The function of the demand for money of Keigan operates during periods of inflation. When the chains in the economy change very quickly, and the purchasing power of money is also rapidly declining, the impact of real income, real interest rate and other real parameters of the economy on money demand becomes insignificant. People react mainly to the rate of inflation. The desire and needs of individuals in money decrease in proportion to the rate of inflation:
where - expected inflation; - demand for real money.
For example, you can consider a function Keigan
Or in the logarithmic form:
where a is the elasticity of demand for money at a rate of interest, defined as follows:
where i - the nominal interest rate; r - The real rate of interest.
The relationship between nominal and real interest rates is determined based on another Fisher formula: , the real rate of interest is found as a nominal rate minus the expected rate of inflation.
The Baumol-Tobin model
Transaction demand for money depends on the interest rate. This is due to the fact that in many firms wages are paid not in cash, but by transferring the appropriate amount to the employee's deposit account. If an employee does not use this money (or part thereof), he receives a percentage of the bank exactly as any other person who opened an account with the bank. But then he will not be able to buy goods and services.
In order to pay for the purchased products, a person must withdraw cash from the account. This requires a trip to the bank, filling out documents, waiting in line (all such expenses are transactional costs for the individual) and, of course, is accompanied by a loss of interest income. Thus, the individual faces a dilemma: not to take money from the account - to have interest income, but not be able to purchase goods and services; or withdraw money from the account - lose interest, but pay for purchased products. What amount to withdraw?
The answer to this question is the model of Baumol-Tobin reserves.
The model considers the financial behavior of an individual under the following assumptions:
- people keep money because they want to use them to pay for purchases of goods and services;
- Under the money is understood primarily cash;
- in the economy there are two competing assets: cash (money) and bonds;
- the household receives a real income Y once a month and spends it evenly over the next month, so that at the end there is a zero amount of money
- money - household income - transferred to a bank account
- a household for one transaction can withdraw from a bank account and transfer the value to,
- the transaction is accompanied by transaction costs of h (brokerage fee).
Have a person receive a monthly salary of Y den. units, which is transferred to his bank account. And let him evenly spend this amount for a month, so at the end of the period his account is reduced to zero (the individual does not make savings). Suppose that k shows how much a person withdraws from his account at the same time (Figure 3.1). Each operation for obtaining cash costs the individual in b den. units, which includes all possible transaction costs.
Fig. 3.1. The distribution of a person's funds within a month
The goal of the household is to minimize the costs associated with owning money.
Since the individual spends the cash evenly for a month, the total number of operations to convert a bank account into cash will be . And the accompanying aggregate transaction costs will be .
The average amount of money for a period (in real terms) is
The average amount of cash on hand in an individual is defined as the average of the amount that he could transfer to cash at the beginning of the period and the amount that remains at the end of the period (but by condition he spends all money on the end). If the bank interest rate on the accounts is r, then the cumulative loss of interest income for the month will be: . The total expenses of a person related to the possession of cash, rather than a bank account, will be equal to
How to minimize the costs associated with cash?
In order to find the optimal amount of cash, we use the condition of the first order of minimization of the cumulative cost function:
From where we find the optimal amount, which the individual must monthly withdraw from the account: . And the optimal amount of cash on hand on average for a month will be half of this amount: Since the individual is interested in the real purchasing power of money, the optimal demand for money should be expressed as the demand for real money money:
The function of demand for nominal money can be represented
in a more convenient form:
This is the transaction demand for Baumol's money - Tobin.
We see that in this case, the demand for money depends not only on the amount of income received by a person, but also on the rate of interest - the higher the interest rate, the greater the percentage income that an individual loses if he transfers the account into cash, and, consequently, the less is the demand for cash.
Note that, unlike the Fisher equation, in this case, the demand for money changes less than proportionally with the growth of income Y. This means that for the economy as a whole, the demand for money will be determined not only by the total value of the generated income, but also by its distribution . The larger the share of income is concentrated in the hands of a few (the more uneven is the distribution of income in the economy), the lower is the demand for cash.
We assumed that the average transaction costs b are constant. However, in reality banks often charge different fees for withdrawing small and large amounts. And other transaction costs also vary depending on the amount of cash. How will the demand for Baumol-Tobin money change if b = f + ck, where f is a constant component of costs , and the parameter c can take two values: 1) c> 0 and 2) with <0?
Let's summarize the main conclusions of the model.
1. Demand for money grows less than proportionally with the growth in the volume of transactions in the economy.
The elasticity of demand for money on income is .
There is a positive scale effect (economies of scale) in the possession of money for an individual. The higher the amount of cash a person has, the less the unit costs of maintaining that monetary value become.
Thus, the demand for money depends on the distribution of income (inequality in the distribution of income). The greater the inequality in the distribution of income in the economy (the greater the proportion of income concentrated in the hands of a few), the lower is the demand for money at each level of income, since one economic agent, with a given amount of transactions, demands a lower amount of cash (money) than two agent with half of his income.
Any change in the money supply will have a greater short-term effect on the economy than if the demand for money were proportional to the level of income. Monetary policy in the Baumol-Tobin model has a more significant impact on the economy than in previous cases.
2. The demand for money is inversely proportional to the interest rate.
The elasticity of the demand for money relative to the interest rate is: . The demand for money is relatively inelastic to a change in the interest rate in the economy.