Econ402_Solutions_Set6 - Economics 402 Winter 2009 Problem...

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Economics 402, Winter 2009 Problem Set 6 Suggested Solutions 1 The Baumol-Tobin Inventory Model of Money Demand (30) Sarah ° your old high school friend ° now works for a living and receives a paycheck of $Y dollars per month (notice that here Y denotes nominal income for the sake of notational simplicity, whereas usually it denotes real income). Sarah is paid at midnight on the last day of the month, with a direct deposit into her interest bearing bank account. She spends all $Y during the month, so at the end of the month she has exactly zero dollars in her account. The account pays - at the end of the month - a nominal interest rate of R on the AVERAGE cash holdings per month, L. Only cash is accepted for transactions, so Sarah needs to walk to an ATM and pay a ±xed cost $F per trip to get cash. Sarah wants to economize on trips, by withdrawing more than she needs for everyday use, but at the same time she does not want to lose interest on her deposit. She turns to you, a professional economist, for advice. You will determine for her the optimal number of trips to the ATM, N*, the optimal average cash holdings, L*, and how it varies with her income and the nominal interest rate. 1. If Sarah holds L units of cash in her pocket on average, what are her opportunity costs of holding this amount of cash, given that the nominal interest rate on the AVERAGE cash holdings per month is R? (1) The opportunity cost of holding cash is the interest foregone, which is RL: 2. What are the total trip costs as a function of N, the total number of trips? (1) Trips to the ATM cost $ F per trip. If Sarah makes N trips, the total trip costs are FN. 3. One can show that under certain assumptions the average money demand, L, in the Baumol-Tobin inventory model as a function of the agent²s nominal income, Y, and the number of trips to the bank, N, is: L= Y 2 N : Express N as a function of L and Y. (1) L = Y 2 N = ) 2 N = Y L = ) N = Y 2 L : 4. What are the total trip costs as a function of L, Y and F? (2) The total trip costs are: FN = FY 2 L : 1
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5. What are the total costs of holding cash as a function of R, L, Y and F? (2) The total costs of holding cash are the sum of the opportunity cost and the trip costs. That is, Total Costs = RL + FY 2 L : 6. Find the cost minimizing nominal cash holdings, L*, as a function of Y, R and F. (9) We want to solve the following minimization problem: min L RL + FY 2 L : Taking the derivative of the total costs with respect to L and setting it to zero, we get: R ° FY 2 L 2 = 0 : Solving for L yields the optimal cash holdings: L ° = r FY 2 R : 7. What is the optimal number of trips to the ATM, N*, again as a function of Y, R and F? (4) Plugging in the optimal cash holdings ( L ° ) to the equation for the number of of trips to the ATM, we get the optimal number of trips to the ATM: N ° = r RY 2 F : 8. Take a natural log on both sides of the expression for L*. What is the derivative of log(money demand) with respect to log(Y) and log(R)? Remark: these two derivatives are a very famous re- sult. They are the elasticity of money demand with respect to income and the nominal interest rate, respectively. As you can see, they lead to a very clear prediction of how money demand varies with
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