Question

Verify that the three properties of elasticity hold for Cobb-Douglas Utility with U= , (Engel Aggregation, Homogeneity, and Slutsky Equation).

Answer #1

Consider the Cobb-Douglas utility function
u(x1,x2)=x1^(a)x2^(1-a).
a. Find the Hicksian demand correspondence h(p, u) and the
expenditure function e(p,u) using the optimality conditions for the
EMP.
b. Derive the indirect utility function from the expenditure
function using the relationship e(p,v(p,w)) =w.
c. Derive the Walrasian demand correspondence from the Hicksian
demand correspondence and the indirect utility function using the
relationship
x(p,w)=h(p,v(p,w)).
d. vertify roy's identity.
e. find the substitution matrix and the slutsky matrix, and
vertify the slutsky equation.
f....

A consumer has utility for protein bars and vitamin water
summarized by the Cobb-Douglas utility function U(qB,qW) =
qBqW.
e. Find the consumer’s Engel curve for vitamin water when PB =
PW = 1.
f. What is the consumer’s optimal bundle when M = 100 and PB =
PW = 1?
Suppose the price of protein bars increases to P’B = 2.
g. Find the new optimal bundle.
h. Find the substitution effect of the price increase on
purchases of...

A consumer has utility for protein bars and vitamin water
summarized by the Cobb-Douglas utility function U(qB,qW) =
qBqW.
a. Show that the consumer’s MRS at a generic bundle (qB,qW) is
MRS = - MUB/MUW = - qW/qB.
b. Show that the consumer’s MRS would equally be - qW/qB. if the
consumer’s utility function was V(qB,qW) = qB0.5qW0.5.
c. Find the consumer’s Marshallian demand for protein bars when
M = 100 and Pw = 1.
d. Is the demand function...

The utility function U(x1, x2) = lnx1 ^5 + lnx2^2 represents
Cobb- Douglas preferences. Show if it is true or false. (without
using graph)

1.54 The n-good Cobb-Douglas utility function is u(x) = A n i=1
x αi i , where A > 0 and n i=1 αi = 1. (a) Derive the
Marshallian demand functions. (b) Derive the indirect utility
function. (c) Compute the expenditure function. (d) Compute the
Hicksian demands.

A consumer has the Cobb-Douglas utility function
u(x1,x2)=x3.51x42u(x_1, x_2) = x_1^{3.5}x_2^{4}
The price of good 1 is 1.5 and the price of good 2 is 3. The
consumer has an income of 11.
What amount of good 2 will the consumer choose
to consume?

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...

Julie has preferences for food, f, and clothing, c, described by
a Cobb-Douglas utility function u(f, c) = f · c. Her marginal
utilities are MUf = c and MUc = f. Suppose that food costs $1 a
unit and that clothing costs $2 a unit. Julie has $12 to spend on
food and clothing.
a. Sketch Julie’s indifference curves corresponding to utility
levels U¯ = 12, U¯ = 18, and U¯ = 24. Using the graph (no algebra
yet!),...

6. Yiwei buys only two goods. Her utility function is
Cobb-Douglas. Her demand functions have which of the following
properties? a. Her demand for one of the two goods does not depend
on income.
b. Her demand for neither good depends on income.
c. Her demand for each of the goods depends on income and on the
prices of both goods.
d. Her demand for each of the two goods depends only on her
income and on the price of...

Homer is a deeply committed lover of chocolate. Assume his
preferences are Cobb-Douglas over chocolate bars (denoted by C on
the x-axis) and a numeraire good (note: we use the notion of a
numeraire good to represent spending on all other consumption goods
– in this example, that means everything other than chocolate bars
– its price is always $1).
a. Homer earns a salary that provides him a monthly income of
$360. Last month, when the price of a...

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