Question

Consider the competitive banking model of Freixas and Rochet where a bank's profit function is given by

π(D,L)=(rₗ​−r)L+(r(1−α)−rD​)D−C(D,L),
where rₗ​ is the rate of interest charged on loans, rD​ is the rate of interest paid on deposits, r is the interbank interest rate and α is the parameter for the reserve requirement. Suppose that rₗ= 7/4 ​,r=3/2​,rD​=11/10 α=1/5​ and the cost function is given by C(D,L)=2(D²+L²)−DL
Derive a bank's equilibrium level of Deposits and Loans

Answer 1

Determine the equilibrium levels of Deposits (D) and Loans (L) by substituting the given interest rates and reserve requirement into the profit function, taking its derivatives with respect to D and L, setting them to zero and solving the resulting equations.

Step-by-step explanation:

To determine the equilibrium level of Deposits (D) and Loans (L) for a bank using the competitive banking model of Freixas and Rochet, we need to analyze the given profit function π(D,L)=(rṓ-r)L+(r(1-α)-rD)D-C(D,L). With the specified values rṓ=7/4, r=3/2, rD=11/10, and α=1/5, we plug these into the profit function. We also consider the cost function C(D,L)=2(D²+L²)-DL. To find the equilibrium, we must take the first-order condition by differentiating the profit function with respect to D and L, and setting these derivatives equal to zero, which will give us a system of equations to solve for D and L.

The details provided about the market for credit card borrowing illustrate how equilibrium works in a financial market. When the interest rate is above equilibrium, a surplus exists, leading firms to lower rates to attract customers. Similarly, when it is below equilibrium, a shortage occurs, causing interest rates to rise. This dynamic equilibrium concept is fundamental in financial markets.