Let U(x; y) = 5x:8y:2 showing all derivation work, find:

(a) the Marshallian demand functions for x and y
(b) the Indirect Utility Function
(c) the compensated demand functions xc and yc

To find the Marshallian demand functions for x and y, we need to maximize the utility function U(x, y) subject to the budget constraint.

The budget constraint is given by px * x + py * y = m, where px represents the price of good x, py represents the price of good y, and m represents the budget or income.

To find the Marshallian demand for x, we differentiate the utility function U(x, y) with respect to x and set it equal to the ratio of the prices of good x and good y:

∂U/∂x = 5:8:2 = px/py

Similarly, to find the Marshallian demand for y, we differentiate the utility function U(x, y) with respect to y and set it equal to the inverse of the ratio of the prices of good x and good y:

∂U/∂y = 5:8:2 = -px/py

Now, let's solve these two equations for x and y.

(a) Marshallian demand functions:
∂U/∂x = 5/8 = px/py
∂U/∂y = 2 = -px/py

From the first equation, we can express px in terms of py:
px = (5/8) * py

Then plug this into the second equation:
2 = -[(5/8) * py] / py

Simplifying the equation:
2 = -5/8

This contradicts the equation. Thus, there is no solution.

(b) The Indirect Utility Function:
The indirect utility function represents the maximum utility level attained at a given income and prices. It can be found by substituting the optimal x and y values into the utility function U(x, y):

V = U(x, y) = 5x:8y:2

Since there is no solution for the Marshallian demand functions, we cannot derive the indirect utility function.

(c) The compensated demand functions, xc and yc, represent the demand for x and y after adjusting the individual's income to maintain the initial level of utility.

Unfortunately, since we were not able to find the Marshallian demand functions, we cannot calculate the compensated demand functions.

To find the Marshallian demand functions for goods x and y, we need to maximize the utility function U(x, y) subject to a budget constraint. The budget constraint is defined as the total expenditure (px * x + py * y) being equal to the income or budget (I).

(a) Marshallian Demand Function for x:
To find the demand for good x (dx/dI), we need to differentiate U(x, y) with respect to x, while holding y constant:
dU/dx = 5:8y

To find the demand for good y (dy/dI), we differentiate U(x, y) with respect to y, while holding x constant:
dU/dy = 5x:-8

We can equate dU/dx to px and dU/dy to py (since they represent the marginal utilities) and solve for x and y.

dU/dx = px => 5:8y = px
dU/dy = py => 5x:-8 = py

Rearranging those equations, we get:
5:8y = px => y = (8/5)px
5x:-8 = py => x = (-8/5)py

So, the Marshallian Demand Function for good x is x = (-8/5)py and for good y is y = (8/5)px.

(b) Indirect Utility Function:
The indirect utility function (V) represents the maximum utility achieved given the prices (px, py) and the income (I).

To calculate the indirect utility function, we substitute the values of x and y from the Marshallian demand equations into the utility function U(x, y):

U(x, y) = 5x:8y:2

Substituting the demand equations:
U(x, y) = 5((-8/5)py):(8/5)px:2
= (-8py):(5py:2px)

Simplifying further, we get:
U(x, y) = (-8py):(5py:2px)
= -8:(5 - 2(px:py))

Hence, the Indirect Utility Function is V = -8:(5 - 2(px:py)).

(c) Compensated Demand Functions:
The compensated demand functions for goods x and y represent the demand changes due to a change in prices but with the consumer's purchasing power held constant at a certain level (compensated income).

To find the compensated demand functions, we differentiate the indirect utility function V with respect to the price of x (px), while holding utility (U) constant:

dV/dpx = dU/dpx

Using the utility function derived earlier:
dU/dpx = dU/dx * dx/dpx + dU/dy * dy/dpx
= (5:8y) * dx/dpx + (5x:-8) * dy/dpx

Setting this equal to zero, we solve for dx/dpx, which represents the compensated demand for good x while holding utility constant:

(5:8y) * dx/dpx + (5x:-8) * dy/dpx = 0

Using the equations for dx/dI and dy/dI, we can substitute and solve for dx/dpx:

(5:8y) * dx/dpx + (5x:-8) * (dy/dI / dx/dI) = 0
(5:8y) * dx/dpx + (5x:-8) * ((-8/5)/(-8/5)*py) = 0

Replacing dy/dI with py, and rearranging the equation, we find the compensated demand for good x:

dx/dpx = -(5:8y) * (5x:-8) * ((-8/5)/(-8/5)*py)
= (-5:64)xy

Similarly, for the compensated demand for good y (dy/dpy), we differentiate V with respect to the price of y (py) while holding utility constant:

dV/dpy = dU/dpy

Using the utility function derived earlier:
dU/dpy = dU/dy * dy/dpy
= (5x:-8) * dy/dpy

Setting this equal to zero, we solve for dy/dpy:

(5x:-8) * dy/dpy = 0

Using the equations for dx/dI and dy/dI, we can substitute and solve for dy/dpy:

(5x:-8) * dy/dpy = (5x:-8) * (dx/dI / dy/dI)
= (5x:-8) * ((-8/5)/(-8/5)*py)
= (-5:64)xy

Hence, the compensated demand function for good x is dx/dpx = (-5:64)xy, and for good y is dy/dpy = (-5:64)xy.