Question

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 p2 = 4. 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.

Answer #1

Graph 1)

graph 2)

Suppose x1 and x2 are perfect substitutes
with the utility function U(x1, x2) =
2x1 + 6x2. If p1 = 1,
p2 = 2, and income m = 10, what it the optimal bundle
(x1*, x2*)?

Suppose Alex only consumes 3 units of x1 with 8 units of x2.
That is, if he is consuming more x1 or x2 in a different ratio, it
does not increase his utility
a) Write down Alex’s utility function. What kind of utility
function does he have?
b) Suppose Alex wants to have a utility 48. If he desires to
make the best use of his money, based on your utility function in
a) how many units of x1 and...

Consider utility function u(x1,x2)
=1/4x12
+1/9x22. Suppose the prices of good
1 and
good 2 are p1 andp2, and income is
m.
Do bundles (2, 9) and (4, radical54) lie on the same
indifference curve?
Evaluate the marginal rate of substitution at
(x1,x2) = (8, 9).
Does this utility function represent
convexpreferences?
Would bundle (x1,x2) satisfying (1)
MU1/MU2 =p1/p2 and (2)
p1x1 + p2x2 =m be an
optimal choice? (hint: what does an indifference curve look
like?)

Consider the following utility function: U(x1,x2)
X11/3 X2
Suppose a consumer with the above utility function faces prices
p1 = 2 and
p2 = 3 and he has an income m = 12. What’s his optimal
bundle to consume?

Suppose a consumer has quasi-linear utility: u(x1,x2 ) = 3x1^2/3
+ x2 . The marginal utilities
are MU1(x) = 2x1^−1/3 and MU2 (x) = 1. Throughout this problem,
assume p2 = 1
1.(a) Sketch an indifference curve for these preferences (label
axes and intercepts).
(b) Compute the marginal rate of substitution.
(c) Assume w ≥ 8/p1^2 . Find the optimal bundle (this will be a
function of p1 and w). Why do
we need the assumption w ≥ 8/p1^2 ?...

Suppose the utility function is given by U(x1,
x2) = 14 min{2x, 3y}. Calculate the optimal consumption
bundle if income is m, and prices are p1, and
p2.

Math/Economics/Optimization problem
Suppose that given quantities x1 and x2 of two items, your
utility is given by u(x1, x2) = x2 + 5 - (x1 - 3)^2.
a Sketch a few level curves of u.
b Suppose your purchasing is subject to a budget
constraint, and the optimal purchase is given by (x1, x2) = (2, 8).
Give the equation of the level curve of u that corresponds to this
purchase.
c Graph the level curve from b and use...

1. A consumer has the utility function U = min(2X, 5Y ). The
budget constraint isPXX+PYY =I.
(a) Given the consumer’s utility function, how does the consumer
view these two goods? In other words, are they perfect substitutes,
perfect complements, or are somewhat substitutable? (2 points)
(b) Solve for the consumer’s demand functions, X∗ and Y ∗. (5
points)
(c) Assume PX = 3, PY = 2, and I = 200. What is the consumer’s
optimal bundle?
(2 points)
2....

2. A consumer has the utility function U ( X1,
X2 ) = X1 + X2 +
X1X2 and the budget constraint
P1X1 + P2X2 = M ,
where M is income, and P1 and P2 are the
prices of the two goods. .
a. Find the consumer’s marginal rate of substitution (MRS)
between the two goods.
b. Use the condition (MRS = price ratio) and the budget
constraint to find the demand functions for the two goods.
c. Are...

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...

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