Question

3. Suppose that a consumer has a utility function given by U(X,Y) = X^.5Y^.5 . Consider the following bundles of goods: A = (9, 4), B = (16, 16), C = (1, 36).

a. Calculate the consumer’s utility level for each bundle of goods.

b. Specify the preference ordering for the bundles using the “strictly preferred to” symbol and the “indifferent to” symbol.

c. Now, take the natural log of the utility function. Calculate the new utility level provided by each bundle.

d. Specify the new preference ordering for the bundles using the “strictly preferred to” symbol and the “indifferent to” symbol.

4. Suppose that you are consuming at a specific bundle of goods, and your marginal rate of substitution at the current bundle is 4. Suppose that the price ratio is currently 3.

a. How should you change the bundle of goods you are consuming to maximize your utility if your preferences are Cobb-Douglas?

b. Suppose now that you are consuming a different bundle of goods, and that your marginal rate of substitution at the new bundle is 3. Suppose that the price ratio has not changed. How should you change the bundle of goods you are consuming to maximize your utility if your preferences are Cobb-Douglas?

5. (5 points) Suppose that a consumer has the utility function U(X,Y) = XY^3 . The price of X is 12, the price of Y is 8, and the consumer has an income of 96. Solve for the consumer’s optimal consumption bundle. How much utility does the consumer receive from this bundle?

Answer #1

Suppose a consumer’s utility function is given by U(X,Y) = X*Y.
Also, the consumer has $360 to spend, and the price of X, PX = 9,
and the price of Y, PY = 1.
a) (4 points) How much X and Y should the consumer purchase in
order to maximize her utility?
b) (2 points) How much total utility does the consumer
receive?
c) (4 points) Now suppose PX decreases to 4. What is the new
bundle of X and...

Suppose a consumer has the utility function U (x, y) = xy + x +
y. Recall that for this function the marginal utilities are given
by MUx(x,y) = y+1 and MUy(x,y) = x+1.
(a) What is the marginal rate of substitution MRSxy?
(b)If the prices for the goods are px =$2 and py =$4,and if the
income of the consumer is M = $18, then what is the consumer’s
optimal affordable bundle?
(c) What if instead the prices are...

Consider a consumer with preferences represented by the utility
function
u(x,y)=3x+6 sqrt(y)
(a) Are these preferences strictly convex?
(b) Derive the marginal rate of substitution.
(c) Suppose instead, the utility function is:
u(x,y)=x+2 sqrt(y)
Are these preferences strictly convex? Derive the marginal rate
of substitution.
(d) Are there any similarities or differences between the two
utility functions?

Consider a consumer with preferences represented by the utility
function:
U(x,y) = 3x + 6 √ y
Are these preferences strictly convex?
Derive the marginal rate of substitution
Suppose, the utility function is:
U(x,y) = -x +2 √
y
Are there any similarities or differences between the two
utility functions?

Suppose the consumer’s utility function is equal to U=3x+5y.
Currently the price of x is $5, the price of y is $15 and the
income the consumer has to spend on these goods is $100.
A) Determine the MRSyx if we consume the bundle of (X,Y) =
(1,2).
B) What if we consume the bundle of (50,2).
C) What is the opportunity cost of X in terms of Y?
D) What quantities of X and Y should this consumer consume...

2. Consider a consumer with preferences represented by the
utility function:
u(x,y)=3x+6sqrt(y)
(a) Are these preferences strictly convex?
(b) Derive the marginal rate of substitution.
(c) Suppose instead, the utility
function is:
u(x,y)=x+2sqrt(y)
Are these preferences strictly convex?
Derive the marginal rate of sbustitution.
(d) Are there any similarities or diﬀerences between the two
utility functions?

Consider a consumer with Cobb-Douglas preferences over two
goods, x and y described by the utility function u(x, y) = 1/3ln(x)
+ 2/3n(y) 1. Assume the prices of the two goods are initially both
$10, and her income is $1000. Obtain the consumer’s demands for x
and y.
2. If the price of good x increases to $20, what is the impact
on her demand for good x?
3. Decompose this change into the substitution effect, and the
income effect....

Consider a consumer with a utility function U =
x2/3y1/3, where x and y are the quantities of
each of the two goods consumed. A consumer faces prices for x of $2
and y of $1, and is currently consuming 10 units of good X and 30
units of good Y with all available income. What can we say about
this consumption bundle?
Group of answer choices
a.The consumption bundle is not optimal; the consumer could
increase their utility by...

Suppose the Utility function of the consumer is given by
U = x + 5y^3
Suppose the price of x is given by p x and the price of y is
given by p y and the budget income of the consumer is given by I.
Price of x, Price of y and Income are always strictly positive.
Assume interior solution.
a) Write the statement of the problem
b) Compute the parametric expressions of the equilibrium
quantity of x &...

Suppose a consumer’s Utility Function
U(x,y) = X1/2Y1/2. The consumer wants to
choose the bundle (x*, y*) that would maximize utility.
Suppose Px = $5 and Py = $10 and the
consumer has $500 to spend.
Write the consumer’s budget constraint. Use the budget
constraint to write Y in terms of X.
Substitute Y from above into the utility function U(x,y) =
X1/2Y1/2.
To solve for the utility maximizing, taking the derivative of U
from (b) with respect to X....

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