Question

There are two goods, Good 1 and Good 2, with positive prices
p_{1} and p_{2}. A consumer has the utility
function U(x_{1}, x_{2}) = min{2x_{1},
5x_{2}}, where “min” is the minimum function, and
x_{1} and x_{2} are the amounts she consumes of
Good 1 and Good 2. Her income is M > 0.

(a) What condition must be true of x_{1} and
x_{2}, in any utility-maximising bundle the consumer
chooses? Your answer should be an equation involving (at least)
these two variables.

(b) Use the answer to (a), along with the budget constraint, to calculate the consumer’s demand functions for both goods.

(c) Is Good 2 a normal good? Explain your answer.

(d) Are the goods complements, substitutes, or neither? Explain your answer.

Answer #1

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...

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. Consider a utility-maximizing price-taking consumer in a two
good world. Denote her budget constraint by p1x1 + p2x2 = w, p1,
p2, w > 0, x1, x2 ≥ 0 (1) and suppose her utility function is u
(x1, x2) = 2x 1/2 1 + x2. (2) Since her budget set is compact and
her utility function is continuous, the Extreme Value Theorem tells
us there is at least one solution to this optimization problem. In
fact, demand functions, xi(p1,...

Suppose you consume two goods, whose prices are given by p1 and
p2, and your income is m.
Solve for your demand functions for the two goods, if
(a) your utility function is given by U(x1, x2) = ax1 + bx2
(b) your utility function is given by U(x1, x2) = max{ax1,
bx2}

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...

Each individual consumer takes the prices as given and chooses
her consumption bundle,(x1,x2)ER^2, by maximizing the utility
function: U(x1,x2) = ln(x1^3,x2^3), subject to the budget
constraint p1*x1+p2*x2 = 1000
a) write out the Lagrangian function for the consumer's
problem
b) write out the system of first-order conditions for the
consumer's problem
c) solve the system of first-order conditions to find the
optimal values of x1, x2. your answer might depend on p1 and
p2.
d) check if the critical point...

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,...

Bilbo can consume two goods, good 1 and good 2 where
X1 and X2 denote the quantity consumed of
each good. These goods sell at prices P1 and
P2, respectively. Bilbo’s preferences are represented by
the following utility function: U(X1, X2) =
3x1X2. Bilbo has an income of m.
a) Derive Bilbo’s Marshallian demand functions for the two
goods.
b) Given your answer in a), are the two goods normal goods?
Explain why and show this mathematically.
c) Calculate Bilbo’s...

Given a utility function for perfect complements: U(x1,x2) =
min{x1,βx2}, where β is a positive num- ber, and a budget
constraint: p1x1 + p2x2 = Y , where p1 and p2 are prices of good 1
and good 2 respectively, Y is the budget for the complements. Find
the demand functions for good 1 and good 2.

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*)?

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