Lecture Notes: BAML-Tobin Model in Macroeconomics

Introduction

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Key Topic: Transaction Money Demand

• Objective: Understand why individuals hold cash instead of earning interest by keeping it in bank accounts.
• Model: BAML-Tobin Model (1952, 1956)

Model Assumptions

• Income: Continuous fixed yearly income (e.g., $50,000/year)
• Savings Account: Income received continuously with an average interest rate (e.g., 5%)
• Cash Withdrawal: Money cannot be directly spent from the savings account; requires ATM or bank branch withdrawal.
• Withdrawal Fee: Example fee is $2 per withdrawal.

Model Explanation

• Cash Holding: Balancing withdrawal frequency affects cash holding and foregone interest.
• Example Scenario:
  • Withdraw twice a year, holding significant cash leading to foregone interest.
  • Formula for average cash holding: Y/2n (Y = Annual Income, n = Number of Withdrawals).
• Foregone Interest Calculation: Multiply average cash holding by interest rate.
• Cost Structure: Sum of withdrawal fees and foregone interest.

Optimization of Withdrawals

• Objective: Minimize total costs (fees + foregone interest).
• Cost Function: Total Costs = (Y/2n * I) + (C * n)
  • I: Interest Rate
  • C: Withdrawal Fee
• Finding Optimal Withdrawals
  • Derivative of cost function set to zero to find optimal number of withdrawals.
  • Optimal withdrawals calculated as sqrt(Y * I / 2C).

Example Calculation

• Optimal Withdrawals: 25 times per year (twice a month)
• Total Cost Calculation: Combination of foregone interest and fees.

Impact of Exogenous Factors

• Interest Rate Changes: Higher interest rates lead to more frequent bank trips.
• Withdrawal Fee Changes: Higher fees reduce the number of trips, increasing cash holding.

Conclusion

• Transaction Money Demand: Affected by income, interest rate, and transaction costs.
• Model Application: Useful for understanding demand for cash and optimal withdrawal strategies.
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