Question

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.

Answer #1

According to the information given the utility function is U = 2X1 + X2

this implies that the marginal rate of substitution which is the ratio of the marginal utilities of the two goats, is -2/1 or -2. Slope of the budget constraint is -p1/p2 or -5/3.

In absolute terms, marginal rate of substitution is greater than the slope of the budget constraint which means X1 generates more utility than X2.

At the optimal consumption bundle only X1 is consumed. It is equal to 100/5 or 20 units

Optimal consumption basket has 20 units of X1 and no units of X2.

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*)?

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)...

7.
Suppose you have the following utility function for two
goods:
u(x1, x2) = x
1/3
1 x
2/3
2
. Suppose your initial income is I, and prices are p1 and
p2.
(a) Suppose I = 400, p1 = 2.5, and p2 = 5. Solve for the
optimal bundle. Graph the budget
constraint with x1 on the horizontal axis, and the
indifference curve for that bundle.
Label all relevant points
(b) Suppose I = 600, p1 = 2.5, and...

Consider the problem of a consumer who must choose between two
types of goods, good 1 (x1) and good 2 (x2) costing respectively p1
and p2 per unit. He is endowed with an income m and has a
quasi-concave utility function u defined by u(x1, x2) = 5 ln x1 + 3
ln x2. 1. Write down the problem of the consumer. 1 mark 2.
Determine the optimal choice of good 1 and good 2, x ∗ 1 = x1(p1,...

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...

1.1 Suppose that the MRS of consuming (x1, x2) is x2/x1,
and that p1, p2 > 0.
Find out (p1, p2) such that (x1, x2) = (0.2, 0.8) is the optimal
bundle chosen by the
consumer.
1.2 Show that as long as p1, p2 > 0, the optimal bundle is
always interior, i.e., it is
impossible to obtain corner solution with the MRS being
x2/x1.

A product is produced using two inputs x1 and x2 costing P1=$10
and P2 = $5 per unit respectively. The production function is y =
2(x1)1.5 (x2)0.2 where y is the quantity of output, and x1, x2 are
the quantities of the two inputs. A)What input quantities (x1, x2)
minimize the cost of producing 10,000 units of output? (3 points)
B)What is the optimal mix of x1 and x2 if the company has a total
budget of $1000 and what...

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...

Consider the following utility function: U(x1,x2)
X11/3 X2
Suppose a consumer with the above utility function faces prices
p1 = 2 and
p2 = 3 and he has an income m = 12. What’s his optimal
bundle to consume?

Find the optimal bundle (x1, x2) (two
numbers). Does Jeremy consume positive amounts of both goods?
(e) Find the optimal bundle given p1 = 2,
p2 = 4 and m = 40 assuming U(x1,
x2) = 2x1 + 3x2. Does Jeremy
consume positive amounts of both goods? Is the optimal bundle at a
point of tangency?

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