This is a question about Microeconomics. Thank you so much the person who can he

Question

This is a question about Microeconomics. Thank you so much the person who can hepls me

I already finish part (a) and (b) that I get y = 2xPx / Py

However, I do not know how to do the part (c) and (d)

1 2 1. Suppose that U(x, y, z) = xy iz a. Find the first order conditions for utility maximization. b. Use the first order conditions along with the budget constraint to find demand functions for x, y and z: x = x(Px, Py, P2, 1), y = y(Px, Py, Pz, 1), and z = z(Px, PY, Pz, 1). This notation means that a demand function should contain only income and prices and should not contain quantities of other goods. Note that while a demand function may contain prices of other goods, it is not required to. c. Verify that when Px = 1, PY= 2, Pz= 1, and 1 = 24, the bundle (4, 4, 12) satisfies the conditions you found for utility maximization. Then, verify that two nearby bundles that also exhaust income yield lower utility than (4, 4, 12). d. How could you be sure that the bundle (4, 4, 12) is the utility maximizing bundle and not a utility minimizing bundle or a local maximum? (Just explain in a sentence or two. No need to actually show this.)

Explanation / Answer

c) Lets put the given values into utility function and alos in the budget constraint.

Px = 1, Py = 2, Pz = 1, Budget Constraint : I = xPx + yPy + zPz

Utility function: U = x1/3y2/3z

For maximizing the utility function First condition is:

dU/dx = 0.5y2/3z(x-0.5) = 0
dU/dy =  (2/3)x1/2(y-1/3)z = 0
dU/dz = x1/2y2/3 = 0

Forming Langrangian function and solving we find:

x = (3/4 yPy)/Px; y = 2xPx / Py and z = 3xPz also z = yPy / 2/3*Pz

Putting values of Px, Py and Pz : x = 3y / 2; y = x and z = 3x also z = 3y

I = xPx + yPy + zPz

24 = 1.5y(1) + y(2) + 3y (1)
y = 24/6.5 = approx 4 units

again,  as y = x thus x = 4 also

now, as z = 3x = 3y we have x = 4 = 4 thus z = 4*3 = 12

thus bundle (4,4,12) gives maximum utilization.

Lets another bundles (4,2,10) and (3,5,8)

Utility at (4,4,12) U = (4)1/3 (4)2/3 (12) = 85.33

Utility at (4,2,10) U = (4)1/3 (2)2/3 (10) = 17.77

Utility at (3,5,9) U = (3)1/3 (5)2/3 (9) = 75

Thus proof that bundle (4,4,12) gives maximum utility.

(d) The bundle (4,4,12) is maxima because while differentiating the utility function second times we will get negative values of x , y and z.

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