Question

utility function u(x,y) = x3 ·y2

I am going to walk you through the process of deriving the optimal quantity of apples and bananas that will make you the happiest. To do this, we are going to apply what we learnt about derivatives.

a) First, you have a budget. You cannot just buy an inﬁnite amount since you would not be able to aﬀord it. Suppose the price of a single apple is Px = 2 while the price of a single banana is Py = 4. If you buy x apples and y bananas, how much would that cost you (your answer should depend on x and y)? Make that equal to 100, meaning that you have a budget of $100 that you are trying to allocate between apples and bananas. The equation that you got is what we call a budget constraint.

a) General form of budget constraint is

P1* X + P2 * Y =m

where P1 = price of good 1 and P2 = price of good 2 and m = income

So, here

Budget constraint is 2X + 4Y = 100

b) Take your budget constraint and solve it for x in terms of y.

2X + 4Y = 100

2X = 100-4Y

X = 50 - 2Y

c) Substitute the expression that you got in part b) into the utility function. You should now have and expression for your utility that depends only of y, that is u(y).

d) Find the value of y that maximizes your utility. You should get more two solutions but explain which one(s) solve(s) your problem and why the other solution is obviously not a maximum.

e) Take that value of y you found in part d), plug it back into your budget constraint and solve for x

Please answer C D E

Answer #1

7.
Suppose you have the following utility function for two
goods:
u(x1, x2) = x
1/3
1 x
2/3
2
. Suppose your initial income is I, and prices are p1 and
p2.
(a) Suppose I = 400, p1 = 2.5, and p2 = 5. Solve for the
optimal bundle. Graph the budget
constraint with x1 on the horizontal axis, and the
indifference curve for that bundle.
Label all relevant points
(b) Suppose I = 600, p1 = 2.5, and...

Jen’s utility function is U (X, Y ) = (X + 2)(Y + 1), where X is
her consumption of good X and Y is her consumption of good Y .
a. Write an equation for Jen’s indifference curve that goes
through the point (X, Y ) = (2, 8). On the axes below, sketch Jen’s
indifference curve for U = 36
b. Suppose that the price of each good is 1 and that Jen has an
income of 11....

5. The utility function of a consumer is u(x, y)
=x2y
i. Find the demand functions x(p1,p2,m)
and y(p1,p2,m).
ii. What is the consumers indirect utility?

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....

(Hajikhameneh & Rietz) Answer the following questions for a
consumer with utility function U(x, y) = x2 y2 and a budget
constraint of g(x, y) = 2x + 4y = 40. You must show all of your
work for full credit. a. What is the marginal utility of x? of y?
b. In one to two sentences, define the economic meaning of the term
“marginal utility.” c. What is the marginal rate of substitution
for the given utility function? d....

. Suppose your utility depends on two goods: x and y. The
utility function is u(x, y) = ln(x) + ln(y) . Suppose you have an
income of $800. Further, assume that the price of x is 8 and the
price of y is 10.
Write down the equation for the budget constraint. Compute the
marginal rate of subsitution between x and y. • Compute the utility
maximizing combination of x and y. •
Suppose your income increases to $1000...

Claraís utility function is u (x; y) = (x + 2) (y + 1) where x
is her consumption of good x and y is her consumption of good
y.
(a) Write an equation for Claraís indi§erence curve that goes
through the point (x; y) = (2; 8).
(b) Suppose that the price of each good is $1 and Clara has an
income of $11. Can Clara achieve a utility level of at least 36
with this budget? (
c)...

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...

Scenario 5
Charlie has the utility function U ( X A , x B ) =
X1/2A X1/2B
where XA stands for the number of apples and
XB stands for the number of bananas consumed.
Refer to Scenario 5.
- Putting apples on the x-axis, and bananas on the y-axis, what
is the equation for the Marginal Rate of Substitution (MRS)?
Refer to Scenario 5.
- His indifference curve passing through 36 apples and 4 bananas
will also pass through...

Sophia’s utility function is as follows: U = 10x0.8
y0.6
Budget constraint: 100 = 6x + 4y
Solve for the utility-maximizing bundle
Find the equation of the indifference curve that contains the
utility-maximizing bundle.
Sketch the solution, labeling all relevant items, x on the
horizontal axis and y on the vertical axis.
At the utility-maximizing bundle, what is the increase in
Mustapha’s utility from the last dollar spent on good X? What about
for good Y?
Mustapha is moving to...

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