Please help LEDA Les L- u 5. (20 points) Lets and y be the member of items of go

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LEDA Les L- u 5. (20 points) Lets and y be the member of items of goods and good B. respectively. bought by a customer. Then a customer's utility (o mentre of the customer's demand) is given by action of the U(X.) - 2xy + 4x. The cost per unit for good A is $and $3 for good B. The consumer's disposable income is $100. (a) Write the consumer's income constraint as a mathematical equation (b) Find the maximum value of U using the method of Lagrange multipliers (c) Estimate the new optimal utility if the consumer's disposable income increases by $6.

Explanation / Answer

a)
1x + 3y = 100

So,
1x + 3y = 100 ----> ANS

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b)
f = 2xy + 4x ---> to be max-ed
g = x + 3y = 100

fx = 2y + 4 , fy = 2x
gx = 1 , gy = 3

fx = m*gx and fy = m*gy...

(2y+4) = m(1)
2x = m(3)

So, we have
x = 3m/2
and y = (m-4)/2

Plug this into the constraint :
x + 3y = 100

3m/2 + 3(m-4)/2 = 100

(6m - 12) = 200

m = 106/3

With this,
x = 106/2 ---> x = 53
y = 47/3

So,
U = 2xy + 4x becomes :

U = 2(106/2)(47/3) + 4(106/2)

U(max) = 5618/3 -----> ANS

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c)
So, we have
x + 3y = 106

with this, we have
6m - 12 = 212

m = 112/3

And with this,
x= 112/2 = 56
y = 50/3

U(max)= 2xy + 4x

2(56)(50/3) + 4(56)

U(max) = 6272/3 -----> ANS

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