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 1
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, Y t , and on the nominal interest rate, i t . Formally, we write M d − + t = L ( i t , Y t ) P t where M d t denotes the demand for nominal money balances in period t ; P t denotes the price level in period t ; i t denotes the nominal interest rate in period t ; Y t 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. 2
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Some Specific Examples of L ( i t , Y t ) – The Baumol-Tobin Money Demand Function � Y t C L ( i t , Y t ) = 2 i t where C is a positive parameter. – A Linear Money Demand Function L ( i t , Y t ) = ai t + bY t 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 ( i t , Y t ) = Y t (1 + i t ) − η with η > 0 3
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, Y t , ∂L ( i t , Y t , ) > 0 ∂Y t – demand for real balances is decreasing in the nominal interest rate, i t (why? — opportunity cost of holding money is i t ) ∂L ( i t , Y t ) ≤ 0 ∂i t 4
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 ∆ L ( i t ,Y t ) L ( i t ,Y t ) Income elasticity = ∆ Y t Y t where ∆ denotes change. As ∆ Y t → 0, we have income elasticity = ∂L ( i t , Y t ) Y t ∂Y t L ( i t , Y t ) or equivalently income elasticity = ∂ ln L ( i t , Y t ) ∂ ln Y t 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. 5
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 ∆ L ( i t ,Y t ) L ( i t ,Y t ) interest semielasticity = ∆ i t As ∆ i t → 0 we have interest semielasticity = ∂L ( i t , Y t ) 1 ∂i t L ( i t , Y t ) or, equivalently interest semielasticity = ∂ ln L ( i t , Y t ) ∂i t 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., ∆ i t = 0 . 01), then the demand for money falls by 2 percent (∆ L ( i t , Y t ) /L ( i t , Y t ) = − 0 . 02). 6
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Examples (1) The QTM Money Demand Function: L ( i t , Y t ) = 1 v Y t ¯ Income elasticity of money demand: ∂ ln L ( i t , Y t ) = 1 ∂ ln Y t Interest-rate Semi Elasticity of money demand: ∂ ln L ( i t , Y t ) = 0 ∂i t 7
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 � Y t C (2) The Baumol-Tobin Money Demand Function: L ( i t , Y t ) = 2 i t Income elasticity of money demand: ∂ ln L ( i t , Y t ) = 1 ∂ ln Y t 2 Interest-rate Semi Elasticity of money demand: ∂ ln L ( i t , Y t ) = − 1 2 i t ∂i t 8
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 (3) The Cagan-style Money Demand Function: L ( i t , Y t ) = Y t (1 + i t ) − η Income elasticity of money demand: ∂ ln L ( i t , Y t ) = 1 ∂ ln Y t Interest-rate Semi Elasticity of money demand: ∂ ln L ( i t , Y t ) η = − < 0 ∂i t 1 + i t 9
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 Y t 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. 10
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 Y t at the beginning of the month and 0 at the end of the month. Therefore, in this case, his average real money balances are Y t / 2. The total cost for the individual is Y t cost = i t 2 + C The cost has two terms. The first term, i t Y t / 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. 11
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Money Holdings With One Trip to the Bank Money holdings Y t Y t Average = Y t 2 2 | 1 Time 12
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 If the individual goes to the bank twice a month and withdraws Y t / 2 each time, then his real money holdings are Y t / 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 Y t / 4. 13
Money Holdings With Two Trips to the Bank Money holdings Y t 2 Average = Y t 4 Y t 4 | | 1 Time 1 2
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 The total cost for the individual is Y t cost = i t 2 × 2 + 2 C 14
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Similarly, if he decides to make 3 withdrawals, his average money holdings are Y t / (2 × 3), and his total cost is Y t cost = i t 2 × 3 + 3 C 15
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: Y t 2 n And his total cost is Y t cost = i t 2 n + nC 16
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Money Holdings With n Trips to the Bank Money holdings Average = Y t 2 n Y t n Y t 2 n | | 1 Time 1 n 17
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 of the cost with respect to n , equate it to zero, and then solve for n . Let’s start with the first step: ∂ cost = − i t Y t 2 n 2 + C ∂n Equating this derivative to zero and solving for n , we get � i t Y t n ∗ = 2 C 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. 18
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Finally, recalling that his average money holdings are Y t / (2 n ), and replacing n by its optimal value n ∗ , we get the Baumol-Tobin money demand function � Y t C L ( i t , Y t ) = 2 i t 19
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 of 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 Y t . So let’s say that C = cY t , where c is a constant parameter in (0 , 1). 20
Econ UN3213 Interm. Macro The Money Demand Function Lecture 19 Then, replacing C with cY t in the Baumol-Tobin money demand function, we obtain the following money demand function � c L ( i t , Y t ) = Y t 2 i t which features a unit income elasticity. 21
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