CALCU-Question 181 ptsThe total differential of U(

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Solution by Steps

step 2

The asksia-ll calculator output for the total differential of

U(x, y)

is

dU(x, y) = dy U^{(0, 1)}(x, y) + dx U^{(1, 0)}(x, y)

step 3

The notation

U^{(0, 1)}(x, y)

represents the partial derivative of

U

with respect to

y

, which is denoted as

U_y

. Similarly,

U^{(1, 0)}(x, y)

represents the partial derivative of

U

with respect to

x

, which is denoted as

U_x

step 4

Therefore, the total differential of

U(x, y)

is

U_x dx + U_y dy

1 Answer

C

Key Concept

Total Differential

Explanation

The total differential of a function

U(x, y)

with respect to

x

and

y

is the sum of the partial derivatives with respect to each variable multiplied by the differential of the respective variable, which is expressed as

U_x dx + U_y dy

.

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Solution by Steps

step 1

To find the optimal level of

x

, we need to maximize the utility function

U(x, y) = e^{x^4} e^{y^4}

subject to the budget constraint

x + 2y = 100

step 2

We express

y

in terms of

x

using the budget constraint:

y = \frac{100 - x}{2}

step 3

Substitute

y

into the utility function to get

U(x) = e^{x^4} e^{\left(\frac{100 - x}{2}\right)^4}

step 4

To maximize

U(x)

, we take the derivative with respect to

x

and set it equal to zero. However, as per the asksia-ll calculator result, no global maxima were found

step 5

We need to check for local maxima or endpoints since the global maximum does not exist. The endpoints are

x=0

and

x=100

due to the budget constraint

step 6

Evaluate the utility function at the endpoints:

U(0) = e^{0} e^{\left(\frac{100}{2}\right)^4}

and

U(100) = e^{100^4} e^{0}

step 7

Compare the utility at the endpoints to find the maximum utility, which will give us the optimal level of

x

[Question 24] Answer

The optimal level of

x

is at one of the endpoints, either

x=0

or

x=100

, based on which gives the higher utility.

Key Concept

Maximization of Utility Function with Budget Constraint

Explanation

To maximize a utility function subject to a budget constraint, we substitute the constraint into the utility function and find the critical points and endpoints.

Solution by Steps

step 1

To find the optimal level of

y

, we follow a similar process as for

x

, maximizing the utility function

U(x, y) = e^{x^4} e^{y^4}

subject to the budget constraint

x + 2y = 100

step 2

We express

x

in terms of

y

using the budget constraint:

x = 100 - 2y

step 3

Substitute

x

into the utility function to get

U(y) = e^{\left(100 - 2y\right)^4} e^{y^4}

step 4

To maximize

U(y)

, we take the derivative with respect to

y

and set it equal to zero. However, as per the asksia-ll calculator result, no global maxima were found

step 5

We need to check for local maxima or endpoints since the global maximum does not exist. The endpoints are

y=0

and

y=50

due to the budget constraint

step 6

Evaluate the utility function at the endpoints:

U(0) = e^{\left(100\right)^4} e^{0}

and

U(50) = e^{0} e^{50^4}

step 7

Compare the utility at the endpoints to find the maximum utility, which will give us the optimal level of

y

[Question 25] Answer

The optimal level of

y

is at one of the endpoints, either

y=0

or

y=50

, based on which gives the higher utility.

Key Concept

Maximization of Utility Function with Budget Constraint

Explanation

To maximize a utility function subject to a budget constraint, we substitute the constraint into the utility function and find the critical points and endpoints.

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