The Baumol–Tobin model is an economic model of the transactions demand for money as developed independently by William Baumol (1952) and James Tobin (1956). The theory relies on the tradeoff between the liquidity provided by holding money (the ability to carry out transactions) and the interest forgone by holding one’s assets in the form of non-interest bearing money. The key variables of the demand for money are then the nominal interest rate, the level of real income that corresponds to the amount of desired transactions, and the fixed transaction costs of transferring one’s wealth between liquid money and interest-bearing assets. The model was originally developed to provide microfoundations for aggregate money demand functions commonly used in Keynesian and monetarist macroeconomic models of the time. Later, the model was extended to a general equilibrium setting by Boyan Jovanovic (1982) and David Romer (1986).
For decades, debate raged between the students of Baumol and Tobin as to which deserved primary credit. Baumol had published first, but Tobin had been teaching the model well before 1952. In 1989, the two set the matter to rest in a joint article, conceding that Maurice Allais had developed the same model in 1947.
Formal exposition of the model
Suppose an individual receives her paycheck of
As a result the total cost of money management is equal to the cost of withdrawals,
The average holdings of money during the period depend on the number of withdrawals made. Suppose that all income is withdrawn at the beginning (N=1) and spent over the entire period. In that case the individual starts with money holdings equal to Y and ends the period with money holdings of zero. Normalizing the length of the period to 1, average money holdings are equal to Y/2. If an individual initially withdraws half her income,
This means that the total cost of money management is equal to:
The optimal number of withdrawals can be found by taking the derivative of this expression with respect to
The condition for the optimum is then given by:
Solving this for N we get the optimal number of withdrawals:
Using the fact that average money holdings are equal to Y/2N we obtain a demand for money function:
The model can be easily modified to incorporate an average price level which turns the money demand function into a demand for liquidity function: