CLASS 5: INFLATION AND THE GROWTH RATE OF MONEY SUPPLY

• we will study a model in which money supply grows at a constant rate
• we will see that different values of money growth imply different values of:
  
  • inflation
  • the nominal interest rate
  • real money holdings
  • welfare of households
• the bottom line of this class will be trying to understand the effect of different monetary policy regimes on inflation, interest rates, real money holdings, and the welfare of individuals
• for the time being, the rate of growth in money supply is supposed to be constant at a given level $\mu$ forever
  
  $$M^s_t=M^s_0(1+\mu)^t$$
  
  
  or
  
  $$\frac{M^s_t-M^s_{t-1}}{M^s_{t-1}}=\mu$$
  
  

• introduce the government budget constraint
• why? if the Central Bank is printing money, then it is also collecting seignorage
• what is seignorage? the amount of real resources obtained by the Central Bank by printing money:
  
  $$\frac{M_{t+1}^s-M_t^s}{p_t}$$
  
  

• seignorage needs to be spent somehow. we will assume that it is given back to the household through lump sum transfers T_t
• notice that if money creation is negative (the Central Bank is buying money back) lump sum transfers become lump sum taxes
• the government budget constraint therefore is:
  
  M_t+1^s-M_t^s=T_t
  
  

the problem faced by the household is the same as before:

$$max_{{M_{t+1},c_t}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})$$

subject to

$$M_{t+1} \leq M_t- p_t c_{t} +p_t y_{t}$$

and subject to an initial given quantity of money, M₀
except that we now have lump sum transfers in the intertemporal budget constraint:

$$c_t \leq y_t-\frac{M_{t+1}-M_t}{p_t}+\frac{T_t}{p_t}$$

since this constraint will hold with the = sign at the optimum, we can substitute it into the objective function, which becomes:

$$max_{{M_{t+1}}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(y_t-\frac{M_{t+1}-M_t}{p_t}+\frac{T_t}{p_t},\frac{M_t}{p_t})$$

notice that the presence of transfers does not affect at all the first order condition with respect to M_t+1, which is:

$$\frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \beta \frac{1}{p_{t+... ...frac{M_{t+1}}{p_{t+1}})+ u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}})]$$

Using the same arguments of the previous class (including transversality) we obtain the condition:

$$\frac{1}{p_t} = \frac{1}{u_c(c_t,\frac{M_t}{p_t})} \sum_{i=1... ...beta^i \frac{1}{p_{t+i}} u_m(c_{t+i},\frac{M_{t+i}}{p_{t+i}})$$

Equilibrium

• in equilibrium it must be that c_t=y_t, all t
• assume that the endowment is constant: y_t=y. Therefore consumption is constant.
• as discussed in the introduction, the money supply is growing at a constant rate, which we denote as $\mu$ (which can possibly be negative): $M^s_t=(1+\mu)^t M_0$
• in equilibrium, money demand equals money supply. it follows that $M_t=(1+\mu)^t M_0$
• substituting for c_t and M_t in the pricing formula, we obtain:
  
  
  $$\frac{1}{p_t} = \frac{1}{u_c(y,\frac{(1+\mu)^t M_0}{p_t})} \... ...^i \frac{1}{p_{t+i}} u_m(y,\frac{(1+\mu)^{t+i} M_0}{p_{t+i}})$$
  
  

• let us use the following trick: let us call $\hat{p}_t=\frac{p_t}{(1+\mu)^t}$ and substitute in the above formula, obtaining:
  
  $$\begin{eqnarray*}\frac{1}{\hat{p}_t}\frac{1}{(1+\mu)^t} = \\ \frac{1}{u_c(y,\f... ...{t+i}} \frac{1}{(1+\mu)^{t+i}} u_m(y,\frac{M_0}{\hat{p}_{t+i}}) \end{eqnarray*}$$
  
  or, simplifying:
  
  $$\frac{1}{\hat{p}_t}= \frac{1}{u_c(y,\frac{M_0}{\hat{p}_t})}... ...})^i \frac{1}{\hat{p}_{t+i}} u_m(y,\frac{M_0}{\hat{p}_{t+i}})$$
  
  

• notice that the pricing formula is again stationary, but this time as a function of $\hat{p}$ and not of p: $\hat{p}_t$ depends on $\hat{p}_{t+1},\hat{p}_{t+2},..$ in the same way that $\hat{p}_{t+1}$ depends on $\hat{p}_{t+2},\hat{p}_{t+3},..$
  consequently, a constant price level, $\hat{p}_t=\hat{p}$, is a solution
• what is the implication? If $\hat{p}_t$ is constant over time in equilibrium, since by definition $p_t=\hat{p}_t{(1+\mu)^t}$, it must be that prices also grow at the constant rate $\mu$, that is:
  in this model inflation is equal to the growth rate in money supply
• to find the equilibrium, we still have to determine what the value of $\hat{p}$ is
• Notice that when the price level grows at the constant rate $\mu$, real money balances $m_t=\frac{M_t}{p_t}$ are constant at a level $m=\frac{M_0}{\hat{p}}$
• with constant prices and constant real money balances the pricing formula becomes:
  
  $$1=\frac{1}{u_c(y,m)}\frac{\beta}{1+\mu-\beta}u_m(y,m)$$
  
  
  or, rearranging terms:
  
  $$u_c(y,m)-\frac{\beta}{1+\mu-\beta}u_m(y,m)=0$$
  
  

• Is there an equilibrium level of real money balances m^* such that
  
  $$u_c(y,m^*)-\frac{\beta}{1+\mu-\beta}u_m(y,m^*)=0\mbox{ ?}$$
  
  
  notice that if we find an equilibrium level of real money balances then we have also found the the value of $\hat{p}$, which would be: $\hat{p}^*=\frac{M_0}{m^*}$
• the argument of why an equilibrium value of real money balances exists and is unique is the same as in the previous class
• also, under the same set of conditions, we have that the equilibrium real money balances increase with an increase in income
• How does the equilibrium real money demand change with changes in the growth rate of money supply $\mu$?
  this can be inferred from the study of:
  
  $$u_c(y,m^*)-\frac{\beta}{1+\mu-\beta}u_m(y,m^*)=0,$$
  
  
  or:
  
  $$u_c(y,m^*)=\frac{\beta}{1+\mu-\beta}u_m(y,m^*)$$
  
  

• if $\mu$ increases, the right hand term decreases. In order to restore equilibrium, m^* has to change. How?
  If real money and consumption are complement, and if the marginal utility from real money balances is decreasing in m, a decrease in m^* decreases the left hand side u_c(y,m^*), and increases the term u_m(y,m^*), thereby restoring equilibrium
• so, if $\mu$ increases, m^* has to decrease to restore equilibrium
• we found that an increase in the rate growth of money supply implies an equal increase in inflation (growth rate of the price level) and a decrease in real money holdings (Bresciani-Turroni effect)

 

Table: Bresciani-Turroni effect

Country	Period	Inflation	real money as % of
			initial value
Greece	1944	85,000,000 %	0.7 %
Hungary	1946	42,000,000,000,000 %	0.25 %
Bolivia	1985	100% monthly	25% (1980)

Notice that low real money demand means tat people do not have enough currency to carry over transactions (that is, they have a lot of pieces of paper, but the real value of those pieces of paper is very low). Is printing pieces of paper at a faster rate a solution? the answer we get from our model is no, because real money balances would go down even further.

The Optimal Quantity of Money

• what is the optimal rate at which the central bank should be printing money?
  
• notice that from our model we obtain that the utility of the household in equilibrium is:
  
  
  $$\sum_{t=0}^\infty \beta^t u(y,m^*)=\frac{1}{1-\beta} u(y,m^*)$$
  
  

• therefore agents utility is larger in equilibrium the larger the amount of real money balances they hold
• we saw that real money balances are a negative function of $\mu$.
  How can we make real money balances as high as possible? by choosing $\mu$ as low as possible
• what is the lowest possible level of $\mu$ compatible with the existence of an equilibrium?
• From the equilibrium condition, which can be rewritten as
  
  $$(1+\mu-\beta)u_c(y,m^*)=\beta u_m(y,m^*)$$
  
  
  , given that both u_c and u_m are non negative, we obtain that:
  
  
  $$\mu \geq -(1-\beta)$$
  
  

• in particular, if when $\mu = -(1-\beta)$ the demand for money can be satiated, that is, $\hat{m}^*=m(-(1-\beta))^*$ (remember that equilibrium real money balances are a function of $\mu$) is such that $u_m(y,\hat{m}^*)=0$, then we can have an equilibrium with $\mu = -(1-\beta)$: this is the Friedman rule
• from the government budget constraint
  
  
  M_t+1^s-M_t^s=T_t
  
  
  
  one can see that the Friedman rule (or any negative growth of money) means that the Central bank should be taxing the household in order to reduce existing money supply.
  Why is this optimal? Because equilibrium consumption is not affected anyway
  (remember $c_t = y_t-\frac{M_{t+1}-M_t}{p_t}+\frac{T_t}{p_t}$, and in equilibrium c_t=y_t)
  and real money balances are higher