CLASS 16: IS MONETARY POLICY NEUTRAL?

• Do changes in monetary policy have an affect on output?
  in particular:
  • Do changes in the level of money supply have an affect on output? (Is monetary policy neutral?)
  • Do changes in the growth rate of money supply have an affect on output? (Is monetary policy superneutral?)
• We will first address these questions in a setting where prices and wages are flexible, and then is a setting where they are rigid
• We will find that ...

Effects of an Unexpected Change in the Level of Money Supply with Flexible Prices

• Before time T money supply is constant at the level M^s₀, and expected to stay so forever.
• Let us suppose that at time T money supply is unexpectedly raised to the level M^s_T: what happens to prices, output, and employment?
• Let us remember the equations determining the equilibrium:
  
  $$u_c(c^*,m^*(0),l^*)-\frac{\beta}{1-\beta}u_m(c^*,m^*(0),l^*)=0$$
  
  
  
  
  $$f'(\bar{l}-l^*)=\frac{u_l(c^*,m^*(0),l^*)}{u_c(c^*,m^*(0),l^*)}$$
  
  
  
  
  $$c^*=f(\bar{l}-l^*)$$
  
  
  
  
  $$p_t=\frac{M^s_0}{m^*(0)}$$
  
  
  
  
  $$\frac{w_t}{p_t}=\frac{u_l(c^*,m^*(0),l^*)}{u_c(c^*,m^*(0),l^*)}$$
  
  

• Note that, as long as prices are flexible, real money balances, leisure (and henceforth employment) and consumption are determined independently of the level of money supply.
• Moreover, the growth rate of money supply is zero both before and after the announcement.
  We conclude that with flexible prices money is neutral: a change in the level of money supply has no effect on the `real' variables.
• What happens to prices and wages? from the equilibrium conditions it is clear that prices increase in proportion to the increase in money supply:
  
  $$p_T=\frac{M^s_T}{m^*(0)}=\frac{M^s_T}{M^s_0}p_{T-1}$$
  
  

• Nominal wages increase proportionally to prices (so that the real wage is constant): given that $\frac{w_T}{p_T}=\frac{u_l}{u_c}$ we have $\frac{w_T}{p_T}=\frac{w_{T-1}}{p_{T-1}}$, or:
  
  $$w_T=\frac{M^s_T}{M^s_0}w_{T-1}.$$
  
  

Effects of an Unexpected Change in the Growth Rate of Money Supply with Flexible Prices

• Again, let us suppose that before time T money supply is constant at the level M^s₀, and expected to stay so forever. At time T it is announced that money supply will grow at rate $\mu$ forever after:
  
  $$M^s_t=M^s_0 (1+\mu)^{t-T}$$
  
  

• When money supply grows at a constant rate the equilibrium conditions become:
  
  $$u_c(c^*,m^*(\mu),l^*)-\frac{\beta}{1+\mu-\beta}u_m(c^*,m^*(\mu),l^*)=0$$
  
  
  
  
  $$f'(\bar{l}-l^*)=\frac{u_l(c^*,m^*(\mu),l^*)}{u_c(c^*,m^*(\mu),l^*)}$$
  
  
  
  
  $$c^*=f(\bar{l}-l^*)$$
  
  
  
  
  $$p_t=\frac{M^s_t}{m^*(\mu)}$$
  
  
  
  
  $$\frac{w_t}{p_t}=\frac{u_l(c^*,m^*(\mu),l^*)}{u_c(c^*,m^*(\mu),l^*)}$$
  
  

• From the equilibrium conditions two things are apparent: in the first place m^*(0) is no longer the equilibrium value for real money balances, which is $m^*(\mu)$.
  
• In the second place, since real money balances enter all three equilibrium conditions, it is also clear that in general output, employment, and consumption will also be affected.
• Money is not superneutral, even with flexible prices.
• In which direction? remember that $\frac{w_T}{p_T}=\frac{u_l}{u_c}$: the direction depends on whether a decrease in real money balances increases or decreases the marginal rate of substitution between leisure and consumption.
• For example, presumably a decrease in real money balances makes consumption less attractive, as people have less of a choice between cash and credit goods.
  If this is the case, the marginal utility of consumption decreases. If the marginal utility of leisure is unaffected by the level of real money balances, real wages have to go up, unless leisure increases. This suggests that output and employment may go down (so with flexible wages/prices it is hard to get a Phillips curve kind of response).

Effects of an Unexpected Change in the Level of Money Supply with Wage Rigidity

• Why should wages be rigid? Staggered contracts: most workers do not rewrite their contracts every two weeks. Most times the terms of the contract do not change for a given period of time (two or three years), until the contracts are renewed.
• Since most of the times contracts define the wage in nominal terms, this implies that nominal wages are fixed for a period of time.
• Okay, but why don't workers negotiate in real terms? To some extent, this happens in (hyper) inflation countries: wage indexation.
• What about prices? there is some evidence (at least for the US) showing some degree of `price rigidity' as well (menu costs).
  However, we are going to assume that prices are flexible, on the ground that they adjust faster than wages.
• To be more specific, we are going to assume that nominal wages are fixed for one period. This implies that, if nominal prices change, the real wage may be different than the one that equilibrates supply and demand for labor. We are going to assume that we will always be on the labor demand schedule: for given real wage, the firm chooses how much work to demand. In other words, we assume that the contract specifies the nominal wage, and that workers agree to work for whatever number of hours the firms requires at that nominal wage.
• Once again, assume that before time T money supply is constant at the level M^s₀, and expected to stay so forever, and that at time T money supply is unexpectedly raised to the level M^s_T.
• First, let us consider what happens for $t \geq T+1$, that is, after wages have adjusted. Of course, after wages have adjusted, the equilibrium is the same one as with flexible wages:
  
  $$p_{T+1}=p_{T+2}=...=\frac{M^s_T}{m^*(0)}=\frac{M^s_T}{M^s_0}p_{T-1}$$
  
  
  and
  
  $$w_{T+1}=w_{T+2}=...=\frac{M^s_T}{M^s_0}w_{T-1}$$
  
  
  and output, employment, and consumption are unchanged from period T-1
• What about period T? we know that in period T nominal wages are fixed:
  
  w_T=w_T-1
  
  
  . We also know we are on the demand for labor schedule, that is:
  
  $$f'(h_T)=\frac{w_T}{p_T}=\frac{w_{T-1}}{p_T}.$$
  
  

• So it all depends on whether p_T is greater or smaller than p_T-1: if greater, then the real wage is smaller than in the previous period, and therefore employment and output are higher. Viceversa, if p_T is smaller than p_T-1 greater, then the real wage is larger, and therefore employment and output are lower.
• To address this question we analyze the usual first order condition for nominal money balances in period T (after substituting the equilibrium condition c_t=y_t=f(h_t)):
  
  $$\frac{1}{p_T} u_c(y_T,\frac{M_T}{p_T},l_T)= \beta \frac{1}{p_{T+1}}[u_m(y_{T+1},\frac{M_{T+1}}{p_{T+1}},l_{T+1})+$$
  
  
  
  
  $$+u_c(y_{T+1},\frac{M_{T+1}}{p_{T+1}},l_{T+1})]$$
  
  

• We know that for $t \geq T+1$ the equilibrium condition for real money balances is
  
  $$u_c(c^*,m^*(0),l^*)-\frac{\beta}{1-\beta}u_m(c^*,m^*(0),l^*)=0$$
  
  
  so we can substitute for u_m(c^*,m^*(0),l^*) and obtain:
  
  $$\frac{1}{p_T} u_c(y_T,\frac{M_T}{p_T},l_T)= \frac{1}{p_{T+1}} u_c(y_{T+1},\frac{M_{t+1}}{p_{T+1}},l_{T+1})$$
  
  
  or
  
  $$p_T=\frac{u_c(y_T,\frac{M_T}{p_T},l_T)}{u_c(y_{T+1},\frac{M_{T+1}}{p_{T+1}},l_{T+1})} p_{T+1}$$
  
  

• Substituting for $p_{T+1}=\frac{M^s_T}{M^s_0}p_{T-1}$, and knowing that M_T+1=M_T, y_T+1=y_T-1, l_T+1=l_T-1, and m_T+1=m_T-1 we get:
  
  $$p_T=\alpha \frac{M^s_T}{M^s_0}p_{T-1}$$
  
  
  where we define $\alpha$ as:
  
  $$\alpha \equiv \frac{u_c(y_T,\frac{M_T}{p_T},l_T)}{u_c(y_{T-1},\frac{M_{0}}{p_{T-1}},l_{T-1})}$$
  
  

• We will show that p_T>p_T-1 by contradiction. Let us say that $p_T \leq p_{T-1}$. Then it must be that: $y_T \leq y_{T-1}$ and $\frac{M_T}{p_T}>\frac{M_{0}}{p_{T-1}}$
• but then (if u_cc<0 and $u_{cm}\geq0$) it must be that
  
  $$u_c(y_T,\frac{M_T}{p_T},l_T) > u_c(y_{T-1},\frac{M_{0}}{p_{T-1}},l_{T-1})$$
  
  
  and
  
  $$\alpha \geq 1$$
  
  
  . Therefore $\alpha \frac{M^s_T}{M^s_0}>1$. Therefore the assumption $p_T \leq p_{T-1}$ contradicts the formula:
  
  $$p_T=\alpha \frac{M^s_T}{M^s_0}p_{T-1}.$$
  
  

• So we have shown that p_T>p_T-1. because of this, we have also shown that y_T > y_T-1: the real wage is lower (since the nominal wage is constant, and prices are higher), so from $f'(h_T)=\frac{w_{T-1}}{p_T}$ the firm is willing to hire more people, and output will rise: money is not neutral with wage rigidity.
• By how much will prices increase? we will argue that they have increased by less than the increase in nominal money balances: that is
  
  $$\frac{p_T}{p_{T-1}}<\frac{M^s_T}{M^s_0}$$
  
  
  which implies $\alpha<1$.
• In fact, say that $\frac{p_T}{p_{T-1}}=\frac{M^s_T}{M^s_0}$, which implies two things: i) $\frac{M_T}{p_T}=\frac{M_{0}}{p_{T-1}}$, ii) $\alpha=1$. But we also know that y_T > y_T-1. This implies that
  
  $$u_c(y_T,\frac{M_T}{p_T},l_T) < u_c(y_{T-1},\frac{M_{0}}{p_{T-1}},l_{T-1})$$
  
  
  (if leisure and consumption are not strong substitutes), which means $\alpha<1$: we have a contradiction.
• In conclusion, prices must rise, but less than the rise in money supply. Since we know that in the next period prices will have risen proportionally to the increase in money supply, it must mean that prices will rise again between T and T+1. (see figure)
• Does the fact that output rises imply that it is welfare improving to raise money supply unexpectedly? That is, is it a good thing that the monetary authority `surprise' the public?

Effects of an Unexpected Change in the Growth Rate of Money Supply with Wage Rigidity

• In this section we will study the effects of a change in $\mu$ on output and employment with rigid wages (and flexible prices).
• We will focus on the case in which the level of real money balances does not affect the marginal rate of substitution between leisure and consumption, that is: m does not affect $\frac{u_l}{u_c}$.
• Why is this case interesting? Let us recall the equilibrium with flexible prices:
  
  $$u_c(c^*,m^*(\mu),l^*)-\frac{\beta}{1+\mu-\beta}u_m(c^*,m^*(\mu),l^*)=0$$
  
  
  
  
  $$f'(\bar{l}-l^*)=\frac{u_l(c^*,m^*(\mu),l^*)}{u_c(c^*,m^*(\mu),l^*)}$$
  
  
  
  
  $$c^*=f(\bar{l}-l^*)$$
  
  
  
  
  $$p_t=\frac{M^s_0}{m^*(\mu)}$$
  
  
  
  
  $$\frac{w_t}{p_t}=\frac{u_l(c^*,m^*(\mu),l^*)}{u_c(c^*,m^*(\mu),l^*)}$$
  
  

• If m does not enter $\frac{u_l}{u_c}$, then it does not affect the labor supply. Therefore it does not affect the equilibrium in the labor market, and therefore it does not affect employment and output. In this case, money is superneutral.
• What happens with wage rigidity? is money still superneutral? we know that wage rigidity lasts only one period in the model. This means that in the long run (after one period) money is superneutral: $\frac{w_{T-1}}{p_{T-1}}=\frac{w_{T+1}}{p_{T+1}}$, h_T-1=h_T+1, y_T-1=y_T+1, etc.
• What happens in the short run (period T)?
• As in the previous case, with fixed nominal wages it all depends on what happens to prices: if prices increase, the real wage decreases and production increases.
• To understand what happens to prices we proceed as in the previous case (increase in the level of money supply). Let us write the FOC for nominal money balances in period T: (after substituting the equilibrium condition c_t=y_t=f(h_t)):
  
  $$\frac{1}{p_T} u_c(y_T,\frac{M_T}{p_T},l_T)= \beta \frac{1}{p_{T+1}}[u_m(y_{T+1},\frac{M_{T+1}}{p_{T+1}},l_{T+1})+$$
  
  
  
  
  $$+u_c(y_{T+1},\frac{M_{t+1}}{p_{T+1}},l_{T+1})]$$
  
  

• As before, we know the equilibrium condition for real money balances for $t \geq T+1$
  
  $$u_c(c^*,m^*(\mu),l^*)-\frac{\beta}{1-\beta+\mu}u_m(c^*,m^*(\mu),l^*)=0$$
  
  
  and we can plug it in the previous relationship, obtaining:
  
  $$\frac{1}{p_T} u_c(y_T,\frac{M_T}{p_T},l_T)= (1+\mu) \frac{1}{p_{T+1}} u_c(y_{T+1},\frac{M_{t+1}}{p_{T+1}},l_{T+1})$$
  
  
  or:
  
  $$p_T=\frac{u_c(y_T,\frac{M_T}{p_T},l_T)}{u_c(y_{T+1},\frac{M_{t+1}}{p_{T+1}},l_{T+1})} \frac1{1+\mu}p_{T+1}$$
  
  

• Now let us recall that l_T-1=l_T+1, y_T-1=y_T+1. Furthermore let us notice that:
  
  $$\begin{array}{l} p_{T+1}=\frac{M_{T+1}}{m^*(\mu)}=\frac{M_{T... ...}p_{T-1} = \frac{m^*(0)}{m^*(\mu)}(1+\mu)p_{T-1} \end{array}$$
  
  
  (remember in fact that $M_{T+1} =M_0 (1+\mu)$ and that M_T=M_T-1=M₀)
• This implies that:
  
  $$p_T=\frac{u_c(y_T,\frac{M_T}{p_T},l_T)}{u_c(y_{T+1},\frac{M_{t+1}}{p_{T+1}},l_{T+1})} \frac1{1+\mu}p_{T+1}=$$
  
  
  
  
  $$=\frac{u_c(y_T,\frac{M_0}{p_T},l_T)}{u_c(y_{T-1},m^*(\mu),l_{... ...ac{m^*(0)}{m^*(\mu)}p_{T-1}=\eta \frac{m^*(0)}{m^*(\mu)}p_{T-1}$$
  
  
  where
  
  $$\eta \equiv \frac{u_c(y_T,\frac{M_0}{p_T},l_T)}{u_c(y_{T-1},m^*(\mu),l_{T-1})}$$
  
  

• As before, we will show that p_T>p_T-1 by contradiction. Let us say that $p_T \leq p_{T-1}$. Then it must be that: $y_T \leq y_{T-1}$ and
  
  $$\frac{M_0}{p_T}>\frac{M_{0}}{p_{T-1}}=m^*(0)>m^*(\mu)$$
  
  
  . This implies that
  
  $$u_c(y_T,\frac{M_0}{p_T},l_T) > u_c(y_{T-1},m^*(\mu),l_{T-1})$$
  
  
  , or $\eta >1$. Since $\frac{m^*(0)}{m^*(\mu)}>1$, it must follow that p_T>p_T-1, which contradicts the original assumption.
• By how much will prices increase? we will argue that they have increased by less than the increase in real money balances: that is
  
  $$\frac{p_T}{p_{T-1}}<\frac{m^*(0)}{m^*(\mu)}$$
  
  
  which implies $\eta <1$.
• In fact, say that $\frac{p_T}{p_{T-1}}=\frac{m^*(0)}{m^*(\mu)}$. This implies that: i) $m^*(\mu)= \frac{p_{T-1}}{p_T}m^*(0)=\frac{p_{T-1}}{p_T}\frac{M_{0}}{p_{T-1}}= \frac{M_T}{p_T}$ ii) $\eta=1$. But we also know that y_T > y_T-1. This implies that $u_c(y_T,\frac{M_0}{p_T},l_T) < u_c(y_{T-1},m^*(\mu),l_{T-1})$ (if leisure and consumption are not strong substitutes), which means $\eta <1$: we have a contradiction.
• Note that, if wages were flexible, $\frac{p_T}{p_{T-1}}=\frac{m^*(0)}{m^*(\mu)}$. Therefore the fact that $\eta <1$ implies that prices grow less than in the flexible wage case, since $\frac{p_T}{p_{T-1}}=\eta \frac{m^*(0)}{m^*(\mu)}$
• Note also that this implies that between periods T and T+1 prices must grow at a faster rate than $\mu$. In fact note that:
  
  $$\frac{p_{T+1}}{p_{T-1}}=\frac{\frac{M_{T+1}}{m^*(\mu)}}{\frac{M_{T-1}}{m^*(0)}}=$$
  
  
  
  
  $$=\frac{\frac{M_{T-1}(1+\mu)}{m^*(\mu)}}{\frac{M_{T-1}}{m^*(0)}} =(1+\mu)\frac{m^*(0)}{m^*(\mu)}$$
  
  

• So we have that:
  
  $$\frac{p_{T+1}}{p_{T-1}}=\frac{p_{T+1}}{p_{T}}\frac{p_{T}}{p_{T-1}} =(1+\mu)\frac{m^*(0)}{m^*(\mu)}.$$
  
  
  But since we know that
  
  $$\frac{p_T}{p_{T-1}}=\eta \frac{m^*(0)}{m^*(\mu)}$$
  
  
  this implies:
  
  $$\frac{p_{T+1}}{p_{T}}=\frac{1+\mu}{\eta}>1+\mu.$$
  
  

Short Run and Long Run Phillips Curve

• This model displays a case of short run and long run Phillips curve (Friedman)
  in the short run, an inflationary surprise leads to an increase in output, and a consequent increase in employment (reduction of unemployment):
  the Phillips curve in the short run is downward sloping.
• In the long run (after wages have adjusted) we are back to the flexible wages case: if changes in real money balances have no impact on the marginal rate of substitution between labor and leisure, the change in the rate of money growth (and inflation) has no impact on output and employment.

The effect of Announcement, and the effect of shocks to fiscal policy

• As we know, the announcement of an increase in the growth rate of money supply in the future triggers an increase in inflation now.
• With wage rigidity, an announcement of a more loose (tight) monetary policy in the future may therefore induce an expansion (contraction) of output now
• one implication of this reasoning is that a shock to government spending (if the public believes that taxes will not increase to cover the deficit) may induce an expansion in output.
  This does not occur through the usual keynesian channel of a higher aggregate demand.
  it may occur through the expectation of future monetization of the budget deficit, which raises inflationary expectations and therefore raises current inflation.
  With rigid wages, an unexpected increase in inflation reduces real wages, stimulates the demand for labor and increases output.
• A second implication is that the recession in Europe could in principle be related to the anticipation of the monetary union (and the consequent decline in inflation).