Question

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.

Answer #1

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.

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!

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...

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...

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

1.) Liz has utility given by u(x2,x1)=x1^7x2^8. If P1=$10,
P2=$20, and I = $150, find Liz’s optimal consumption of good 1.
(Hint: you can use the 5 step method or one of the demand functions
derived in class to find the answer).
2.) Using the information from question 1, find Liz’s optimal
consumption of good 2
3.) Lyndsay has utility given by u(x2,x1)=min{x1/3,x2/7}. If
P1=$1, P2=$1, and I=$10, find
Lyndsay’s optimal consumption of good 1. (Hint: this is Leontief
utility)....

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.

1. Al Einstein has a utility function that we can describe by
u(x1, x2) = x21 +
2x1x2 + x22
. Al’s wife, El Einstein, has a utility function v(x1,
x2) = x2 + x1.
(a) Calculate Al’s marginal rate of substitution between
x1 and x2.
(b) What is El’s marginal rate of substitution between
x1 and x2?
(c) Do Al’s and El’s utility functions u(x1,
x2) and v(x1, x2) represent the
same preferences?
(d) Is El’s utility function a...

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...

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