Question

Solve the following using Optimization

1) Using substitution: U = 2 ln(x) + ln(y) s.t. 2x + y = 20

2) Using LaGrangian: U = √x + ln(y) s.t. x+y=10

3) Using both: U = 5 ln(x) + 3 ln(y) s.t. 4x + 3y = 100

Answer 1

1) U = 2ln(x) + ln(20-2x)

dU/dX = 2/x + [1/(20-2x)]*(-2) = 0

So, 1/x = 1/(20-2x)

20-2x = x

20 = 3x

X = 20/3

Y = 20-2*(20/3)

= 20 -40/3

= 20/3

So (X*,Y*) = (20/3 , 20/3)

2) lagrangian

L =√x + ln(y) + m(10-x-y)

m : lagrangian multiplier

So FOC,:

dL/dX =0

1/2*√x - m = 0

1/2√x = m

dL/y =0

1/y - m = 0

So 1/2√x = 1/y

so 2√x = y

now from dL/dm = 0

so, 10 = x + 2√x

X* = 5.367

So Y* = 10-5.367 = 4.633

3) with substitution

U = 5ln(x) + 3ln( (100-4x)/3)

dU/dx = 5/x +[ 3*3/(100-4x) ](-4/3)

Put dU/dx = 0

So, 5/x = 12/(100-4x)

500 - 20x = 12x

500 = 32x

X* = 500/32 = 15.625

Y* = (100-4*15.625)/3

= 12.5

(x*,y*) = (15.625, 12.5)

lagrangian

L = 5ln(x) + 3ln(y) + m (100-4x-3y)

dL/dx = 5/x -4m = 0

m = 5/4x

dL/dy = 3/y -3m = 0

m = 1/y

So, 5/4x = 1/y

5y = 4x

From Constraint:

Put 4x = 5y

8y = 100

Y* = 12.5, so X* = 15.625