Question

Utility cobb douglas function = 2X^{.5}Y^{2}

MUx =Y^{2}/X^{.5} MU_{y}
=4X^{.5}Y Px=1 PY=2 and M=100

1.Graph the consumer optimization problem in(X,Y) space. Clealy label the precise location of the optimal bundle, the budget constraintm, and the shape of the furthest obtainable indifference curve.

2.Assume Px increase to 2. What is the total effect of the price change in terms of X and Y.

3. What is the precise location of the bundle used to decompose the substitution and income effect?

4.What is the substitution effect due to the change in Px? State your answer in terms delta Xand delta Y .

5. What is the income effect due to the change in Px? State your answer in terms Delta X and Delta Y .

6.What is the compensating Variation assciated with this price change?

Answer #1

The utility function U(X,Y)=XaY1-a where
0≤a≤1 is called the Cobb-Douglas utility function.
MUx=aXa-1Y1-a
MUy=(1-a)XaY-a
(note for those who know calculus
MUx=∂U∂x and
MUy=∂U∂y)
Derive the demand functions for X and Y
Are X and Y normal goods? If the quantity of the good increases
with income a good is a normal good. If the quantity decreases with
income the good is an inferior good.
Describe in words the preferences corresponding to a=0, a=1,
a=.5

Consider a consumer with Cobb-Douglas preferences over two
goods, x and y described by the utility function u(x, y) = 1/3ln(x)
+ 2/3n(y) 1. Assume the prices of the two goods are initially both
$10, and her income is $1000. Obtain the consumer’s demands for x
and y.
2. If the price of good x increases to $20, what is the impact
on her demand for good x?
3. Decompose this change into the substitution effect, and the
income effect....

Assume that Sam has following utility function: U(x,y) =
2√x+y
MRS=(x)^-1/2, px = 1/5, py = 1 and her income I = 10. price
increase for the good x from px = 1/5 to p0x = 1/2.
(a) Consider a price increase for the good x from px = 1/5 to
p0x = 1/2. Find new optimal bundle under new price using a graph
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a) (4 points) How much X and Y should the consumer purchase in
order to maximize their utility?
b) (4 points) How much utility does the consumer receive?
c) (4 points) Now suppose PX increases to$6. What is the new
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How...

Amy has income of $M and consumes only two goods: composite good
y with price $1 and chocolate (good x) that costs $px per unit. Her
util- ity function is U(x,y) = 2xy; and marginal utilities of
composite good y and chocolate are: MUy = 2x and MUx = 2y.
(a) State Amy’s optimization problem. What is the objective
function? What is a constraint?
(b) Draw the Amy’s budget constraint. Place chocolate on the
horizontal axis, and ”expenditure all other...

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