The Baumol Model The Baumol Model is based on the Economic Order Quantity (EOQ)

The Baumol Model The Baumol Model is based on the Economic Order Quantity (EOQ)

model developed for inventory management. 2 We will see it applied to inventory in Chapter 20. Applied to the management of cash, the EOQ model determines the amount of cash that minimizes the sum of the holding cost and transactions cost. The holding cost includes the costs

2 William J. Baumol, “The Transactions Demand for Cash: An Inventory Theoretic Approach,” Quarterly Journal of Economics (November 1952).

Management of Cash and Marketable Securities

of administration (keeping track of the cash) and the opportunity cost of not investing the cash elsewhere. The transaction cost is the cost of getting more cash—either through selling marketable securities or through borrowing. The economic order quantity is the level of cash infusion (from selling marketable securities or borrowing) that mini- mizes the total cost associated with cash.

Suppose each time our cash balance is zero we generate $100,000 (borrowing or selling securities). Further suppose that our opportunity cost for holding cash is 5%—we could have invested the cash in some- thing that earns 5% instead of holding it. Our holding costs are the product of the average cash balance and the opportunity cost. If we start with $0 cash and end up with $100,000 after an infusion, our average cash balance is = $50,000, so our holding cost is:

0.05  $100,000   -------------------------  = $2,500

Holding cost =

If we did not hold $50,000 of cash on average, we could have earned $2,500 by investing it.

Now suppose we need $1,000,000 cash for transactions over a given period. If we need $1,000,000 in total and we get $100,000 cash at a time, we need to make 10 transactions during the period. If it costs us $200 every time we make a cash infusion our transactions cost is $2,000:

$1,000,000  Transaction cost = $200 per transaction  -----------------------------------------------------------------   $100,000 per transaction 

↑ Cost per transaction Number of transactions = $200(10) = $2,000

The total cost associated with cash is the sum of the holding cost and the transactions cost:

Total cost = $2,500 + 2,000 = $4,500

Will cash infusions of $100,000 at a time produce the lowest cost of getting cash? We can’t control the cash needed for transactions purposes or the cost per transaction. But we can control how many cash infusions we make. And that number affects both the holding cost and the trans- actions cost.

MANAGING WORKING CAPITAL

The holding cost is a function of the amount of the cash infusion: With larger cash infusions, we hold more cash. Holding more cash, we have a greater opportunity cost to holding it. The transactions cost is also

a function of the amount of cash infusion: The larger the cash infusion, the fewer the transactions, and therefore the lower our transactions costs. Let’s use these considerations and what we know about economic order quantity to determine the minimum cost of cash. If we get cash in the amount of Q at the beginning of a period and wait until the cash balance is zero before we get more cash, the average cash balance over the period is Q/2. The cost of holding cash during this period is determined by the average cash balance, Q/2, and the opportu- nity cost of holding the cash, k:

Holding cost = -----

But each time we get cash, we have to make a transaction. If we demand a total of S dollars of cash each period, we end up making S/Q transactions per period. If it costs K to make a transaction, the transac- tions cost for the period is:

S Transactions cost = K -----

Putting the holding cost and the transaction cost together, the total cost associated with the cash balance is:

Total cost = Holding cost + Transaction cost = k Q ----- + K ----- S

The total cost associated with any given level of inventory ordering Q is:

Total cost = k ----- + K -----

To calculate the minimum total cost with respect to the amount of inventory we get each time, we:

1. Calculate the first derivative of the total cost equation with respect to Q.

Management of Cash and Marketable Securities

2. Set this first derivative equal to zero.

3. Solve for Q. The first derivative of the total cost with respect to Q (where “d” indi-

cates “change”) is:

---------------------------------- ( d Total cost )

--- k = S – -------- K

dQ ()

Setting the first derivative equal to zero:

--- k 0 S = – -------- K

Solving for the level of Q that minimizes the total cost, Q*,

Q* = ----------------------------------------------------------------------------------------------------------------------------- ( 2 Cost per transaction ) Total demand for cash ( )

Opportunity cost of holding cash or,

Q* = 2 ----------- KS k

What does this mean? If we look at the relations among Q* and K, S, and k in this equation, we see that:

■ The larger the cost per transaction, K, the greater the amount of cash, Q*, infused in a single transaction—the larger the transaction cost, the fewer transactions we make.

■ The larger the demand for cash, S, the larger the amount of cash, Q*, infused in a single transaction. ■ The larger the opportunity cost of holding cash, k, the smaller the amount of cash, Q*, infused in a single transaction.

In our example, K = $200 per transaction, S = $1,000,000, k = 5%, and

Q* = --------------------------------------------------------- ( 2 $200 ) $1,000,000 ( ) = $89,443

MANAGING WORKING CAPITAL

If every time we need a cash infusion, we get $89,443, the costs associated with cash will be minimized. We can check our work by looking at the total costs of cash for lev- els of Q on either side of Q* = $89,443. If Q = $100,000,

Total cost = $2,500 + $2,000 = $4,500

as we saw before. If Q = $50,000:

--------------------- $50,000

Total costs = 0.05 

If Q = $89,443,

 $1,000,000  Total costs = 0.05  ---------------------  + $200  ------------------------------  

We can see in Exhibit 19.3 that the minimum of the total cost curve is at

a cash infusion level of $89,443, which corresponds to a total cost of $4,472. If the level of cash infusion is less than or more than $89,443, the cost of cash will be higher.

EXHIBIT 19.3 Costs of Cash for Different Levels of Cash Infusions

Management of Cash and Marketable Securities

The EOQ model can be applied to any time framework—whether the period is a year, a month, a week, or any other unit of time. It is only necessary to make sure that all the elements that depend on the unit of time—the holding costs, k, and transactions demand, S—are in that same unit of time.

The economic order quantity model can be modified to suit the cir- cumstances of different cash situations. For example, the EOQ model for cash can be modified to include a safety stock—a balance of cash for precautionary purposes. The safety stock is a level of cash balance that acts as a cushion in case our cash needs are suddenly greater than expected.