Financial Economic

Homework 2 MF640

1. What is the meaning of Lagrange Multiplier?

Ans The Lagrange multipliers method is one of methods for solving constrained extrema problems. Assume that we have a function f of n variables {y = f (x1,..., x n)} and we would like to find the value of x1,..., xn that will maximize or minimize this function subject to constraint g (x1,..., xn). The Lagrange multipliers method is based on setting up the new function (the Lagrange function)

                        L(x1,..., x n,λ) = f(x1,..., x n)  +λg (x1,..., x n) 

where λ is an additional variable called the Lagrange multiplier. From the Lagrange equation, the conditions for a critical point are
                                        L´ x1 = f´x1 +λg´ x1
                                        L´ x2 = f´x2 +λg´ x2
                                                …
                                        L´ xn = f´xn +λg´ xn

                                        L´ λ = g (x1,..., xn)

Rearrange the first n equations and set it equal to zero as

                                λ or    = λ[pic 3][pic 4]

The Lagrange multipliers can interpret that the value of optimal f will change λ unit if the exogenous variable (g(x)) change 1 unit. This show the marginal effect of changing in exogenous variable affect to the value of optimal objective variable (f). If the Lagrange multipliers are too high, the value of f will change more than the constraint which the Lagrange multiplier is low.

3. What is the optimum point of investment? If the utility function is quadratic utility function and

Begin with the efficient Frontier (Calculate from excel), we separate weight of investment into 11 groups.

[pic 6]

After that we create the quadratic utility function as
and assume that a = 0 (it doesn’t play any role). .
However, we have to create the conditions of b and c to support the theory.[pic 7][pic 8]

We find that the optimum point of investment depend on preference of investor (b and c). If you change value of b and c, the optimum point will change as table below

1. b = 0.3 and c = -0.7

1. b = 0.3 and c = -0.6