Question

4. Suppose a consumer has perfect substitutes preference such that good x1 is twice as valuable as to the consumer as good x2.

(a) Find a utility function that represents this consumer’s preference.

(b) Does this consumer’s preference satisfy the convexity and the strong convex- ity?

(c) The initial prices of x1 and x2 are given as (p1, p2) = (1, 1), and the consumer’s income is m > 0. The prices are changed, and the new prices are (p1,p2) = (4,1). Decompose the changes of the demand of x1 into substitution effect and income effect.

(d) Draw indifference curves and budget lines to show the substitution effect and income effect in a diagram.

Answer #1

Suppose x1 and x2 are perfect substitutes
with the utility function U(x1, x2) =
2x1 + 6x2. If p1 = 1,
p2 = 2, and income m = 10, what it the optimal bundle
(x1*, x2*)?

Suppose the two goods, X1 and X2, are perfect substitutes at the
ratio of 1 to 2 – each unit of X1 is worth, to the consumer, 2
units of X2. The consumer had an income of $100. P1=5, and P2=3.
Find the optimal basket of this consumer.

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 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?)

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...

A consumer’s preferences over two goods
(x1,x2)
are represented by the utility function
ux1,x2=5x1+2x2.
The income he allocates for the consumption of these two goods is
m. The prices of the two goods are p1
and p2, respectively.
Determine the monotonicity and convexity of these preferences
and briefly define what they mean.
Interpret the marginal rate of substitution
(MRS(x1,x2))
between the two goods for this consumer.
For any p1, p2,
and m, calculate the Marshallian demand functions of
x1 and...

Suppose a consumer has quasi-linear utility: u(x1,x2 ) = 3x1^2/3
+ x2 . The marginal utilities
are MU1(x) = 2x1^−1/3 and MU2 (x) = 1. Throughout this problem,
assume p2 = 1
1.(a) Sketch an indifference curve for these preferences (label
axes and intercepts).
(b) Compute the marginal rate of substitution.
(c) Assume w ≥ 8/p1^2 . Find the optimal bundle (this will be a
function of p1 and w). Why do
we need the assumption w ≥ 8/p1^2 ?...

There are two goods, Good 1 and Good 2, with positive prices
p1 and p2. A consumer has the utility
function U(x1, x2) = min{2x1,
5x2}, where “min” is the minimum function, and
x1 and x2 are the amounts she consumes of
Good 1 and Good 2. Her income is M > 0.
(a) What condition must be true of x1 and
x2, in any utility-maximising bundle the consumer
chooses? Your answer should be an equation involving (at least)
these...

There are two goods, Good 1 and Good 2, with positive prices
p1 and p2. A consumer has the utility
function U(x1, x2) = min{2x1,
5x2}, where “min” is the minimum function, and
x1 and x2 are the amounts she consumes of
Good 1 and Good 2. Her income is M > 0.
(a) What condition must be true of x1 and
x2, in any utility-maximising bundle the consumer
chooses? Your answer should be an equation involving (at least)
these...

Suppose that a consumer has preferences over bundles of
non-negative amounts of each two goods, x1 and x2, that can be
represented by a quasi-linear utility
function of the form
U(x1,x2)=x1 +√x2.
The consumer is a price taker who faces a price per unit of good
one that is equal to $p1 and a price per unit of good two that is
equal to $p2. An- swer each of the following questions. To keep
things relatively simple, focus only on...

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