Question

1.Suppose the Baumol-Tobin model of money demand is correct. Everyone is alike and earns money income of $30,000/year. Brokers charge a fee of $2 for every transaction. The money supply is $1000 per person. What is the equilibrium nominal interest rate?Suppose the Fed wants to reduce the interest rate to 2% (.02). How much of an increase in the money supply per person is necessary to do so?2. In the Baumol-Tobin model, show that the optimal solution entails equality between the foregone interest costs and toal brokerage costs. (Use the fact that in the model, n, the number of times you sell bonds per period , is equal to PY/2M, where PY is money income and M is average money demand

Answer 1

1).

Consider the given problem here everyone earns money income of “$30,000/yr”, => Y=$30,000. The brokers charge is “$2” for every transection, => F=$2 and “i” be the interest rate, => the individual money demand is given by.

=> Md = [YF/2i]^0.5 and money supply per person is given by, “Ms=$1,000”. So, at the equilibrium “Md=Ms”.

=> [YF/2i]^0.5 = 1,000, => [30,000*2/2i]^0.5 = 1,000, => [30,000/i]^0.5 = 1,000.

=> [30,000/i] = 1,000^2, => i = [30,000/1,000^2] = [3/100] = 0.03 = 3%. So, the equilibrium interest rate is “i=3%”.

Now, let’s assume that the Fed wants to reduce the interest rate to “i=2%=0.02”. So, at the equilibrium, => Md=Ms.

=> [YF/2i]^0.5 = Ms, => [30,000*2/2i]^0.5 = Ms, => [30,000/0.02]^0.5 = Ms.

=> Ms = [30,000/0.02]^0.5 = 1,500,000^0.5 = 1,224.75, => Ms = $1,224.75. So, the required money supply per person to reduce the interest rate at “2%” is “Ms=$1,224.75”.

2).

Now, here “N=number of time you sell bonds”, “F= brokerage fee”, “i=interest rate=opportunity cost of holding money” and “PY=money income”, => average income hold is given by, “PY/2N”.

So, the “forgone interest rate” is given by “PYi/2N” and the brokerage cost is given by “FN”.

So, the total cost function is given by.

=> TC = PYi/2N + FN, => FOC require “dTC/dN = 0”.

=> (-1)*iPY/2N^2 + F = 0, => iPY/2N^2 = F, => iPY/2F = N^2, => N = [iPY/2F]^0.5, be the optimum value of “N”.

So, “forgone interest cost” is given by, => iPY/2N = iPY/2*(1/N) = iY/2*(2F/iY)^0.5.

=> “forgone interest cost” = (iY/2)*(2F/iY)^0.5 = (iYF/2)^0.5.

Now, the brokerage cost is given by.

=> FN = F*[iY/2F]^0.5 = [iYF/2]^0.5 = forgone interest cost.

So, at the optimum “forgone interest cost” is same as “brokerage cost”.