CLASS #20: MONEY SUPPLY AND MONEY DEMAND

There have been three great inventions since the beginning of time: fire, the wheel, and central banking.
Will Rogers

• The money supply: the Balance Sheet of the Central Bank.
• How do Central Banks change the Money Supply in practice?
• The Demand for Money: a simple model.

The money supply: the Balance Sheet of the Central Bank

• Assets
  • Government securities
  • Foreign reserves
  • Gold
  • other (loans to banks)
• Liabilities
  • Currency
    • circulation + vault cash
  • Deposits of depository institutions
  • Other and net worth

In practice, the Balance Sheet of the Central Bank can be summarized as follows:

• Assets
  • Foreign reserves
  • Net Domestic Credit
• Liabilities
  • Monetary Base

How do Central Banks change the Money Supply in practice?

• Changes in reserve requirement
  (prehistory of central banking)
• Open Market Operations
  buy Government paper (bond, bills) and inject currency in the system
  need a liquid market for government debt
• Discount Window Lending at the discount rate
  the discount rate is the crucial instrument of monetary policy now in the US, and in most of the world
  lend resources to banks at the discount rate
  if the discount rate increases $\rightarrow$ borrowing from the Central Bank is costly for banks $\rightarrow$ banks borrow less $\rightarrow$ the monetary base decreases.
• Currently in Mexico the Central Bank controls the money supply via ``cortos''. If the Central Bank puts the banking system in a ``corto'' -short- position, it provides liquidity to the system, but at a higher interest rate. Given that the interest rate is higher the public will demand less of it.

The Demand for Money

• The demand for money is a portfolio decision (money as store of value): people have to choose how much of their wealth they want to hold as money.
• However the nominal return to holding money is zero: money is a dominated asset (when the nominal interest rate is positive).
  Why do people hold money?
• because money is also a mean of exchange.

Using a simple two period model we will obtain:

• the Fisher equation: the nominal interest rate is equal to the real interest rate minus -expected- inflation:
  
  $$i=r+\pi^e$$
  
  

• the demand for money depends positively on ``output´´ and negatively on the nominal interest rate:
  
  $$\frac{M}{P}=L(Y,i)$$
  
  

The Model

The model has the following characteristics:

• It is a -one good- endowment economy (no production), like the one studied in the first part of this class.
• M1 (M2) is the quantity of money (billions of Pesos) held by individuals in period 1 (2); P1 (P2) is the price of the good expressed in monetary terms in the period 1(2); B is the nominal amount of one period bonds held by individuals.
• The utility function depends -separably- on both consumption and real money holdings:
  
  U(C1,C2,M1,M2)=
  
  
  
  
  $$=U(C1)+U(\frac{M1}{P1})+\beta(U(C2)+U(\frac{M2}{P2}))$$
  
  
  • Do we eat money?
  • No, but we derive utility from holding money, as it makes our life (transactions) easy (better, but slightly more complicated algebraically, to have transaction costs explicitly).
  • Why real money holdings, and not nominal?
    Because the amount of stuff you can buy with a given quantity of nominal money depends on the price level.
• The budget constraint of the -representative- household is:
  
  $$C1+\frac{B}{P1}+\frac{M2}{P1}=Y1-T1+\frac{M1}{P1}$$
  
  
  
  
  $$C2=(1+i)\frac{B}{P2}+\frac{M2}{P2}+Y2-T2$$
  
  
  where T is taxes, and i is the nominal interest rate.
• The budget constraints of the government in periods 1 and 2 are (assuming initial government debt is zero):
  
  $$G1-T1=\frac{M2-M1+B}{P1}$$
  
  
  
  
  $$T2-G2=\frac{(1+i)B+M2}{P2}$$
  
  
  Note that the government can finance the deficit in period 1 by either issuing interest bearing debt or by printing money (seignorage).

The household's problem then is:

$$max_{\{C1,C2,B,M2\}} U(C1)+U(\frac{M1}{P1})+\beta(U(C2)+U(\frac{M2}{P2}))$$

subject to:

$$C1+\frac{B}{P1}+\frac{M2}{P1}=Y1-T1+\frac{M1}{P1}$$

$$C2=(1+i)\frac{B}{P2}+\frac{M2}{P2}+Y2-T2$$

or equivalently choose the M2 and B that solve:

$$max_{\{B,M2\}} U(Y1-T1+\frac{M1}{P1}-\frac{B}{P1}-\frac{M2}{P1})+U(\frac{M1}{P1})+$$

$$+\beta(U((1+i)\frac{B}{P2}+\frac{M2}{P2}+Y2-T2)+U(\frac{M2}{P2}))$$

The FOCs with respect to B and M2 are respectively:

$$-\frac{U'(C1)}{P1}+\beta \frac{U'(C2)(1+i)}{P2}=0$$

$$-\frac{U'(C1)}{P1}+\beta \frac{U'(C2)}{P2}+\beta \frac{U'(m)}{P2}=0$$

where m is real money balances, i.e. $m=\frac{M2}{P2}$ From the FOC wrt B we obtain:

$$(1+i)=\frac{U'(C1)}{\beta U'(C2)}\frac{P2}{P1}$$

remember that the real interest rate r in this models satisfies the following equation (you can show it by introducing ``real'' government bonds in the analysis):

$$(1+r)\beta U'(C2)=U'(C1)$$

Substituting we obtain:

$$(1+i)=(1+r)\frac{P2}{P1}$$

Taking logs we obtain the Fisher relationship:

$$i \simeq r + \pi^e$$

where $\pi=\ln{P2}-\ln{P1}$ is inflation.
nominal interest rate = real interest rate + (expected) inflation

The demand for real balances

Rearranging the FOC wrt M2 we get:

$$\frac{U'(m)}{U'(C2)}=\frac{U'(C1)P2}{\beta U'(C2)P1}-1$$

or

$$\frac{U'(m)}{U'(C2)}=i$$

In order to get an explicit solution, let us assume that utility is logarithmic: $U(X)=\ln{X}$.
Let us also note that, as we are in a closed endowment economy, consumption has to be equal to the available resources (you can get this by substituting the government's constraint in the household's constraint):

C2=Y2-G2

This implies:

$$\frac{M2}{P2}=m=\frac{Y2-G2}{i}$$

We obtained that the desired amount of real balances is an increasing function of output( minus government spending), and a decreasing function of the nominal interest rate:

$$\frac{M}{P}=L(Y,i)$$

What is the intuition behind?

• Money as a mean of exchange:
  real money demand is an increasing function of output because the higher is output the higher the demand for money for transactions.
  
• Money as a store of value:
  real money demand is a decreasing function of the nominal interest rate because as the nominal interest rate increases the opportunity cost of holding money increases as well.