Question

2. A simple banking model. A bank raises funds by accepting deposits D in order to make loans L. The central bank requires that the bank hold a minimum quantity of reserves R in proportion to its deposits: where ρ is the required reserve ratio. You should assume that the bank does not hold excess reserves and has no capital. The hank is competitive ad takes the leding rate r, and the competitive and takes the lending rate rL and the deposit rate ro as given. The supply of deposits is perfectly elastic at ro. The bank chooses a quantity of deposits to accept, loans to make, and reserves to hold in order to maximize its profit. (a) (3 poins) Writeon the lak's Ialane sheet constraint (b) (3 points) Write down an expression for the bank's profit.j (c) (3 points) Solve the bank's profit maximization problem to find the equilibrium relation- ship between the deposit rate r, the lending rate rL, and the required reserve ratio p. Show your work.

Answer 1

A).

As we know that a balance sheet have two side one is “Assets” and the “Liabilities” side. Consider the following table shows the balance sheet of the bank.

Balance Sheet of Bank Assets Liabilities REp*DD Total D

So, here “D” be the total amount deposited, => the total reserve is “p*D”, => “p fraction” of “deposit” are being kept as reserved, => total amount left for loan is given by “L=(1-p)*D”.

So, the total of “Assets” and the “Liability” must be same.

B, C).

Now, given the problem here the bank chooses a quantity of deposits to accept loans to make and reserves to hold in order to maximize its profits. So, here the profit function is given by.

=> A1 = r_L*L – r_D*D, where “r_D” and “r_L” are the “deposit” and “lending” rates respectively.

=> A1 = r_L*(1-p)*D - r_D*D, => the FOC for maximization require “dA1/dD = 0”.

=> r_L*(1-p) - r_D = 0, => r_L = r_D/(1-p). So, at the optimum the “lending rate” is given by “r_D/(1-p)”.