\tau_t$$

(again, after imposing the equilibrium conditions $M^s_t=M_t\mbox{, }B^s_t=B_t$), you obtain the condition:

$$c_t=y^*_t-g_t=y_t-\phi \frac{\tau^2_t}{y_t}-g_t$$

If debt is by assumption equal to zero in all periods, the government budget constraint becomes:

$$\frac{M^s_{t+1}-M^s_t}{p_t}+\tau_t=g_t$$

where the term $\frac{M^s_{t+1}-M^s_t}{p_t}$ is seignorage. Under the assumption that money supply grows at a constant rate, seignorage is equal to

$$\mu m^*$$

, where m^* is equilibrium real money balances (let us assume for a second that real money balances are constant over time; we will see in a few lines that this is indeed the case). This implies that in all periods:

$$\tau_t=g-\mu m^*$$

. The first order condition with respect to money are again unchanged. The equilibrium condition for real money balances is therefore:

$$(1+\mu-\beta)u_c(y-\phi \frac{\tau^2_t}{y}-g ,m^*)=\beta u_m(y-\phi \frac{\tau^2_t}{y}-g,m^*)$$

where we used the equilibrium condition for c_t. From the above expression we have an expression for $\tau$, which we can substitute in and obtain:

$$(1+\mu-\beta)u_c(y-\phi \frac{(g-\mu m^*)^2}{y}-g ,m^*)=\beta u_m(y-\phi \frac{(g-\mu m^*)^2}{y}-g,m^*)$$

This equation can be solved for its only unknown, m^*, thereby obtaining all equilibrium conditions. Note that none of the terms in this equation depends on time. So our claim that equilibrium real money balances m^* does not depend on time is verified.