CLASS 7: THE BENEFITS FROM REDUCING INFLATION

In this class we are going to address the following questions:

• What are the benefits from reducing inflation?
• Are the benefits worth the costs.

• when we assume that money supply is growing at a constant rate and output is constant, we obtain that the utility of the household in equilibrium is:
  
  $$\sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})=\sum_{t=0}^\infty \beta^t u(y,m^*)=\frac{1}{1-\beta} u(y,m^*)$$
  
  

• remember also that real money balances are a -decreasing- function of the rate of money growth $\mu$
• in order to study the effect on utility of a small variation in the rate of money growth, we differentiate the equilibrium condition, which is
  
  $$(1+\mu-\beta)u_c(y,m^*)=\beta u_m(y,m^*)$$
  
  
  and obtain:
  $\begin{array}{l} u_c(y,m^*)+(1+\mu-\beta)u_{cm}(y,m^*)\frac{dm^*}{d\mu}=\\ =\beta u_{mm}(y,m^*)\frac{dm^*}{d\mu}\end{array}$
• so we get that:
  
  $$\frac{dm^*}{d\mu}=-\frac{u_c(y,m^*)}{(1+\mu-\beta)u_{cm}(y,m^*)-\beta u_{mm}(y,m^*)}$$
  
  

• and the overall effect on utility is therefore:
  $\frac{dU}{d\mu}=\frac{1}{1-\beta} u_m(y,m^*)\frac{dm^*}{d\mu}=\\ -\frac{1}{1-\beta} u_m(y,m^*)\frac{u_c(y,m^*)}{(1+\mu-\beta)u_{cm}(y,m^*)-\beta u_{mm}(y,m^*)}$
  an increase in money growth reduces real money balances, which enter positively in the utility function, and therefore reduces overall utility.
• Of course, utility is hard to quantify. Therefore a more useful tool is the compensating variation in consumption: by how much do I have to increase consumption (for one period) to compensate the household for the loss in utility due to lower real money balances (in all future periods)?
  This is the same as asking, how big should dy be such that:
  
  $$\frac{dU}{dy}dy+\frac{dU}{dm^*}\frac{dm^*}{d\mu}d\mu=0?$$
  
  

• the answer is:
  
  $$\frac{dy}{d\mu}=-\frac{\frac{dU}{dm^*}}{\frac{dU}{dy}}\frac{d... ...frac{\frac{1}{1-\beta} u_m(y,m^*)}{u_c(y,m^*)}\frac{dm^*}{d\mu}$$
  
  

• This answer still depends on marginal utilities, which are hard to quantify. But notice first that:
  
  $$\frac{u_m(y,m^*)}{u_c(y,m^*)}=\frac{(1+\mu-\beta)}{\beta}=R$$
  
  
  from the equilibrium condition for the nominal exchange rate.
• Also, we can use the money demand function, m=L(y,R), which in equilibrium can be written as:
  
  $$m=L(y,R)=L(y,\frac{(1+\mu-\beta)}{\beta})$$
  
  
  given that $R=\frac{(1+\mu-\beta)}{\beta}$ in equilibrium. Differentiating this function with respect to $\mu$ we obtain:
  
  $$\frac{dm^*}{d\mu}=\frac{1}{\beta}\frac{dL}{dR^*}=\frac{1}{\beta}m^*\frac{d\ln{L}}{dR^*}$$
  
  
  but the component $\frac{d\ln{L}}{dR^*}$, which we call $-\eta$, is the semi-elasticity of money demand with respect to the interest rate, which can be estimated empirically from the regression:
  
  $$\ln{\frac{M_t}{p_t}}=\gamma y_t-\eta R_t+\epsilon_t$$
  
  
  so that we get:
  
  $$\frac{dm^*}{d\mu}=-\frac{1}{\beta}m^*\eta$$
  
  

• Using these results we can rewrite the expression for the compensating variation in consumption as:
  
  $$\frac{dy}{d\mu}=-\frac{1}{1-\beta} \frac{(1+\mu-\beta)}{\beta}\frac{1}{\beta}m^* \eta$$
  
  

• but we know $\beta$, since from the equilibrium real interest rate we get
  
  $$\beta=\frac{1}{1+r}$$
  
  
  we know $\mu$, which is the rate of inflation, we know real money balances, and we know $\eta$, which can be inferred from empirical studies.
  So we have a way to quantify the benefits from reducing inflation.
• Friedman applies this formula to find that the benefits from reducing inflation from 2% to 1% is $2.3\%$ of GDP (assuming $r=5\%$ which implies $\beta=.952$, m^*=0.15GDP, $\eta=10$). In fact:
  $\begin{array}{lll} \frac{dy}{d\mu}(-.01)&=& -\frac{1}{1-\beta}\frac{(1+\mu-\be... ...-21\cdot.07\cdot1.05\cdot.15GPD\cdot10\cdot(-.01)\\ &=& 2.3\%GDP \end{array}$
• Notice that as inflation goes down the benefits from reducing inflation decrease, because the term $(1+\mu-\beta)$ goes down (although real money balances m^* increase, but we know that there is a cap for that). If you assume that real money balances are constant, you can see that reducing inflation from 1% to 0% would have a benefit of less than 2% of GDP.
• A main criticism to Friedman approach stems from the fact that reducing inflation, if prices are not completely flexible, may have a -temporary- cost in terms of lost output: Robert Gordon (Northwestern) estimates such cost to be 6% of GDP. If his estimation are correct, then one may ask the question: what is the level of inflation such that the marginal cost of reducing inflation by 1% is equal to the marginal benefit? That is, one would know the level of $\mu$ such that:
  
  $$-\frac{1}{1-\beta}\frac{(1+\mu-\beta)}{\beta}\frac{1}{\beta}m^* \eta (-.01)=.06GDP$$
  
  
  which is roughly 13% (always assuming that real money demand is 15% of GDP). If these calculations are correct, the message is that if inflation is below 13% the Central Bank should not bother to reduce inflation
  of course, the cost from reducing inflation crucially depend on how the disinflationary plan is carried out, and may well be different than 6%.
• Regardless of the costs of disinflation, a second argument against the Friedman rule (or against reducing inflation until it becomes negative) comes from the Ramsey principle of optimal taxation, which says that the government should tax in such a way that the marginal distortion from the different taxes are equal
• notice that the amount of seignorage collected by the government in the model is equal to:
  
  $$\sigma=\frac{M_{t+1}^s-M_t^s}{p_t}=\frac{\mu M_t^s}{p_t}=\mu m^*$$
  
  
  so that the impact on seignorage of a marginal change in the rate of money growth is:
  $\begin{array}{lll} \frac{d \sigma}{d\mu} &=& m^*+\mu\frac{dm^*}{d\mu}=\\ &=& m^*-\mu\frac{1}{\beta}m^*\eta=\\ &=& m^*(1-\mu\frac{1}{\beta}\eta)\end{array}$
• When $\mu$ is small enough ( $\mu<\frac{\beta}{\eta}\simeq 10\%$), seignorage grows with an increase in money growth, and in inflation
• Also, when money growth is negative seignorage is also negative. The negative seignorage is financed through lump sum taxes, according to the model.
  But if those taxes are distortive, one is introducing inefficiencies in the economy by following Friedman's rule.
• In general, for countries like Italy, or Mexico, where tax collection is inefficient (tax evasion) it may be best to collect some revenues through seignorage, and therefore it may make sense to have positive rates of inflation.