Question

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 demands

2. Find the indirect utility function

Answer #1

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

A consumer has utility function
U(x1,x2)= x1x2 / (x1 + x2)
(a) Solve the utility maximization problem. Construct the
Marshallian demand function D(p,I) and show that the indirect
utility function is
V (p, I) = I / (p1+ 2 * sqrt (p1*p2) + p2)
(b) Find the corresponding expenditure function e(p; u). HINT:
Holding p fixed, V and e are inverses. So you can find the
expenditure function by working with the answer to part (a).
(c) Construct the Hicksian...

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 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....

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...

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...

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*)?

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?)

Given a utility function for perfect complements: U(x1,x2) =
min{x1,βx2}, where β is a positive num- ber, and a budget
constraint: p1x1 + p2x2 = Y , where p1 and p2 are prices of good 1
and good 2 respectively, Y is the budget for the complements. Find
the demand functions for good 1 and good 2.

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