Question

Find the optimal bundle (x_{1}, x_{2}) (two
numbers). Does Jeremy consume positive amounts of both goods?

(e) Find the optimal bundle given p_{1} = 2,
p_{2} = 4 and m = 40 assuming U(x_{1},
x_{2}) = 2x_{1} + 3x_{2}. Does Jeremy
consume positive amounts of both goods? Is the optimal bundle at a
point of tangency?

Answer #1

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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*)?

Determine the optimal quantities of both x1 and
x2 for each utility function. The price of good 1
(p1) is $2. The price of good 2 (p2) is $1.
Income (m) is $10.
a.) U(x1,x2) =
min{2x1, 7x2}
b.) U(x1,x2) =
9x1+4x2
c.) U(x1,x2) =
2x11/2 x21/3
Please show all your work.

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.

Each individual consumer takes the prices as given and chooses
her consumption bundle,(x1,x2)ER^2, by maximizing the utility
function: U(x1,x2) = ln(x1^3,x2^3), subject to the budget
constraint p1*x1+p2*x2 = 1000
a) write out the Lagrangian function for the consumer's
problem
b) write out the system of first-order conditions for the
consumer's problem
c) solve the system of first-order conditions to find the
optimal values of x1, x2. your answer might depend on p1 and
p2.
d) check if the critical point...

There are two goods, Good 1 and Good 2, with positive prices
p1 and p2. A consumer has the utility
function U(x1, x2) = min{2x1,
5x2}, where “min” is the minimum function, and
x1 and x2 are the amounts she consumes of
Good 1 and Good 2. Her income is M > 0.
(a) What condition must be true of x1 and
x2, in any utility-maximising bundle the consumer
chooses? Your answer should be an equation involving (at least)
these...

There are two goods, Good 1 and Good 2, with positive prices
p1 and p2. A consumer has the utility
function U(x1, x2) = min{2x1,
5x2}, where “min” is the minimum function, and
x1 and x2 are the amounts she consumes of
Good 1 and Good 2. Her income is M > 0.
(a) What condition must be true of x1 and
x2, in any utility-maximising bundle the consumer
chooses? Your answer should be an equation involving (at least)
these...

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?

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...

Bilbo can consume two goods, good 1 and good 2 where
X1 and X2 denote the quantity consumed of
each good. These goods sell at prices P1 and
P2, respectively. Bilbo’s preferences are represented by
the following utility function: U(X1, X2) =
3x1X2. Bilbo has an income of m.
a) Derive Bilbo’s Marshallian demand functions for the two
goods.
b) Given your answer in a), are the two goods normal goods?
Explain why and show this mathematically.
c) Calculate Bilbo’s...

The utility function is given by u (x1,x2) = x1^0.5 + x2^0.5
1) Find the marginal rate of substitution (MRSx1,x2 )
2) Derive the demand functions x1(p1,p2,m) and x2(p1, p2,m) by
using the method of Lagrange.

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