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The Microfoundations of the Money Demand Function The Baumol-Tobin Model Humberto Barreto (barretoh@wabash.edu) How do people decide how much of their wealth to hold as money? This is the question underlying the money demand function. The answer, as usual, is that each individual solves an optimization problem. Developed in the 1950s, the Baumol-Tobin Model is a transactions theory of money demand because it emphasizes the role of money as a medium of exchange. Holding money makes transactions (buying and selling) more convenient—you do not have to go to the bank every time you want to buy something. The cost of this convenience is the foregone interest you would have received on the funds had they been in an interest bearing asset. Assume you plan to spend a given, fixed amount of Y dollars gradually over the course of a year. How much money should you hold throughout the year? What, in other words, is the optimal size of average money holdings? After Setting Up the Problem, we'll find Find the Initial Solution, then do Comparative Statics. We're especially interested in how money demand responds to the interest rate and income because the money demand function we used in the IS/LM—AD/AS Model used money demand function L(i, Y), with dL/di < 0 and dL/dY > 0, remember? Setting Up the Problem: Goal: min total costs Endogenous Variables: N, the number of trips to the bank, which then determines average cash holdings, or money demand Exogenous Variables: i, the nominal interest rate Y, the amount of spending the individual plans to do F, the cost of each trip to the bank Further Understanding the Problem See sheets Idea1 and Idea2 in BaumolTobin.xls Idea 1: Average Cash Holdings equals how much cash the person has on hand each day divided by 365 days in a year Formula for Day 1 cell in sheet Idea1: =$B$4-(A10-0.5)/365*$B$4 582678c7-49dc-44ce-8115-a558cbcb566a.doc Page 1 of 5 Idea 2: Average Cash Holdings equals Y/2N Idea 2B: This means that total foregone interest equals i x Y/2N Set the Number of Trips to the Bank in Idea2 sheet. So, as N increases, the cost of foregone interest falls AND the total cost of going to the bank rises. The separate graphs look like this: The Cos t of Foregone Income depends on i, Y, a nd N, like this: Y $ 10 ,0 00 amo unt of d ol lars to be spen t grad ual l y over the year i 10 % i nterest ra te p er year Cos t of Foregone Incom e i x Y /2 N N Co st o f forego ne i nco me $600.00 1 $ 50 0.00 $500.00 5 $ 10 0.00 $400.00 73 $ 6.85 $300.00 36 5 $ 1.37 $200.00 $100.00 $- 0 100 200 300 400 N The Cos t of Going to the bank depe nds on F and N, like this: F $ 5.00 pe r un it cost o f each tri p to the b ank FN N Co st o f Goi ng to the Ba nk Costs of Going to the Ba nk 1 $ 5.00 5 $ 25 .0 0 $2,000.00 73 $ 36 5.00 36 5 $ 1,82 5.00 $1,500.00 $1,000.00 $500.00 $- 0 100 200 300 400 N The tension in the problem is easy to see. As N rises, the Cost of Foregone falls, but the Cost of Going to the Bank rises. How many trips should be taken? The cost minimizing amount. 582678c7-49dc-44ce-8115-a558cbcb566a.doc Page 2 of 5 Finding the Initial Solution: • Graphically That’s easy. Put the two graphs together. See Underlying Graph sheet. The Total Cos t is the s um of the tw o indiv idua l costs . Y $ 10 ,0 00 amo unt of do ll ars to be spen t grad ual ly ove r the ye ar i 10 % i nterest ra te , co mpou nde d dai l y so that i/36 5 is d ai ly i nterest rate F 5 co st per tri p to the ba nk i x Y /2 N FN i x Y /2 N + FN Figure 18 -2 : The Cos t of Holding Mone y Co st o f $140.00 Tota l Fo re gon e Co st o f Goi ng N Inco me to th e Bank To tal Cost $120.00 Cost 5 $ 10 0.00 $ 25 .0 0 $ 12 5.00 $100.00 7.5 $ 66 .6 7 $ 37 .5 0 $ 10 4.17 Cost of Going $80.00 10 $ 50 .0 0 $ 50 .0 0 $ 10 0.00 to the Bank 12 .5 $ 40 .0 0 $ 62 .5 0 $ 10 2.50 $60.00 15 $ 33 .3 3 $ 75 .0 0 $ 10 8.33 $40.00 Cost of 17 .5 $ 28 .5 7 $ 87 .5 0 $ 11 6.07 Foregone 20 $ 25 .0 0 $ 10 0.00 $ 12 5.00 $20.00 Intere st $- 0 5 10 15 20 25 N BaumolTobin.xls • Excel’s Solver See sheet Optimization in BaumolTobin.xls Execute Tools: Solver. The Solver dialog box has been configured for you. The goal to Min Total Costs in cell B6 by choosing N in cell B14, given the exogenous variables, Y, i, and F. Excel's Solver gets very close, but not exactly equal to 1. That's a numerical algorithm for you. When little is gained, it stops hunting for a better solution and announces that it has converged to a solution. You can tighten the convergence criterion by clicking on the Options button in the Solver dialog box. Explore Convergence, Precision, and Tolerance. 582678c7-49dc-44ce-8115-a558cbcb566a.doc Page 3 of 5 • Calculus Here's the analytical solution. iY min TC FN 2N dTC iY 2 F dN 2N At a min, this derivative is equal to zero, so iY *2 F 0 2N iY *2 N 2F iY N* 2F At i = 10%,Y = $10,000, and F = $5/trip, 0.10 * $10,000 N* 10 2 *5 Y Notice that N* is not money demand. Money demand is average cash holdings, M d , and 2N must be calculated like this: Y YF M d* iY 2i 2 2F At i = 10%, Y = $10,000, and F = $5/trip, (10,000)* (5) M d* $500 2 * (0.1) d* YF The equation for optimal money holding, M , is the money demand function, L(Y,i) that 2i we were trying to derive. Notice how it behaves as expected—increases in Y lead to increases in money demand and increases in i lead to decreases in money demand. • Comparing Excel and Calculus Although Solver didn't give us exactly 1, for all intents and purposes we are getting the same answer (as shown by the data in the Optimization sheet). 582678c7-49dc-44ce-8115-a558cbcb566a.doc Page 4 of 5 Comparative Statics There are three exogenous variables in the model and they all appear in the optimal money demand equation. We can do comparative statics from the perspective of Y, i, and F. In each case, we can proceed analytically, taking the derivative of the optimal money demand function, or numerically, changing the exogenous variable by an arbitrary, finite amount and recalculating the optimal solution. Of course, the Comparative Statics Wizard add-in makes the latter approach much easier. 582678c7-49dc-44ce-8115-a558cbcb566a.doc Page 5 of 5

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posted: | 3/1/2010 |

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