Question

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. How big is each?

Answer #1

3. Suppose that a consumer has a utility function given by
U(X,Y) = X^.5Y^.5 . Consider the following bundles of goods: A =
(9, 4), B = (16, 16), C = (1, 36).
a. Calculate the consumer’s utility level for each bundle of
goods.
b. Specify the preference ordering for the bundles using the
“strictly preferred to” symbol and the “indifferent to” symbol.
c. Now, take the natural log of the utility function. Calculate
the new utility level provided by...

Consider a consumer whose preferences over the goods are
represented by the utility function U(x,y) = xy^2. Recall that for
this function the marginal utilities are given by MUx(x, y) = y^2
and MUy(x, y) = 2xy.
(a) What are the formulas for the indifference curves
corresponding to utility levels of u ̄ = 1, u ̄ = 4, and u ̄ = 9?
Draw these three indifference curves in one graph.
(b) What is the marginal rate of substitution...

Consider a consumer whose utility function is
u(x, y) = x + y (perfect substitutes)
a. Assume the consumer has income $120 and initially faces the
prices px = $1 and py = $2. How much x and y would they buy?
b. Next, suppose the price of x were to increase to $4. How
much would they buy now?
c. Decompose the total effect of the price change on demand
for x into the substitution effect and the...

1. Suppose utility for a consumer over food(x) and clothing(y)
is represented by u(x,y) = 915xy. Find the optimal values of x and
y as a function of the prices px and py with an income level m. px
and py are the prices of good x and y respectively.
2. Consider a utility function that represents preferences:
u(x,y) = min{80x,40y} Find the optimal values of x and y as a
function of the prices px and py with an...

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 preferences represented by the utility
function
u(x,y)=3x+6 sqrt(y)
(a) Are these preferences strictly convex?
(b) Derive the marginal rate of substitution.
(c) Suppose instead, the utility function is:
u(x,y)=x+2 sqrt(y)
Are these preferences strictly convex? Derive the marginal rate
of substitution.
(d) Are there any similarities or differences between the two
utility functions?

A consumer has utility for protein bars and vitamin water
summarized by the Cobb-Douglas utility function U(qB,qW) =
qBqW.
e. Find the consumer’s Engel curve for vitamin water when PB =
PW = 1.
f. What is the consumer’s optimal bundle when M = 100 and PB =
PW = 1?
Suppose the price of protein bars increases to P’B = 2.
g. Find the new optimal bundle.
h. Find the substitution effect of the price increase on
purchases of...

A consumer likes two goods; good 1 and good 2. the consumer’s
preferences are described the by the cobb-douglass utility
function
U = (c1,c2) =
c1α,c21-α
Where c1 denotes consumption of good 1, c2
denotes consumption of good 2, and parameter α lies between zero
and one; 1>α>0. Let I denote consumer’s income, let
p1 denotes the price of good 1, and p2
denotes the price of good 2. Then the consumer can be viewed as
choosing c1 and c2...

Consider a consumer with preferences represented by the utility
function:
U(x,y) = 3x + 6 √ y
Are these preferences strictly convex?
Derive the marginal rate of substitution
Suppose, the utility function is:
U(x,y) = -x +2 √
y
Are there any similarities or differences between the two
utility functions?

2. Consider a consumer with preferences represented by the
utility function:
u(x,y)=3x+6sqrt(y)
(a) Are these preferences strictly convex?
(b) Derive the marginal rate of substitution.
(c) Suppose instead, the utility
function is:
u(x,y)=x+2sqrt(y)
Are these preferences strictly convex?
Derive the marginal rate of sbustitution.
(d) Are there any similarities or diﬀerences between the two
utility functions?

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