CLASS 4: THE VALUE OF MONEY: THE SIDRAUSKI-BROCK MODEL

• we will study the determinants of the price level in an economy where the money supply is constant
• we will also point out at analogies, but also at crucial differences, between money and durable goods

The Sidrauski-Brock Model

we will start exactly where we left last time; that is, the problem faced by the household is as follows:

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

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
notice that the intertemporal budget constraint can be rewritten as:

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

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

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

since the term M_t+1 appears only in the following two terms of the infinite sum:
$\beta^t u(y_t-\frac{M_{t+1}-M_t}{p_t},\frac{M_t}{p_t})+ \\ \beta^{t+1} u(y_{t+1}-\frac{M_{t+2}-M_{t+1}}{p_{t+1}}, \frac{M_{t+1}}{p_{t+1}})$
the FOC with respect to M_t+1 is:

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

As in the previous classes, we use the FOC's for M_t+2,M_t+3,.. to substitute repeatedly for $\frac{1}{p_{t+1}}u_c(c_{t+1},\frac{M_{t+1}}{p_{t+1}}),\\ \frac{1}{p_{t+2}}u_c(c_{t+2},\frac{M_{t+2}}{p_{t+2}}),..$ and obtain:

$$\begin{eqnarray*}& \frac{1}{p_t} u_c(c_t,\frac{M_t}{p_t})= \\ &\sum_{i=1}^{T} ... ... \beta^T\frac{1}{p_{t+T}}u_c(c_{t+T},\frac{M_{t+T}}{p_{t+T}}) \end{eqnarray*}$$

if $T \rightarrow \infty$, and if the transversality condition

$$lim_{T \rightarrow \infty} \beta^T\frac{1}{p_{t+T}}u_c(c_{t+T},\frac{M_{t+T}}{p_{t+T}})=0$$

holds true (we will discuss it later), we obtain a formula for (the inverse of) the price level:

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

• the similarities between this formula and the one for the price of durable goods is striking: both the value of durable goods and the value of money (the inverse of the price level) is a present discounted value of all the future services that they respectively provide.
• however, there are also notable differences between the durable goods and money: while the service provided by a durable good does not depend on their price (the service provided by a refrigerator does not depend on its price), the service provided by money balances depends crucially on the future price levels (with a given amount of nominal money balances you can buy more consumption goods if prices are low than if prices are high)
• moreover, one can see that in a finite horizon economy money is worthless, while durable goods are not.
  Let us suppose that the economy ends at time T. Then at time T money is worthless. But that will imply that at time T-1 nobody will accept money as a mean of payments. Because of this, money is worthless at time T-1 as well. So nobody will accept it at time T-2, and so on... the conclusion is that money is worthless in all periods (why is this not the case for a durable good?)

Equilibrium

In equilibrium it must be that c_t=y_t, all t
For the time being let us assume that the Central Bank keeps the money supply constant at the level M^s in all periods. since in equilibrium money demand must equal money supply: M_t=M^s, all t.
Substituting for c_t and M_t in the pricing formula, we obtain:

$$\frac{1}{p_t} = \frac{1}{u_c(y_t,\frac{M^s}{p_t})} \sum_{i=1... ...y} \beta^i \frac{1}{p_{t+i}} u_m(y_{t+i},\frac{M^s}{p_{t+i}})$$

Unlike in the previous models, we are not home yet! the formula does not deliver a solution for the price level, because the price in period t depends on the prices in all future periods. In order to obtain a formula for the price level we have to make the further assumption that output is constant in all periods: y_t=y all t.
under this assumption, the pricing formula becomes:

$$\frac{1}{p_t} = \frac{1}{u_c(y,\frac{M^s}{p_t})} \sum_{i=1}^{\infty} \beta^i \frac{1}{p_{t+i}} u_m(y,\frac{M^s}{p_{t+i}})$$

Notice that the pricing formula is stationary, in the sense that p_t depends on p_t+1,p_t+2,.. in the same way that p_t+1 depends on p_t+2,p_t+3,..
consequently, a constant price level, p_t=p, is a solution (it may not be the only solution. However, we will focus on stationary equilibria).
Notice also that when the price level is constant, real money balances $m_t=\frac{M_t}{p_t}$ are constant as well at a level $m=\frac{M^s}{p}$. How is the equilibrium price level determined?
with constant prices and constant real money balances the pricing formula becomes:

$$1=\frac{1}{u_c(y,m)}\frac{\beta}{1-\beta}u_m(y,m)$$

rearranging terms this formula becomes:

$$u_c(y,m)-\frac{\beta}{1-\beta}u_m(y,m)=0$$

Is there an equilibrium level of real money balances m^* such that:

$$u_c(y,m^*)-\frac{\beta}{1-\beta}u_m(y,m^*)=0$$

notice that if we find an equilibrium level of real money balances then we have also found the equilibrium price level, given that:

$$p^*=\frac{M^s}{m}$$

To answer this question let us draw (figure 1) the function:

$$f(m)=u_c(y,m)-\frac{\beta}{1-\beta}u_m(y,m)$$

If the function crosses the horizontal axis at least once, then there is an equilibrium level for real money balances. if it crosses the axis only once, then the equilibrium is unique.

• when real money balances tend to zero, the marginal utility from holding money is likely to be higher (possibly $\infty$) than the marginal utility from income, therefore:
  
  $$\begin{eqnarray*}&\lim_{m \rightarrow 0} f(m)=\\ &\lim_{m \rightarrow 0} u_c(y,m)-\frac{\beta}{1-\beta}u_m(y,m)<0 \end{eqnarray*}$$
  
  
• conversely, when real money balances are infinity, the marginal utility from holding money is likely to be lower (possibly 0, remember the ``cash in advance'' model) than the marginal utility from income, therefore:
  
  $$\begin{eqnarray*}&\lim_{m \rightarrow \infty} f(m)\\ &=\lim_{m \rightarrow \infty} f(m)u_c(y,m)-\frac{\beta}{1-\beta}u_m(y,m)>0 \end{eqnarray*}$$
  
  
• if the function f(.) is continuous, it must be that it crosses the x-axis at least once
• furthermore, if the function f(.) is always increasing, that is, if:
  
  $$\frac{df}{dm}=u_cm(y,m)-\frac{\beta}{1-\beta}u_mm(y,m)>0$$
  
  
  then it crosses the axis only once, and the equilibrium is unique (one can show that this condition has to do with convexity of the utility function)
• How does the equilibrium real money demand change with changes in income y (figure 2)?
  if real money and consumption are complement, then an increase in y increases the marginal utility of money (increase u_m(y,m), for given m). on the other hand, if the marginal utility from consumption is decreasing, an increase in income will decrease the marginal utility of consumption (decrease in u_c(y,m)). the overall effect is that the function f(.) shift down: the equilibrium real money balances increase with an increase in income
• finally, notice that the equilibrium real money balance do not depend on the level of the money supply. so doubling money supply in this model (assuming that it remains constant afterwards) only implies that the price level double, since $p^*=\frac{M^s}{m^*}$

What have we got?

• the Quantity Theory of Money
  
  $$\frac{M}{p}=vy$$
  
  
  • when money supply doubles, the price level doubles
  • when income increases real money balances increase as well. For given money supply, this means that prices decline
• perhaps more importantly, this model will be the workhorse to study more interesting economies that the one we analyzed in this class (constant money supply, constant price level): this model provides us with the tools to understand inflations, currency crises, and other interesting phenomena