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Keeping that in mind, we have also recently launched the YouTube Membership Support Scheme, so please check the links in the description if you would like to to consider supporting us either via Patreon or via YouTube memberships. My name is Sava, and today we're investigating a key concept and a key model in macroeconomics and financial economics that shed some light on how transaction money demand is formed. Why do individuals hold cash on their hands, whereas they could earn interest by keeping it in their bank accounts? This model is called the BAML-Tobin model, in honor of BAML and Tobin who discovered this model independently in 1952 and 1956, respectively.
And the assumptions and the concept of the model are beautifully simple. Imagine an individual who earns continuously throughout the year some fixed income, for example, let's assume our individual makes 50 000 dollars a year, and by default their salary is continuously being received on their savings account, and the savings account has some interest rate that the bank provides, on the average remainder of money on their account. So the bank calculates how much money on average is there throughout the year on the account and charges some interest rate that the holder of the accounts or our individual benefits from.
And let's assume that interest rate is 5%, quite high by modern standards, but whatever. And the individual cannot directly spend the money from their bank account, they have to either go to the ATM and retrieve the money that way, withdraw the money that way, or go to the... bank's branch and withdraw the money at the telecounter.
And that's quite outdated of an assumption, you might think, but if we consider that not to be a bank account, but for example an account at a retail trading app, and that is not an interest rate but some expected rate of return on their investment, and they do not withdraw cash from the bank, but they rather withdraw the money from their retail trading app onto their bank account, that makes it much more realistic, given the fact that many retail trading apps do charge fixed withdrawal rates for whatever amount you wish to withdraw. And let's assume that our withdrawal fee is two dollars, which is quite low compared to the income of our individual. However, it can accumulate quite a lot if you decide to go to your bank or withdraw money from your retail app, as we discussed, quite frequently during the year.
And the idea is to calculate what is the optimal number of withdrawals or trips to the bank, as they are called in the original 1950s formulation of the model, that an individual would like to do to maximize their gain or minimize their costs. And to understand what the costs are and what is the trade-off in this model, we have to consider some baseline scenario. So let's assume that the individual goes to the bank, or withdraws cash very unfrequently, very rarely, for example, only twice a year. That would mean that they have got, on average, quite a lot of cash on their hands throughout the year.
They withdraw it twice a year, so they withdraw 25 grand at the start of January and at the start of July, as can be seen with this visualization over here, and they pay quite a small amount in fees. They only pay two times two, four dollars in fees. However, they do forego a lot of interest that could have been charged on their savings account or on their retail trading app, they forego some expected return, that is not there because they called a lot of cash on their hands, and it means that the average remainder of money on their account is lower.
And to figure it out, we can simply consider the following. As our salary is charged or received continuously, then the average holding of cash in hands and the average remainder of cash at the account should sum up to our annual income. And it means that if we calculate the average cash holding that our individual has, then we can simply calculate the foregone interest by multiplying the interest rate we would have received onto this average cash holding.
And considering that the individual is assumed to spend this amount uniformly across the time period so we can see that they continuously spend this 25 000 they have withdrawn and then they are left with no cash they a pretty good plan i assume until the 1st of July and then when they are run out of cash, then they go to the bank or make another withdrawal, and they are left with another 25 grand that they then continuously spend until the new year. So the average cash hold in this case would be our annual income divided by the number of withdrawals, which is the size of the withdrawal, and then we also need to divide it by two simply because we start with this amount at the start of the period when we withdraw, and then we're left with almost nothing, next to nothing, before our next withdrawal. So it means that the average cash holding we have got throughout the year with our individual can be expressed with a formula of Y, Y being the annual income, divided by 2 times the number of withdrawals.
And it means that we've multiplied by the interest rate I, we will have calculated the foregone interest. And that's what we're going to do now. So we can multiply this average cash holding, which is the source of our foregone interest, by the interest rate, and we see that we forego more than 1%. than $600 in forgone interest if we only go to the bank twice. And in terms of the fees that we pay, we have already discussed that in that setting it is quite a minor amount, quite a negligible amount, it's only two times two, meaning it's four dollars, and we can calculate the total cost.
It is our accounting cost, in a sense, the money that we physically pay to the bank teller or to the retail app, whatever our assumptions are. And we also add up our economic cost, our alternative cost of forgoing the interest or the return on the cash that we're holding in our hands and not at the account. So we can see that the total cost that we endure in this scenario is $629.
So let's see what happens if we increase the number of trips to the bank or the number of withdrawals. So if we go to the bank three times instead of twice, our forgone interest plummets. significantly, and the fees that we pay increases somewhat. But our total cost does get reduced quite a lot. We can see that we hold much less cash on average throughout the year, and that contributes to a lower foregone interest.
However, it does not continue indefinitely. If we go to the bank four times, we're still better off than three times or twice. But consider if we go to the bank some ridiculous amount of times, like a hundred times. we go to the bank or withdraw cash almost every three days, and in that sense, we forgo very few interest because the average cash holding we have got at every particular point in time is very low, but we pay more than $200 in fees. And it's easy to envision that if we go to the bank every single day, 365 times in a year of 2021, there are 365 days in that particular year, we pay more in fees than we would have.
lost from both sources if we go to the bank twice. And that means that there should be some optimal number of trips to the bank that balances out this trade-off between foregone interest and fees paid to the broker, to the bank, or whatever our assumption, our model framework is. And to elaborate on that, to formalize this, we can specify the function of our total costs, which is the...
sum of our forlorn interest that we have discussed to be equal to our average cash holding, which is y divided by 2n, again y being the annual income and n being the number of withdrawals, multiplied by the interest rate i. And then we only have to add up our accounting costs, the total fees paid to the bank teller or the accountants. broker, which is our withdrawal fee, which is two dollars now, in this case, times the number of withdrawals.
And this would be our cost function that we seek to minimize. We seek to vary n, so this function is as low as we can make it. And here we can simply calculate the first derivative of this particular function f with respect to n, and that is quite an easy task. Well, the derivative of cn with respect to n would simply be c, as it's a linear function, we can just get the constant here and n gets reduced to 1. Here the second component of our function also is pretty easy to differentiate.
First of all, y times i over 2 is a constant that can be simply preserved when we take the derivative, and the derivative of 1 over n is minus 1 over n squared, so here the sign becomes negative, and we've got n squared in the denominator. And we can look at this function and see how it behaves. First of all, we can see that if n is quite small, then this expression is quite large. So it's very likely that this expression is negative, meaning that if our n is small, then our costs reduce if we increase n. So if we go to the bank two times, which is a small amount of trips to the bank or a small amount of withdrawals, we can benefit, we can reduce our total costs by increasing the number of trips.
So for example, if we go from two to three, our total cost plummets quite substantially. And this is how it's visualized, how it's proven. using the derivative. However, if n is very large, for example, if n is 365, then the denominator is very large, because it's n squared, and this amount is negligible, and as c is positive, then we have got a positive value of the derivative, meaning that if we increase the number of trips to the bank, our costs would increase, meaning that we are better off reducing the number of trips to the bank.
So if we go down from 365 to something more tolerable, like 52 every single week, we can see that our costs do get a lot smaller. And that means that our optimal number of withdrawals would be the point where the derivative is equal to zero, because it is negative initially, so we are reducing the cost by going to the bank more frequently, and then it becomes positive, so we are losing. if we're going to bank more frequently. And this sweet spot is exactly where the value of the derivative is precisely zero. So we can do some simple algebra, just rearranging the terms, and then taking the square root, we can express the optimal number of withdrawals and star as a square root of y times i in the numerator, which is the expression of our foregone interest, isn't it?
And in the denominator, we've got 2C, which is the expression of the other side of the trade-off, the accounting cost of withdrawal fees we pay to the bank teller or to the broker. So let's plug in our numbers and see what happens with our optimal N. So we can just take the square root of our annual income times the interest rate, and we can divide it by 2 times the withdrawal fee C.
And we can specify that the optimal number of withdrawals, optimal number of trips to the bank, is equal to 25, so roughly twice a month. And we can see that the total costs, foregone interest plus paid fees, sums up to $100, and that's the lowest amount we can achieve. If we go to the bank 24 times, then our total cost increases slightly, and if we go to the bank 26 times, then the story is the same.
So 25 is indeed the optimal amount. And moreover, looking at this function, we can see how does an increase in or a decrease in some of the exogenous parameters in our model, so income, interest rate, transaction costs, affect the number of optimal trips to the bank, number of optimal withdrawals, and how it ultimately affects our money demand as our average cash holding, which is the transaction money demand, the amount of money you hold in our hands to facilitate daily transactions, which is expressed by y over 2n, as we previously discussed, how is it affected by those parameters? So if our interest rate i increases, we are less incentivized to save on the withdrawal fees, and we are more incentivized to think about the foregone interest, so we would go to the bank more frequently.
So, for example, if our interest rate increases to something tremendous like 20%, we will go to the bank 50 times as an optimal amount. And we can see that here it's indeed the case. That's where our total cost is the lowest.
However, if, for example, the cost does increase quite a lot, let's imagine that our withdrawal fee increases from $2 to $8, the number of optimal trips to the bank does reduce. it is now 12 and a half. And well, what to do with this 12 and a half? You cannot go to the bank half a time, isn't it?
You cannot withdraw half a time. Well, in that case, you can simply compare what is going on with n equals 12 and n equals 13, and select the best of the two. So we can see what is the total cost at 12, and we can see what it is at 13, and figure out that 13 is optimal.
as the cost at 13 is slightly lower than the cost at 12. So that's what you might do if your optimal n is not a whole number, but a fraction. But returning to our starting conditions, we can see how the transaction money demand is being affected by income. So well, if income goes up, then the number of optimal trips to the bank goes up.
If the interest rate goes up, everything else held constant, then the number of trips to the bank goes up, meaning that the demand for money goes down, and that's quite a staple finding in introductory macroeconomics, isn't it? the reverse is true if the transaction cost goes up. If the transaction cost goes up, then you're incentivized to go to the bank less frequently, meaning that you would hold more cash on your hands at every single point in time, meaning that your demand for cash would increase. And that's all there is for the Bermal-Torben model and using its logic to figure out how demand for money is affected by exogenous factors and how to calculate the optimal number of withdrawals if you have the same situation with your...
retail trading app, for example. Please leave a like on this video if you found it helpful. In the comments below, I'm eager to see any further suggestions for videos in business, finance, or economics you would like me to record. And please don't forget to subscribe to our channel or consider supporting us on Patreon.
Thank you very much and stay tuned.