CLASS 3: THE ROLE OF MONEY: A ``CASH IN ADVANCE´´ MODEL

• What is money?
• Why do people hold money?
• A formal model of why people hold money: a ``cash in advance´´ model

What is money, and why do we use it?

Example: cigarettes as money in WWII Prisoner-of-War camps.
Money as a device for making transactions.
Barter economy is inefficient because of: 1) search costs 2) lack of specialization Why cigarettes. Cigarettes satisfy a number of criteria for being ``good money":

• cigarettes are portable, and ``divisible" - an individual cigarette has small enough value (medium of exchange)
• a cigarette is a fairly standardized commodity, whose value in terms of other goods is easy to ascertain (unit of value)
• cigarettes do not spoil quickly (store of value)

but cigarettes (like gold) have an alternative use - resource cost

What is money for?

• Money as a medium of exchange for transactions:
  money makes trade less costly
• Money as unit of value:
  makes the value of other commodities or services easy to ascertain...
  when the ``real value" of money is not uncertain
  in countries with hyperinflation money prices are changed frequently and unpredictably: money, while still sometimes used as medium of exchange, is no longer used as unit of account (instead, dollars are used; Chile: ``unidad de fomento", Mexico: UDI(?))
• Money as store of value:
  money as an asset: keep your savings under the mattress (why is it inefficient?)

The Monetary Aggregates

Assets differ in their ``moneyness", that is, in their ability to ``substitute" currency as a medium of exchange (and unit of value)

• M1: currency, checkable deposits (travelers' checks)
  most liquid (you can pay with checks... almost all the time)
  
• M2: M1 + savings deposits, time-deposits (1 day, 28 days), money market mutual funds
  less liquid (wait time, cannot write checks for less than...)
  changes in monetary aggregate due to ATM machines
• M3 and L: M2 + short term assets (treasury bill, commercial paper)
  ``liquidity": readily exchangeable, small default risk, etc.

what is the relevant monetary aggregate?

The Cash-in-Advance Model

In the rest of the course we will simply assume that money enters in the utility function of agents. In particular, we will study objective function of the type:

$$\sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})$$

(note the similarity with the model for durable goods)
however, in the remainder of this class we will see that such a utility function can be derived from a model which has some features that recall some of the ideas discussed in the first part of this class, and in particular the fact that people need to use cash in order to purchase some kinds of goods (for example, a cappuccino) Try to model an ``exchange technology´´ which justifies why people use money. The idea is that there are credit goods (say, a car), which can be purchased using credit (say, American Express), and cash goods, which have to be purchased using money.
reference: Clower, R.W.,1967, ``A reconsideration of the Microfoundations of Monetary Theory,´´ Western Economic Journal,6, pp. 1-9
Lucas, Robert E. Jr., and Nancy L. Stokey, 1987, ``Money and Interest in a Cash-in-Advance Economy´´, Econometrica, Vol.55, pp. 491-513 Assumptions

• at the beginning of each period t the household has a quantity of money M_t inherited from the previous period, and receives an endowment y_t
• the households enjoys consuming both cash and credit goods (which are not perfect substitutes). its objective is to maximize the utility function:
  
  $$U= \sum_{t=0}^\infty \beta^t v(c_{1t},c_{2t})$$
  
  

• there is specialization within the household. One member of the household (say, the wife) is the seller: she stays in the shop and sells the endowment y_t. Another member of the household (say, the husband) is the buyer: he goes to the market and buys both cash and credit goods
• cash and credit goods have the same price expressed in monetary terms: p_t. This is because the two goods are perfect substitutes in production, in the sense that the seller can divide the endowment y_t into cash goods (y_1t) and credit goods (y_2t) subject to the constraint:
  
  $$y_{1t}+y_{2t} \leq y_t$$
  
  

• the seller sells cash goods obtaining cash, and credit goods obtaining credit, which will be settled at the end of the day
• in the meanwhile, the buyer is buying at the market both cash (c_1t) and credit goods (c_2t). A crucial feature of the model is that the buyer can buy cash goods only using the money M_t held at the beginning of period t. In other words, there is a cash in advance constraint:
  
  $$p_t c_{1t}\leq M_t$$
  
  
  the credit good can be purchased for credit, which again will be settled at the end of the day
• at the end of the day all credit transactions are settled, and the budget constraint for the household is:
  
  $$M_{t+1} \leq (M_t- p_t c_{1t}) +p_t y_{1t}+p_t (y_{2t}-c_{2t})$$
  
  
  or
  
  $$M_{t+1} \leq M_t- p_t (c_{1t}+c_{2t}) +p_t (y_{1t}+y_{2t})$$
  
  
  where the first term is cash left from the buyer, the second is cash obtained from selling cash goods, and the last term is the settlement of credit transactions

wrapping up, the household's problem is:

$$max_{{c_{1t},c_{2t}}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t v(c_{1t},c_{2t})$$

subject to:
$p_t c_{1t}\leq M_t \mbox{ or } c_{1t}\leq \frac{M_t}{p_t} \\ y_{1t}+y_{2t} \leq y_t\\ M_{t+1} \leq M_t- p_t (c_{1t}+c_{2t}) +p_t (y_{1t}+y_{2t})$
now, notice that the amount of money the household saves for next period depends only on total consumption, and not on the division between cash and credit goods. that is, if we call

c_t=c_1t+c_2t

the intertemporal budget constraint becomes:

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

We can divide the household's problem in two steps, an intratemporal problem and an intertemporal problem:
Step 1 (intratemporal): Given the initial quantity of money M_t and the price level p_t, and for a given choice of c_t (which in turn determines next period's money M_t+1 through the budget constraint), the household decides how to split consumption between cash and credit goods. The problem is then:

max_{c<<86>>1t,c<<87>>2t} v(c_1t,c_2t)

subject to

$$c_{1t}\leq \frac{M_t}{p_t} \mbox{ ,} c_{1t}+c_{2t} \leq c_t$$

notice that the solution to this problem is a function of the two variables determining the constraints, namely c_t and $\frac{M_t}{p_t}$. we can then define an indirect utility function,

$$u(c_t,\frac{M_t}{p_t})$$

which is defined as:

$$u(c_t,\frac{M_t}{p_t}) = max_{c_{1t},c_{2t}} v(c_{1t},c_{2t})$$

subject to

$$c_{1t}\leq \frac{M_t}{p_t} \mbox{ ,} c_{1t}+c_{2t} \leq c_t$$

The following example may clarify things a little:
say that

$$v(c_1,c_2)=\ln{c_1}+\delta \ln{c_2}$$

(notice we can drop the time index because we are studying the intratemporal problem)
call $m=\frac{M_t}{p_t}$
the problem is:

$$max_{c_1,c_2} \ln{c_1}+\delta \ln{c_2}$$

subject to

$$c_1\leq m, c_1+c_2 \leq c$$

how do we solve this problem? we ignore the first constraint (for the time being) and we substitute the second constraint in the objective function:

$$max_{c_2} \ln{c-c_2}+\delta \ln{c_2}$$

and obtain the solution: $c_1=\frac{1}{1+\delta}c,c_2=\frac{\delta}{1+\delta}c$
if $\frac{1}{1+\delta}c \leq m$ (that is, the cash in advance constraint is not binding), then this is the solution of the problem. Otherwise, the solution is:
c₁=m,c₂=c-m
so that the indirect utility function is:

$$u(c,m)=\{ \begin{array}{c} \ln{m}+\delta \ln{c-m} \mbox{ if ... ...delta}c} \mbox{ if } \frac{1}{1+\delta}c \leq m \end{array}$$

Step 2 (intertemporal): give the indirect utility function, the consumer chooses the total consumption c_t and the money balances M_t that solve:

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

subject to

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

and for given M₀
In the remainder of the course we will forget about cash and credit goods, and work only with the indirect utility function. However, the payoff from doing all this math is that we know why real money balances enter in the utility function.

What did we learn?

• we discussed the 3 roles of money (unit of account, store of value, medium of exchange)
• we studied a model (Cash-in-Advance Model) where money is needed by the household as a medium of exchange in order to buy some kinds of goods (``cash goods´´)
  but it is also store of value and unit of account!
• derived an indirect utility function which we are going to use in the rest of the class