Prof. Bryan Caplan

Econ 816

Spring, 2000


Week 3: Basics of Monetary Economics

I.                    The Monetary Base

A.                There are numerous measures of"the" money supply; the least controversial and the most fundamentalis known as the monetary base.

1.                 In a fiat regime, the monetary base iscurrency, coins, and deposits with the Fed.

2.                 Under a commodity standard, the monetarybase is the (non-industrial?) metallic stock.

B.                A simple test for base money: base moneyis net wealth; non-base money is not. If I hold $1 in currency and the price level declines, my wealth isgreater, and no one else's is smaller. In contrast, if a price level decline merely increases the holder'swealth but reduces someone else's wealth by an equal amount, the item held isnot base money.

1.                 Ex: Under the gold standard, a fall inthe price level enriched the holders of Federal Reserve notes, but increasedthe Fed's debts to the public by the same amount (since the Fedpledges to redeem a fixed quantity of gold per dollar).

2.                 Why does going off the gold standard turncurrency into net wealth?

3.                 In modern fiat monetary systems, themonetary base is normally the only thing under the complete control of thecentral bank/government.  (In practice,responsibilities are divided between agencies; e.g., in the U.S., both the Fedand the Treasury play a role, although the Fed is clearly dominant).

4.                 In consequence, if you are just lookingat the market for base money, it probably makes sense to draw the money supplycurve as a vertical line.

II.                  Broader Measures of the Money Supply +Money Substitutes

A.                Other - broader - measures of the money supply often receive attention, e.g.M1, M2, M3, etc.

B.                At the same time, there are many items thatare nowherecounted as money (e.g. credit cards), that still seem to be very closesubstitutes for money.

1.                 If you wanted to somehow include creditcards in a monetary aggregate, how would you do it?

C.               These two facts give rise to pointlessdisputes about what is/is not money. What does it matter to an agent if e.g. a mutual fund ismoney, or is merely a close substitute for money?

1.                 How would you toggle the labels andgraphical S&D representations?

D.               The tendency to classify some items asmoney is closely connected to "required reserves" laws, which e.g.require banks to hold base money reserves of 2% against savings accounts.  But this is really no different than e.g.imposing a requirement that mutual funds hold at least 2% of their fund's valuein the form of base money.

III.                 Money in the utility function

A.                Individuals must allocate their wealthbetween current consumption, future consumption, and their cash balances.  So this looks like a standard constrainedoptimization problem.

B.                But, there is a key difference: unlikeconsumption and investment, money is normally valued for its purchasing power,not for its quantity alone.

1.                 Ex: How many drachmas do you want to holdwhen you're in Greece?  You have no ideauntil you find out the real value of a drachma.

C.               In short, what people want is realmoney, not nominal money.  Inconsequence, a price of money has to appear in the utility function - as itnormally would not.  (Of course, pricesdo normally appear in the indirect utility function).

IV.              A Simple Model of Money Demand

A.                Objective function: ; an infinitely-lived agentderives utility from consumption and real balances.

B.                Agents can consume their wealth, hold itas money, or invest it in bonds.  Bondssell for st and pay 1 next period. (They could also just hold real goods between periods, but since realgoods pay no interest and give no utility unless consumed, this would make nosense).

C.               Budget constraint: ; consumption this period must equal the agent's per-periodendowment (W) plus the agent's transfer from the government (TR - note: thiscan be negative, i.e., a tax) plus the current real value of bonds purchasedlast period, minus the current real value of bonds purchased this period, minusadditions to his cash balance.

D.               Note the important assumption that money earns no interest.  The present value of consumption equals thepresent value of endowments less the present value of interestlost as a result of holding non-interest-bearing money.

E.                It is intuitively clear that agents willequalize marginal utility along their 3 margins; it will be impossible toincrease utility by re-allocating between consumption, bonds, and money.  (But note: Bonds are only valued as a meansof future consumption).

F.                More formally, we can set up the so-calledBellman equation, which just says that we can maximize our lifetime utility bysimply maximizing today's utility plus our lifetime utility tomorrow:

G.               Taking first derivatives (the argument isindicated by the numerical subscript):

H.                The envelope theorem implies that:

I.                    Substituting, and doing a little morework, we can derive two optimality conditions:

J.                 The first equation just says that youcan't gain in utility by buying or selling bonds to shift consumption betweenpresent and future.  The second equationbasically says that you can't gain in utility by swapping current or futureconsumption for current money.

V.                Money Demand and Nominal Interest Rates

A.                Assume a closed economy so net supply ofbonds is zero, and consumption equals endowments.  Assume further that all transfers are financed by printing money- so  - and that transfersare proportional to pre-existing real balances, so .  This implies a constantmoney growth rate of .

B.                Note further the (definitional)relationship between the nominal interest rate i and the bond price s:.

C.               You can re-arrange the above equations tofind that:  .  Proof: According tothe first equation in IV.I, .  According to thesecond equation in IV.I, .  Then just sub infor st.

D.               When the nominal interest rate  rises, 1-stby definition rises; U2 tends to rise and U1 tends tofall.  (Slippage: Intertemporalmargin).  From the principle ofdiminishing MU, this means that when the nominal interest rate rises, agents want tohold less money.

E.                In a real economy with fixed endowments, the realinterest rate has to be just the inverse of the discount factor,

F.                Since nominal money grows at a constantrate, and real money balances are constant under constant conditions, itfollows that the inflation rate must be exactly equal to the money growth rate:.

G.               The nominal rate follows immediately fromthe real rate and the inflation rate: .  It then followsthat real money demand falls when the rate of money creation rises, AND whenthe discount factor falls (i.e., people get more impatient).

1.                 Complication:  High inflation often accompanied by interest rate ceilings.  In such cases inflation rather than nominalinterest rates may be the crucial influence on real money demand after all.

2.                 Have to assume that money earns nointerest to get these conclusions.  Orrather, these conclusions only apply to non-interest-bearing money.  For money with sticky nominal rates in theshort-term, these results apply only in the short-term (note importance of e.g.spread between savings rates and commercial paper rates).

VI.              Money Demand and Other Factors

A.                Price level.  Immediate conclusion from the above analysis: Nominal moneydemand is precisely proportional to the price level, since real money demand isindependent of the price level (though not the inflation rate!).

1.                 N.B. It is a standard micro conclusion that e.g. doubling income and allprices leaves the optimal allocations unchanged.  This conclusion is stronger: doubling the supply of just onegood leaves both arguments in the utility function unchanged.

2.                 What has to happen to the price level ifthere is a discrete, unexpected change in the rate of growth of the moneysupply?  (Discrete "jump" inreal balances must be generated by a discrete "jump" in the pricelevel). 

3.                 Romer, p.393 offers excellent graphicaltreatment (attachment).

B.                Real income and real wealth.  It can be easily derived from the abovemodel (relaxing the assumption of constant endowments) that so long as money isa normal good, you will hold more in real balances as your real income and realwealth rise.

C.               Substitute assets.  Take bonds away - then the only way to dointer-temporal substitution is with money. Money demand will rise when there are fewer substitutes for money, andfall when there are more substitutes.

D.               Substitute transactions technology.  Since it is costly to hold money, alternatemeans of doing transactions (e.g. credit cards) will reduce money demand.  Or changes in the technology of transactionsthemselves could affect money demand - e.g. cheaper cars means fewer trips tothe store, making it more convenient to hold less money.  This isn't explicitly in the model, but youcould just add a parameter in front of real balances to capture the quality ofsubstitute transactions technology.

E.                Others?

VII.             Circularity, or Coordination?

A.                It has occasionally been argued that thepreceding analysis is circular.  Youneed nominal money demand to have prices, but prices are necessary to havenominal money demand.

B.                Even if their were a circularity problemwith price level determination, you could still figure out your realmoney demand.  But as the above sectionmakes clear, once you've got the real money demand function, you can quicklyderive what the price level has to be for a given money supply.  How is this possible?

C.               Suppose all dollars disappeared.  Could I just pass out play money, and derivethe value of play money by figuring out what price level is consistent withyour former level of real balances?

D.               What if we were on a spaceship thatcrash-landed on a new planet with unknown production possibilities?  Would a dollar-based monetary system breakdown before the spaceship doors open, just because we no longer have any idea whatthe real value of our nominal dollar holdings is?

E.                The problem is not circularity, butcoordination.  The problem isn't thatthere is no way to figure out what the equilibrium price level would be if anew money were introduced; the problem is getting everyone to make a"joint deviation" to a new money. Any credible way of doing this works, whether it's:

1.                 An historically-evolved money.

2.                 An old fiat money.

3.                 A new fiat money tied to some real good.

4.                 A new fiat money that is"focal."

F.                This coordination aspect of money meansthat sub-optimal monetary systems are possible, so it makes sense todeliberate over the advantages and disadvantages of different conceivablemonetary regimes (as will be done throughout the course).

VIII.           The Equation of Exchange

A.                Famous identity: MV=PY.  Like all identities, it says nothing aboutcausality unless you make more assumptions, but it can still be a good way toorganize thought.

B.                M is money - measured however you likeit.

C.               PY is nominal GDP; Y is real GDP, and Pis the price level.

D.               V, velocity, is the number that makes theequation true; its value depends on your choice of monetary aggregate.  (V1 is velocity for M1, etc.)  V is only "turnover" in a worldwith no resales.

E.                What's your personal V1?  V2? What does your personal velocity look like over time?

IX.              Velocity, Money Demand, and More

A.                Velocity is just the inverse of the fraction ofyour income that you keep in the form of cash.

B.                One interesting way to think aboutvelocity: income-adjustedmoney demanded.

C.               If money demand increases/decreases, whathappens to velocity?  If nominalinterest rates rise/fall?

D.               Constant V implies that only M mattersfor nominalGDP. 

1.                 If V is constant, then fiscal policymakes no difference for nominal output.

E.                Constant P implies that increases inmoney or velocity increase real output.

F.                Constant Y implies that increases inmoney or velocity increase the price level.

G.               Constant M means that only V matters fornominal GDP determination.

H.                Fun fact: V2 was in fact been nearlyconstant at around 1.7 from early '50's to 1991.  V1 much more unstable - large rise to mid-80's, followed bydecline.

1.                 Question: Why did V1 rise for so long,and then fall?  Hint: Inflation andregulated interest rates.

X.                Cash-in-Advance Models of Money Demand

A.                Some people think that it doesn't makesense to put money in the utility function: people want money for what it willbuy, not as a final good.

B.                Alternate modeling strategy:"cash-in-advance" models. These get essentially the same results as models with money in theutility function.

C.               Skeletal cash-in-advance model, usingsame notation as before.  Objectivefunction removes real balances, leaving only consumption: .

D.               Constraints change.  There is the cash-in-advance constraint thatthis period's purchases of consumption and bonds cannot exceed this period'smoney holdings (you need money to buy stuff; barter and autarky assumed away): .  There is also asimilar second constraint, that next period's money holdings equal thisperiod's sales of endowment and bonds: .  Then just maximizeO.F. subject to both constraints.

E.                All of the standard results work.  The only unusual feature is that dependingon timing assumptions, you can get a short-run "liquidity effect" -i.e., within a period, a burst of new money reduces real interest rates.  Somewhat interesting, since this model hasfully flexible prices (but not fully flexible portfolios - they take a periodto adjust).

F.                My judgment: cash-in-advance adds muchmore complexity without changing conclusions. Money in the utility function seems like a better approach to me.