Question

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 demand function. HINT: use Shephard’s lemma.

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

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....

1. Using the following utility function, U(x1,x2) =
x1x2+x1+2x2+2 Find the demand functions for both x1 and x2 (as
functions of p1, p2, and m).
Thank you!

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...

Suppose an individual consumers two goods, with utility function
U (x1; x2) = x1 + 6(x1x2)^1/2 + 9x2. Formulate the utility
maximization problem when she faces a budget line p1x1 + p2x2 = I.
Find the demand functions for goods 1 and 2.
(b) Now consider an individual consumers with utility function U
(x1; x2) = x1^1/2 + 3x2^1/2. Formulate the utility maximization
problem when she faces a budget line p1x1 + p2x2 = I. Find the
demand functions for...

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...

Consider a two good economy. A consumer has a utility function
u(x1, x2) = exp (x1x2). Let p = p1 and x = x1.
(1) Compute the consumer's individual demand function of good 1
d(p).
(2) Compute the price elasticity of d(p).
Compute the income elasticity of d(p).
Is good 1 an inferior good, a normal good or neither?
Explain.
(3) Suppose that we do not know the consumer's utility function
but we know that the income elasticity of his...

Suppose the indirect utility functions is: v(p1, p2, m) = ( ln(m
/p2) , if p1 ≥ m. (m−p1)/ p1 + ln(p1/ p2 ), if p1 < m.
a) Compute the Marshallian demand for both goods x1 and x2 for
the different values of m.
b) Based on your answers from (a), can you guess the type of the
original utility function u(x) (Hint: It is one of the 5 common
utility functions we have taken in the course)? Explain...

Suppose the indirect utility functions is: v(p1, p2, m) = ln (m
/p2) , if p1 ≥ m.( m−p1)/ p1 + ln (p1/ p2) , if p1 < m.
a) Compute the Marshallian demand for both goods x1 and x2 for
the different values of m.
b) Based on your answers from (a), can you guess the type of the
original utility function u(x) (Hint: It is one of the 5 common
utility functions we have taken in the course)?...

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