Question

Alice’s preferences over two goods are described by the utility function u(x1, x2) = 2x1+ 4x2. Her income is m= 100 and p1= 4, p2= 5. Assume now that the price of good 1 falls to p01= 2.

a) Find the substitution, income, and total effect for good 1.

b) Find the substitution, income, and total effect for good 2.

c) Verify that the Slutsky equation holds for both goods

Answer #1

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*)?

A consumer’s preferences over two goods
(x1,x2)
are represented by the utility function
ux1,x2=5x1+2x2.
The income he allocates for the consumption of these two goods is
m. The prices of the two goods are p1
and p2, respectively.
Determine the monotonicity and convexity of these preferences
and briefly define what they mean.
Interpret the marginal rate of substitution
(MRS(x1,x2))
between the two goods for this consumer.
For any p1, p2,
and m, calculate the Marshallian demand functions of
x1 and...

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.

Qin has the utility function U(x1, x2) = x1 + x1x2, where x1 is
her consumption of good 1 and x2 is her consumption of good 2. The
price of good 1 is p1, the price of good 2 is p2, and her income is
M.
Setting the marginal rate of substitution equal to the price
ratio yields this equation: p1/p2 = (1+x2)/(A+x1) where A is a
number. What is A?
Suppose p1 = 11, p2 = 3 and M...

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.

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.

Consider the utility function:
u( x1 , x2 ) = 2√ x1 +
2√x2
a) Find the Marshallian demand function. Use ( p1 ,
p2 ) to denote the exogenous prices of x1 and
x2 respectively. Use y to denote the consumer's
disposable income.
b) Find the indirect utility function and verify Roy's
identity
c) Find the expenditure function
d) Find the Hicksian demand function

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 the following Constant Elasticity of Substitution
utility function U(x1,x2) =
x1^p+x2^p)^1/p
a. Show that the above utility function corresponds to (hint:use
the MRS between good 1 and good 2. The ->refers to the concept
of limits.
1. The perfect substitute utility function at p=1
2. The Cobb-Douglas utility function as p -->0
3. The Leontiff (of min(x1,x2) as p--> -infinity
b. For infinity<p<1, a given level of income I and prices
p1 and p2.
1. Find the marshallian...

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 ?...

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