CLASS 6: INFLATION AND THE NOMINAL INTEREST RATE

• we introduce nominal bonds in the model
• we will study the relationship between money growth, inflation, and the nominal interest rate
• we will derive a -textbook- equation for real money demand that depends positively on output and negatively on nominal interest rates
• we will discuss the benefits from reducing inflation

• introduce nominal bonds in the government budget constraint
• the government now has the option of financing budget deficits (the difference between government expenditures and taxes) either by issuing bonds to the private sector, or by having the Central Bank printing money and use it to purchase government bonds (thereby allowing the government to collect the seignorage)
• the government budget constraint is:
  
  M_t+1^s+B_t+1^s=M_t^s+B_t^s(1+R_t)+T_t
  
  
  where B_t^s is the supply of nominal (one period) bonds, R_t is the it nominal interest rate, and T_t is -net- transfers to the private sector
• note that the government budget constraint can be rewritten as:
  
  M_t+1^s-M_t^s+B_t+1^s-B_t^s=B_t^sR_t+T_t
  
  
  How can the government finance its expenditures (interest on previous debt plus transfers B_t^sR_t+T_t)? By printing money ( M_t+1^s-M_t^s) or by issuing new debt ( B_t+1^s-B_t^s).

• The objective function of the household remains the same: $max_{\{M_{t+1},c_t\}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})$
• but the budget constraint changes:
  
  $$M_{t+1} +B_{t+1}+p_t c_{t}\leq M_t+B_t(1+R_t)+T_t+p_t y_{t}$$
  
  
  and subject to an initial given quantity of liabilities of the government (which represent wealth of the private sector) W₀=M₀+B₀(1+R₀)
• the private sector can in principle borrow from the government ( $B_{t+1} \leq 0$), but we want to avoid the so called Ponzi games on the side of the private sector, that is, having the private sector accumulating more and more debt thereby increasing consumption. Therefore we impose the constraint $B_{t+1} \geq -\bar{B}$ where $\bar{B}$ is some positive number: ( $\bar{B}= \sum_{\tau=t+1}^\infty {\frac{p_{\tau}y_{\tau}+T_{\tau}} {\Pi_{s=t+1}^{\tau=0} (1+R_s)}} +M_{t+1}$) (no Ponzi game on the side of the government can be also ruled out)

• substituting the intertemporal budget constraint of the agent into the objective function, we obtain:
  $max_{\{M_{t+1},B_{t+1}\}_{t=0}^{\infty}} \\ \sum_{t=0}^\infty \beta^t u(y_t-\... ...t+1}-M_t}{p_t}-\frac{B_{t+1}-B_t(1+R_t)}{p_t} +\frac{T_t}{p_t},\frac{M_t}{p_t})$

• now the agent is facing not only the decision of how much to save, but also of how to save: portfolio choice between money and bonds
• notice however that the presence of bonds does not affect at all the first order condition with respect to M_t+1, which is the usual:
  $\begin{array}{l} \frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})=\\ \beta \frac{1}{p_{t... ...+1},\frac{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})] \end{array}$
• the first order condition with respect to B_t+1 is:
  
  $$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{(1+R_{t+1})}{p_{t+1}}u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$
  
  

• as far as the determination of the equilibrium price (inflation) level is concerned, this first order condition with respect to M_t+1 is enough.
  in equilibrium, c_t=y_t (just substitute the government budget constraint into the household budget constraint) and M_t=M_t^s, so that the FOC with respect to money becomes
  $\begin{array}{l} \frac{1}{p_t} u_c(y_t,\frac{M^s_t}{p_t})=\\ \beta \frac{1}{p_... ...\frac{M^s_{t+1}}{p_{t+1}})+ u_c(y_{t+1},\frac{M^s_{t+1}}{p_{t+1}})] \end{array}$
• the above relationship says that the price level depends only on current and future output, and on current and future money supplies.
  In other words for a given path of the money supply, the choice between taxes and borrowing (the time pattern of taxes) does not affect the equilibrium path of prices

• the FOC with respect to B_t+1 helps to pin down the equilibrium nominal interest rate. In fact, dividing the FOC wrt money for the FOC wrt bonds, and rearranging terms, one obtains:
  
  $$u_m(c_{t+1},\frac{M_{t+1}}{p_{t+1}})=R_{t+1}u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})$$
  
  
  which can be rewritten as:
  
  $$u_m(c_t,\frac{M_t}{p_t})=R_tu_c(c_t,\frac{M_t}{p_t})$$
  
  

• this is the relationship that determines the portfolio choice between bonds and money for the household (for given total wealth). The left hand side in fact represents the cost of investing one more unit of wealth into bonds (the fact that one loses the services that money provides) while the right hand side represents the benefit (the fact that one earns the nominal interest rate by investing in bonds, and nothing by keeping the money under the mattress). If this equation is not satisfied, the portfolio composition of the household is not optimal.
• after substituting for the equilibrium consumption and for the equilibrium real money balances, this relationship determines the equilibrium nominal interest rate as:
  
  $$R_t=\frac{u_m(y_t,\frac{M_t^s}{p_t})}{u_c(y_t,\frac{M^s_t}{p_t})}$$
  
  

• notice that the equilibrium nominal interest rate depends only on current output, and on real money balances. We showed above that the price level (and therefore real money balances) does not depend on the choice between taxes and borrowing (for a given path of the money supply). Therefore, for a given path of the money supply, the choice between taxes and borrowing (the time pattern of taxes) does not affect the equilibrium nominal interest rate either.
• The above equation establishes a relationship between the demand for real money balances, consumption, and the nominal interest rate. In equilibrium consumption is equal to output, and the demand for real money balances has to equal the supply of real money balances. This means that the equation
  
  $$R_t=\frac{u_m(y_t,\frac{M_t^s}{p_t})}{u_c(y_t,\frac{M^s_t}{p_t})}$$
  
  
  can be reinterpreted as the standard textbook relationship between money demand and money supply (so called ``liquidity preference"):
  
  $$\frac{M_t^s}{p_t}=L(y_t,R_t)$$
  
  

• notice that
  
  $$R_t=\frac{u_m(y_t,\frac{M_t^s}{p_t})}{u_c(y_t,\frac{M^s_t}{p_t})}$$
  
  
  can be rewritten as:
  
  $$R_t =\frac{u_m(y_t,m_t)}{u_c(y_t,m_t)}$$
  
  
  where m_t is real money balances, that is m_t=L(y_t,R_t)
  from the study of the function $f(m_t)=\frac{u_m(y_t,m_t)}{u_c(y_t,m_t)}$ one can notice that:
  • as long as the function is continuous, and $lim_{m_t \rightarrow \infty} \frac{u_m(y_t,m_t)}{u_c(y_t,m_t)}=0$, $lim_{m_t \rightarrow 0} \frac{u_m(y_t,m_t)}{u_c(y_t,m_t)}=+\infty$, the demand for money function is defined for all y and R
  • as long as $\frac{d \frac{u_m(y_t,m_t)}{u_c(y_t,m_t)}}{dm}<0$, it is defined in a unique way, and it is increasing in y, and decreasing in R

• Let us define the real interest rate r_t using the Fisher relationship
  
  $$(1+r_t)=(1+R_t)\frac{p_{t-1}}{p_t}$$
  
  
  (if we had `real' bonds in the model we would obtain the Fisher relationship from the first order conditions)
• in equilibrium the real interest rate is equal to:
  $\begin{array}{lll} (1+r_t)&=&(1+R_t)\frac{p_{t-1}}{p_t}\\ &=&\frac{u_c(y_t,\f... ..._{t-1},\frac{M^s_{t-1}}{p_{t-1}})}{\beta u_c(y_t,\frac{M^s_t}{p_t})}\end{array}$
  where in the last line we used the first order condition with respect to money holdings
  
  $$u_c(y_{t-1},\frac{M^s_{t-1}}{p_{t-1}})=\beta\frac{p_{t-1}}{p_t} [u_c(y_t,\frac{M^s_t}{p_t})+u_m(y_t,\frac{M_t^s}{p_t})]$$
  
  

• in other words we have the standard relationship between the real interest rate and the marginal rate of substitution between consumption in period t-1 and t:
  
  $$(1+r_t)=\frac{u_c(y_{t-1},\frac{M^s_{t-1}}{p_{t-1}})}{\beta u_c(y_t,\frac{M^s_t}{p_t})}$$
  
  
  (one of the implications of this relationship is that the real interest rate should be counter-cyclical) except that real money balances (and therefore monetary policy) now affect the real interest rate as they affect the marginal utility from consumption.
• once again for a given path of the money supply, the choice between taxes and borrowing (the time pattern of taxes) does not affect the equilibrium real interest rate either.

• as the first order condition with respect to money holdings is not affected by the introduction of bonds, if we assume that money supply is growing at a constant rate $\mu$ ( $M^s_t=(1+\mu)^t M_0$), and that output is constant (y_t=y, all t), the equilibrium real money balances, and the equilibrium path of prices is exactly the same as last time:
  • inflation is equal to the growth rate in money supply
  • the equilibrium condition for real money balances is again:
    
    $$(1+\mu-\beta)u_c(y,m^*)=\beta u_m(y,m^*)$$
    
    

• but this implies that under these assumptions the nominal interest rate is constant and equal to:
  
  $$R=\frac{u_m(y,m)}{u_c(y,m)}=\frac{(1+\mu-\beta)}{\beta}$$
  
  
  and the real interest rate rate is also constant and equal to the inverse of the household's discount rate:
  
  $$(1+r_t)=\frac{1}{\beta}$$
  
  

• it also implies that the Friedman rule
  ( $\mu = -(1-\beta)$) becomes implementable. In fact it implies that nominal interest rate are zero in equilibrium
  This means that the Central Bank does not need to know anything about the discount rate of households, but simply needs to follow the rule ``shrink the money supply at a rate at which the nominal interest rate is zero"
• why? from $u_m(c_t,\frac{M_t}{p_t})=R_tu_c(c_t,\frac{M_t}{p_t})$ one sees that if the nominal interest rate is zero bonds have no greater return than money, so that individuals can maximize the money balances they hold without fearing to lose the return on bonds