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EC3115 :: L.2 : The demand for money Almaty, KZ :: 11 September 2015

EC3115 Monetary EconomicsLecture 2: The demand for money

Anuar D. Ushbayev

International School of EconomicsKazakh-British Technical University

https://anuarushbayev.wordpress.com/teaching/ec3115-2015/

Tengri Partners | Merchant Banking & Private [email protected] – www.tengripartners.com

Almaty, Kazakhstan, 11 September 2015

ISE – KBTU A.D. Ushbayev (2015)

https://anuarushbayev.wordpress.com/teaching/ec3115-2015/

EC3115 :: L.2 : The demand for money - 2 / 31 -

Relevant reading

Book treatment

F. Mishkin. (2010). Economics of Money, Banking & Financial Markets, 9thed., Chapter 19.

R. Dornbush and S. Fischer. (1993). Macroeconomics, 6th ed., Chapter 13.

Recommended articles

W. Baumol. (1952). “The transactions demand for cash: an inventorytheoretic approach”, Journal of Econometrics, 66, pp. 545–56.

J. Tobin. (1956). “The interest elasticity of transactions demand for cash”,The Review of Economics and Statistics, 38(3), pp. 241–47.

J. Tobin. (1958). “Liquidity preference as behaviour towards risk”, Reviewof Economic Studies, 25(1), pp. 65–86.

S. Goldfeld. (1976). “The Case of the Missing Money,” Brookings Paperson Economic Activity, Vol. 3, pp. 683–730.


EC3115 :: L.2 : The demand for money - 3 / 31 - Motives for holding money

Section 1

Motives for holding money



The three basic motives

Keynes, in his magnum opus1, classified the different motives for holdingreal money balances as follows:

1. The transactions motive – arising from the need to make regularpayments.

2. The precautionary motive – arising from the need for capacity to meetunforeseen expenditure.

3. The speculative motive – arising from the uncertainty about the futuremonetary value of other assets.

The first two motives are mostly responsive to actual changes in thegeneral economic activity and the level of incomes.

1J. M. Keynes. (1936). The General Theory of Employment, Interest and Money. Macmillan.ISE – KBTU A.D. Ushbayev (2015)


Liquidity preference

The third motive focuses on the observation that individuals and firms willcontinuously choose to shift their stock of monetary wealth between liquidand illiquid assets (e.g. between money, bonds and stocks) as theopportunity cost of holding money – the interest rate – changes.

Bearing in mind the contemporaneous relationship between interest ratesand bond prices (which we will cover later), Keynes argued that individualsand firms will want to store their wealth in bonds if the expected return onthem is greater than that of cash.

If, however the market expects future interest rates to rise, for example,then it will also anticipate the future fall in bond prices, and, – dependingon whether this expected fall causes a capital loss that is greater or smallerthan the coupon interest income on the bonds, – may decide to hold morecash or bonds, respectively.

This is why Keynes called his money demand theory liquidity preference.ISE – KBTU A.D. Ushbayev (2015)


Modelling approaches

The three components have traditionally been modelled separatelydepending on their various degrees of dependence on the followingfactors:

1. interest rate differentials – difference in yield between cash (~0%) andother financial assets.

2. cost of transfers – transaction/brokerage costs associated with thepurchase and sale of income-bearing assets as opposed to cash.

3. future asset price uncertainty – risk of adverse valuation effects onincome-bearing assets as opposed to pure inflation risk on cash.

4. expected patterns of incoming and outgoing cash flows – level ofcertainty in receipts and level predictability of expenses.

The transactions demand and the precautionary demand for moneybalances are typically modelled as a function of factors 1, 2 and 4.The speculative demand for money usually focuses on factor 3.


EC3115 :: L.2 : The demand for money - 7 / 31 - The Baumol-Tobin model

Section 2

The Baumol-Tobin model



Inventory-theoretic approach to transactions demand for money

The transactions demand for money arises from its use in making regularpayments for goods and services.

In order to be to able to facilitate these payments as they come due, allindividuals and firms have to keep an inventory of cash (the logic of theso-called cash-in-advance models) and (perhaps) other assets.

The key idea in this approach is the fact that there exists a trade-offbetween the level of interest rates and transactions demand for money.

The reason for this trade-off lies in the opportunity cost associated withholding money balances – which is the foregone interest income that anagent could earn by holding their wealth in interest-bearing assets insteadof cash.



Transactions demand mechanicsConsider an individual who receives $1,800 of disposable (post-tax)monthly income. Assuming they spend their income evenly over thecourse of the 30 days in the month, their daily expenditure rate must equal$60/day.

The individual has a spectrum of options for the allocation of his wealth inthis period:

1. Leave all of $1,800 in cash (or a current account paying no interest)and spend $60 daily.

2. Leave $60 in cash (or c/a) and deposit the remaining $1,740 in asavings account that pays interest, and then make daily withdrawals of$60.

3. Anything in between.

The benefit of keeping wealth in interest-bearing assets such as savingsaccounts (or bonds) is the interest income accrued daily, while the costs ofsuch is the inconvenience of making trips to the bank (or brokerage fees).



Baumol-Tobin transactions demand model

The two papers by Baumol (1952) and Tobin (1956) were among the first todevelop an inventory-theoretic model for the transactions demand formoney.

Their framework assumes that:1. Money balances pay zero interest.

2. Timing and amounts of an agent’s receipts and expenditures areknown with certainty.

3. An agent always possesses the funds needed to finance expenditure,and all payments are made in money balances.

4. There exist transaction/brokerage costs for transferring wealthbetween bonds (or savings accounts) and money balances.



Let’s denote:Y – annual spending, smoothed over the year.

i – interest rate on bonds (or savings accounts).

N – number of transactions (in which an agent converts bonds intomoney balances).

T – transaction/brokerage costs.

We also assume for convenience that:the agent’s monthly income is initially paid directly into his savingsaccount or in the form of bonds, and

the agent wishes to minimize his total costs of money management(transaction costs + foregoing of interest on cash balances).

In this way, the average holding of money balances for an agent willdepend negatively on the number of transactions (trips to the bank).



1 transaction per year



2 transactions per year



3 transactions per year



Thus, we observe that average money holdings for an agent is equal to

= Y /2N

and hence:the amount of foregone interest income is= i ×

�

Y2N

�

the total transaction costs are= T × N

This provides us with the objective function of total cost:

C = i ×� Y

2N

�

+ T × N

Assuming that Y ,i and T are exogenous for any given agent at least in theshort-term, they choose N so as to minimize total cost, C .



The solution to the model is achieved via a simple exercise inunconstrained optimization2:

minN

C =iY

2N+ T N

C is minimized when:

dCdN= 0 and d2C

dN2 > 0

Hence we need to solve:

− iY2N2 + T = 0

�

also check that d2CdN2 =

iYN3 > 0

�

which gives:

Noptimal = N ∗ =

√

√ iY

2T2Note that this commits the logical error of assuming that N is a continuous variable

whereas in fact it can only take integer values, and so the end result cannot be exactly correct.ISE – KBTU A.D. Ushbayev (2015)


Using N ∗ =Ç

iY2T

and the fact that average money holdings equal= Y /2N , we can calculate the optimal demand holding of money:

M doptimal =

Y

2N ∗=

√

√Y T

2iThis result is known as the famous Baumol-Tobin square-root formula.

It reveals the elasticities of money demand with respect to income (0.5),transaction costs (0.5) and the interest rate (-0.5):

M d = f (+Y ,+T ,−i )

e.g. if you log-linearize3 the Baumol-Tobin formula for money demandyou get:

ln M =1

2ln Y +

1

2ln T −

1

2ln i − ln

p2

3This is the version of the model that usually gets tested. Exercise: calculate the elasticitiesdirectly using the definition: Er =

∆M∆r ×

rM and similarly for EY .



Note that the elasticities mentioned above refer to the demand for realmoney balances and not nominal money balances. There is no moneyillusion in the Baumol-Tobin model, meaning that when all prices double,both income and transaction costs also double. This will double thedemand for nominal money balances while leaving the demand for realmoney balances unchanged.

The general findings of a positive relationship between money demandand income and a negative relationship between money demand andinterest rates are relatively robust with respect to the use of more realisticmodelling assumptions such uneven periodic spending patterns, initialincome payments directly in cash and uncertainty in the amounts andtiming of cash flows.

The suggested sensitivities of money demand with respect to income andinterest rates are under- and overestimated compared to empirical tests4.

4Although, once the integer constraints on the values of N mentioned above are takeninto account, it can be shown that the income and interest elasticities of money demandsuggested by inventory-theoretic models in fact lie between 1

2and 1, and between − 1

2and 0,

respectively.ISE – KBTU A.D. Ushbayev (2015)


Other models for money demand

Miller-Orr inventory model – tries to relax the Baumol-Tobinassumption of cash-flow certainty.

Clower cash-in-advance model – a simpler version of theBaumol-Tobin model that recognizes the cash-on-hand constraint onconsumption.

Double-coincidence of wants models – models the demand formoney as an instrument of reducing the search costs in a bartereconomy.

Cash and credit models – models that recognize that some goods canalso be bought on credit, and thus the distribution of goods mattersfor money demand.

and others.

Many of these models can be generalized in what is called money-in-utilityframework, which we will later start to use for general equilibrium analysis.


EC3115 :: L.2 : The demand for money - 20 / 31 - Tobin’s portfolio balance model

Section 3

Tobin’s portfolio balance model



Portfolio considerations

Tobin (1958) expanded on the famous work of Markowitz on portfoliotheory5 – which you will have undoubtedly met if you’ve done any financemodules – by adding a risk-free asset to the analysis.

This allowed for the possiblity to leverage or deleverage portfolios on theefficient frontier, and later led to the development of CAPM6 by BillSharpe.

Tobin’s insights into portfolio theory also add to our understanding ofdemand of money balances as opposed to Keynes’ original speculativedemand formulation.

One problem with Keynes’ original ideas of the micro level demand formoney was that individual investors held expectations of interest ratemovements with certainty, and thus each one would make the binarychoice of holding either cash or bonds depending on their view.

5H. Markowitz. (1952). “Portfolio selection”, Journal of Finance, Vol. 7, No. 1, pp. 77-91.6W. Sharpe. (1964). “Capital asset prices: A theory of market equilibrium under conditions

of risk”, Journal of Finance, Vol. 19, No. 3, pp. 425-442.ISE – KBTU A.D. Ushbayev (2015)


This assumption was clearly restrictive and was relaxed by Tobin throughthe introduction of a utility maximization problem for a risk-averse7 agentwho allocates portfolio wealth between cash and bonds in the presence ofrisk and uncertainty:

Money balances are once again assumed to have a zero rate of returnand also a zero risk of capital loss (or gain) – meaning that the varianceof the return on money is also zero.

Money balances∼ Prob(0, 0)

Bonds, however, pay a positive rate of return, but carry the risk ofcapital loss (or gain) if liquidated before maturity8 – meaning that thereturn on bonds is variable (expectations of future interest rates areuncertain).

Bonds∼ Prob(µB,σ2B)

7That is, they prefer less risk to more risk at a given rate of return.8Note that this does not mean they carry credit risk of default.



Next we note that the probability distributional assumptions for theindividual assets imply that a portfolio consisting of the above two assetswith proportion B for bonds and (1− B) for cash will be distributed as:

Portfolio∼ Prob�

µB × B + 0× (1− B),σ2B × B2 + 0× (1− B)2

�

∴

Portfolio∼ Prob(µBB︸︷︷︸

=µP

,σ2BB2

︸︷︷︸

σ2P

)

Noting that σ2P = σ

2BB2, we can write B =

σp

σB, and thus

µp = µBσP

σB= σP

µB

σB

Assuming that agents are risk-averse, this means that the indifferencecurves will be convex and upward sloping.



Tobin’s portfolio balance model



Tobin (1958) showed that the theory of risk-avoiding behaviour can serveas a basis for liquidity preference and for an inverse relationship betweenthe demand for money balances and interest rates.

The derivation doesn’t require the assumption of inelasticity ofexpectations of future interest rates, and instead uses the assumption thatthe expected value of capital gain (or loss) from holding interest-bearingassets is always zero.

It also explains the empirically observed diversification in individualportfolios, where the same agent holds both bonds and money balances,unlike the Keynes (1936) view that implied a binary choice for each agent.

At low interest rates, the expectation of capital loss may push the optimalposition to (all cash, no bonds). At high interest rates, the expectation ofcapital gain would increase the observed frequency of (no cash, all bonds)positions.

The more inflexible the agent’s expectations, the more sensitive theirdemand for cash will tend to be to the changes in the interest rates.


EC3115 :: L.2 : The demand for money - 26 / 31 - Macroeconomic modelling of money demand

Section 4

Macroeconomic modelling of money demand



The function of demand for real money balances takes the general form:�

M d

P

�

= f (Y,Θ)

where:Y – is the level of nominal income,

P – is the general price level, and

Θ – is a vector of variables measuring the various opportunity costs ofholding money balances, which traditionally include:

domestic (and foreign) interest rates

expected changes in exchange rates

expected inflation rates

expected return on bonds

expected return on stocks



The famous debate between Keynesians and Monetarists is reflected intheir view regarding the interest elasticity of money demand whenincorporated into the basic IS-LM framework.

A low value would imply a steeper LM curve and, other things being equal,monetary policy would have a larger effect on output than fiscal policy.

A high value would imply a flatter LM curve and, other things being equal,fiscal policy would have a larger effect on output than monetary policy.



Stability of money demand

Money demand function used to be thought of as a relatively stablerelation up until the early 1970s9, when central banks learned that:

attempts to control a particular form of financial liability or creditorganization tend only to encourage the invention of other forms notyet controlled,

many current accounts now pay (small) interest

long-dates assets can relatively quickly be converted into currentaccounts

mutual funds (baskets of stocks) are relatively easy to redeem

mortgages allow to draw money relatively easily against property value

money and near-money are close substitutes, switchable at cost.switches can be large and unpredictable

9Up until 1974, when the conventional M1 money demand function began to severelyoverpredict the demand for money. Goldfeld (1976) called this phenomenon of instability ofthe money demand function “the case of the missing money”.



If the money demand function was stable, and thus, say, the M2 velocitywas stable or predictable, central banks could use the money demandfunction for setting monetary policy: choose a path of M2 and thusproduce the desired path of nominal GDP.

The growth in financial innovation has revealed the inherent instability ofthe money demand function, and corresponding unpredictability of thevelocity of money.

This meant that if the monetary authorities attempted to target monetaryaggregates then an unstable money demand function could mean thatthese actions may lead to unexpected and/or adverse changes in nominaland potentially real economic variables.

Since then, central banks realized that the liability side of the balance sheetof commercial banks is the wrong side to be attempting to control.

Thus they became price-setters instead of quantity-setters: monetaryaggregates targeting has been replaced by interest rate and inflationtargeting, i.e. fix the interest rate exogenously, and let the system decideon the quantity of money it requires.



Goodhart’s lawIt is often observed that when a stable relationship is discovered and startsto be used, it breaks down. This behaviour is known as Goodhart’s law10.The disappearances of stable M1 and M2 money demand relations can be regarded asinstances of Goodhart’s law in action.

10Original formulation: “any observed statistical regularity will tend to collapse oncepressure is placed upon it for control purposes” – C. Goodhart. (1975). “Problems of MonetaryManagement: The U.K. Experience”, Papers in Monetary Economics, Reserve Bank of Australia.


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