Columbia University Department of Economics Lecture 19 Economics - - PowerPoint PPT Presentation

Columbia University Department of Economics Lecture 19 Economics UN3213 Intermediate Macroeconomics Professor Mart ´ ın Uribe Spring 2019

Announcements Homework 5 due now Homework 6 posted, due April 15 in class. Recitations: – review of hwk5 – review concepts of income elasticity and interest semielasticity of money demand (i.e., review of slides 2-9 in this lecture).

Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Topics Today

• The Demand for Money
• Interest semielasticity of money demand.
• Microfoundations of money demand: The Baumol-Tobin Model

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

The Money Demand Function

Thus far, we have postulated that the demand for real money bal- ances depends on real income, Yt, and on the nominal interest rate,

• it. Formally, we write

Md

t

Pt = L(

− +

it, Yt) where Md

t denotes the demand for nominal money balances in period

t; Pt denotes the price level in period t; it denotes the nominal interest rate in period t; Yt denotes real output in period t We gave reasons why it makes sense that the demand for money depends negatively on the interest rate and positively on income. In this lecture we dig deeper and derive a money demand function like the one shown above that results from the optimizing behavior of individual agents. Before doing that, we turn to two common ways of measuring the sensitivity of the demand for money to the interest rate and income.

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Some Specific Examples of L(it, Yt)

– The Baumol-Tobin Money Demand Function L(it, Yt) =

• YtC

2it where C is a positive parameter. – A Linear Money Demand Function L(it, Yt) = ait + bYt Note that the money demand function postulated by the QTM is a special case of the linear money demand function in which a = 0 and b = 1/¯ v. – A Cagan-Style Money Demand Function L(it, Yt) = Yt(1 + it)−η with η > 0

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Properties of the money demand function – demand for real balances is increasing in real activity, Yt, ∂L(it, Yt, ) ∂Yt > 0 – demand for real balances is decreasing in the nominal interest rate, it (why? — opportunity cost of holding money is it) ∂L(it, Yt) ∂it ≤ 0

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

The Income Elasticity of Money Demand:

gives the percentage increase in the demand for money in response to a 1 percent in income. That is Income elasticity =

∆L(it,Yt) L(it,Yt) ∆Yt Yt

where ∆ denotes change. As ∆Yt → 0, we have income elasticity = ∂L(it, Yt) ∂Yt Yt L(it, Yt)

• r equivalently

income elasticity = ∂ ln L(it, Yt) ∂ ln Yt Empirical studies find that the income elasticity of money demand is about 1. That is, if income increases by 1 percent, the demand for money also increases by 1 percent.

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

The Interest Semi-Elasticity of Money Demand: gives

the percentage increase in the demand for money in response to an increase in the nominal interest rate. That is interest semielasticity =

∆L(it,Yt) L(it,Yt)

∆it As ∆it → 0 we have interest semielasticity = ∂L(it, Yt) ∂it 1 L(it, Yt)

• r, equivalently

interest semielasticity = ∂ ln L(it, Yt) ∂it Empirical studies estimate the interest semielasticity of money de- mand to be about −2. This means that if the interest rate increases by 1 percentage point (i.e., ∆it = 0.01), then the demand for money falls by 2 percent (∆L(it, Yt)/L(it, Yt) = −0.02).

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Examples (1) The QTM Money Demand Function: L(it, Yt) = 1

¯ vYt

Income elasticity of money demand: ∂ ln L(it, Yt) ∂ ln Yt = 1 Interest-rate Semi Elasticity of money demand: ∂ ln L(it, Yt) ∂it = 0

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

(2) The Baumol-Tobin Money Demand Function: L(it, Yt) =

• YtC

2it

Income elasticity of money demand: ∂ ln L(it, Yt) ∂ ln Yt = 1 2 Interest-rate Semi Elasticity of money demand: ∂ ln L(it, Yt) ∂it = − 1 2it

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

(3) The Cagan-style Money Demand Function: L(it, Yt) = Yt(1 + it)−η Income elasticity of money demand: ∂ ln L(it, Yt) ∂ ln Yt = 1 Interest-rate Semi Elasticity of money demand: ∂ ln L(it, Yt) ∂it = − η 1 + it < 0

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Microfoundations of the Money Demand Function The Baumol-Tobin Model The Baumol-Tobin Money Demand Function is the solution to a cash-management problem faced by a typical consumer (or firm). Consider an individual who earns income Yt in real terms. His em- ployer deposits his income in an interest-bearing bank account at the beginning of each month. The individual spends his income at a constant rate through the month. Each money withdrawal has a real cost C (withdrawal fee, time cost of going to the bank or to the ATM etc.). The problem of the individual is how many withdrawals to make per month.

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

If he goes to the bank only once a month and withdraws all of his monthly income, then his real money holdings are Yt at the beginning

• f the month and 0 at the end of the month.

Therefore, in this case, his average real money balances are Yt/2. The total cost for the individual is cost = it Yt 2 + C The cost has two terms. The first term, itYt/2, reflects the inter- est earnings forgone for holding money instead of keeping it in the interest-bearing bank account. The second term, C, is the cost of the withdrawal.

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Money Holdings With One Trip to the Bank Yt Yt 2 Average = Yt 2 Time 1 Money holdings

|

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

If the individual goes to the bank twice a month and withdraws Yt/2 each time, then his real money holdings are Yt/2 at the beginning of the month and 0 in the middle of the month. And the same pattern in the second half of the month. Therefore, his average real money holdings are Yt/4.

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Money Holdings With Two Trips to the Bank Yt 2 Yt 4 Average = Yt 4 Time 1 1 2

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Money holdings

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

The total cost for the individual is cost = it Yt 2 × 2 + 2C

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Similarly, if he decides to make 3 withdrawals, his average money holdings are Yt/(2 × 3), and his total cost is cost = it Yt 2 × 3 + 3C

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

In general, if he goes n times to the bank to withdraw money, his average money holdings are: Yt 2n And his total cost is cost = it Yt 2n + nC

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Money Holdings With n Trips to the Bank Yt n Yt 2n Average = Yt 2n Time 1 1 n

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Money holdings

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

The individual wishes to find the optimal number of withdrawals, which we will denote by n∗. To this end, he must take the derivative

• f the cost with respect to n, equate it to zero, and then solve for
• n. Let’s start with the first step:

∂cost ∂n = −itYt 2n2 + C Equating this derivative to zero and solving for n, we get n∗ =

• itYt

2C This expression is quite intuitive. The individual makes more trips to the bank the higher the interest rate (in this way he can keep his unspent income in the bank longer), the higher is his income, and the lower the cost of each withdrawal.

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Finally, recalling that his average money holdings are Yt/(2n), and replacing n by its optimal value n∗, we get the Baumol-Tobin money demand function L(it, Yt) =

• YtC

2it

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

This money demand function implies that the income elasticity is 1/2. However, an empirically realistic value of the income elasticity

• f money demand is unity.

But this unsatisfactory feature of the model is relatively easy to fix. The reason is that the most important part of the cost of a with- drawal C, is the opportunity cost of time of the individual involved in going to the bank. The cost of his time, in turn, is his hourly

• wage. We therefore have that C is, in fact, some fraction of Yt. So

let’s say that C = cYt, where c is a constant parameter in (0, 1).

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Econ UN3213 Interm. Macro The Money Demand Function Lecture 19

Then, replacing C with cYt in the Baumol-Tobin money demand function, we obtain the following money demand function L(it, Yt) = Yt

c

2it which features a unit income elasticity.

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