Question

A utility maximizing saver has u(f₁, f₂) = f₁^1/2f₂^1/2 and earns m₁ = 90, m₂ = 90. She can save at an interest rate of 25 percent. If she hires an investment advisor she can save at an interest rate of 80 percent. What is the most that she would pay in fees to this advisor?

Answer 1

solution:

Given data

m1=90

m2=90

Lets form budget constraint for this problem.

Period 1 :

f1 + s = m1 => s = 90 - f1

where s = saving in period 1

Period 2 :

f2 = m2 + s + rs = 90 + (1 + r)(90 - f1) - m where m is the amount he is willing to pay to advisor

=> (1 + r)f1 + f2 = 90 + (1 + r)*90 = (2 + r)*90 - m

So here we have to find m such that utility after hiring advisor is equal to utility before hiring the advisor

Maximize : u =

Subject to : (1 + r)f1 + f2 = (2 + r)*90 ------------(1)

Legrange us given by :

where u = legrange multiplier

First order condition :

Dividing (2) from (3) we get :

f2/f1 = 1 + r => f2 = f1(1 + r)

Putting this in (1) we get :

(1 + r)f1 + (1 + r)f1 = (2 + r)*90 - m

=> f1 = [(2 + r)*90 - m]/(1 + r)

=> f2 = f1(1 + r) = (2 + r)*90 - m

So, Before hiring the advisor we have r = interest rate = 25% = 0.25 and m = 0

Thus, Utility before advisor is : u = ((2 + 0.25)*90 + 0)/(1 + 0.25) -------------(4)

After hiring the advisor we have r = 0.8, thus, Utility after hiring the advisor is u = ((2 + 0.8)*90 - m)/(1 + 0.8) ------(5)

Thus equating (4) and (5) we get :

((2 + 0.8)*90 - m)/(1 + 0.8)1/2 = ((2 + 0.25)*90 + 0)/(1 + 0.25)

Solving above equation we get : m = 9

Hence, The most that she would pay in fees to this advisor is 9.