Question

(A). Find the maximum of the following utility function with respect to x;

U= x^2 * (120-4x).

The utility function is U(x,y)= sqrt(x) + sqrt(y) . The price of good x is Px and the price of good y is Py. We denote income by M with M > 0. This function is well-defined for x>0 and y>0.

(B). Compute (aU/aX) and (a^2u/ax^2). Is the utility function increasing in x? Is the utility function concave in x?

(C). Write down the maximization problem with respect to x and y.

(D). Write down the Langrangean function.

(E). Write down the first order conditions for this problem with respect to x, y, and (lambda).

a = a symbol that appears to look like a.

Answer #1

A consumer has utility function U(x, y) = x + 4y1/2 .
What is the consumer’s demand function for good x as a function of
prices px and py, and of income m, assuming a
corner solution?
Group of answer choices
a.x = (m – 3px)/px
b.x = m/px – 4px/py
c.x = m/px
d.x = 0

Assume that we have following utility maximization problem with
quasilinear utility function:
U=2√ x + Y
s.t. pxX+pyY=I
(a)derive Marshallian demand and show if x is a normal good, or
inferior good, or neither
(b)assume that px=0.5, py=1, and I =10. Then the price x
declined to 0.2. Use Hicksian demand function and expenditure
function to calculate compensating variation.
(c)use hicksian demand function and expenditure function to
calculate equivalent variation
(e) briefly explain why compensating variation and equivalent
variation are...

Suppose a consumer’s Utility Function
U(x,y) = X1/2Y1/2. The consumer wants to
choose the bundle (x*, y*) that would maximize utility.
Suppose Px = $5 and Py = $10 and the
consumer has $500 to spend.
Write the consumer’s budget constraint. Use the budget
constraint to write Y in terms of X.
Substitute Y from above into the utility function U(x,y) =
X1/2Y1/2.
To solve for the utility maximizing, taking the derivative of U
from (b) with respect to X....

Jane’s utility function has the following form: U(x,y)=x^2
+2xy
The prices of x and y are px and py respectively. Jane’s income
is I.
(a) Find the Marshallian demands for x and y and the indirect
utility function.
(b) Without solving the cost minimization problem, recover the
Hicksian demands for x and y and the expenditure function from the
Marshallian demands and the indirect utility function.
(c) Write down the Slutsky equation determining the effect of a
change in px...

Let income be I = $90, Px = $2, Py = $1, and utility U = 4X½Y.
a.[12] Write down and simplify the two conditions required for
utility maximization. b.[6] Compute the optimal consumption bundle
for the consumer. What is the level of utility at the optimum?

2. For Each of the following situations,
i) Write the Indirect Utility Function
ii) Write the Expenditure Function
iii) Calculate the Compensating Variation
iv) Calculate the Equivalent Variation
a) U(X,Y) = X^1/2 x Y^1/2. M = $288. Initially, PX= 16 and
PY
= 1. Then the Price of X changes to PX= 9.
i) Indirect Utility Function: __________________________
ii) Expenditure Function: ____________________________
iii) CV = ________________
iv) EV = ________________
b) U(X,Y) = MIN (X, 3Y). M = $40. Initially,...

A consumer's preferences are given by the utility function
u=(107)^2+2(x-5)y and the restrictions x>5 and y>0 are
imposed.
1. Write out the Lagrangian function to solve the consumer's
choice problem. Use the Lagrangian to derive the first order
conditions for the consumer's utility maximizing choice problem.
Consider only interior solutions. Show your work.
2. Derive the Optimal consumption bundles x*(px,py,w) and
y*(px,py,w)
3. Use the first order condition from 1 to calculate the
consumer's marginal utility of income when w=200,...

An agent has preferences for goods X and Y represented by the
utility function U(X,Y) = X +3Y
the price of good X is Px= 20, the price of good Y is
Py= 40, and her income isI = 400
Choose the quantities of X and Y which, for the given prices and
income, maximize her utility.

Suppose Rajesh has a utility function resulting in an MRS = Y /
X (from U = √XY) and he has an income of $240 (i.e. M = 240).
Suppose he faces prices PX = 8 and PY = 10. If the price of good Y
goes down to PY = 8, while everything else remains the same, find
Rajesh’s compensating variation (CV).
The answer is CV = -25.34, please show your work

The consumer’s Utility Function is
U(x,y) = X1/2Y1/2. Further Px = $5 and Py =
$10 and the consumer has $500 to spend. The values of x* = 50 and
y* = 25 maximizes utility.
The dual to the utility maximization
problem is expenditure minimization problem where the consumer
choose x and y to minimize the expenditure associated with
achieving a specified level of utility. That is,
Choose x and y to Minimize
Expenditure 5x + 10y subject to U...

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